goal functions and ecosystem constraints ...

GOAL FUNCTIONS AND ECOSYSTEM CONSTRAINTS: THERMODYNAMIC GOAL FUNCTIONS, LOCAL STABILITY, MAXIMAL RESILIENCE, AND PERMANENCE

By

Nadiah Pardede Kristensen, B.Eng. (Hons I)(Env Eng)

A thesis submitted to the University of Queensland for the degree of Doctor of Philosophy School of Life Sciences October 2007

ii

Except where acknowledged in the customary manner, the material presented in this thesis is, to the best of my knowledge, original and has not been submitted in whole or part for a degree in any university.

Nadiah Pardede Kristensen, B.Eng. (Hons I)(Env Eng)

Acknowledgements I would like to acknowledge the contribution of my supervisor, Hugh Possingham at the University of Queensland. His time, effort, expertise, and encouragement were invaluable. I would also like to thank my ex-supervisor at Griffith University, Albert Gabric, for his time and effort while I was there, and continued support after I left. Special thanks is due to Edward Laws at the University of Hawaii, for generously providing me with his model and expertise, and for collaborating with me on manuscripts. He has been very kind and patient, and I have learnt a lot from him. I would also like to thank Carlo Hamalainen for proof-reading the thesis, providing moral support, being a wonderful sys-admin, and for making sure that I kept my backups up-to-date. Financial assistance was provided by an Australian Postgraduate Award, and a Griffith University Research Development Grant.

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Abstract Cropp & Gabric (2002) used a three-compartment food web model to demonstrate a similarity between the parameter set that maximised resilience and the parameter set that maximised the thermodynamic goal functions: ‘maximise autotroph biomass’, ‘maximise heterotroph biomass’, ‘maximise the flux of nutrients’, and ‘maximise the flux to biomass ratio’. Laws, Falkowski, Smith, Ducklow & McCarthy (2000) used a 10 compartment food web model to demonstrate that the ratio of nutrient loading rate to total primary production could be predicted accurately when free parameters in the model were selected such that the model ecosystem had maximal resilience. Cropp & Gabric (2002) suggested that ecosystems maximise resilience as a consequence of maximising other goal functions. I use Cropp & Gabric’s (2002) model to demonstrate that, while maximising resilience offers a compromise between the thermodynamic goal functions, maximising the thermodynamic goal functions does not necessarily optimise resilience. I explore the possibility that maximal resilience could be used as a heuristic – a way to compromise between the thermodynamic goal functions. It is found that maximal resilience does offer a compromise between the thermodynamic goal functions, however that is not a sufficient reason to recommend the use of resilience as a goal function. The Laws et al. (2000) model is used to demonstrate that the predictive ability of maximal resilience in their model is independent of the relationship between maximal resilience and the thermodynamic goal functions. A system-level feedback mechanism is explored as a potential explanation for this observation. It is found that the feedback mechanism does not maximise resilience. However, given certain assumptions, the system will have high resilience. The assumptions are: the system is small, the system is simple, and the perturbation strength and frequency is moderate. ‘Maximise resilience’ can be used to predict the attributes of the system when the mapping from attribute space to resilience is peaked. The predictions of the feedback mechanism are tested on the Laws et al. (2000) model and field data. Evidence for high resilience is found, however the model and field data do not show evidence of the existence of the feedback mechanism. The maximal resilience goal function is applied to the Fasham, Ducklow & McKelvie (1990) model to iv

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find the reason behind the success of ‘maximise resilience’. It is found that the feasible-stable region shows predictive ability in the Fasham et al. (1990) model. Therefore, because the Laws et al. (2000) model preserves the feasible-stable region in the peaks of the resilience surface, it also preserves the predictive ability of the Fasham et al. (1990) model. Thus, the resilience hypothesis is reformulated: maximal resilience is an effective goal function when peaks in the resilience surface correspond to the feasible-stable region in the real system. To explore the hypothesis that stability and feasibility constrain ecosystems, a food web building algorithm is created using permanence as a constraint, and the attributes of the model webs resulting from the algorithm are compared with the attributes reported in the literature for real systems. Evidence is found for the restriction of food web attributes by a permanence constraint for: maximal chain length, link density, and basal fraction. Attributes such as basal-top link-type fraction require more than permanence to explain their patterns.

List of publications The following works have arisen from this thesis. Relevant chapters are indicated. 1. Kristensen N. P. Gabric A. Braddock R. and Cropp R. (2003) Is maximizing resilience compatible with established ecological goal functions? Ecological Modelling 169:61–71. (a) Chapter 4; and (b) Chapter 5. 2. Kristensen N. P. (2002) What is the role of resilience in ecosystems? Talk presented at the Annual Meeting of the Society of Mathematical Biology, Knoxville, Tennessee, July 2002. (a) Chapter 6. 3. Kristensen N. P. Laws E. and Gabric A. (2004) Maximal resilience as an independent goal function, (submitted to Ecological Modelling). (a) Chapter 7. 4. Kristensen N. P. Laws E. and Possingham H. P. (2004) Are stable systems more common in a stochastic world? (unpublished manuscript). (a) Chapter 8. 5. Kristensen N. P. Gabric A. and Possingham H. P. (2004) Structure of permanent food webs resulting from an assembly algorithm, (submitted to The American Naturalist, and presented at the AMSI Winter School, Brisbane, Queensland, July 2004). (a) Chapter 11.

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Statement of contribution of others The following contributions by others have been included in this thesis: 1. Roger Cropp, Griffith University (a) Matlab code for the original Cropp & Gabric (2002) model, including the genetic algorithm used. 2. Edward Laws, University of Hawaii (a) Matlab code for the original Laws et al. (2000) model; and (b) Matlab code for the first version of the Fasham et al. (1990) model.

vii

Contents Acknowledgements

iii

Abstract

iv

List of publications

vi

Statement of contribution of others

vii

1 Introduction

22

1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

1.2

Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2 Goal functions 2.1

28

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.2

The goal function approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.3

Sum of parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.3.2

Example: Lotka’s maximal flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.3.3

Evaluating validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Emergent properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.4.2

Example: ‘Thermodynamic’ goal functions

. . . . . . . . . . . . . . . . . . . . . .

34

2.4.3


39

System-level adaptation and feedback mechanisms . . . . . . . . . . . . . . . . . . . . . .

40

2.5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.5.2

Example: Stability measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.5.3


41

Succession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.4

2.5

2.6

1

2

Contents

2.6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.6.2

Clementsian succession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.6.3

The Odum brothers and succession . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

2.6.4

Reinterpreting succession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2.7

Further examples of reinterpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.8

Goal functions of specific interest in this thesis . . . . . . . . . . . . . . . . . . . . . . . .

51

2.8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

2.8.2

Maximise biomass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

2.8.3

Maximise flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

2.8.4

Maximise Production to biomass ratio . . . . . . . . . . . . . . . . . . . . . . . . .

55

2.8.5

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.9

3 Stability 3.1

3.2

3.3

3.4

59

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Types of stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.2.1

What is stability? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.2.2

Local stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.2.3

Sector Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

3.2.4

Permanence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

3.2.5

Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

Stability as a constraint upon ecosystems . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.3.1

Stability versus complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.3.2

Resilience and ecosystem survival . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4 The resilience hypothesis 4.1

78

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.2

Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

5 The resilience heuristic 5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 84

3

Contents

5.1.1 5.2

5.3

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

5.2.1

Variations in the CG Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.2.2

Concordance - A measure of agreement between goal functions . . . . . . . . . . .

86

5.2.3

Hypothesis tested . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

5.2.4

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

6 Resilience and the generalised Lotka-Volterra 6.1

6.2

6.3

6.4

6.5

6.6

95

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

6.1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

6.1.2

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

The generalised Lotka-Volterra model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

6.2.1

Formalisation of the generalised Lotka-Volterra model . . . . . . . . . . . . . . . .

96

6.2.2

Comments about building the model from first-principles . . . . . . . . . . . . . .

98

Expressions for goal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.3.1

Biomass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3.2

Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3.3

Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Analytic results for the generalised Lotka-Volterra model . . . . . . . . . . . . . . . . . . . 101 6.4.1

The diagonal elements of the Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . 101

6.4.2

Routh-Hurwitz criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4.3

Ecosystem 3-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4.4

Sector stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4.5

Permanence of chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Investigation of random-coefficient generalised Lotka-Volterra models . . . . . . . . . . . . 113 6.5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5.2

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.5.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Investigation of Lotka-Volterra chains using a genetic algorithm . . . . . . . . . . . . . . . 119 6.6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.6.2

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.6.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.6.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.6.5

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4

Contents

6.7

Investigation of non-chain Lotka-Volterra models . . . . . . . . . . . . . . . . . . . . . . . 127 6.7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.7.2

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.7.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.7.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.8

Chapter discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.9

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7 Laws Model: Independence and robustness of resilience 7.1

133

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2

The ef-ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.3

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.4

7.5

7.6

7.7

7.3.1

Laws Model details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.3.2

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.3.3


‘Maximise resilience’ as a compromise between the traditional goal functions 7.4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.4.2

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.4.3


Is the success of the Laws Model independent of the other goal functions? . . . . . . . . . 147 7.5.1

Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.5.2


7.5.3


Variations on the Laws Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.6.2

The successful variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8 Quantitative stability and system-level selection 8.1

. . . . . . . 141

154

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8.1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.2

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.3

The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.3.2

Model for exploring resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.3.3

Model for exploring resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5

Contents

8.4

How reasonable are the model assumptions? . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

9 Laws Model and peaks in the resilience surface 9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.1.1

9.2

9.3

9.4

9.5

170

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Peaks in the resilience surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

9.2.2

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

9.2.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

9.2.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Predicting general trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 9.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

9.3.2

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

9.3.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.3.4

Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Field data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.4.1

Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.4.2


Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

10 The Fasham Model and the feasible-stable region

191

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 10.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 10.2 Fasham Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 10.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 10.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.3 Fasham-Laws Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10.3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 10.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 10.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 11 Permanence as a food web building algorithm constraint

214

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

6

Contents

11.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 11.1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 11.1.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 11.1.4 Generalisations concerning food web structure . . . . . . . . . . . . . . . . . . . . 219 11.1.5 Criticisms of generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.2.1 Selected food web data-set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.2.2 Food web building algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 11.2.3 The control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 11.2.4 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11.3.1 Selected food web data-set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11.3.2 Algorithm predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 11.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 11.4.1 Food web size and structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 11.4.2 Link density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 11.4.3 Trophic relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 11.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 12 Conclusion

254

Bibliography

261

Appendices

275

A Appendices relating to Chapter 2

276

A.1 Two necessary conditions for organised behaviour in dissipative systems . . . . . . . . . . 276 A.1.1 Item 1: Open systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 A.1.2 Item 2: Far from equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 A.2 The general evolution criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 A.3 The assumptions behind the theorem of minimum entropy production . . . . . . . . . . . 279 A.4 The least specific dissipation principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 A.5 Lotka’s maximal flux hypothesis as a Prigoginean concept . . . . . . . . . . . . . . . . . . 281 A.6 Patten’s evidence for linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 A.7 The history of the exergy goal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 A.8 An example ascendency calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 A.9 Response of P/B to perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

Contents

B Appendices relating to Chapter 3

7

291

B.1 Lyapunov function example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 B.2 MacArthur stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 B.3 An error in the work of Yodzis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 C Appendices relating to Chapter 5

294

C.1 Differential equations governing models and goal functions . . . . . . . . . . . . . . . . . . 294 C.1.1 Model LL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 C.1.2 Model LH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 C.1.3 Model HL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 C.1.4 Model HH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 C.2 Finding resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 C.2.1 General Jacobian Matrix for CG Model variations . . . . . . . . . . . . . . . . . . 298 D Appendices relating to Chapter 6

316

D.1 Detailed derivation of the Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 D.2 Routh-Hurwitz criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 D.3 Trends between resilience and biomass in the nutrient compartment . . . . . . . . . . . . 318 D.4 An example of a feasible, unstable chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 D.5 Feasibility in closed chains does not imply stability . . . . . . . . . . . . . . . . . . . . . . 322 D.6 Normalised value of goal functions for chains sized 3-10 . . . . . . . . . . . . . . . . . . . 323 D.7 Values and normalised value of goal functions for non-chain structures . . . . . . . . . . . 325 E Appendices relating to Chapter 7 E.1 Field results used to determine the ef-ratio

327 . . . . . . . . . . . . . . . . . . . . . . . . . . 327

E.2 Solving the Laws Model at steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 E.3 Correspondence with Edward Laws regarding the parameter restrictions . . . . . . . . . . 329 E.4 Variations on the Laws Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 E.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 E.4.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 E.4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 E.5 A note on the sinking rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 F Appendices relating to Chapter 9

338

F.1 Making ef-ratio the free parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 F.2 The ef-ratio isocline key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 G Appendices relating to Chapter 10

340

8

Contents

G.1 Comparing the Fasham Model and the Laws Model . . . . . . . . . . . . . . . . . . . . . . 340 G.2 Documentation of the Fasham Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 G.3 Fasham Model comparisons between goal functions . . . . . . . . . . . . . . . . . . . . . . 369 G.3.1 Comparison between goal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 G.4 Variations on the Fasham Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 G.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 G.4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 G.4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 G.4.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 G.5 Maximal resilience versus ef-ratio profile – Fasham Model 6 . . . . . . . . . . . . . . . . . 379 G.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 G.5.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 G.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 H Appendices relating to Chapter 11

383

H.1 Documentation of the permanence food web building code . . . . . . . . . . . . . . . . . . 383 H.2 All GLV chains are permanent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 H.3 The effect of increasing productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 H.3.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 H.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 Index

490

List of Figures 1.1

A flow diagram of the thesis. Arrows indicate the logical progression between chapters, and the dotted-line boxes indicate the primary questions addressed in each of the chapters. 25

3.1

The phase space of a two-compartment ecological system, where the types of steady-state solutions are indicated.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

A diagrammatic representation of local stability (adapted from Pahl-Wostl 1995).

. . . .

3.3

A diagrammatic representation of permanence for one species. Xi is the population of the species, and t is time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

63 63

69

The horizontal axis indicates the goal function being maximised. P: phytoplankton biomass, Z: zooplankton biomass, F: flux of nutrients, and F/B: flux to biomass ratio. The stacked bar shows the normalised value of each of the other goal functions when the goal function indicated on the horizontal axis is maximised.

5.1

. . . . . . . . . . . . . . . . . . . . . . . .

Resilience verses eZ for Model LL1. All parameters except eZ are held constant. Note the maxima of resilience is at the transition from real to complex conjugate eigenvalues. . . .

6.1

81

94

A simple trophic system with a nutrient compartment. Numbered nodes are compartments, n + 1 is the nutrient compartment. Solid arrows indicate a flow of nutrients due to a predator-prey relationship, and dotted arrows indicate a flow of nutrients from biotic compartments to the nutrient compartment due to waste or mortality. . . . . . . . . . . .

6.2

97

Gerschgorin Disks from a hypothetical three heterotroph, one autotroph system. The eigenvalues of the Jacobian Matrix will lie within these disks. Although all Gershgorin Disks are centred around non-positive values, there is no guarantee that all eigenvalues will be negative, and therefore stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3

Diagrammatic representation of Ecosystem 3-1. Solid arrows indicate the direction of nutrient transfer due to predator-prey interactions. Dotted arrows show nutrient transfer from each biotic compartment to the nutrient compartment via death. 9

. . . . . . . . . . 106

10

List of Figures

6.4

A diagram showing subsystems of a chain food web, and the notation used in the text.

6.5

All possible arrangements of compartments (excluding omnivory) for n = 1 . . . 4. Rank(B) < n are displayed.

6.6

. 111

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

The probability that resilience, R, is greater than some value, M , found empirically and compared to predicted value 1/(8M ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.7

Flux versus biomass in the nutrient compartment. Five hundred randomly chosen points.

6.8

Comparison between Resilience versus biomass in the nutrient compartment for the 5 biotic

118

compartment chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.9

The number of compartments in the system versus the maximum resilience (as found with the genetic algorithm).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.1

The Laws Model.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.2

Model ef-ratio versus observed ef-ratio for the Laws Model using ‘maximise resilience’ as a goal function. Numbers correspond to ocean regions given in Table 7.4. The diagonal line indicates the position of a perfect prediction. . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.3

The location of the parameter set that maximises each of the goal functions in the Laws Model. The boundary of the feasible and stable region is marked as a dotted line. The key is B: ‘maximise biomass’, P: ‘maximise production’, P/B: ‘maximise production to biomass ratio’, and R: ‘maximise resilience’.

7.4

. . . . . . . . . . . . . . . . . . . . . . . . . 143


7.5

. . . . . . . . . . . . . . . . . . . . . . . . . 144


7.6

. . . . . . . . . . . . . . . . . . . . . . . . . 145


7.7

. . . . . . . . . . . . . . . . . . . . . . . . . 146

A box-plot of the ef-ratios in the feasible and stable region for each of the oceans versus the observed ef-ratios. Laws Model. The diagonal line indicates the position of a perfect prediction.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

11

List of Figures

7.8

Observed versus model ef-ratio for each of the goal functions, indicated as follows: ‘maximise biomass’ ×, ‘maximise new production’ ◦, ‘maximise production to biomass ratio’ +, and ‘maximise resilience’ . The diagonal line indicates the position of a perfect prediction. 149

7.9

A comparison between the parameter values that maximised resilience in the Laws Model when parameter relationships were removed, and the equations used by Laws et al. (2000) to describe those relationships. Numbers indicate the ocean region. The point in parameter space that maximises resilience is indicated by a ‘◦’, and the equation by a dotted line.

. 151

7.10 Observed versus model ef-ratio for each of the goal functions, indicated as follows: ‘maximise biomass’ ×, ‘maximise new production’ ◦, ‘maximise production to biomass ratio’ +, and ‘maximise resilience’ . Modelled such that all f values were free parameters. Details in Appendix E.4 ‘Laws Model 2’. The diagonal line indicates the position of a perfect prediction.

8.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

A typical trajectory of the resistance (solid line), r¯ of the model compared to a control (dotted line). Trajectory shown is for a 3 species compartment.

8.2

The effect of the algorithm upon the mean resistance versus the (fixed) size of the system. Each data point is for 10000 time steps, or invasions.

8.3

. . . . . . . . . . . . . . 159

. . . . . . . . . . . . . . . . . . . . 159

An arbitrarily chosen peak-shaped function used to related the single species-attribute a to the contribution to resilience r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.4

The constant of proportionality, f , affects the range of probability of replacement. . . . . 164

8.5

Resilience verses time where resilience is a function of the attribute space. One species. . 165

8.6

A histogram of the attribute space for a single species system where resilience is a function of the one-dimensional attribute space.

8.7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Histograms of the attribute space for a 2, 4 and 10 species system where resilience is a function of the one-dimensional attribute space. The tendency of the system to fluctuate about the point in attribute space corresponding to maximal resilience decreases with increasing system size.

9.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

The ef-ratios predicted by ‘maximise resilience’ for variations on the model. Model variations differ by which parameter is a free parameter: when the ef-ratio is used as a proxy for total production ×, and when the ef-ratio is used as a proxy for loading rate . The diagonal line indicates the position of a perfect prediction. Labels correspond to ocean region ID numbers as given in Table 7.1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 174

12

List of Figures

9.2

The maximum resilience contours in total production versus loading rate space. The blue regions are unstable. Resilience increases as the contour colours move from blue to dark red. Three markers are placed on the contours to indicate the observed data , the prediction of Laws Model L ×, and the prediction of Laws Model TP . Ocean regions are: (a) Bermuda Atlantic Time Series, (b) Hawaiian Ocean Time-series, (c) North Atlantic Bloom Experiment, (d) Pacific Equatorial Upwelling during normal conditions, (e) Pacific Equatorial Upwelling during El Nino, (f) Arabian Sea.

9.3

. . . . . . . . . . . . . . . . . . . 175

The maximum resilience contours in total production versus loading rate space. The blue regions are unstable. Resilience increases as the contour colours move from blue to dark red. Three markers are placed on the contours to indicate the observed data , the prediction of Laws Model L ×, and the prediction of Laws Model TP . Ocean regions are continued from Figure 9.2: (g) Ross Sea, (h) Subarctic Pacific Station P, (i) Peru upwelling during normal conditions, (j) Peru upwelling during El Nino, (k) Greenland Polynya.

9.4

The maximum resilience surfaces in total production versus loading rate space for the Bermuda Atlantic Time-series, and the North Atlantic Bloom Experiment.

9.5

. . . . . . . . 177

The maximum resilience surfaces in total production versus loading rate space for Subarctic Pacific Station P.

9.6

. . 176

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

The maximum resilience contours in total production versus loading rate space when the mixed layer depth is 80m. Temperatures are: (a) 0◦ C, (b) 5◦ C, (c) 10◦ C, (d) 15◦ C, (e) 20◦ C, (f) 25◦ C. The blue regions are unstable. Resilience increases as the contour colours move from blue to dark red. Black lines indicate ef-ratio isoclines of 0.1, 0.2, ...0.7. . . . . 182

9.7

The maximum resilience surface in total production versus loading rate space for two temperatures: 0◦ C, and 25◦ C.

9.8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

The maximum resilience contours in total production versus loading rate space when the temperature is 15◦ C. Mixed layer depths are: (a) 30m, (b) 50m, (c) 80m, (d) 110m, (e) 150m. The blue regions are unstable. Resilience increases as the contour colours move from blue to dark red. Black lines indicate ef-ratio isoclines of 0.1, 0.2, ...0.7.

9.9

. . . . . . . 184

The maximum resilience surface in total production versus loading rate space for two mixed layer depths: 30m, and 150m.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

9.10 Temperature versus ef-ratio. A least squares linear regression is shown, where ef = 0.63 − 0.02T . r2 = 0.87. Labels correspond to ocean region ID numbers as given in Table 7.1.

. 188

9.11 Mixed layer depth versus ef-ratio. A least squares linear regression is shown, where ef = 0.54 − 0.003z. r2 = 0.44. Labels correspond to ocean region ID numbers as given in Table 7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 9.12 Loading rate versus ef-ratio. A least squares linear regression is shown, where ef = 0.24 + 1 × 10−3 L. r2 = 0.19. Labels correspond to ocean region ID numbers as given in Table 7.1. 189

13

List of Figures

10.1 The Fasham et al. (1990) model.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

10.2 Observed versus model ef-ratio for each of the goal functions in the Fasham Model, indicated as follows: ‘maximise biomass’ ×, ‘maximise new production’ ◦, ‘maximise production to biomass ratio’ +, and ‘maximise resilience’ . The diagonal line indicates the position of a perfect prediction.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

10.3 Box-plots of the ef-ratios in the feasible and stable region for each of the oceans versus the observed ef-ratios. Fasham Model. The diagonal line indicates the position of a perfect prediction.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

10.4 Maximal resilience for the Fasham Model. The free parameter, ef-ratio, is a proxy for total production. Ocean regions are: (a) Bermuda Atlantic Time Series, (b) Hawaiian Ocean Time-series, (c) North Atlantic Bloom Experiment, (d) Pacific Equatorial Upwelling during normal conditions, (e) Pacific Equatorial Upwelling during El Nino, (f) Arabian Sea.

. . 198

10.5 Maximal resilience for the Fasham Model. The free parameter, ef-ratio, is a proxy for total production. Ocean regions are continued from Figure 10.4: (g) Ross Sea, (h) Subarctic Pacific Station P, (i) Peru upwelling during normal conditions, (j) Peru upwelling during El Nino, (k) Greenland Polynya. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 10.6 Fasham-Laws Model.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

10.7 Observed versus model ef-ratio for each of the goal functions in the Fasham-Laws Model, indicated as follows: ‘maximise biomass’ ×, ‘maximise new production’ ◦, ‘maximise production to biomass ratio’ +, and ‘maximise resilience’ . The diagonal line indicates the position of a perfect prediction.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

10.8 Box-plots of the ef-ratios in the feasible and stable region for each of the oceans versus the observed ef-ratios. Fasham-Laws Model. The diagonal line indicates the position of a perfect prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 10.9 A comparison between the maximal resilience profile versus the ef-ratio for the Fasham Model, the Fasham-Laws Model, and the Laws Model. Bermuda Atlantic Time-series Study.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

10.10A comparison between the maximal resilience profile versus the ef-ratio for the Fasham Model, the Fasham-Laws Model, and the Laws Model. Greenland Polynya. . . . . . . . . 209 11.1 An example of an interval and non-interval niche overlap graph. Adapted from May (1983). 218 11.2 (a) An example of a rigid-circuit niche-overlap graph that also has the interval property, (b) and a rigid-circuit niche-overlap graph that does not have the interval property.

. . . 218

11.3 A flow-chart showing the major steps in the food web building algorithm, and the postprocessing of the results of the food web building algorithm.

. . . . . . . . . . . . . . . . 224

14

List of Figures

11.4 A singular system: competitive exclusion such that the system will either collapse to {1, 2} or {1, 3}.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

11.5 A flow-chart showing the major steps in the control food web building algorithm.

. . . . 231

11.6 Systems generated by the GLV permanence algorithm may consist of several independent food webs.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

11.7 Total number of compartments versus Time. Arrows indicate places where the algorithm was restarted.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

11.8 Maximum chain length versus number of compartments. Permanence algorithm and control. Means reported in Table 11.2 indicated by the range shown on the left-hand side of the graph, where upper limit of range at Maximum Chain Length = 12. Note that the median value reported by Cohen (1989b) is 2.

. . . . . . . . . . . . . . . . . . . . . . . . 235

11.9 Example of a system generated by the permanence algorithm. Numbers approximate order of arrival, with lowest numbers being the most recent additions.

. . . . . . . . . . . . . . 236

11.10Total number of links per compartment versus the total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. Means reported in Table 11.2 indicated by the range shown on the left-hand side of the graph, where upper limit of range at Total Links/Total Compartments = 13.6.

. . . . . . . . . . . . . . . . . 237

11.11Number of prey versus number of predators. For the permanence algorithm, the regression for the predator vs prey relationship is prey= −0.2 + 1.6 pred, which has a coefficient

of determination of r2 = 0.8. For the control, the regression for the predator vs prey relationship is prey= −0.9 + 1.1 pred, which has a coefficient of determination of r2 = 0.9.

Range of mean slopes reported in Table 11.2 indicated by dotted-line cone. . . . . . . . . 240 11.12Basal fraction versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 11.13Intermediate fraction versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 11.14Top fraction versus Total number of compartments Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

15

List of Figures

11.15Fraction of basal-intermediate links versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph. . . . . . . . . . . . . . . . . . . . . . . 244 11.16Fraction of basal-top links versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 11.17Fraction of intermediate-intermediate links versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph. . . . . . . . . . . . . . . . . . . . . 246 11.18Fraction of intermediate-top links versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph. . . . . . . . . . . . . . . . . . . . . . . 247 11.19Iteration 27 of 500. Numbers approximate order of arrival, with lowest numbers being the most recent additions. Note the absence of intermediate compartments. . . . . . . . . . . 250 A.1 Entropy flux and entropy production in an open system. Reproduced from Nicolis & Prigogine (1977).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

A.2 Figure 1 of Ulanowicz (1980), reproduced to provide an example of how ascendency is calculated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 A.3 Time averaged production to biomass ratio in response to a reduction in the biomass. Both the maximum and minimum resilience models are shown. . . . . . . . . . . . . . . . . . . 290 D.1 Comparison between Resilience verses biomass in the nutrient compartment for a number of models.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

D.2 Comparison between Resilience verses biomass in the nutrient compartment for a number of models.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

D.3 Nearing the cycle of the example 5-compartment chain described in the text. . . . . . . . 321 E.1 Model versus observed ef for Laws Model variations.

. . . . . . . . . . . . . . . . . . . . 333

E.2 Ocean 3. Variation in resilience and ef with changes in the parameter gp , for models Laws 2 and Laws 3. All other parameters held constant at the values that maximised resilience. 334 E.3 Ocean 7. Variation in resilience and ef with changes in the parameter gp , for models Laws 2 and Laws 3. All other parameters held constant at the values that maximised resilience. 334

16

List of Figures

E.4 Ocean 8. Variation in resilience and ef with changes in the parameter gp , for models Laws 2 and Laws 3. All other parameters held constant at the values that maximised resilience. 334 E.5 Ocean 9. Variation in resilience and ef with changes in the parameter gp , for models Laws 2 and Laws 3. All other parameters held constant at the values that maximised resilience. 335 E.6 Ocean 10. Variation in resilience and ef with changes in the parameter gp , for models Laws 2 and Laws 3. All other parameters held constant at the values that maximised resilience. 335 E.7 Ocean 11. Variation in resilience and ef with changes in the parameter gp , for models Laws 2 and Laws 3. All other parameters held constant at the values that maximised resilience. 335 E.8 Ocean 11. The resilience surface in f3 × gp parameter space.

. . . . . . . . . . . . . . . . 336

E.9 Plateau in attribute space about which resilience is maximised. V = 1m d−1 . Ocean ID 1. 337 F.1 A graph showing the position of ef-ratio isoclines in total production versus loading rate space. For use with Figure 9.6 and Figure 9.8.

. . . . . . . . . . . . . . . . . . . . . . . . 339

G.1 The location of the parameter set that maximises each of the goal functions in the Fasham Model. The boundary of the feasible and stable region is marked as a dotted line.

. . . . 371


. . . . 372


. . . . 373


. . . . 374

G.5 Fasham Model variations. Dotted lines indicate nutrient flows that were varied in the variations investigated. See Table G.9 for a summary. . . . . . . . . . . . . . . . . . . . . 375 G.6 Maximal resilience for the Fasham Model 6.

. . . . . . . . . . . . . . . . . . . . . . . . . 381

G.7 Maximal resilience for the Fasham Model 6.

. . . . . . . . . . . . . . . . . . . . . . . . . 382

H.1 Intermediate fraction versus total number of compartments. Comparison between food webs with differing productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 H.2 Basal fraction versus total number of compartments. Comparison between food webs with differing productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 H.3 Fraction of basal-intermediate links versus Total number of compartments. Comparison between food webs with differing productivity.

. . . . . . . . . . . . . . . . . . . . . . . . 482

H.4 Fraction of basal-top links versus Total number of compartments. Comparison between food webs with differing productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 H.5 Fraction of intermediate-intermediate links versus Total number of compartments. Comparison between food webs with differing productivity.

. . . . . . . . . . . . . . . . . . . 483

17

List of Figures

H.6 Fraction of intermediate-top links versus Total number of compartments. Comparison between food webs with differing productivity.

. . . . . . . . . . . . . . . . . . . . . . . . 484

H.7 Histogram of relative frequency of food web sizes. Comparison between food webs with differing productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 H.8 Maximum chain length versus number of compartments. Comparison between food webs with differing productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 H.9 Proportion of food webs that are chains. Comparison between food webs with differing productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 H.10 Proportion of food webs containing a cycle. Comparison between food webs with differing productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 H.11 Proportion of food webs that have rigid-circuit niche-overlap graphs. Comparison between food webs with differing productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 H.12 Links per compartment versus number of compartments. Comparison between food webs with differing productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

List of Tables 1.1

Cropp (1999) result. The parameter values that maximised each of the selection pressures. Those that were maximised at the same vertex as maximal resilience are indicated in bold. 23

2.1

Hypothesised general principles of system development, reproduced from Fontaine (1981).

2.2

“Trends observed in autotrophic succession of an aquatic microecosystem.” Adapted from Cooke (1967) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

29

46

“A tabular model of ecological succession: trends to be expected in the development of ecosystems.” (Odum, 1969) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.4

A summary of the history of the exergy goal function. . . . . . . . . . . . . . . . . . . . .

53

3.1

A summary of food web building experiments in the literature. . . . . . . . . . . . . . . .

75

4.1

Model HL4 results. Parameters that the goal functions were independent of are marked with an *. Parameter values that agree with ‘maximise resilience’ are highlighted.

. . . .

81

5.1

A summary of attributes of models ran. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

5.2

Base parameter values. Range used was ±50%. No was taken to be a constant 500 mg/m2 . 88

5.3

Concordance (Equation 5.2) for each goal function in each model, relative to all goal functions including resilience. F/B is maximised.

5.4

. . . . . . . . . . . . . . . . . . . . . .

Concordance (Equation 5.2) for each goal function in each model, relative to all goal functions including resilience. F/B is minimised. . . . . . . . . . . . . . . . . . . . . . . .

5.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Concordance (Equation 5.3) for each goal function in each model, relative to traditional goal functions only. F/B is minimised.

6.1

91

Concordance (Equation 5.3) for each goal function in each model, relative to traditional goal functions only. F/B is maximised.

5.6

90

Random runs and failures.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 18

19

List of Tables

6.2

Mean goal function values for randomly chosen points in parameter space. Concordance of random points calculated with respect to traditional goal functions only. Maximum possible concordance is 3. Maximise and minimise F/B scenarios shown.

. . . . . . . . . 116

6.3

Values for goal functions (chains sized 3-10).

6.4

Summarising statistical data for the attributes of maximally resilient chains for system sizes n = 2 . . . 22.

6.5

. . . . . . . . . . . . . . . . . . . . . . . . . 122

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Concordance of chains, relative to all goal functions. The maximum concordance possible is 3. F/B is maximised. Highest concordance indicated in bold. . . . . . . . . . . . . . . . 124

6.6

Concordance of chains, relative to all goal functions. The maximum concordance possible is 3. F/B is minimised. Highest concordance indicated in bold. . . . . . . . . . . . . . . . 124

6.7

Concordance of chains, relative to the traditional goal functions only. The maximum concordance possible is 3. F/B is maximised. Highest concordance indicated in bold. . . . 125

6.8

Concordance of chains, relative to the traditional goal functions only. The maximum concordance possible is 3. F/B is minimised. Highest concordance indicated in bold. . . . 125

6.9

Concordance of models, relative to all goal functions including resilience. Maximum concordance possible is 3. F/B is maximised. Highest concordance indicated in bold.

. . . . 129

6.10 Concordance of models, relative to all goal functions including resilience. Maximum concordance possible is 3. F/B is minimised. Highest concordance indicated in bold.

. . . . 129

6.11 Concordance of models, relative to the traditional goal functions only. Maximum concordance possible is 3. F/B is maximised. Highest concordance indicated in bold. . . . . . . 129 6.12 Concordance of models, relative to the traditional goal functions only. Maximum concordance possible is 3. F/B is minimised. Highest concordance indicated in bold. 7.1

. . . . . . 129

Ocean regions and observed ef-ratio. Adapted from Table 3 of Laws et al. (2000), with citations therein. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.2

Notation used in the Laws Model. Adapted from Laws et al. (2000).

7.3

Parameter values used in the Laws Model (Laws et al. 2000). . . . . . . . . . . . . . . . . 138

7.4

The parameter set that maximises resilience for the Laws et al. (2000) model, and the ef-ratio predicted by the ‘maximise resilience’ goal function.

9.1

. . . . . . . . . . . 137

. . . . . . . . . . . . . . . . 139

The ef-ratios predicted by the goal function maximise resilience for two different freeparameter sets: total production as a free parameter, and loading rate as a free parameter. Most maximal resilience indicated in bold.

. . . . . . . . . . . . . . . . . . . . . . . . . . 173

10.1 Notation used in the Fasham-Laws Model.

. . . . . . . . . . . . . . . . . . . . . . . . . . 204

10.2 Parameter values used in the Fasham-Laws Model.

. . . . . . . . . . . . . . . . . . . . . 205

20

List of Tables

11.1 Scale-variance literature. Increases and decreases only reported if P < 0.05, or monotonic trends in the case of nonlinear regressions, else marked 0. Slope given where regression was used, else simply Inc or Dec for increase and decrease respectively. No data indicated by ‘-’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.2 Selected literature. References are presented as Recommender and Recommended. Web sizes are presented as a range “[min, max]”. Attribute are presented as “mean[min, max]”. A ‘-’ indicates that a value is not meaningful (e.g. a mean is omitted because of a relationship between that attribute and web size), or not reported. The fraction of webs containing cycles does not include cannibalistic cycles.

. . . . . . . . . . . . . . . . . . . 233

11.3 Comparison between algorithm and literature results. The fraction of cycles is taken from (Cohen 1989b). The proportion of food-webs with rigid-circuit niche-overlap graphs is taken from (Sugihara, Schoenly & Trombla 1989). . . . . . . . . . . . . . . . . . . . . . . 238 11.4 Comparing scale-variance and invariance between the control, the permanence algorithm, and the literature reporting scale-variance.

. . . . . . . . . . . . . . . . . . . . . . . . . . 239

A.1 A summary of the ascendency calculation in Ulanowicz (1980). . . . . . . . . . . . . . . . 289 C.1 Model LL1 and LL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 C.2 Model LL3 and LL4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 C.3 Model LL5 and LL6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 C.4 Model LH1 and LH2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 C.5 Model LH3 and LH4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 C.6 Model LH5 and LH6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 C.7 Model HL1 and HL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 C.8 Model HL3 and HL4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 C.9 Model HL5 and HL6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 C.10 Model HH1 and HH2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 C.11 Model HH3 and HH4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 C.12 Model HH5 and HH6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 C.13 Normalized Goal Function Values. Model LL . . . . . . . . . . . . . . . . . . . . . . . . . 312 C.14 Normalized Goal Function Values. Model LH . . . . . . . . . . . . . . . . . . . . . . . . . 313 C.15 Normalized Goal Function Values. Model HL . . . . . . . . . . . . . . . . . . . . . . . . . 314 C.16 Normalized Goal Function Values. Model HH . . . . . . . . . . . . . . . . . . . . . . . . . 315 D.1 Normalised values for goal functions (chains sized 3-10). . . . . . . . . . . . . . . . . . . . 324 D.2 Values for goal functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 D.3 Normalised values for goal functions.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

21

List of Tables

E.1 Field data adapted from Table 3 of (Laws et al. 2000).

. . . . . . . . . . . . . . . . . . . 327

E.2 A summary of variations on the Laws Model investigated.

. . . . . . . . . . . . . . . . . 330

E.3 Parameter values that maximise resilience for Laws 2. . . . . . . . . . . . . . . . . . . . . 331 E.4 Parameter values that maximise resilience for Laws 3. . . . . . . . . . . . . . . . . . . . . 332 G.1 Comparing the differential equations governing the Fasham Model and the Laws Model. G.2 Comparing the parameter values in the Fasham Model and the Laws Model.

341

. . . . . . . 342

G.3 Comparing the nutrient flow terms in the Fasham Model and the Laws Model. . . . . . . 343 G.4 List of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 G.5 Parameters used for finding the steady state of detritus.

. . . . . . . . . . . . . . . . . . 361

G.6 Parameters used for finding the steady state with the zooplankton differential equation G.7 Parameters used for finding the steady state with the bacteria differential equation G.8 Parameters used for finding the steady state for the nutrient compartments. G.9 A summary of variations on the Fasham Model investigated.

. 361

. . . 362

. . . . . . . 362

. . . . . . . . . . . . . . . . 377

G.10 Residual sum of square error, comparing model predictions to field data. A summary of linear regression on Fasham models. (x, y) =(Observed ef, Model ef). Maximise resilience. 379 H.1 A summary of trends in various food web attributes with increasing productivity range. . 490

Chapter 1

Introduction 1.1

Motivation

The purpose of this thesis is to investigate stability as an organising constraint in ecosystems. We begin by testing if maximal resilience can successfully be used as method for choosing free parameters in food web models. Two recent works have suggested that resilience is maximised in ecosystems: Cropp & Gabric (2002), and Laws et al. (2000). We begin the thesis by investigating these works. Cropp & Gabric (2002) describes a three-compartment food web consisting of a nutrient compartment, a phytoplankton compartment, and zooplankton compartment (Cropp 1999). A genetic algorithm (GA) was used to “simulate individual adaptation in response to selection pressures over evolutionary time scales” (Cropp & Gabric 2002). The selection pressures were: 1. Maximise phytoplankton biomass; 2. Maximise zooplankton biomass; 3. Maximise the flux of nutrient through the system; and 4. Maximise the flux to biomass ratio. The selection pressures chosen were also described as goal functions, and were based upon thermodynamic goal functions from the systems ecology literature: maximum exergy (Jørgensen & Mejer 1977), maximum ascendency (Ulanowicz & Kemp 1979), maximum power (Odum & Pinkerton 1955), and maximum entropy production (Johnson 1988). We will refer to the four selection pressures used in Cropp & Gabric (2002) as traditional goal functions. The parameter set that maximised each of the traditional goal functions was compared to the parameter set that maximised resilience. Resilience is a measure of ecosystem stability (DeAngelis 1992). It has no strong precedent as a goal function in the literature. 22

1.1 Motivation

23

Cropp & Gabric (2002) identifies a similarity between the parameter set that maximised each of the goal functions and the parameter set that maximised resilience. In Cropp & Gabric (2002) it states: “The most interesting aspect of the simulation [of the maximisation of each of the goal functions] is that the biotic attributes that optimize the thermodynamic goal functions also maximize the system’s resilience. The probability of nine independent functions being optimized at the same vertex of a five-dimensional parameter space is 2.8×10−14 and substantially less if optima at interior points are considered.” Table 1.1 reproduces their result (adapted from Cropp (1999)). It can be seen that flux is maximised at exactly the same vertex as resilience, and all other goal functions have three parameter values in common with ‘maximise resilience’. In Cropp & Gabric (2002) it is noted that: “There is also no a priori indication that the optimization of an ecosystem’s response to thermodynamic or ecological imperatives should result in a maximally resilience ecosystem. This outcome, however, is reasonable, given that all ecosystems exist within the constraints of thermodynamic laws and that highly resilient ecosystems are more likely to remain extant than ecosystems with low resilience.” Thus, the work concludes that “The principal implication of our simulations is the hypothesis that ecosystems evolve to maximize resilience.” Laws et al. (2000) provides empirical support for the Cropp & Gabric (2002) resilience hypothesis: that ecosystems maximise resilience. Laws et al. (2000) investigates a ten compartment marine pelagic food web model. They were interested in predicting the ef-ratios (the ratio of loading rate to total primary production) reported in the literature for different ocean regions. We call this the Laws Model. It

Table 1.1: Cropp (1999) result. The parameter values that maximised each of the selection pressures. Those that were maximised at the same vertex as maximal resilience are indicated in bold. Goal function Maximal Parameter set that maximised goal function maximised value eZ µP kP dZ ηZ Phytoplankton biomass 60.191 0.003 0.92 289.8 0.07 0.59 Zooplankton biomass 267.79 0.003 1.33 140.4 0.03 0.32 Flux 44.79 0.003 1.32 141.5 0.07 0.60 Flux/Biomass 0.15 0.008 1.29 171.1 0.07 0.60 Resilience 0.056 0.003 1.35 139.1 0.07 0.60 Note: eZ is the consumption per day of phytoplankton mass per zooplankton mass µP is the maximum phytoplankton nutrient uptake rate kP is the nutrient half saturation concentration for phytoplankton dZ is the zooplankton mortality ηZ is the efficiency of zooplankton conversion of nutrient into biomass

24

Introduction

was found that choosing free-parameter values such that the resilience of the model steady state was maximised led to excellent agreement between the model and the field data. In this thesis we explore this model and determine if the results are evidence for maximal resilience in ecosystems.

1.2

Structure of the thesis

Figure 1.1 shows a flow diagram of the thesis. We address four general issues in this thesis: 1. The resilience hypothesis: Do ecosystems maximise resilience? If they do, then ‘maximise resilience’ can be used as a goal function. The resilience hypothesis is proposed in Cropp & Gabric (2002) on the basis that maximising the traditional goal functions also maximises resilience. Laws et al. (2000) provides empirical support for the resilience hypothesis, however no attempt was made to determine why ‘maximise resilience’ was an effective goal function in the Laws Model. 2. The resilience heuristic: Does maximising resilience offer a compromise between the traditional goal functions? If, by maximising resilience, we can come close to simultaneously maximising the traditional goal functions, then the use of ‘maximise resilience’ could be justified as a heuristic. Alternately, if we also assume that the resilience hypothesis is correct, then we have an assurance that using ‘maximise resilience’ as a goal function will also reasonably satisfy the other goal functions. 3. Independence of resilience: Is ‘maximise resilience’ an effective goal function independent of its relationship with the traditional goal functions? If the interpretation of the results in Cropp & Gabric (2002) is found to be inaccurate, the resilience hypothesis may still be true for reasons unrelated to the traditional goal functions. The work of Laws et al. (2000) is evidence in support of this. If this is the case, then the reason for why ‘maximise resilience’ worked well in the Laws et al. (2000) study should be formulated independent of the traditional goal functions. 4. Stability/survival hypothesis: If stability is a proxy for ecosystem survival, then ecosystems that are stable are more likely to exist, and that ecosystems that have higher stability are more likely to survive. Does this imply that the attributes of real ecosystems are restricted by the constraint that they possess some level of stability? If the various measures of stability used in ecological models accurately measure some aspect of ecosystem survival likelihood, then this implies that the real ecosystem and the stable model ecosystems should have some attributes in common.

25

1.2 Structure of the thesis

The Cropp & Gabric (2002) resilience hypothesis assumes that the selection pressures are valid. In Chapter 2, we will undertake a critical review of the literature on goal functions, and assess the validity of Cropp & Gabric’s (2002) approach. Cropp & Gabric (2002) also supports the hypothesis with the statement that “highly resilient ecosystems are more likely to remain extant than ecosystems with low resilience”. In Chapter 3, we define resilience and other stability measures which will be used in the rest of the thesis. We propose that stability has been interpreted as a proxy for the survival of an ecosystem. This will also be important for the theory developed in Chapter 8.

Resilience heuristic Resilience hypothesis

Chapter 2 Goal functions

Stability/survival hypothesis

Chapter 3 Stability

Chapter 4 The resilience hypothesis

Chapter 5 The resilience heuristic

Chapter 8

Chapter 11

Quantitative stability and

Permanence as a food

system−level selection

web building constraint

Chapter 6 Resilience and the generalised Lotka−Volterra

Chapter 7

Chapter 9

Chapter 10

Laws Model: independence

Laws Model and peaks in

Fasham Model and the

and robustness of resilience

the resilience surface

feasible−stable region

Independence of resilience goal function

Figure 1.1: A flow diagram of the thesis. Arrows indicate the logical progression between chapters, and the dotted-line boxes indicate the primary questions addressed in each of the chapters.

26

Introduction

In Chapter 4, we directly test Cropp & Gabric’s (2002) statement that “the biotic attributes that optimize the thermodynamic goal functions also maximize the system’s resilience”. We do this using the original Cropp & Gabric (2002) ecosystem model. We find that maximising the thermodynamic goal functions does not necessarily maximise resilience, but maximising resilience does offer a compromise between the thermodynamic goal functions. We call this the resilience heuristic. It implies that, if one were willing to accept the validity of the thermodynamic goal functions to begin with, maximising resilience would be a sensible way to also achieve high values for the traditional goal functions. In Chapter 5, we test the resilience heuristic to variations in the original Cropp & Gabric (2002) model. We find that the resilience heuristic is generally robust to variations in the model, however the parameter range employed affects the result. How much agreement there is between the goal functions depends upon where the edges of the parameter range (which is where the traditional goal functions tend to maximise) are relative to the interior point at which resilience is maximised. In Chapter 6, we create generalised Lotka-Volterra models of varying structures to further test the robustness of the resilience heuristic. The models are created such that the parameter range is less restricted than in Chapter 5. We find that maximal resilience continues to offer a compromise between the thermodynamic goal functions, however, unless one is willing to accept that resilience is a valid, independent goal function, one would be better off choosing one of the thermodynamic goal functions to achieve this compromise instead. In Chapter 7, we investigate the results of Laws et al. (2000). We determine if the predictive ability of ‘maximise resilience’ in this model is merely a consequence of its ability to offer a compromise between the thermodynamic goal functions, or if maximal resilience’s predictive ability is independent of this relationship. It is found that maximal resilience acts independently of the other goal functions, however it is also found that the predictive ability of maximal resilience is greatly reduced when certain parameter relationships assumed in Laws et al. (2000) are removed. In Chapter 8, we are interested in developing a theory for why maximal resilience should be an effective goal function, independent of its relationship with the thermodynamic goal functions. We focus upon Cropp & Gabric’s (2002) statement: “highly resilient ecosystems are more likely to remain extant than ecosystems with low resilience”. This statement has been echoed by other researchers (Pimm & Lawton 1977, Jørgensen & Mejer 1977, May 1973, Dunbar 1960). We identify assumptions that would allow the above statement to provide a mechanism by which maximal resilience is an effective goal function. The key assumptions are: the system is small, the system is simple, the perturbations are weak, the perturbations have a suitable frequency, and that the mapping from the attribute space to resilience is peaked.

1.2 Structure of the thesis

27

In Chapter 9, we revisit the model of Laws et al. (2000) and use it to test the predictions of the theory developed in Chapter 8. In Chapter 8, we find that ‘maximise resilience’ will only be an effective goal function if the mapping from the attribute to resilience contains a single peak. We predict that the efficacy of ‘maximise resilience’ will decrease as the mapping deviates from this requirement. However, in Chapter 9 we do not find any evidence of this, which means that the Laws Model does not support the hypothesis of Chapter 8. In Chapter 10, we investigate another marine pelagic food web model, the model of Fasham et al. (1990). We create the Fasham Model by including a temperature-dependent feeding-rate, as was done in the Laws Model. We find that the ef-ratio predicted by the parameter values of the entire feasible-stable region in the Fasham Model show qualitative agreement with the ef-ratio reported in the field data. We hypothesise that the predictive ability of the feasible-stable region of the Fasham et al. (1990) model reflects a constraint upon the real food web: that it possesses a stable steady state, and that the steady state concentrations of all compartments must be greater than zero. We also find that the ef-ratio versus maximal resilience profile in the Fasham Model has a similar shape to the maximal resilience surface of the Laws Model. We hypothesise that because the changes between the Fasham Model and the Laws Model preserves the feasible-stable region of the Fasham Model in the resilience peaks in the Laws Model, maximising resilience in the Laws et al. (2000) model also possesses some predictive ability. Chapter 10 supports the hypothesis that local stability is an ecosystem constraint. This hypothesis has a long history, which is discussed in Chapter 3. In Chapter 11, we investigate permanence as a stability constraint. Permanence is a stability measure, which requires that the trajectories of the populations are bounded within the feasible region, below some finite value. A food web building algorithm is created using permanence as a constraint, and the system-level structural attributes of resulting model webs are compared to food webs in the literature. By identifying agreement and disagreement between the model and the literature, we identify which food web patterns may be the result of stability constraints, and which require another explanation.

Chapter 2

Goal functions 2.1 2.1.1

Introduction Motivation

In Chapter 1, we described the Cropp & Gabric (2002) resilience hypothesis: that ecosystems, in the course of maximising thermodynamic goal functions, will also maximise resilience. The purpose of this chapter is to review the goal function literature, and to determine the validity of the thermodynamic goal functions used in Cropp & Gabric (2002).

2.2

The goal function approach

When a model is created, it is often the case that there is uncertainty about the value of particular parameters, or worse still, that the modeller is unable to measure the unknown parameter values at all. Therefore, the modeller is faced with the problem of finding the best parameter value for a given situation. The most common approach is to fit the model to one data set, and then test it against another, however this is not always possible. An alternative approach is to fit unknown or uncertain parameters to some goal function, which is thought to reflect the structure or development of ecosystems (e.g. Fontaine 1981, Jørgensen & Mejer 1981). The work of Laws et al. (2000) is an example of the goal function approach, which we will explore in more detail in Chapter 7 and Chapter 9. Laws et al. (2000) developed a ten-compartment pelagic food web model to investigate the factors that regulate the ratio of export or new production to total primary production, which is called the ef-ratio. In creating the model, parameter values were sourced from 28

29

2.2 The goal function approach

the literature, and from analysis of the relationships between parameter values within the model. After this process was completed, two parameter values remained unknown: the steady state biomass of the filter feeder compartment, and the fraction of maximal uptake achieved by the large phytoplankton compartment. Maximal stability, as measured by resilience (Section 3.2.2), was chosen as the goal function. To assign the parameter values, a grid search was employed to locate the point in parameter space which maximised resilience. Once the parameter values were assigned in this manner for each ocean area investigated, the ef-ratio was then predicted. In Chapter 1, I discussed some examples of goal functions, which were investigated in Cropp & Gabric (2002). They were ‘maximise biomass’, ‘maximise flux’, and ‘maximise production to biomass ratio’. These three goal functions are a small subset of the goal functions available in the literature, which include those given in Table 2.1 (Fontaine 1981).

Table 2.1: Hypothesised general principles of system development, Proposed for Principle (Objective function) All systems at all Maximum energy flow temporal spatial, and organisational levels of resolution Minimum entropy

reproduced from Fontaine (1981). Reference Lotka (1922); Odum and Pinkerton (1955) Glansdorff and Prigogine (1970)

Maximum energy

Mejer and Jørgensen (1979)

Ecological systems at organisational level of resolution greater than populations

Optimum ascendency

Ulanowicz (1980)

Maximum persistent organic matter

Whittaker and Woodwell (1971); O’Niell et al (1975)

Ecological systems; applicable scale unclear

Maximum biomass: maintenance metabolism ratio

Margalef (1968)

Economic systems

Maximum profit (specific growth rate)

Various authors

There are many definitions of what a goal function is, and what they mean (Christensen 1995, Jørgensen 2000, Fath, Patten & Choi 2001, Fontaine 1981). However, common to many of the descriptions is an assumption that ecosystems develop over time, and that there is a direction to this development. The goal function identifies which attribute can describe this development (e.g. biomass, production, etc.), and what the direction of the development is (i.e. maximise or minimise). For example, Jørgensen (2000) states

30

Goal functions

“Goal functions are understood as functions that can describe the direction of ecosystem development.” The concept that ecosystems develop in some direction can sound teleological, particularly when the word “goal” is used. Jørgensen (2000) continues the quotation above by addressing this issue. “This should not be interpreted that ecosystems have predetermined goals, but rather that the self-organisational abilities of ecosystems make it possible to meet perturbations by directive reactions, which can be described by goal functions.” This statement may appear contradictory. On one hand, Jørgensen (2000) is saying that ecosystems do not have predetermined goals, yet at the same time, the direction of their reactions are predetermined by the goal functions. However, I interpret the intention of his statement as being, not to deny that directional development occurs, but to emphasise that it occurs for mechanistic, natural reasons, without any ‘foresight’ on the part of the ecosystem. In the example above, Jørgensen (2000) cites perturbations as the driver behind the organisation in ecosystems. Other authors cite different mechanistic causes behind the goal functions they use. This emphasis in Jørgensen (2000) on perturbation reflects the author’s interest in the goal function exergy (see Section 2.8.2, and Appendix A.7). For example, Fath et al. (2001) refer to Prigoginean (Prigogine & Stengers 1984) concepts (see Section 2.4.2), and Christensen (1994) refers to succession theory (see Section 2.6), as the central causes behind the goal functions that they are interested in. However, despite these differences, the assumption of directional development remains. It is simply a matter of identifying where this directional change springs from, and the mechanistic cause behind its direction. There are a large number of goal functions in the literature, and a similarly large number of mechanisms by which these goal functions are thought to operate. In order to evaluate the goal functions used in Cropp & Gabric (2002), we will need to investigate these mechanisms in detail. This task will become the focus of the remaining sections of this chapter. However, before we begin a detailed discussion of each individual goal function, I would like to propose a central theme to the discussion, to provide a framework through which to discuss the theory of goal functions. I have already mentioned that there is an assumption of direction and development underlying the goal function approach. This reflects an assumption that there is some mechanism by which ecosystems are compelled to develop in an orderly manner. For thermodynamic goal function theorists in particular (Section 2.4.2), understanding the source of this order is the key to predicting the direction of ecosystem development. In turn, this is the key to predictive modelling. Consider, for example, the following quote from Schrodinger’s (1956) book What is life 1 1 Schrodinger’s book has honoured status among systems ecologists, and goal function modellers. See, for example, key citations by Prigogine & Stengers (1984, pp. 142–145), Odum (1971, pp. 38–39), and Schneider & Kay (1994).

2.2 The goal function approach

31

“It is by avoiding the rapid decay into the inert state of ‘equilibrium’, that an organism appears so enigmatic; so much so, that from the earliest times of human thought some special non-physical or supernatural force ... was claimed to be the operative in the organism, and in some quarters is still claimed.” If this is true for a single organism, it is even more true for the system of organisms. While natural selection explains the ‘fit’ of organisms to their environments, there are aspects of the ecosystem’s ‘fit’ to the environment that transcend the scope of natural selection. How does one explain the apparent predetermination of succession (see Section 2.6), or the apparent stability of the ecosystem (see Section 2.5.2) despite its low probability (see Section 3.3.1)? These are pressing questions for ecosystem modellers, who are asked not only to explain, but to predict, ecosystem behaviour, often under the influence of novel anthropogenic forces. Therefore, the framework that I propose to work through is one of viewing goal functions as an attempt to identify and explain order in ecosystems. For convenience, I have devised three categories into which system-level explanations of ecosystem order can be organised. 1. Sum of parts: Section 2.3. In this category, ecosystem level trends are taken to be the sum of the trends of their constituent parts. For example, in Section 2.3.2, we explore the hypothesis that natural selection will favour individuals that increase the flux of energy through the ecosystem, and as a consequence, the flux of energy through the entire ecosystem will increase with time. 2. Emergent properties: Section 2.4. In this category, ecosystem level trends are taken to be the result of the actions of their constituent parts, but not necessarily in the predictable way described in Item 1. The unpredictable element arises from the interaction of constituent parts, so that no single part possesses the attribute, but when they act together, the attribute manifests on the system level. An example from physical systems is the effect of randomised particulate motions to disperse and homogenise gas mixtures in a volume. We will find that emergent properties of physical systems are often applied by analogy to ecology, sometimes with questionable results. We will discuss this approach in Section 2.4, focusing upon the preeminent example of this: dissipative systems (Section 2.4.2). 3. System-level adaptation: Section 2.5. This category groups all principles that assume that ecosystems can adapt to their environment at a system level. This is often justified by appealing to feedback mechanisms between the system and its environment. For example, Clements (1916, p. 6) describes a series of actions upon the environment by the ecosystem resulting in positive, then negative, reactions by the environment

32

Goal functions

upon the ecosystem, causing system change, until a steady state is produced. Similarly, Lovelock (1995) conceives of a system structured such that negative feedback between the biotic and abiotic environment regulate the system. The oft-repeated criticism of such constructions is that the focus upon organising, rather than disorganising, feedbacks, amounts to teleology. For example, while a species may facilitate the invasion of another species sensu Clements, it may also inhibit it (Drury & Nisbet 1971). While negative feedback can regulate a system sensu Lovelock (1995), one must assume that a positive feedback is equally likely, and will destroy it (Kirchner 1989).

2.3

Sum of parts

2.3.1

Introduction

Theorists are interested in describing system-level attributes and identifying their trends. The simplest way to do this is what I have called the “sum of parts” approach. By this approach, trends at the system level are direct manifestations of the trends of the constituent parts of the system. So, for example, if we can determine that all populations will change such that attribute X increases, we might suppose that the whole ecosystem, being a sum of populations, will also change such that attribute X is increased on the system level. In this section, we will discuss one example: the theory of maximal flux. It has retained popularity with goal functions theorists, and is widely cited as an early example of a goal function (e.g. Choi, Mazumder & Hansell 1999). It has also been reinterpreted in the context of thermodynamic goal functions with mixed success (Section 2.7).

2.3.2

Example: Lotka’s maximal flux

The maximal flux hypothesis was first described in Lotka (1922): “In every instance considered, natural selection will so operate as to increase the total mass of the organic system, to increase the rate of circulation of matter through the system, and to increase the total energy flux through the system, so long as there is presented an unutilized residue of matter and available energy. This may be expressed by saying that natural selection tends to make the energy flux through the system a maximum, so far as compatible with the constraints to which the system is subject.”

2.3 Sum of parts

33

The central assumption is that organisms are competing for energy. Those that have a greater ability to capture energy and utilise it for the purposes of survival and reproduction gaining the competitive advantage, and natural selection will “give relative preponderance (in number or in mass)” (Lotka 1922) to those species. Consequently, the energy flow through the entire system will increase. Lotka (1922) elaborates upon how this principle operates2 . First, Lotka (1922) recognises efficiency as a means to gain competitive advantage. Lotka (1922) states that “so long as their remains an unutilised margin of available energy, sooner or later the battle, presumably, will be between two groups of species equally efficient, equally economical, but the one more apt than the other in tapping previously unutilised sources of available energy”. He argues that natural selection will favour these organisms, and in doing so, the total energy flux through the system will be maximised. He states: “If sources are presented, capable of supplying available energy in excess of that actually being tapped by the entire system of living organisms, then an opportunity is furnished for suitably constituted organisms to enlarge the total energy flux through the system. Whenever such organisms arise, natural selection will operate to preserve and increase them. The result, in this case, is not a mere diversion of the energy flux through the system of organic nature along a new path, but an increase of the total flux through that system.”

2.3.3

Evaluating validity

To evaluate the validity of goal functions based upon mechanisms of this type, two key assumptions must be addressed: 1. That the trend predicted for the constituent parts is accurate; and 2. That the trend of the constituent parts translates up the hierarchy and manifests as system level trend. The trend predicted for the constituent parts in the maximum flux hypothesis is a simplification. For example, there is no attempt to address selection pressures aside from resource capture ability, which will obviously complicate the mapping from the ability to tap novel sources of energy to the survival and reproduction of a species. Further, it is assumed that there is a relatively simple mapping from resource use efficiency to reproductive success. This is obviously a generalisation, and does not always hold. That flux maximisation on an individual level will translate into flux maximisation of a system level is also a simplification. Resources shared by two or more competitors, both of whom are aiming to maximise their resource use, does not necessarily lead to the maximum possible resource use for the system. The 2 Lotka (1922) also gives the hypothetical example of humans who, being restricted in the amount of land for (and hence biomass of) crops, increase the turnover of crops to allow expansion of human populations. As this necessarily acknowledges the need for intelligent direction, something ecosystems do not possess, it is omitted it from further discussion.

34

Goal functions

‘tragedy of the commons’ is a well-known example where competitors end up with lower yield than if they had shared the resource fairly (Gersani, Brown, O’Brien, Maina & Abramsky 2001, Hardin 1968). This kind of unexpected system level effect is the subject of the next section. The proposal in Lotka (1922) can be argued on purely individualistic terms, however later authors have revisited the maximal flux hypothesis within the context of succession and thermodynamic goal functions. As will be discussed later (Section 2.7), this serves as an example of how goal functions can be classified into several of the types described in Section 2.2, obscuring the theoretical basis behind them.

2.4

Emergent properties

2.4.1

Introduction

We have previously described the tragedy of the commons, in which the tendency of competing individuals to maximise their own yield can result in a suboptimal yield for all players (Gersani et al. 2001, Hardin 1968). This is an example of an emergent property. It is where a property of a system occurs by the interactions of its constituent parts, even though the constituent parts do not possess the property when alone. Emergent properties occur in many different types of systems in different fields. While the study of emergent properties is relatively new, there are common principles being discovered about them, which may prove useful in ecology. Cropp & Gabric (2002) describe the goal functions they use as “thermodynamic” goal functions. As the name suggests, they are ecological goal functions inspired by the emergent properties of systems studied in thermodynamics. While borrowing ideas from other fields is a source of many intriguing ideas, the transfer of ideas from one field to another can lead to misuse, confusion, and obfuscation. In this section, we describe the history of thermodynamic goal functions, and identify the key shortcomings of the theory.

2.4.2

Example: ‘Thermodynamic’ goal functions

The history of ‘thermodynamic’ thinking in the goal function literature begins, as with most goal functions, with a desire to understand the source of order in ecosystems, and a desire to predict ecosystem behaviour. Schneider (1988, pp. 116) states “Ecology is a young, data rich science with a weak theoretical base. Except for well defined concepts in biology, such as biological limits and population dynamics, ecology offers little theory to allow one to predict future system functions or states. Thermodynamics, and in particular non-equilibrium thermodynamics, appears to offer ecology a framework for the development of such a theory.”

2.4 Emergent properties

35

The non-equilibrium thermodynamics to which Schneider (1988) refers began with Schrodinger (1956, pp. 71), who tried to reconcile the natural order in biological systems with the universal tendency to disorder, or increasing entropy. He states “Thus a living organism continually increases its entropy – or, as you may say, produces positive entropy – and thus tends to approach the dangerous state of maximum entropy, which is death. It can only keep aloof from it, i.e. alive, by continually drawing from its environment negative entropy ... What an organism feeds upon is negative entropy. Or, to put it less paradoxically, the essential thing in metabolism is that the organism succeeds in freeing itself from all the entropy it cannot help producing while alive.” The concept of life as a negative entropy pump was further developed by Ilya Prigogine (Prigogine & Stengers 1984, pp. 142–145). Prigogine identified a group of systems, which he named dissipative systems, which possessed this special self-organising property. They share in common an ability increase their own internal organisation at the expense of the order of their external environment. They do this by exporting or dissipating the entropy out of the system (Heylighen 1999). Two necessary conditions for organised behaviour in dissipative systems are identified. We discuss each of these conditions in more detail in Appendix A.1. They are: 1. The dissipative system is an open system The system is open to entropy transfer with the greater environment, through energy or matter transfer (Nicolis & Prigogine 1977, pp. 24). 2. The dissipative system operates far from thermodynamic equilibrium This requires (Nicolis & Prigogine 1977, pp. 60): • an initial fluctuation away from thermodynamic equilibrium, • feedback mechanisms that enhance, rather than dampen, such fluctuations (Nicolis & Prigogine 1977, pp. 158–159), and • the existence of non-equilibrium steady states, which are possible for complex, nonlinear dynamical systems.

Of interest to us are predictive attributes of dissipative systems: does there exist a quantifiable3 trend common to all dissipative systems that can be used to predict its behaviour (Glandsdorff & Prigogine 1971, pp. 106)? Two trends have been identified: 3 While I acknowledge the role that dissipative structures have played as a conceptual framework in the life sciences (e.g. Schneider & Kay 1994, Brooks, Collier, Maurer, Smith & Wiley 1989), we restrict our attentions to quantifications specifically, for it is these that have become the goal functions that we are interested in.

36

Goal functions

1. the theorem of minimum entropy production (Glandsdorff & Prigogine 1971, Nicolis & Prigogine 1977); and 2. the general evolution criterion (Nicolis & Prigogine 1977, pp. 37). I have not been able to find any reference to the general evolution criterion in the systems ecology literature, so a brief explanation of it is reserved for Appendix A.2, and we now focus upon the theorem of minimum entropy production. The theorem of minimum entropy production states that (see Glandsdorff & Prigogine (1971, pp. 30–43) and Nicolis & Prigogine (1977, pp. 31–45)): dP < 0 away from steady state dt dP = 0 at steady state dt

(2.1) (2.2)

where P = dS/dt is the entropy production. The theorem of minimum entropy production relies upon some simplifying assumptions, which are summarised in Appendix A.3. The key assumption is that of local equilibrium thermodynamics. Local equilibrium thermodynamics is the assumption that although the macroscopic system exists far from thermodynamic equilibrium, the microscopic scale behaves as though it were at thermodynamic equilibrium. It is also assumed that the linear expansion of the flows, which is performed during the derivation of the minimum entropy production principle, holds (see Appendix A.3). Any theorist who wishes to make use of the minimum entropy production theorem in a novel context, such as ecosystems, faces two challenges: 1. to provide some proxy for measuring entropy or entropy production; and 2. to show that the assumptions behind the theorem of minimum entropy production are reasonably satisfied in the novel context. Many proxies for entropy production have been suggested in the ecology literature. Examples include: energy flux (e.g. Choi et al. 1999), respiration to biomass ratio (e.g. Choi et al. 1999, Fath et al. 2001), production to biomass ratio (e.g. Johnson 1988), genetic information content (e.g. Jørgensen, Nielsen & Mejer 1995), and diversity information content (Wicken 1988). This diversity indicates some confusion in the literature as to what entropy and entropy production should be measured by (Goodman 1975, Wicken 1988). Even with ‘simple’ systems, authors struggle to identify, let alone quantify, processes generating entropy4 . Without a clear description of how to quantify entropy production, the use of the theorem of minimum 4 For example, compare Zotin & Zotina’s (1967) respiration and entropy production equivalence, to Briedis & Segrave’s (1984) more extensive list of sources (Kay 2002).


37

entropy production is not possible. Further, the justification of the assumptions behind the theorem of minimum entropy production becomes more difficult. It is difficult enough to justify local equilibrium thermodynamics in the context of simple chemical systems (Nicolis & Prigogine 1977, pp. 32, 37), let alone in the context of ecosystems. For chemical systems, Nicolis & Prigogine (1977) states that the conditions of local equilibrium thermodynamics lead to “to situations corresponding to the immediate vicinity of equilibrium or to reactions proceeding with an extremely low activation energy”. It was the impracticality of this strict assumption that led them to develop the ‘general evolution criteria’ (Appendix A.2). The incompatibility between life and local thermodynamic equilibrium in most contexts causes many authors to treat the application of the theorem of minimum entropy production to the ecological context with scepticism (e.g. Kay 2002). Despite this, theorists have used, and continue to use, the theorem of minimum entropy production in the ecological context. Further, the justification for doing this has been given scant attention by most practitioners. The difficulty of satisfying the assumptions is often dismissed by some reference to the theory being an ‘approximation’. This is unsatisfying for those who want to assess the validity of applying the theorem to this novel context. For example, consider the justification given in Fath et al. (2001): “Although this least specific dissipation principle5 was developed for systems near thermodynamic equilibria, we contend that even if the global system is “far from equilibrium”, subsystems at finer spatio-temporal-organisational scales may be considered to be in some proximity to a quasi-local steady state – close enough at least for the principle to provide an understanding of “how a system should change”.” The scale at which local equilibrium thermodynamics is assumed is not described, nor the way in which the subsystem satisfies local equilibrium thermodynamics. Are we stating that the population dynamic interactions approximate local equilibrium thermodynamics, or is it the biochemistry of the organisms that is the subsystem of interest? Unlike Prigogine and coworkers, who specifically describe the subsystem and the conditions for which local equilibrium thermodynamics is a reasonable approximation (Nicolis & Prigogine 1977, Eqs 3.24, 3.28), Fath et al. (2001) simply asserts that there does exist some relevant subsystem, that it does possess some sufficient approximation to local equilibrium thermodynamics, and that this subsystem is relevant to the hierarchical level at which proxy for entropy production is measured. Similar difficulties arise in justifying the related assumption that a linear expansion can be used for the ecological analogue of the relationship between chemical flows and forces (Appendix A.3). Two studies are often cited as supporting evidence for linearised approximations of ecological and biological systems: Zotin & Zotina (1967), and Patten (1983) (e.g. Choi et al. 1999). 5 ‘Least specific dissipation’ is this group of researchers’ name for the application of the theorem of minimum entropy production to the decrease in the respiration to biomass ratio in ecosystems. See Appendix A.4

38

Goal functions

Zotin & Zotina (1967) applies the theorem of minimum entropy production to the ageing of an individual organism. They used heat production, which is an indicator of the metabolic rate, as a proxy for entropy production. They used weight gain as an analogue for flow, and time to maximal age as an analogue for force. They justified local equilibrium thermodynamics by observing that the following linear relationship was sufficient to explain the empirical data 1 dP = K(tm − t), P dt

(2.3)

where P is weight of the individual, t is age, tm maximal age, and K some empirically derived constant. Therefore, they concluded that the system was sufficiently close to linearity for the theorem of minimum entropy production to apply. Irrespective of the validity of the Zotin & Zotina (1967), the application of the theorem of minimum entropy production, and the justification of the assumptions behind it, is specific to the particular system that is studied in the work: an individual organism. Therefore, whether it is correct or not is irrelevant to the discussion of the application of the theorem of minimum entropy production to an ecosystem. The linearisation in Patten (1983) builds upon a previous work: Patten (1975). Patten (1975) contends that that ecosystem are linear in the sense that inputs scale in a linear fashion to outputs. For example, precipitation scales linearly to cation output. The central justification given in Patten (1975) is that, by assuming that a system is at or near steady state, the population dynamics may be approximated by linearising the system (other justifications are noted in Appendix A.6). In effect, the nonlinear ecosystem behaves in a linear fashion near the steady state (see Section 3.2.2 for details). The assumption of steady state population dynamics is disputed by many ecologists (e.g. Ehrlich & Birch 1967). Patten (1975) acknowledges this by stating that the group of ecologists from “animal and population ecology [as opposed to plant and community ecology], sees ecosystems as irregular, erratic, unpredictable, unreliable, poorly behaved, unstable or only locally stable, uncertain, and fluctuating wildly”. We will discuss perspectives on the stability of ecosystems further in Chapter 3. However, once again, irrespective of the validity of the hypothesis in Patten (1975) hypothesis, the linearisation of population dynamics about a steady state is irrelevant to the question of applying the theorem of minimum entropy production to ecosystems, for example, in terms of system-level respiration to biomass ratio sensu Choi et al. (1999). They are completely different systems. If an analogy may be made between population dynamics and chemical clocks such as the Brusselator (as Nicolis & Prigogine (1977, pp. 452) themselves suggest), to assume steady state is to preclude the organised behaviour of the dissipative system, namely the coherent of the limit cycle or other structured trajectories. But it is apparent that limit cycles are not the self-organised behaviour of interest to systems ecologists. To take Choi et al. (1999) as an example, they are interested in organised behaviour in the form of increasing energy flux, through changes in species attributes sensu Lotka (1922). Therefore, linearity must be justified for fluxes


39

and forces relevant to the Lotka (1922) hypothesis, which are increased nutrient flux and novel energy sources, invasion, and the mutation forces driving the Lotka (1922) mechanism.

2.4.3

Evaluating validity

The observed patterns in nature prompt the search for a holistic theory of ecosystems; one in which the unmanageable complexity of interactions between biotic and abiotic elements might be reduced to the investigation of a few aggregate variables, likened to the reduction of particulate motions in gases to the perfect gas law (McIntosh 1981, Ulanowicz & Kemp 1979). Finding inspiration in self-organisation and dissipative systems, and noting the organisation of ecological systems, theorists have applied directive principles from dissipative systems to the ecological context. However doing so deviates from the methodology, which is the key to understanding emergent behaviour. The key to understanding emergent behaviour is to build the theory from the microscopic scale up. Successful examples in the literature, such as the Brusselator (Appendix A.1), have been successful because they have demonstrated that, by applying simple rules to the behaviour of constituent interacting parts, a system-level organised behaviour emerges. Although the behaviour may be unexpected, it’s mechanism can be traced from the microscopic levels up, and so there is minimum confusion as to what organised behaviour we are studying, and what the specific assumptions are that must be made. In contrast, the application of thermodynamic principles to ecological goal function theory has been built from the top down. It is assumed that, because the organised behaviour exists, it is a result of dissipative structures. However, by not specifying the mechanism by which the organised behaviour emerges, it is impossible to assess the validity of the assumptions. Further, the very meaning of the goal function has become obscured: what is this ecological entropy production that is being minimised? What is the manifestation of organisation that we are seeking to explain?

We began this section with Schneider’s (1988) lament that the ability of ecological theory to predict ecosystem behaviour was limited. Schneider (1988) points to thermodynamics as a potential source for such theory, as it has demonstrated ability to predict and explain organised behaviour. I would share this optimistic outlook, but only if the methodology which produces these exciting results is adhered to.

40

Goal functions

2.5

System-level adaptation and feedback mechanisms

2.5.1

Introduction

The most famous emergent property in biology is adaptation, and its mechanistic cause, natural selection. Its ability to provide an explanation of design in natural systems has made adaptationism (GodfreySmith 2001) a fertile source of inspiration for systems ecologists, despite its shortcomings (Gould & Lewontin 1979). A perception among systems ecologists that ecosystems are suited to the environment in which they live, in a way which cannot be explained by the adaptation of the constituent parts alone, has led to the hypothesis that ecosystems adapt to their environment on the system level. Such adaptation is usually explained in terms of preferential extinction of ecosystems, and feedback from the system level to its constituent parts. This section discusses examples of this. We will develop our own theory of system-level adaptation in Chapter 8.

2.5.2

Example: Stability measures

Several works have suggested that ecosystems adapt to increase their stability. For example, Dunbar (1960) states that, because “oscillations are bad for any system and that violent oscillations are often lethal”, Darwinian selection may be applied to the level of the ecosystem, to minimise these oscillations. “For instance, suppose an ecosystem, locally defined, begins to develop oscillations to a lethal degree, a degree such that one or more vital parts are not able to survive; the resulting empty environmental space, as in Cuvierian cataclysms 6 , is available for occupation by communities from the adjacent regions; and these adjacent systems, as their survival suggests, are not of precisely the same constitution as the extinguished system. ... and if the difference is favourable to the continued survival of the system, its chances of survival are enhanced. In this way the system dominant in any geographical region changes, and changes (if the present assumptions are correct) in the direction of greater stability” (Dunbar 1960). As evidence of this, Dunbar (1960) observes that polar systems are less stable (more oscillatory) than tropical systems, and also have a shorter history than tropical systems. The work proposes that “steady systems of the tropics are the result of long evolution and that oscillations observed in the higher latitudes are systems of non-adaptation”. Another example is the exergy goal function proposed in Jørgensen & Mejer (1977). This was first formulated as a measure of the ‘buffer capacity’ of an ecosystem, which is the ability of an ecosystem 6 A reference to Georges Cuvier [1769–1832], who was a proponent of the idea that the fossilised remains were those of species now extinct due to of mass-extinctions of natural cause, rather than the remains of living species or species yet to be found in remote areas. Notably, he demonstrated that Indian and African elephants were distinct species, and that they were also distinct from fossil mammoths.

2.5 System-level adaptation and feedback mechanisms

41

to maintain the value of a state-variable (e.g. phosphorus concentration) in spite of changes in external forces effecting that variable (e.g. phosphorus loading rate). Because this ability improves the survival of the ecosystem, Jørgensen & Mejer (1979) states “A species which can adapt itself to changed environment will survive – this is a well accepted theory (compare with Darwin) – and, in accordance with the theory presented here, the same is equally valid for the ecosystem. The ecosystem structure which can adapt itself to changed circumstances will survive.” It is proposed that the survival of an ecosystem will feed back to the species in the system, such that those species with attributes that promote the survival of the system will be favoured. Jørgensen et al. (1995) states that the goal function approach is an extension of natural selection to the level of the ecosystem. “... those [organisms and species] with the properties best fitted to prevailing conditions, including the conditions determined by the presence of other species, will survive. If, however, we have to account for the simultaneous struggle of survival by all the species present in an ecosystem, we have to consider the level of the ecosystem.”

2.5.3

Evaluating validity

System level adaptive fit theories identify some attribute of systems that are beneficial to the survival of the system (“fitness”), and reason that this benefit is sufficient to encourage system development in the direction of this attribute’s increase. In this example, we have discussed stability measures, but the lessons drawn from it may be applied to any analogous measure of system fitness. While the instability of a system may be attributed to one or more component parts, the instability itself manifests on the level of the system, and hence affects the entire system. Because the deleterious effect of instability caused by any part(s) of the system will affect all parts of the system, the feedback mechanism will not target the particular parts causing the instability. This ‘feedback misdirection’, which is the subject of Chapter 8, is overlooked by the proponents of system-level adaptation mechanisms. For example, when considering the apparent self-regulation of the reproductive rate of oceanic birds, Dunbar (1960) states: “I suggest that these are highly evolved stable populations which have in the past been subjected to the stress of oscillation in an oscillating system, and that they have responded to selection for this self-regulating character of a restricted breeding rate, tending toward stability.” However, the deleterious effect of over-breeding will feed back to both the individuals that breed too much, and those that breed less, presumably without bias. It is only by luck (or by some unspecified

42

Goal functions

mechanism) that the deleterious effects of oscillation will select those individuals that will reduce those oscillations (see also Chapter 8). A similar error is apparent in the development of the exergy goal function, as discussed in Appendix A.7. In addition to feedback misdirection, there can exist organising tendencies at lower hierarchical levels which will conflict with the system-level objectives. Therefore, in order to justify the preeminence of system-level goal functions, one must be able to demonstrate that the feedback from the system-level, already weakened by misdirection, can over-ride other selection pressures on individuals and populations.

2.6 2.6.1

Succession Introduction

A typical ‘textbook definition’ of succession is “the non-seasonal, directional and continuous pattern of colonisation and extinction on a site by species populations” (Begon, Harper & Townsend 1996, pp. 692-693). Such a definition is often accompanied by examples of succession, such as the pattern of vegetative reclamation of abandoned fields (see, for example, Whittaker (1975, p. 171), Colinvaux (1993, p. 419-420), Billings (1938)). During succession, the properties of the ecosystem as a whole have been observed to change in a predictable and repeatable way (see Table 2.3). These patterns are observed in a variety of ecosystems, suggesting that they may be indicative of some universal attribute of ecosystems as wholes. This has inspired the search for unifying principles and mechanisms driving and governing these observed patterns (Shugart 1984, p. 15-16). Systems ecologists have used succession trends to formulate goal functions, and speculate as to their meaning and interpretation (Christensen 1994, Christensen 1995, Fontaine 1981, Jørgensen & Mejer 1981). Conversely, with the underlying assumption of some driver of directive development in ecosystems (e.g. adaptation, emergent properties, stability or Prigoginean concepts), goal function theorists have interpreted the changes that occur during succession as being a result of these drivers. In this section, we will review succession, from its beginnings with Clements, through to the Odum brothers, and its reinterpretation as a ‘thermodynamic’ theory. We find that there is a close relationship between early succession theory and goal function theory, both philosophically, and in the attributes thought to exhibit developmental trends.

2.6 Succession

2.6.2

43

Clementsian succession

The Clementsian theory F. E. Clements was one of the earliest ecologists to construct a theoretical framework for succession, which has come to be known as Clementsian succession (e.g. McIntosh 1981). Clementsian succession is based upon the analogy between the development of the ecosystem (governed by climate) and the development of the individual (Clements 1916, p. 3). This implies a predetermined route from bare habitat to the climax. The climax is described as having the following attributes (Clements 1936): 1. Unity: The climax formation is an expression and indicator, as well as response to, the overarching driving force of climate. The dominant species associated with a particular climax have the most definite relation to the climate, and the greatest influence upon habitat for subordinate species. 2. Stability: The climax formation is stable, in the sense that it persists and is self renewing. The stable climax is a result of the balancing of reactions between habitat and population, and is maintained by the dominant species by their influence upon the habitat, and the ability of the dominant species to control environmental factors (Clements 1916, p. 98). 3. Evolutionary history: Each climax has evolved from a previous climax, hence a climax has an ontogeny and phylogeny. Climaxes with a common ancestral climax, an eoclimax, are termed panclimaxes. They have developed distinct attributes due to the separation of continents and subsequent isolation allowing differentiation, however, they will have similar life-form and dominants. 4. Inevitability: The climax is the ultimate expression of the predetermined life-form suitable for a particular climate. Ecosystems that do not have the predicted attributes suited to their climate are considered to be deviations and aberrations from the norm. Such situations (such as the subclimax, disclimax) are generally caused by disturbances. The above attributes describe to a perfectly adapted and attuned ecosystem. Its stability is partly due to the balance of reactions, and partly due to the influence of the dominant species. This stability requires some (undefined) absence of instability in the environment, which may cause the ideal life-form to regress to a less mature state. Most importantly, the ecosystem acts as a contiguous whole. The Clementsian theory proposes that “the character of successional development depends more upon the nature of the climax than upon anything else” (Clements 1916, p. 5), and that the climax formation is the one “which is in entire harmony with the climate” (Clements 1916, p. 79). The correlation between dominant species - coniferous, deciduous, and broad-leaved evergreen, and their respective climate zones - boreal, temperate, and tropical is a result of the importance of climate as the overarching organising

44

Goal functions

principle (Clements 1936). The nature of the equilibrium is predetermined by the climate alone, and atypical climax formations are considered aberrations resulting from some force that has inhibited the attainment of maturity, such as environmental perturbations (Clements 1936). The climax is described as the “highest stage possible under the climatic conditions present” (Clements 1916, p. 4), with the process of succession itself being the “change from lower to higher life-forms” (Clements 1916, p. 6). The implication is that the ecosystem strives to increase complexity, and that the climate provides the bounds within which this may be achieved. The climax is characterised by the attainment of relative stability and changes in the environment allowing the growth of a wider range of species due to less extreme environmental conditions (Clements 1916, pp. 95–100). Each stage in the succession improves the conditions for growth of a greater number of species by its reactions (e.g. soil-formation), which reduce the extreme conditions of nudation at the start of the succession (Clements 1916, p. 79, 98). Firstly, the pioneers react with the environment until the reactions are either unfavourable to the pioneers or favourable to new invaders, who then produce reactions (e.g. competition) that are unfavourable to the pioneers (Clements 1916, p. 79). As succession continues, “habitat and population act and react upon each other, alternating as cause and effect until a state of equilibrium is reached” (Clements 1916, p. 6). This process, the preparation of the site by one species for colonisation by another (often to the detriment of the first species), has come to be known as facilitation (Connell & Slatyer 1977). Such a perspective has proved fruitful in motivating theorists to ask intriguing questions about ecosystem function and form. For example, by making an analogy between the mature ecosystem and the mature individual, the question of whether or not an ecosystem ‘ages’ with the same detrimental effect as individuals is raised (Odum 1969). Similarly, Margalef (1968, p. 30) suggests that the developing ecosystem gathers ‘information’ about the environment, lending it stability and resistance, however this same information causes the ecosystem to become locked into a particular form. However, despite the interesting implications of the organismic analogy, theorists have questioned its theoretical validity. We will discuss some of these questions below.

Criticisms of the Clementsian theory Clementsian succession relies upon feedback between the biotic and abiotic compartments of an ecosystem, such that the ecosystem develops toward a climax with perfect attunement to the climate. Drury & Nisbet (1971) describe Clementsianism as “the idea that the members of a community react among themselves while the community as a whole reacts with the physical environment”. By analogy with the development of an individual, Clementsian theory associates succession with an increase in complexity toward a stable

45

2.6 Succession

“higher life-form” (Clements 1916, pp. 4,6). This has been challenged by the individualist theory, which states that each species’ attributes and actions are determined by natural selection. Individualism is the statement “that every species of plant is a law unto itself” (Gleason 1926), and it follows that “generalisations about the behaviour of communities should be viewed with caution unless they can be reconciled with the action of natural selection on the individual organism” (Drury & Nisbet 1973). Facilitation of competing later species by early successional species can run contrary to natural selection (Drury & Nisbet 1973), and counter-examples such as allelopathy have been found (Whittaker & Feeny 1971, Drury & Nisbet 1973). Focusing upon facilitation alone ignores the full variety of effects of early successional species upon later species. In addition to facilitation, there can be: 1. inhibition; a negative effect from early species upon later species, and 2. tolerance; a negligible effect from early species upon later species (Connell & Slatyer 1977). The predictability of the temporal ordering of species can be explained without relying upon facilitation alone. For example, Horn (1976) described a matrix of replacement probabilities of one species over another during succession. The probabilities represent facilitation, inhibition, and tolerance. It is modelled as a Markov process: Xt+1 = AXt

(2.4)

where Xn is a vector of number individuals from each species at time step t, and A is a matrix of elements ai,j (0 ≤ ai,j ≤ 1) which describe the interactions between early and later species. The implication of this is that succession only depends upon the species distribution at a given time, rather than some future climax. Further, the elements of A, which include competition, tolerance, and facilitation, demonstrate that such a system can behave in a predictable way without having to assume that the only reaction between species is one of facilitation. As another example, the life strategies hypothesis (Drury & Nisbet 1973, MacArthur & Wilson 1967, Grime 1979) states that “certain adaptive strategies are mutually exclusive: species whose seeds travel far and grow fast in harsh conditions cannot also grow large and live long” (Drury & Nisbet 1973). Methods of arrival or persistence during and after disturbance, the ability to establish and grow to maturity, and the time taken to reach certain life stages (e.g. juvenile, reproductively mature, etc.), are used to predict a plants position in the successional sequence (Noble & Slatyer 1980). Once again, neither facilitation nor the organismic perspective is required to explain the orderly behaviour of the system.

2.6.3

The Odum brothers and succession

The properties of ecosystems have been observed to change in a predictable way during succession. Under the Clementsian framework, the changes were primarily described in terms of changes in species

46

Goal functions

composition. In contrast, later work on succession described changes in terms of system level properties. Common properties were production, production to biomass ratio, and stability. For example, Cooke (1967) described the succession occurring of a laboratory microcosm in terms of properties listed in Table 2.2. Odum (1969) summarises the succession trends in the empirical literature (including the work of Cooke (1967)) in a “tabular model of ecological succession”. This table is reproduced in Table 2.3.

Table 2.2: “Trends observed in autotrophic succession of an aquatic microecosystem.” Adapted from Cooke (1967) Characteristic Developmental stage immature mature Gross community photosynthesis high low Net community photosynthesis high low Community Night community respiration high low energetics Production/Respiration above one below one Biomass/Gross photosynthesis low high Gross photosynthesis/Chlorophyll low high Community Biomass (organic matter) low high structure Chlorophyll high low Biochemical diversity (pigment ratio) low high low System maintenance low high Efficiency Production high low Chlorophyll (assimilation number) low high Stability Metabolism low high Structure (organic matter) low high

Odum (1969) defines succession as follows. The first point acknowledges that patterns occur in ecosystem succession that are predictable, but it is the latter two points that assign these patterns meaning. “(i) It is an orderly process of community development that is reasonably directional and, therefore, predictable. (ii) It results from modification of the physical environment by the community; that is, succession is community-controlled even though the physical environment determines the pattern, the rate of change, and often sets limits as to how far development can go. (iii) It culminates in a stabilized ecosystem in which maximum biomass (or high information content) and symbiotic function between organisms are maintained per unit of energy flow.” Point (ii) recalls Clementsian concepts of the environment and the biota acting and reacting upon one another, until an equilibrium is reached. It is apparent in a later work that E.P. Odum still considers facilitation an important part of this process. He states “Species replacement in the sere occurs because populations tend to modify the physical environment, making conditions favourable for other populations until an equilibrium between biotic and abiotic is achieved (Odum 1971, pp. 251).”

47

2.6 Succession

Table 2.3: “A tabular model of ecological succession: trends to be expected in the development of ecosystems.” (Odum, 1969) Ecosystem attributes

1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11.

12. 13. 14.

15. 16. 17.

Developmental stages Community energetics Gross production/community Greater or less respiration (P/R ratio) than 1 Gross production/standing crop High biomass (P/B ratio) Biomass supported/unit energy Low flow (B/E ratio) Net community production (yield) High Food chains Linear, predominantly grazing Community Total organic matter Inorganic nutrients Species diversity - variety component Species diversity - equitability component Biochemical diversity Stratification and spatial heterogeneity (pattern diversity)

Niche specialization Size of organism Life cycles

Growth form

19.

Production

23. 24.

Low Poorly organized

Life History Broad Small Short, simple

Nutrient cycling Mineral cycles Open Nutrient exchange rate, between Rapid organisms and environment Role of detritus in nutrient Unimportant regeneration

18.

20. 21. 22.

Structure Small Extrabiotic Low Low

Selection pressure For rapid growth (“r-selection”) Quantity

Overall homeostasis Internal symbiosis Undeveloped Nutrient conservation Poor Stability (resistance to external Poor perturbations) Entropy High Information Low

Mature stages

Approaches 1 Low High Low Weblike, predominantly detritus

Large Intrabiotic High High High Well-organized

Narrow Large Long,complex

Closed Slow Important

For feedback control (“K-selection”) Quality

Developed Good Good Low High

48

Goal functions

However, Point (iii) shows that the emphasis has moved from the Clementsian climactic control to stability as the major organising force. E.P. Odum’s definition is continued with the theme of homeostasis (which was reiterated by his brother, H.T. Odum) “... the “strategy” of succession as a short-term process is basically the same as the “strategy” of long-term evolutionary development of the biosphere, namely, increased control of, or homeostasis with, the physical environment in the sense of achieving maximum protection from its perturbations” (Odum 1969, Odum 1983, p. 251).

2.6.4

Reinterpreting succession

The E.P. Odum Tabular Model has been repeatedly cited in the goal function literature as an early attempt to predict ecosystem development, and as a source for goal functions. Examples include the use of Odum’s (1969) tabular model of ecological succession to determine the success of newer goal functions (Christensen 1994, Christensen 1995), and the use of proposed climax attributes as goal functions (e.g. Fontaine 1981, Jørgensen & Mejer 1981). One can identify the goal functions employed in Cropp & Gabric (2002) in Table 2.3. From reading the goal function literature, one may get the impression that the E.P. Odum Tabular Model was derived from Prigoginean concepts. Schneider (1988), for example, states that “perhaps no other ecologists have advanced the principles of thermodynamics in ecology farther than the brothers Eugene and Howard Odum”. Fath et al. (2001) acknowledges the similarity between goal functions (which they call “orientors”) and the trends described in the E.P. Odum Tabular Model. “In a large part, these orientors follow from the seminal work of Odum (1969) in which he hypothesized on the trends to be expected in ecosystem development.” It is true that the Odum brothers linked their succession trends to thermodynamic concepts, however this was done after the trends were observed in real systems. For example, on one hand, E.P. Odum describes the respiration to biomass ratio (R/B) in thermodynamic language (Odum 1971, pp. 38–39). “As Schrodinger has shown, the continual work of pumping out “disorder” is necessary if one wishes to maintain internal “order” in the presence of thermal vibrations in any system above absolute zero temperature. In the ecosystem the ratio of total community respiration to the total biomass (R/B) can be considered as the maintenance of structure ratio, or as a thermodynamic order function. ... If R and B are expressed in calories (energy units) and divided by the absolute temperature, the R/B ratio becomes the ratio of entropy increase of maintenance (and related work) to the entropy of ordered structure.”

2.7 Further examples of reinterpretation

49

However, the thermodynamic meaning is used as a description of the trend, rather than a predictive hypothesis “Theoretically, an increase in community respiration ... should be the first early-warning sign of stress since repairing damage caused by the disturbance requires diverting energy from growth and production to maintenance. Hence, the R/B ratio ... increases. Odum (1967) speaks of this response as an “energy drain”, or the process of “pumping out the disorder”” (Odum 1985). The mechanism by which the trends in the value of P/B manifest, as described by Odum (1971, pp. 253), does not require thermodynamic interpretations “As long as P [production] exceeds R [respiration], organic matter and biomass (B) will accumulate in the system, with the result that the ratio P/B will tend to decrease ... Theoretically, then, the amount of standing crop biomass supported by the available energy flow (E) increases to a maximum in the mature or climax stages.” Once the assumption was made that there was some directional development occurring in ecosystems, heuristics and empirical observations such as those made by Odum (1969) become likely candidates for goal functions describing this development. Depending upon the bias of the author, successional trends can be interpreted as the result of adaptive or thermodynamic trends.

2.7

Further examples of reinterpretation

The interpretation of early systems ecology theories into the context of thermodynamic theory is not restricted to successional trends. ‘Maximise flux’ from Lotka (1922), and the similar ‘maximise production’ from the E.P. Tabular Model, have been particularly popular subjects for reinterpretation as newer goal functions. For example, Schneider (1988), states that Morowitz maximum order principle7 is materially identical to Lotka’s maximal flux principle, and Choi et al. (1999) makes a direct comparison between what they term ‘the order through fluctuation principle’ and Lotka’s goal function (Appendix A.5). In a few instances, the goal function has become the cause rather than the effect. Consider, for example, the following statement from Loreau (1995) “... the fact that consumers can maximize energy flow and production in ecosystems is, of course, no guarantee that they will necessarily do so. ... In the long run, however, natural selection should tend to optimize the functioning of natural ecosystems and maximize energy flow or power output (Lotka 1922; Odum and Pinkerton 1955; Odum 1983).” 7 Morowitz

maximum order principle: A dissipative system selects stable states with the largest possible stored energy.

50

Goal functions

But the justification for the tendency to maximise energy flow is not because of the benefit to the individuals, as hypothesised in Lotka (1922), but because of the benefits to the whole system. These benefits are then passed down the organisational hierarchy from the system to the individuals. “Increased production in an ecosystem is beneficial to all its functional components, and traits that result in increased productivity can be selected for in heterogeneous environments even when these traits entail costs for those individuals that bear them (Wilson 1976, 1980).” This theory is reminiscent of the system-level adaptation theories discussed in Section 2.5. As a result of the above reasoning, Loreau (1995) proposes that consumption rates will serendipitously settle upon a moderate value, such that the benefit to the consumers is preserved, yet consumption will not be so high as to reduce the biomass of producers such that the whole-system matter circulation is constrained. Another example is H.T. Odum’s ‘maximal power’. He defines power as “the energy flow per time” and conceptualises power as “the rate of flow of energy into useful work” (Odum 1983, p. 6). He interpreted other goal functions8 as special cases that lead to the criteria that power is maximised. M˚ ansson & McGlade (1993) critiques H.T. Odum’s work, stating “In view of the fundamental importance of the maximum power principle to Odum’s framework, it should be discomfiting to its proponents that the mechanisms which are supposed to result in the maximisation of power remain to be elucidated.” Odum (1983, p. 6) states by way of mechanism that “A major design principle observed in natural systems is the feedback of energy from storages to stimulate the inflow pathways as a reward from receiver storage to inflow source.” This is reminiscent of Lotka’s (1922) maximal flux, where natural selection rewards those compartments that increase the energy flow through the system. However, once again, the concept has been extended outside of its original context. Maximal flux no longer just to individuals, but has become a “general design principle of self-organising systems” (Odum 1983, p. 101).

8 Such as maximisation of biomass, reproduction rate, minimum entropy generation, maximum profit, stability, and efficiency.

2.8 Goal functions of specific interest in this thesis

2.8

51

Goal functions of specific interest in this thesis

2.8.1

Introduction

Having discussed the types of goal functions, and identified their themes and shortcomings, we now discuss the goal functions used in Cropp & Gabric (2002). The use of strict maximisation of the goal functions in Cropp & Gabric (2002) is inconsistent with later works in the goal function literature. For example, Fath et al. (2001) describes the ‘maximise’ and ‘minimise’ of the goal functions as indicators of the direction that goal functions are thought to act, rather than absolute imperatives. Ulanowicz & Abarca-Arenes (1997) also shares this perspective. This work states “If there is a propensity for B to follow A in the system, then after most times that A has been observed, B ensues – but not each and every time. On occasions C, D or some other circumstance might result. In the same vein, it is not being claimed that ecosystems maximize or optimize their ascendencies (although Ulanowicz (1980) had prematurely suggested this).” The only constraint that was placed upon the goal functions in Cropp & Gabric (2002) was that parameter values were varied by 50% around the mean values reported in the literature. Using strict maximisation of goal functions, irrespective of other constraints upon the system, is difficult to justify. It also changes the goal functions from being the result of lower level causes, to being the causes themselves, because the lower level causes which would restrict the extent of the maximisation of the goal functions are not included in the model (c.f. adaptive dynamics below). Cropp & Gabric (2002) states that the investigations used “a stochastic GA that simulates individual adaptation in response to selection pressures over evolutionary time scales.” This statement is misleading. First, the genetic algorithm, while borrowing from the principles behind natural selection, is not meant to simulate natural selection in real ecosystems. Other models are available for this, notably adaptive dynamics models (Dieckmann 1996). Adaptive dynamics takes into account the trade-offs that exist between the attributes of an individual. For example, a prey may pay for increased foraging ability with an increased probability of being found by its predator and eaten. In contrast, the genetic algorithm puts no constraints upon the changes that can occur to sets of parameter values, allowing the organisms to simultaneously optimise attribute pairs that may be mutually exclusive due to architectural constraints. Second, the phytoplankton and zooplankton compartments modelled in Cropp & Gabric (2002) are composed of many species of plankton. It is generally held (e.g. Jørgensen 1986b) that goal functions act to change the species composition of a compartment via feedback from the system level, rather than affecting the fitness of individuals, and causing evolution of populations, in the familiar Darwinian way.

52

Goal functions

As the dynamics permitting the coexistence of many species of plankton remains an open question in marine ecology (Hutchinson 1961, Peterson 1975, Allen 1977, Harris & Smith 1977, Connell 1978, Chesson & Huntly 1997, Sommer 1999, Huisman & Weissing 1999), it is difficult to describe the mechanisms that lead to the dominance of particular species of plankton, and the effect of this upon the aggregate attributes of the compartment. Table 2 of Cropp & Gabric (2002) lists the goal functions employed: 1. Maximise autotroph biomass; 2. Maximise heterotroph biomass; 3. Maximise gross primary production; and 4. Maximise primary production per unit biomass. In the remaining sections of this chapter, we discuss each of these goal functions in turn.

2.8.2

Maximise biomass

Exergy Cropp & Gabric (2002) cites the exergy goal function as justification for using maximal autotrophic and heterotrophic biomass as goal functions. We are interested, therefore, in what exergy is, and how readily it may be applied in this context. Table 2.4 shows the key developments of the exergy principle over time. A full description of these works may be found in Appendix A.7. Initially, exergy was used as a goal function because of its association with the buffer capacity of the ecosystem. Buffer capacity is the ability of an ecosystem to withstand changes in the face of environmental perturbations (Jørgensen & Mejer 1977). Originally, it was proposed that exergy was maximised by a mechanism similar to system selection sensu Dunbar (1960) (Jørgensen 1986b). Recent works continue to refer to associations between stability properties and exergy, and refer to a Darwinian system-level feedback mechanism that maximises this stability (Jørgensen et al. 1995, Jørgensen 1999, Jørgensen 2000). As the maximal exergy goal function has been built upon a system-level selection concept, all of the criticisms previously discussed in Section 2.5.3 apply here, namely feedback misdirection, and the assumption that the feedback from the selection pressure on the system-level will successfully over-ride other lower-level selection pressures. Later works in the exergy literature focus upon a coevolutionary mechanism. Higher organisms are given a higher exergy weighting to account for their higher genetic information. This genetic information allows them to exploit lower organisms into seemingly altruistic behaviour. As such, models using exergy often change attributes in only one compartment to maximise the survival of other compartments. Whether or


53

not such informational advantage is sufficient to explain this apparent altruism has not been investigated. Further, some works report the opposite affect (Marrow, Law & Cannings 1992). Several works describe exergy as a thermodynamic imperative (Svirezhev 2000, Jørgensen 1986b, Jørgensen 1986a). It is proposed that exergy is a measure of how far a system is from thermodynamic equilibrium, and hence the principle of the maximisation of exergy represents a new goal function, in which the distance from thermodynamic equilibrium is maximised. However, this is not a theory based in Prigoginean concepts, as there is nothing in the original Prigoginean framework that suggests that the distance from thermodynamic equilibrium (something which is created by the appearance of dissipative structures) is maximised. Table 2.4: A summary of the history of the exergy goal function. Authors Developments Jørgensen & Mejer (1977) Note that sediment gives ecosystem buffer capacity, that is, the ability to withstand changes in nutrient concentration. Buffer capacity increases with two measures of complexity: diversity, and exergy. State that exergy also measures “mechanical energy equivalent of the distance from thermodynamic equilibrium.” Jørgensen & Mejer (1981) Exergy as a measure of information incorporated in the structure of the ecosystem, and ability to meet perturbations. Jørgensen (1986b) Suggest that radical changes to an ecosystem may be buffered by changes in ecosystem structure and species composition. Selection is based upon species’ ability to increase biomass and organisation, which is measured by exergy. Exergy described as distance from thermodynamic equilibrium. Jørgensen et al. (1995) Links selection of species to natural selection, “ however, we have to account for the simultaneous struggle of survival by all the species present in an ecosystem, we have to consider the level of the ecosystem”. Modified exergy to incorporate higher organisation of higher also Marques et al. (1997) organisms. Jørgensen (1999) Provides detailed explanation of exergy. Expands upon natural selection link. By measuring species survival in terms of biomass, suggest exergy as a method of simultaneously optimising all biomass. Biomass of higher organisms are given higher weighting due to their ability to use the higher information in their genes to “obtain a better survival”. The use of biomass as a proxy for exergy in Cropp & Gabric (2002) is not true to the original formulation, neither in the way that it is measured, nor in the mechanism by which it is supposed to act. Maximising exergy is not equivalent to maximising biomass sensu Cropp & Gabric (2002). While biomass is a contributor to the exergy of the system, exergy quantifies more than simply biomass. A simple maximisation of autotroph and heterotroph biomass does not measure the distance of the system from the thermodynamic equilibrium sensu Jørgensen & Mejer (1977). Further, the formulation in Cropp & Gabric (2002) makes no attempt to quantify the information content of the compartments sensu Marques et al. (1997), thus

54

Goal functions

ignoring later developments in exergy as well. ‘Maximise biomass’ sensu Cropp & Gabric (2002) would be better described as a heuristic, based upon E.P. Odum’s Tabular Model.

2.8.3

Maximise flux

The use of ‘maximise flux’ is described in Cropp & Gabric (2002) as being based upon the maximum ascendency principle in Ulanowicz (1980). Ulanowicz (1980) quantifies ascendency as follows (equations and a worked example are given in Appendix A.8): A = throughput × average mutual information

(2.5)

The average mutual information measures the information content in the flows, or how constrained those flows are (Ulanowicz & Kemp 1979). This is an indicator of development. Nutrient throughput is used to scale the equation. Its increase implies growth of the system (Ulanowicz 1980, Ulanowicz 1997, pp. 63,73–73). Despite the inclusion of throughput in Equation 2.5, Ulanowicz (1997, pp. 66–71) states that the main purpose of the equation is to measure flow organisation. Ulanowicz (1997, pp. 62) states “There may or may not be a mechanism that gives rise to the observed probabilities. If not, we could still weigh whether the probabilities might be the result of a formal or final agency. What we cannot do under any circumstances, however, is dismiss a propensity calculated on such a material basis as being devoid of causal content. The transfers palpably occurred; whenever they happen with significant frequency, the underlying propensity demands our serious consideration.” Ascendency is an indicator of organisation, not the mechanism or cause of the organisation itself. However, much theory has been developed to explain why ascendency increases in ecosystems. For example, Ulanowicz (1997, pp. 76–77) describes “the chance discovery that each of these trends [in the E.P. Tabular Model] was a separate manifestation of increasing mutual information in trophic networks”, which he explained with “heuristic arguments centred on autocatalysis”. The quotation from Ulanowicz (1997) below summarises the argument (cf. Odum 1969, Odum 1983, p. 6) “In response to stimuli experienced by the individual, certain connections [in the developing brain] are facilitated, and others inhibited. The neural system develops. Although the overall capacity of the brain remains more or less constant, the fraction of connections entrained by structured responses (ascendency) continues to climb until senility.” Ascendency has also been described as a measure of the (thermodynamic) efficiency of energy utilisation (Nielsen & Ulanowicz 2000), and as a measure of cycling and other succession heuristics of the E.P. Odum Tabular Model (Ulanowicz & Abarca-Arenes 1997). However, the fact that it is listed as a goal


55

function in Cropp & Gabric (2002) suggests that what was once proposed as an indicator of directional development has become interpreted as the directional development itself. Aside from the fact that ascendency is not a goal function, there are several other problems with how it was used in Cropp & Gabric (2002). First, the flux measure in Cropp & Gabric (2002) differs significantly from the ascendency measure. The flux in Cropp & Gabric (2002) does not incorporate the the organisational aspect of ascendency, nor are changes in structure investigated. However, the measurement of these aspects is the very purpose for which ascendency was created, with the nutrient flow term merely added in order to scale the expression after ascendency was developed (Ulanowicz 1997, pp. 66–71). The flux measure in Cropp & Gabric (2002) does not correspond to the throughput measure used by Ulanowicz (1980). Ulanowicz (1980) measures total flows as a sum of throughput for each compartment, whereas Cropp & Gabric (2002) measures throughput through the nutrient compartment only. Comparing the worked example in Appendix A.8 and the description in Cropp & Gabric (2002) makes this difference apparent. Fortunately, this makes no material difference at the point of maximisation, as the point in parameter space that maximises throughput is the same as the point that maximises flux (due to the model in Cropp & Gabric (2002) being closed to nutrient transfer outside of the system).

2.8.4

Maximise Production to biomass ratio

The following works hypothesise that the production to biomass ratio (P/B) of an ecosystem decreases as ecosystems develop: works making observations about succession (Odum 1969, Cooke 1967), works that interpret P/B as a proxy for entropy production (e.g Choi et al. 1999, Fath et al. 2001), and works that subscribe to Margalef’s (1968) cybernetic perspective. Cropp & Gabric (2002) acknowledges this, yet states “We elect to maximize P/B in our selection pressures according with the hypothesis of maximum power capacity (Lotka 1922, Odum and Pinkerton 1955) and entropy production (Johnson 1990).” Lotka (1922) and Odum & Pinkerton (1955) consider the increase in P/B the key to competitive advantage. Lotka (1922) describes a situation in which the biomass of a system had reached its upper physical limit. It was argued that, in such a situation, flux would still be maximised because of the competitive advantage energy provided to those who could utilise it for “application to their life tasks and contests”. However, this justification is inconsistent with the observation in Cropp & Gabric (2002) that “the predicted herbivore attributes ... are both counterintuitive and contrary to the attributes expected from individual-based evolution. The herbivore attributes result in organisms that are less fit to compete for limiting resources at the individual level”.

56

Goal functions

The way in which P/B is maximised in Cropp & Gabric (2002) is not consistent with the theory described in Lotka’s (1922). To apply maximum flux sensu Lotka (1922), one would need to allow structural changes in the model, such that new nutrient compartments, representing the novel sources of energy, could be added incrementally, and new compartments exploiting these energy sources and their associated flow pathways could develop. This would change the structure of the food web diagram, which was not explored in Cropp & Gabric (2002). Johnson (1988) attempts to reconcile the differing perspectives on the expected direction of development of P/B by stating that P/B decreased over successional time, and increased over evolutionary time. The work states that “... it is the species, individually, that proceed to a ‘least value’ of the P/B ratio while the interaction between species tends to increase the value of the P/B ratio.” The justification for this is based upon results of a studies on Arctic lakes, where the presence of the dominant species stimulated energy flow at lower hierarchical levels. Irrespective of its validity, Johnson (1988) states that the trend of increased production is masked by the species’ trend of decreasing the production to biomass ratio. Cropp & Gabric (2002) states that the “... two views are not incompatible, as our model ecosystem maximizes P/B over evolutionary time, but minimizes P/B over shorter time scales, such as when responding to a perturbation.” An earlier work, Cropp (1999, pp. 52), presents a figure showing the response of the time averaged P/B ratio after a perturbation. In response to biomass removal, the time averaged P/B ratio is found to increase, and then decrease (see Appendix A.9). However, this property is not peculiar to the maximally resilient system, as implied in Cropp & Gabric (2002). When biomass is reduced, the system must move from low biomass toward steady state. It does this by temporarily increasing the flow into the biotic compartments, thus increasing the production to biomass ratio. As the biomass accumulates, the production to biomass ratio decreases, until steady state. It can be shown that the minimally resilient system has a similar response to perturbation, as this response is a result of the dynamics, not the specific parameter values (Appendix A.9). As previously mentioned (Section 2.8.3), the model does not have the capacity to simulate novel energy sources, and new flow pathways, and therefore cannot model the maximisation of flux sensu Lotka (1922). The use of P/B in Cropp & Gabric (2002) is most consistent with the citation of Odum (1983), who considered ‘maximise power’ to be a general property of self-organising systems. As such, the use of ‘maximise production to biomass ratio’ in Cropp & Gabric (2002) is subject to the same criticisms (Section 2.6.3).

2.9 Conclusion

2.8.5

57

Concluding remarks

We have investigated the four goal functions used in Cropp & Gabric (2002). We have found that the way that these goal functions were used in Cropp & Gabric (2002) was often inconsistent with their original meaning, and inconsistent with how they are used in the goal function literature. The strict maximisation and minimisation of the goal functions is inconsistent with recent goal function theory, and the constraints upon the parameter ranges were arbitrary. Further, the implication that the genetic algorithm simulated ecosystem adaptation was inaccurate. The way in which the goal functions were used was not consistent with the literature that was cited to justify their use. The ‘maximise biomass’ goal function was justified with reference to ‘maximise exergy’ of (Jørgensen & Mejer 1977). However, the Cropp & Gabric (2002) work did not quantify the distance from ‘thermodynamic equilibrium’, which is usually done using an estimate of the genetic information content associated with the biomass of each compartment. The ‘maximise flux’ goal function was justified with reference to ‘maximise ascendency’ of Ulanowicz (1980). However, the Cropp & Gabric (2002) work did not include the average mutual information term, which is the central purpose of ascendency. The ‘maximise production to biomass ratio’ goal function was justified with reference to ‘maximise flux’ sensu Lotka. However, the Cropp & Gabric (2002) model has no capacity for the structural changes in the energy flows, which are the central mechanism of the hypothesis in Lotka (1922). The goal functions were most consistent with the succession trends reported in the E.P. Odum Tabular Model. As such, they might be interpreted as succession heuristics, rather than ecological imperatives.

2.9

Conclusion

Goal functions are functions that are thought to reflect the directional development of ecological systems over time. They are an attempt to explain and predict the orderly behaviour of ecosystems. In this chapter, we have found that the theoretical basis for goal function theory is controversial. Difficulties include: the simplifications involved in developing goal functions, the obfuscation of the assumptions underlying the application of thermodynamic concepts to ecosystems, an unclear description of the mechanism by which goal functions operate, and the imposition of particular goal function concepts upon ecosystem processes that could be explained by more parsimonious mechanisms. The latter includes the interpretation of successional trends as being caused by non-equilibrium thermodynamical concepts, when such a hypothesis is not tested, and is difficult to justify when the exact mechanisms by which it operates are unclear. Often, goal functions are borrowed from fields outside of ecology. While such exchanges of ideas have the potential to generate creative new ideas, the theories can also be mistranslated, leading to their misuse,

58

Goal functions

and obfuscation of their meaning. It has been assumed that ecosystems conform to non-equilibrium thermodynamics. As a consequence, their organised behaviour is often assumed to be a result of this property, irrespective of whether or not this hypothesis can be tested. For example, the minimisation of the production to biomass ratio has been interpreted as a result of non-equilibrium thermodynamics, where the ratio is a proxy for entropy production. However, there are other, simpler, explanations for this minimisation. Further, there is no way of verifying that the patterns are the result of thermodynamic imperatives when it is not clear how the assumptions inherent in non-equilibrium thermodynamics apply. Succession theory has provided both theoretical and empirical inspiration for goal function theory. As discussed above, the interpretation of successional trends as being evidence for thermodynamic imperatives in Odum (1983) continues in recent works (e.g. Fath et al. 2001), and the Clementsian emphasis on stability, reiterated by the Odum brothers, is reflected in newer goal functions such as exergy (Jørgensen & Mejer 1979). The goal functions in Cropp & Gabric (2002) are subject to the criticisms above. Further, with these criticisms set aside, the way in which the goal functions are used in Cropp & Gabric (2002) is not consistent with the goal function literature. Cropp & Gabric (2002) describes the goal functions as “thermodynamic and ecological imperatives”, however we would suggest that a better description for them would be ‘heuristics based upon succession theory’, in light of their similarity to the trends described in the E.P. Odum Tabular Model. We will refer to the goal functions used by Cropp & Gabric (2002) as traditional goal functions, in light of the difference between them and the current goal function literature, and in recognition of their historical links to the Clementsian succession theory and the E.P. Odum Tabular Model. In addition to citing thermodynamic goal functions as their justification for proposing ‘maximise resilience’ as a goal function, Cropp & Gabric (2002) states that “highly resilient ecosystems are more likely to remain extant than ecosystems with low resilience”. We have briefly discussed the use of stability as a goal function in this chapter. In the next chapter, we will discuss the stability literature, and the place of resilience and goal function theory within it.

Chapter 3

Stability 3.1

Introduction

3.1.1

Motivation

The primary reason given in Cropp & Gabric (2002) for proposing ‘maximise resilience’ as a goal function is the relationship between thermodynamic goal functions and maximal resilience. In Chapter 2, we discussed the thermodynamic goal functions used and found that the theory behind them was incomplete, and that the way in which the goal functions were used in Cropp & Gabric (2002) was inconsistent with the theory. As a secondary justification for proposing ‘maximise resilience’ as a goal function, Cropp & Gabric (2002) makes the following statements: “Although there is little thermodynamic or ecological evidence to suggest [resilience] is a legitimate selection pressure, ecological networks that develop stabilizing feedbacks are considered to be more likely to remain extant than those that do not (Lenton 1998).” Further, the work states: “There is ... no a priori indication that the optimization of an ecosystem’s response to thermodynamic or ecological imperatives should result in a maximally resilient ecosystem. This outcome, however, is reasonable, given that all ecosystems exist within the constraints of thermodynamic laws and that highly resilient ecosystems are more likely to remain extant than ecosystems with low resilience.” In this chapter, we will discuss the meaning of stability, and resilience specifically. We are interested 59

60

Stability

in the use of stability as a measure of survival likelihood, in the hypothesis that ecosystems must be stable, and in its implication that the attributes necessary for stability restricts the attributes that real ecosystems may possess.

3.2 3.2.1

Types of stability What is stability?

Ecological ‘stability’ is one of those overused terms that, although everyone has some intuitive understanding of what it means, there is no generally agreed-upon definition of the term. Grimm, Schmidt & Wissel (1992), for example, identifies 140 different definitions of stability in the literature. Obviously, how stability is defined will depend upon the biases of the author. Peterman, Clark & Holling (1979) refers to stability measures as “myths, simplified views of the world that are not necessarily true but which bring order to the variety of our experience. No single myth applies to all ecological systems, yet each embodies a perspective that has proved rewarding under certain circumstances in the past”. I will therefore present my own bias, and define stability as follows: stability is a proxy measure for ecosystem survival in the face of environmental perturbations. This definition is suitable for the motivation of this thesis, which is to explore stability as a measure of the likelihood with which ecosystems will remain extant, and as an organising constraint upon them. There are several authors who put forward a similar definition to mine (some of which we have discussed in Section 2.5). For example, after reviewing the stability literature, Grimm et al. (1992) named ecosystem persistence as the the ‘central property’ of his three stability properties1 . Obviously, which stability property is ‘central’ depends upon the interests of the author making the judgement, however I argue that there are strong links between stability measures commonly used in the literature and the concept of ecosystem persistence. For example, aspects of a system’s response to perturbation is a common way of measuring stability (e.g. Westman 1986). However, the response to perturbation may be considered an aspect of persistence, in that the “objective of ecosystem persistence requires resistance to and recovery from perturbations” (O’Niell 1976). As another example, stability is often interpreted as some measure of how “well-behaved” a system is (Kay 1991), often in terms of regularity in population densities (e.g. Dunbar 1960, Patten 1975). However once again, a system that is well-behaved is often interpreted as being better able to survive. For example, Soodak & Iberall (1978) state that “complex systems do not act chaotically. Instead they exhibit well-defined chains of behaviour that have been regarded as purposeful, even historical and evolutionary. Everyday language affords many common descriptive usages which mix 1 Grimm et al.’s (1992) three stability properties were: 1. Staying essentially unchanged, 2. Returning to a referential state, 3. Persisting through time. Grimm et al. (1992) considers persistence the most important ecological property, because “the persistence of populations is closely related to the fitness of individuals”.

3.2 Types of stability

61

up teleological purpose with the physical actions that systems must perform in order to survive”. Several authors consider ‘ecological stability’ another phrase for the ‘balance of nature’ (Egerton 1973, Pimm 1991, p. 4), which has been a part of Western thinking since antiquity (DeAngelis & Waterhouse 1987). The modern manifestation of the ‘balance of nature’ is often equated with steady state stability, that is, one that assumes constant populations (e.g. DeAngelis & Waterhouse 1987, Ehrlich & Birch 1967). However I suggest that it is a belief in stability, in its intuitive ‘proxy for survival’ sense, that is the modern ‘balance of nature’. As steady-state measures of stability become less tenable in the literature, new measures, such as permanence, have been developed. Each comes closer to defining what it is that allows ecosystems to survive, but the underlying approach is not changed, it is assumed that ecosystems must be stable in order to exist. The existence of the ecosystem is sufficient proof of survival, at least up until this point in time; if it had not survived, then it would not exist (cf. Ehrlich & Birch 1967, p. 97)2 . The error lies in presupposing a particular mechanism of survival. For example, in response to the hypothesis that plants are resource limited, which promotes system stability, Ehrlich & Birch (1967) state “Having first defined “limited”, it would then be necessary to sample a wide range of plant populations to see how they are “limited””. In other words, just because the system has survived, does not mean that the mechanism by which it survived is the same stability mechanism that happens to be the flavour of the day. Similarly, one must be very specific about the (inevitably inadequate) definition of stability being used, and seek to separate the possible effects of that stability from other pattern-forming influences in the ecosystem (Chapter 11).

We have defined stability as a proxy for ecosystem survival in the face of environmental perturbations. Within this definition, there is considerable ambiguity: Which aspect of the ecosystem is surviving? What are the perturbations that the ecosystem must withstand? Although we could have a very detailed and lengthy discussion about these questions, how they are answered in the context of ecological modelling is often determined by what is mathematically convenient. In the remaining subsections, we will discuss the mathematical definitions of stability that have been used in the ecology literature. We will not be reviewing all stability concepts, but only those that find expression in the types of models with we explore in this thesis.

2 “The existence of a supposed balance of nature is usually argued somewhat as follows. Species X has been in existence for thousands or perhaps millions of generations, and its numbers have never increased to infinity or decreased to zero. ... Such “observations” are made the basis for the statement that population size is “controlled” or “regulated”, and that drastic changes in size are the results of upsetting the “balance of nature”” (Ehrlich & Birch 1967, p. 97).

62

Stability

3.2.2

Local stability analysis

Steady state Ecological systems may be described by the general equation x˙ i = fi (x),

(3.1)

where xi is the nutrient concentration (biomass, or population number) of compartment i, and f is some function describing the rate of change of the compartment. For example, Cropp & Gabric (2002) use f (P, Z)Z = eZ (1 − ηZ )P Z − dZ Z. The steady state, also known as the equilibrium, is the state of a system in which variables do not change with time. The steady state, x⋆ , is described by x˙ i = fi (x⋆ ) = 0.

(3.2)

Equation 3.2 leads to a system of equations that may be solved for x⋆ . If the system of equations is linear (for example, Chapter 6), linear algebra can be used to determine the solution. However if the system of equations is nonlinear, additional techniques may be required (for example, Laws’s (n.d.) method in Chapter 7), and finding a solution may be quite difficult. For ecosystems, there may be several solutions for Equation 3.2. The solutions are grouped into two types: trivial and non-trivial. A trivial steady state is one in which one or more compartments of the system have zero values at steady state. A non-trivial steady state is one in which all compartments in the system have non-zero values. The non-trivial solutions can again be divided into two types: feasible and infeasible. The feasible steady state is one in which all of the steady state values are greater than zero. The infeasible solution is one for which at least one compartment has a negative value. This is shown in Figure 3.1.

Locally stable Consider a ball in two possible positions on a two dimensional landscape, as shown in Figure 3.2. The ball, in both instances, is (stationary or) at steady state, however only the left configuration is locally stable. This means that, while a small push to the ball on the right will send the ball rolling down either slope, the ball on the left will return to its original position after some time. In an ecological context, it is the constancy of population numbers rather than the position of the ball that is the subject of local stability. This type of stability is variously called neighbourhood stability (e.g. May 1973, pp. 13) or Lyapunov stability (e.g. Lewontin 1969).

For a nonlinear system, the local

63


Compartment 1

1111111111111111 0000000000000000 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 Feasible region 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111

Compartment 2

Infeasible region Trivial solutions along axis

Figure 3.1: The phase space of a two-compartment ecological system, where the types of steady-state solutions are indicated.

Locally unstable

Locally stable

Figure 3.2: A diagrammatic representation of local stability (adapted from Pahl-Wostl 1995).

stability can be determined by linearising the system and then analysing the eigenvalues of the system (Kreyszig 1999).

Consider a nonlinear system that is described by two differential equations (this can be generalised for more than two).

dx1 = f1 (x1 , x2 ), dt dx2 = f2 (x1 , x2 ), dt

(3.3)

where fi (x1 , x2 ) is some nonlinear function of x1 and x2 . These state variables could represent the populations of two species, or nutrient concentrations of two compartments. Let the steady state be

64

Stability

(x∗1 , x∗2 ). A multi-dimensional Taylor Series can be used to expand Equation 3.3 to give ∂f1 (x∗1 , x∗2 ) ∂f1 (x∗1 , x∗2 ) (x1 − x∗1 ) + (x2 − x∗2 ) f1 (x1 , x2 ) ≈ f1 (x∗1 , x∗2 ) + ∂x1 ∂x2 1 ∂ 2 f1 (x∗1 , x∗2 ) 2∂f1 (x∗1 , x∗2 ) ∂ 2 f1 (x∗1 , x∗2 ) ∗ 2 ∗ ∗ ∗ 2 + (x1 − x1 ) + (x1 − x1 )(x2 − x2 ) + (x2 − x2 ) , 2! ∂x21 ∂x1 ∂x2 ∂x2 (3.4) and similarly for f2 (x1 , x2 ). If xi − x∗i is small, the higher order terms may be ignored. Given that f1 (x∗1 , x∗2 ) = 0, the system in Equation 3.3 can be approximated by a linear form ∂f1 (x∗1 , x∗2 ) ∂f1 (x∗1 , x∗2 ) d(x1 − x∗1 ) (x1 − x∗1 ) + (x2 − x∗2 ), = dt ∂x1 ∂x2 d(x2 − x∗2 ) ∂f2 (x∗1 , x∗2 ) ∂f2 (x∗1 , x∗2 ) (x1 − x∗1 ) + (x2 − x∗2 ). = dt ∂x1 ∂x2

(3.5)

The ‘coefficient matrix’ is now the Jacobian Matrix, which is defined (for size n) as 

∂  ∂x1

 ∂  ∂x1 J = ..   .  ∂ ∂x1

dx1 dt dx2 dt

∂ ∂x2

∂ ∂x2

dxn dt

∂ ∂x2

.. .

dx1 dt dx2 dt dxn dt

···



∂ ∂xn

··· .. .

∂ ∂xn

···

∂ ∂xn

.. .

dx1 dt  dx2  dt  

.   dxn

(3.6)

dt

The eigenvalues, λi , of the Jacobian matrix, evaluated at equilibrium, can be used to test whether the steady state is stable, and to describe the behaviour of the region around the steady state (but see Section 3.2.4) (Kreyszig 1999). The existence of complex parts in the eigenvalues make the return to steady state oscillatory. The real parts of the eigenvalues determine whether the steady state is locally stable or unstable, and the strength of this attraction or repulsion (in the direction of the eigenvectors). If • Re(λi ) < 0, for all eigenvalues, the steady state is stable, • Re(λi ) > 0, for all eigenvalue, the steady state is unstable, • some λi < 0 and some λj > 0, the steady state is an unstable saddle; and if • λi = 0, for all eigenvalues, the steady state is a limit cycle. Quantifying local stability: Resilience Local stability analysis provides a binary ‘yes’ or ‘no’ answer to whether or not a perturbed system will return to its original state, and some information about the behaviour of the return. However we may be interested in quantifying how strongly a stable steady state attracts trajectories toward it. After DeAngelis (1992), resilience is defined as the inverse of the return time of the ecosystem back to its


65

steady state after a perturbation3 . In a linear system, the steady state is dominated by the eigenvalue with the largest (that is, most positive) real part, max{Re(λi )}. This can be used to quantify the resilience, R. R = −max{Re(λi )}.

(3.7)

For nonlinear systems, the resilience may be quantified with respect to a small perturbation by approximating the system by a linear system near the steady state. The real part of eigenvalue of the Jacobian matrix that is most positive (Equation 3.6), evaluated at equilibrium, quantifies of the resilience around the steady state.

Shortcomings of local stability analysis Shortcoming 1: Linearisation The linearisation of the system about the steady state assumes that the higher order terms are sufficiently small that they may be ignored. While this may be true for a small distance around the steady state, the assumption breaks down at larger distances, causing the information captured by the eigenvalues to no longer hold (Pimm 1982, pp. 21). As we shall see in Section 6.5.4 (and Appendix D.5), the local stability analysis may fail to capture the significant information about the system’s transient and long-term behaviour (Neubert & Caswell 1997). Shortcoming 2: Basin size Returning to our analogy in Figure 3.2, it can be seen that if the ball on the left is pushed sufficiently hard, it may escape the small valley in which it resides. In a dynamical system, the analogue of this valley is called the basin of attraction. While local stability analysis answers whether or not a system will return to its steady state after an arbitrarily small perturbation, it does not measure the extent to which the system may be perturbed and still return (Pimm 1982, pp. 57). It may be possible that the basin of attraction is infinitesimally small. In such a case, within an ecological context, local stability becomes meaningless, as a large enough perturbation can disrupt the system’s function, and thus destroying the aspect of ecosystem survival which we sought to qualify. It is only under certain conditions that local stability implies global stability (Tuljapurkar & Semura 1975, Tuljapurkar 1976), otherwise additional methods are required (Section 3.2.3). 3 This is different to Holling Resilience, which is the perturbation required to disrupt a system and cause it to move to a new equilibrium state (Kay 1991).

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Stability

3.2.3

Sector Stability

Resilience provides a measure of an ecosystem’s response to small perturbations, so there is no assurance that the applicable perturbation range (that is, that for which the linearisation holds) is not infinitesimally small (as it is defined). Hence, the basin of attraction for the equilibrium may also be very small, rendering local stability analysis of limited value. One way to resolve this problem is to provide some assurance that the stable point is stable for some reasonably sized region. Lyapunov’s direct method (we follow Borrelli & Coleman (1996, pp. 414–412)) is often used in the study of nonlinear dynamical mechanical systems to derive a (subset) region of attraction for an equilibrium point. The idea of Lyapunov’s direct method is to find a (strict) Lyapunov function, which is a function that satisfies: 1. V (x⋆ ) = 0; 2. V (x) > 0 ∀x 6= x⋆ in ω; and 3. V˙ (x) < 0 ∀x 6= x⋆ in ω, where ω is a subset of the basin of attraction, and x⋆ is the steady state of interest. The presence of such a function is assurance that the steady state is stable for the region ω specified. An example is given in Appendix B.1. Unfortunately, there is no set way of determining what form the Lyapunov function has for a particular system. In the following subsections, we discuss some of the Lyapunov functions that have been used in the study of ecological systems.

Rouche’s Lyapunov function Rouche, Habets & Laloy (1977, p. 260-264) proposed a Lyapunov function of the form V =

n X i=1

βi x⋆i

xi xi − ln ⋆ x⋆i xi

(3.8)

where the βi term expresses the asymmetry of the interactions. Equation 3.8 can be applied to a generalised Lotka-Volterra system by setting βi = 1 ∀i (see Section 6.4.4). Porati & Granero (2000) used this Lyapunov function to show that trees with n − 1 connections (in the B matrix) are always globally stable, provided that they were feasible and of their specific form.

Goh’s Lyapunov function Goh & Jennings (1977) and Goh (1977) proposed Lyapunov functions with a form similar to that proposed


by Rouche et al. (1977) n X xi ⋆ ⋆ ci xi − xi − xi ln V = , x⋆i i=1

67

(3.9)

which gives 1 dV = (x − x⋆ )T B T C + CB (x − x⋆ ) , dt 2

(3.10)

where C is a positive definite diagonal matrix, also of size n × n. In order to satisfy V˙ ≤ 0 (V˙ < 0), C must be found such that the symmetric (Hermitian) matrix B T C + CB is negative definite (negative semidefinite), or the negative of Equation 3.10 is positive definite (Takeuchi & Adachi 1980). Various conditions for positive definiteness can be found in standard texts (see Strang (1976, pp. 237–244), Hohn (1972, pp. 281–282), Jacob (1995, pp. 235-238), and Leon (1994, pp. 347-353)). A symmetric matrix M is positive definite if any of the following are satisfied: • xT M x > 0 for a nonzero vector x; • All of the eigenvalues of M are greater than zero; • All of the sub-matrices of M have positive determinants; • For M = P DP T where D is a diagonal matrix, all of the diagonal elements of D are greater than zero. If P is constructed such that the columns of P are normalised eigenvectors of M , so P ˜ is chosen such that x ˜ = P T x, then x ˜T P M P T x ˜ = ni=1 λi x˜i 2 ; P T = P −1 , and x

• M has a Cholesky factorisation LLT such that the elements of L are real.

Definition 1 A system is called Goh-Lyapunov Stable4 if it has a feasible steady state that satisfies the requirements described in Equation 3.9. Goh-Lyapunov Stability is a sufficient condition for the equilibrium point, x⋆ , to be sector stable: globally asymptotically stable for Rn+ (Goh 1978). This means that any system that has initial conditions within the positive orthant of the phase space will asymptotically approach the equilibrium point x⋆ with time. A major problem associated with the use of the Goh-Lyapunov Function is that of finding C in Equation 3.10. Cross (1978) gives necessary and sufficient conditions for Goh-Lyapunov stability of matrices of order 2 and 3, however beyond this size, it is difficult to construct a suitable C (Goh 1980, p. 191). We will find C for a Lotka-Volterra system in Section 6.4.4.

The deeper meaning of the Lyapunov function? In physical systems, the Lyapunov function is often equivalent to the energy function, and the steady state the point at which this function is a minimum (Goh & Jennings 1977). In this sense, the Lyapunov 4 This

is also called Volterra-Lyapunov Stable in the literature (e.g. Cross 1978).

68

Stability

function has a real interpretation; that physical systems will tend to dissipate energy, and move to a state of minimum potential energy. An interesting question is whether or not an analogous interpretation may be made of ecological Lyapunov functions (Lewontin 1969). Goh (1980, pp. 101) states that the Lyapunov function is a biologically meaningful principle that implies that “a viable population must absorb energy at low densities and it must dissipate energy and high densities”. This describes the negative feedback characteristic of return to steady state: that when the population is lower than steady state, the change in population is positive representing an increase in matter (energy storage) in the compartment, and when the population is higher than steady state, the change in is negative, representing a redistribution of matter from the compartment to the rest of the system.

3.2.4

Permanence

Introduction The existence of a Goh-Lyapunov function provides a sufficient condition for the global asymptotic stability of a given ecosystem’s steady state. However, is it necessary for an ecosystem to tend toward constant populations in order for it to be considered stable? Jansen & Sigmund (1998) use the familiar example of the lynx-hare limit-cycle to demonstrate that an ecosystem may fit our intuitive understanding of what it is to be stable without being locally stable (an example of this is investigated in Section 6.4.5). With the lynx-hare cycle, the system is stable because all species persist through time. While both the lynx and the hare increase and decrease in number, neither drives the other to extinction, and given sufficiently small perturbations, the species will remain extant. This concept of stability is called permanence.

Finding permanence A dynamical system with state variables x is permanent if there exists a δ > 0 such that δ < lim inf xi (t) ∀i, t→∞

whenever xi (t = 0) > 0 ∀i and lim xi (t) ≤ M < ∞.

t→∞

(Hofbauer & Sigmund 1988). This means that an initially feasible ecosystem will stay within some bound between its limit inferior, δ and limit superior M for all time (Figure 3.3). No species will stay close to zero population (extinction) or grow unimpeded to some infinitely large population (as prescribed by M ).

69


xi M

lim sup x(t)

lim inf x(t)

t Figure 3.3: A diagrammatic representation of permanence for one species. Xi is the population of the species, and t is time.

Permanence has two advantages over local stability analysis. First, it describes the behaviour of the dynamical system beyond some arbitrarily small region around which a linearisation holds (Pimm 1982, p. 57). Second, it places no restrictions upon the dynamics of the feasible ecosystem, allowing for the complex limit cycles and chaotic dynamics. These are excluded from steady-state definitions of stability (Grimm et al. 1992, Jansen & Sigmund 1998). A convenient artifact of realistic ecological models is that the limit superior imposed upon the population may often be assumed. This is true when all autotrophs are self-limiting, and interactions are of the predator-prey type (Law & Blackford 1992). It is also true of closed-nutrient systems, such as the nutrient cycling food webs investigated in Chapter 6. Another way to interpret the limit inferior restriction is to say that the boundaries of the positive cone, bd(Rn+ ), are repellors. It can be shown (Hofbauer & Sigmund 1988, pp. 166–169) that a sufficient condition for permanence for a dissipative Lotka-Volterra system is that the equilibrium points upon the boundaries are repellors. In this way, one does not have to test every point upon the boundary, just the steady states. The transversal eigenvalue test is the test used to ensure that all boundary steady states are repellors. We discuss this test in more detail in Chapter 11. Briefly, for a dynamical system of the form x˙ i = xi f (x),

(3.11)

the transversal eigenvalues, γi , associated with a boundary steady state are γi = fi (x⋆ ),

(3.12)

where x⋆ is the vector of the solution to the boundary steady state. If at least one transversal eigenvalue satisfies γi > 0,

(3.13)

70

Stability

for a particular boundary steady state, then that boundary steady state is a repellor. If this is true for every boundary steady state, and if Equation 3.11 is a dissipative Lotka-Volterra system, this is a sufficient condition for the system to be permanent.

3.2.5

Concluding remark

Each of the stability measures discussed above are attempts to quantify or qualify the survival of the system. However, these attempts are constrained by the mathematical tools available, and so each stability measure can only address a certain aspect of what it means for the system to survive in the face of perturbations. Local stability requires that the system will return to a steady state after a small perturbation. However, this may not be a useful measure of stability when perturbations are large relative to the size of the basin of attraction. To address this, one may choose to use sector stability instead, which requires that the basin of attraction is at least as large as the entire positive orthant. However, this measure of stability is also inadequate, in that it ignores limit cycles and more complicated persistent behaviour. So once again to address this, one may choose to measure stability by permanence, which allows for non-steady-state persistence of populations. However, once again, there are shortcomings to this approach. In particular, the measure is only convenient for certain types of models (e.g. dissipative Lotka-Volterra systems), and thus results derived from permanence models are only as reliable as the models themselves. I have proposed that the interest in stability in ecosystems is motivated by an assumption that ecosystems must be able to survive in the face of environmental perturbations. If stability is used as a proxy for the ability of a system to survive, model ecosystems must possess stability in order to reflect this assumption. In the process of generating stable model ecosystems, one may observe that stable model ecosystems have certain attributes in common. An interesting question, which we will discuss in the next section, is whether or not these generalisations about stable model ecosystems are also true for real ecosystems. Put another way, does the requirement that real ecosystems possess stability constrain the attributes that they may possess?

3.3 Stability as a constraint upon ecosystems

3.3

71

Stability as a constraint upon ecosystems

3.3.1

Stability versus complexity

May’s proposal and subsequent studies During the 1970s, motivated by the work of MacArthur and others (Appendix B.2), ecologists became interested in the relationship between complexity and stability. Perhaps out of mathematical convenience, investigations started by using local stability as the stability measure. Gardner & Ashby (1970) investigated the change in the probability that a system was stable as the size and connectance of the system increased. The system used was linear x = Ax

(3.14)

where the matrix A had elements chosen from a random uniform distribution between −1 and +1. Connectance was defined as the proportion of non-zero elements in A, size was the size of matrix A, and stability determined by Routh-Hurwitz criterion (see Section 6.4.2). Gardner & Ashby’s (1970) results implied that connectance and local stability were negatively related. May (1972) built upon the work of Gardner & Ashby (1970). First, May (1972) recognised the system as being the linearised form of a nonlinear system, and that this made it suitable for describing ecosystems. Second, the elements ai,j of matrix A were reinterpreted as the effect of species j upon species i near equilibrium. May (1972) assumed ai,i = −1, which may be interpreted as species being self-regulating. Based upon the work of Wigner (1957), May (1972) discovered sharp transitions in the probability of stability relative to the connectance and interaction strength (mean square ai,j ). Two corollaries of May’s (1972) work have been made (May 1973, p 67): 1. To be stable, an ecosystem cannot simultaneously have high connectance and high interaction strength; and 2. Large systems can be made more stable by breaking the system into clusters of relatively higher connectance bridged by a few connections. These results were significant because they were counter to the traditional idea that ecosystem complexity was positively related to stability (DeAngelis 1975), and the long-standing belief that ecosystems developed toward greater stability and complexity simultaneously (e.g. see Clementsian succession discussed in Section 2.6.2). Subsequent authors sought to remedy this discrepancy. Roberts (1974) suggested that the counter-intuitive result was an artifact of the systems being predominantly infeasible. Roberts (1974) showed that, when only feasible systems were taken into account, systems tended to be stable. However, this result is not robust (Gilpin 1975). Further investigations

72

Stability

have demonstrated that feasible systems do not tend to be more stable than infeasible systems (Goh & Jennings 1977). Specifically, when the requirement that all species are autotrophs (di < 0) is removed, and not all species are self regulating (ai,i < 0) Roberts’s (1974) observation does not hold. DeAngelis (1975) restricted the original Gardner & Ashby (1970) matrix by taking into consideration the ecological meaning of the coefficients in A. DeAngelis (1975) imposed four assumptions: 1. An assimilation efficiency term was introduced, which was less than 1. This reflects the fact that biomass transfer from on species to another involves some wastage. 2. A hierarchical pattern was imposed upon the coefficient positions in the matrix. This reflects food web structure. 3. Species were divided into autotrophs and heterotrophs by allowing lower trophic levels to increase in the absence of predators, and higher trophic levels to decrease in the absence of prey. 4. Rates of increase or decrease of a species increases with an increase in the number of species in its predator-prey interactions. It was found that a positive relationship between the probability of stability and connectance could be obtained for three cases: 1. When the assimilation efficiency term was low; 2. When the death rate of higher trophic species was high; and 3. When the model is donor-dependant, that is, the interaction terms were more sensitive to changes in prey numbers than predator numbers. Various authors (May 1972, McMurtrie 1975, Pimm 1982) have observed that the probability of stability may be increased by compartmentalising the system, such that small blocks of highly interacting species are spanned by few, weak connections. However, there is no consensus as to whether or not real food webs have this attribute (May (1979), but see Solé & Montoya (2001)). Lawlor (1978) suggested some further constraints to increase the probability of theoretically sound locally stable ecosystems: 1. Less than 5-7 trophic levels (based on thermodynamic constraints); 2. No loops (apart from decomposers); and 3. The existence of primary producers with all interaction matrix elements less than or equal to zero. Stuart Pimm has also done many studies into generalising the attributes of locally stable systems. He summarises these findings with the following predictions (Pimm 1982, p. 187):


73

1. Food chains are more likely to be short; 2. Omnivory (feeding at several different trophic levels) is unlikely to be widespread; and 3. Omnivory is more likely higher in the food chain.

Empirical studies for and against May May’s (1973) observation forms part of the long-running complexity-stability debate (MacArthur 1955, Hairston et al. 1968, Gardner & Ashby 1970, May 1972, Yodzis 1981, Paine 1992, Tilman 1996, Doak et al. 1998, Tilman et al. 1998), in which theorists and empiricists struggled to reconcile May’s (1973) observation with the complexity observed in real ecological systems. Theoreticians involved in expanding May’s (1972) work agree that, in general, food webs tend to have a low probability of local stability for increasing connectance and size. However, field studies (McCann 2000, Doak et al. 1998, and citations within) continued to show the opposite relationship. Recent criticism of these field studies has questioned their validity. Doak et al. (1998) demonstrated that when stability is measured by averaging of biomass fluctuations, this averaging process can cause the appearance of a positive relationship between diversity and stability where none exists. This is commonly known as the ‘portfolio effect’. This result is particularly significant because, as Doak et al. (1998) notes, many studies supporting positive diversity-stability relationships use measure of stability that are subject to this effect. Tilman et al. (1998) acknowledged the presence of this effect within their earlier study, and clarified the conditions required for this effect to manifest. Some empirical studies support May’s (1972) proposal. For example, Yodzis (1981) showed that models structured in the same way as real ecosystems were more likely to be stable if the community matrix entries were of a similar magnitude to the interaction strength expected in the real system (but see Appendix B.3 for details of an error in this work). Also, Paine (1992) found that weak interaction strengths predominated (5 consumers of 7) in a rocky inter-tidal community, and Solé & Montoya (2001) found that real food webs have a scale-free distributions of links5 .

5 Scale free is when “the frequency of nodes with k connections follows a power law distribution P (k) ≈ k −γ , where most units are connected with few nodes and very few nodes are highly connected” (Sol´ e & Montoya 2001).

74

Stability

Food web building algorithms In response to the ensuing debate May (1973, pp. 3–4) stated that “...theoretical work should not try to prove any general theorem that “complexity implies stability”, but instead should focus on elucidating the very special sorts of complexity, the singular strategies, which may promote such mathematically atypical stability.” One candidate for these “singular strategies” is the use of food web assembly algorithms. There are two approaches to building food webs. The first is to create a particular sized food web with randomly selected attributes, and to proceed to delete compartments according to some specified rules until the desired stability criteria are satisfied. The second is to start with some system satisfying the specified stability criteria, and to add and delete compartments according to rules. The results of previous food web building experiments are summarised in Table 3.1. The main finding was that, if a food web was built via a food web building algorithm that used local stability as its constraint, atypically large and complex systems could result (Drake 1990, Post & Pimm 1983, Taylor 1988). Law & Blackford (1992) offers an interesting twist on the original MacArthur (1955) reasoning (Appendix B.2). Allowing permanence to be the stability measure, Law & Blackford (1992) suggest that complex systems are more likely to be able to reassemble than simpler ones because of the larger number of possible paths back to the original system. However, Law & Morton (1996) suggests that, while this may be true, the transient species required for reassembly may have since disappeared from the pool of invaders, making reassembly back to the original system impossible.

[1]

Stability attribute(s) Invasibility Feasibility Local stab’y

[2]

Addition

Permanence

[3]

Subtraction then addition

Permanence

[4]

Addition

Local stab’y

[5]

Addition and subtraction

[6]

Subtraction

Feasibility but others by implication

Feasibility

Rules Add species, if can invade, check feasible, if feasible check stable, if stable repeat If unfeasible, remove negative spp, if unstable, remove invader. Select invader from pool of species. If it can invade, reduce system to permanent subsystem. Permanent systems are disassembled and permitted to reform in any order.

If spp can invade, test for trophic loop, if no loops, check feasibility, if feasible check local stability. If cannot invade, or unstable remove invader. If not feasible, remove most negative species. Use results of numerical integration for compartment subtractions.

For unfeasible, remove compartment with most negative steady-state population. No rule for unstable.

Major result Invasion resistant communities had similar properties to real system in literature, including proportion of herbivores and specialists, ratio of prey to pred., mean food chain length connectance, and fraction of pred. with only one prey. Some final systems could not be reassembled from their components. Invasion resistance increases with invading species pool size. Systems with high connectance are most adept at re-assemblage after species loss, where ability to reassemble is measured by the number of possible paths back to the original system. Invasibility frequency decreased with time, as did resilience.

Supports conjecture that rare stable systems readily formed by food web building algorithms.

No assemblage collapsed to less than 21 compartments. Such large systems have a low probability of formation randomly.

Local stab’y

Note that the ‘if unstable, remove invader assumes that a system level measure acts on a particular compartment. Predators could only predate upon smaller organisms. The result follows logically from the fact that complex systems have more subsystems than simple.

Avoids computation saving assumptions (e.g. Drake 1990) that have doubtful basis. The observation that no feasible system was unstable negated the need for a rule for instability (but see Gilpin’s (1975) criticism).

75

[1]: Drake (1990). [2]: Law & Morton (1996). [3]: Law & Blackford (1992). [4]: Post & Pimm (1983). [5]: Taylor (1988) [6]: Tregonning & Roberts (1979)

Comments


Table 3.1: A summary of food web building experiments in the literature. Compartment addn/subtrctn Addition

76

Stability

3.3.2

Resilience and ecosystem survival

Because ‘maximise resilience’ is one of the goal functions considered by Cropp & Gabric (2002), this section highlights the few instances where resilience has been proposed as an ecosystem constraint. May (1973) was the first to propose that resilience constrains ecosystem properties. He conceptualised the community’s population as a probability cloud in phase space, subject to two opposing forces: stabilising forces, and diffusing forces. The stabilising forces were determined by the resilience of the system, and made the probability cloud more dense, while the diffusing forces were a consequence external random environmental fluctuations, and made the probability cloud more diffuse. In order for the system to remain extant, the probability cloud had to be contained within the positive sector of the phase space. If the probability cloud was too diffuse, parts of the cloud would overlap the axis of the phase space, and species in the system would eventually suffer extinctions. Therefore, May (1973) proposed that the system had to possess a resilience greater than some minimal value, as determined by the environmental fluctuations upon the system. Resilience was constrained by R > σ2

(3.15)

where R is the resilience of the system, and σ 2 is the variance of the stochastic term (white noise) in the system. This would ensure that the stabilising forces could counter-act the diffusing forces, keeping the population safely within the positive sector in the phase space. May’s (1973) hypothesis has been criticised, as has the theory of limiting similarity based upon it (Abrams 1983). Turelli (1978) found that the conditions for which Equation 3.15 was true were strictly limited. For some models, while the stabilising forces were approximately equal to the intrinsic rate of increase of the population, the diffusing forces would increase with the square of this term. Equation 3.15 is true only in certain restricted contexts such as “if independent white noise perturbations with equal variance parameters are added directly to the per capita growth rates [but not the limiting capacity] of a general model [and] all species have equal populations sizes at the deterministic equilibrium” (Turelli 1978). The second example of a link between resilience and ecosystem survival is the work of Pimm & Lawton (1977). Building upon the hypothesis that “long return times imply an increased probability of extinction in the real world”, Pimm & Lawton (1977) hypothesised that ecosystems would tend to have short maximum chain lengths, as these were more likely to have a high resilience (but see Sterner, Baipai & Adams (1997)). Pimm (1991, p. 144) reiterated this reasoning, stating that “a population that recovers slowly from a severe reduction in density will remain longer at risk of extinction from demographic accidents”, presumably from staying too long near the axis of the phase space, increasing the probability that an environmental perturbation will push the population outside of the positive orthant. However he also acknowledged that high resilience could lead to higher variability in populations, because of the highly

3.4 Conclusion

77

resilience population’s ability to track changes in the position of environmental equilibrium (Pimm 1991, pp. 44–52).

3.4

Conclusion

We have discussed the meaning of stability, and various aspects and definitions of stability as they occur in the ecological literature. These definitions will now be used to test the claim that there is a link between resilience and the thermodynamic goal functions that they employed (Cropp & Gabric 2002). All of these definitions capture some aspect of what it means for an ecosystem to be able to survive in a fluctuating environment. The link between stability and survival has led to stability (usually local stability) being used as a constraint upon model ecosystems. This has led to some fundamental questions about the attributes of food webs being asked (e.g. the complexity-stability debate), and some verifiable predictions about real food webs being made. We will make our own contribution to this body of work in Chapter 11. We acknowledge that stability measures can never fully describe and measure the survival likelihood of an ecosystem, no matter how advanced a measure of stability is used. Therefore, while the statement that “highly resilient ecosystems are more likely to remain extant than ecosystems with low resilience” (Cropp & Gabric 2002) is reasonable when resilience is considered a proxy for survival likelihood, it is still a simplification. Assumptions underlying the mapping from resilience to survival are yet to be elucidated. We will explore this question in Chapter 8.

Chapter 4

The resilience hypothesis 4.1 4.1.1


The resilience hypothesis is that “ecosystems will evolve to maximise resilience” because “the biotic attributes that optimize the thermodynamic goal functions also maximize resilience” (Cropp & Gabric 2002). The latter conclusion was made after a similarity between the parameter values that maximised each of the goal functions in a three-compartment model and the parameter values that maximised resilience was made. In Chapter 2 we discussed the validity of the goal function approach, and the theoretical basis for these particular goal functions. Our objectives in this chapter are to revisit the Cropp & Gabric (2002) model, and to test the interpretation of its results. While it may be true that the parameter values that maximise each of the goal functions are similar, the goal functions will be more sensitive to some parameters than others. Therefore, observing a similarity between the parameter values that maximise each of the goal functions does not necessarily mean that the effect of maximising one of those goal functions, or a subset of those goal functions, will optimise the others. In this chapter, we test the claim that maximising the goal functions investigated in Cropp & Gabric (2002) also maximises resilience by measuring the effects of the goal functions upon one another.

78

4.2 Approach

4.2

79

Approach

Cropp & Gabric (2002) describes a three-compartment aquatic food web model (henceforth the CG Model ), which is governed by three differential equations: dP N − eZ P Z, = µP P dt N + kP dZ = eZ (1 − ηZ )P Z − dZ Z, dt N dN , = dZ Z + eZ ηZ P Z − µP P dt N + kP

(4.1a) (4.1b) (4.1c)

where P , Z and N are the nutrient concentrations in the phytoplankton, zooplankton and nutrient compartments respectively, eZ is the consumption per day of phytoplankton mass per zooplankton mass, dZ is the zooplankton mortality, ηZ is the efficiency of zooplankton conversion of nutrient into biomass, kP is the nutrient half-saturation concentration for phytoplankton, and µP is the maximum phytoplankton nutrient uptake rate. The system is closed with respect to the input and output of nutrients, hence the total sum of nutrients, P + Z + N = No , is a constant. The CG Model describes a Lotka-Volterra interaction between zooplankton and phytoplankton, and a Holling Type II interaction between the nutrient compartment and phytoplankton (DeAngelis 1992). In this section, we replicate the Cropp & Gabric (2002) result. We are interested in determining the validity of the claim that maximising the goal functions above optimises resilience. The goal functions we use are the same as those used in Cropp & Gabric (2002). We refer to them as the traditional goal functions: 1. Maximise phytoplankton biomass at steady state, P ; 2. Maximise zooplankton biomass at steady state, Z; 3. Maximise flux of nutrients through the system at steady state, F ; 4. Maximise flux to biomass ratio at steady state, F/(P + Z); and 5. Maximise resilience at steady state, R. We use the same method as used in Cropp & Gabric (2002). A genetic algorithm is used to find the parameter values that maximise each of the goal functions, and the parameter values are compared to one another. Parameter values are varied by ±50% of the base values. The base values used are slightly different to those used in Cropp & Gabric (2002) (however we will find that this does not alter the vertex at which the goal functions are maximised – see Sections 4.3 and 4.4). This is done so that variations of the model, which will be investigated in Chapter 5, possess a feasible and stable equilibria. The base values are: eZ = 0.006,

(4.2a)

80

The resilience hypothesis

µP = 3,

(4.2b)

kP = 277,

(4.2c)

dZ = 0.05,

(4.2d)

ηZ = 0.4.

(4.2e)

To quantify the agreement between goal functions, they are normalised against their maximum and minimum values for the parameter range investigated. Let the parameter set be denoted by a parameter vector, α. Each goal function is a function of the parameter set, Γi (α). Let the parameter set that maximises goal function j be denoted αmax j , and the parameter set that minimises goal function j be denoted αmin j . The normalised value of goal function i using the parameter values that maximise goal ˆ i (αmax j ), is found by function j, Γ ˆ i (αmax j ) = Γi (αmax j ) − Γi (αmin i ) , Γ Γi (αmax i ) − Γi (αmin i )

ˆ i (αmax j ) ≤ 1 ∀i. 0≤Γ

(4.3)

In summary, for each goal function j, we use a genetic algorithm to find the parameter values that maximise that goal function, αmax j . Then, we find the value of each of the other goal functions when that set of parameter values is applied to the system, Γi (αmax j ). To facilitate comparisons, the value Γi (αmax j ) is normalised against its range, as described in Equation 4.3. We are interested in determining if maximising the traditional goal functions gives a high value for resilience.

4.3

Results

The results of the rerun of the CG Model experiment are shown in Table 4.1. Comparing with Table 1.1, each goal function is maximised at the equivalent vertex in parameter space to that found in Cropp & Gabric (2002). Table 4.1 highlights the parameter values that ‘maximise resilience’ and the traditional goal functions have in common. Figure 4.1 shows the normalised goal function values found using the results in Table 4.1. ‘Maximise flux’ optimises resilience. However, ‘maximise zooplankton biomass’, ‘maximise phytoplankton biomass’1 , and ‘maximise flux to biomass ratio’ do not optimise resilience. The goal functions ‘maximise flux’ and ‘maximise resilience’ lead to high values of all of the other traditional goal functions.

1 The goal function ‘maximise phytoplankton biomass’ is not a function of µ and k , however these parameters can be P P set such that both resilience and phytoplankton biomass are maximised.

81

4.3 Results

Normalised Goal Function Values

Table 4.1: Model HL4 results. Parameters that the goal functions were independent of are marked with an *. Parameter values that agree with ‘maximise resilience’ are highlighted. Parameter Value of Goal Functions eZ µP kP dZ ηZ P Z F F/B R αmax Maximise Peq 0.003 3* 277* 0.075 0.6 62.5 311.9 58.5 0.16 0.16 Maximise Zeq 0.003 4.5 138.5 0.025 0.2 10.4 433.3 13.5 0.03 0.090 Maximise Feq 0.003 4.5 138.5 0.075 0.6 62.5 389.0 72.9 0.16 0.56 Maximise (F/B)eq 0.009 4.5 138.5 0.075 0.6 20.8 289.2 54.2 0.18 0.060 Maximise Req 0.003 4.5 138.5 0.075 0.6 62.5 389.0 72.9 0.16 0.56 αmin Minimise Peq 0.009 3* 277* 0.025 0.2 3.5 178.2 5.6 0.03 0.0041 Minimise Zeq 0.009 1.5 415.5 0.075 0.6 20.8 81.5 15.3 0.15 0.0098 Minimise Feq 0.009 1.5 415.5 0.025 0.2 3.5 83.1 2.6 0.03 0.0016 Minimise (F/B)eq 0.003 1.5 415.5 0.025 0.2 10.4 203.8 6.4 0.03 0.0066 Minimise Req 0.009 1.5 138.5 0.025 0.2 3.5 121.7 3.8 0.03 0.0014

CG Model 4

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

3 11111 00000 00000 11111 00000 11111

2

1 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

11111 00000 00000 11111 00000 11111 11111 00000

Max P

Max Z

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

Max F

111 000 000 111 000 111 000 R 111

F/B 11111 00000 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

Max F/B

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

F

111 000 000 111 000 111 000 111 000 Z 111 000 111 000 111 000 111 000 P 111 000 111

Max R

Goal Functions

Figure 4.1: The horizontal axis indicates the goal function being maximised. P: phytoplankton biomass, Z: zooplankton biomass, F: flux of nutrients, and F/B: flux to biomass ratio. The stacked bar shows the normalised value of each of the other goal functions when the goal function indicated on the horizontal axis is maximised.

82

4.4

The resilience hypothesis

Discussion

Although there were slight differences between the parameter range used and that used in Cropp & Gabric (2002), we have been able to verify the Cropp & Gabric (2002) result regarding which vertex in parameter space each of the goal functions were maximised at. The major difference between this study and Cropp & Gabric (2002) is that the Cropp & Gabric (2002) study only considered similarities between the parameter values that maximised each of the goal functions. Looking at Table 4.1 alone, considering only the agreement between the parameters, may lead one to conclude that each of the traditional goal functions will do a fair job of optimising resilience. However, Figure 4.1 tests this claim directly, by exploring the effects of each goal function upon the other. Figure 4.1 reveals that, despite the agreement in parameter values, the biotic attributes that optimise the thermodynamic goal functions do not maximise resilience. The statement in Cropp & Gabric (2002) that “the biotic attributes that optimize the thermodynamic goal functions also maximize resilience” is only true for ‘maximise flux’. A more accurate statement of the similarity between resilience and other goal functions would be to state that, for the CG model, maximising resilience optimises the traditional goal functions. For the CG model, ‘maximise resilience’ offers a compromise between the traditional goal functions.

4.5

Conclusion

Cropp & Gabric (2002) proposes ‘maximise resilience’ as a goal function because “the biotic attributes that optimize the thermodynamic goal functions also maximize resilience”. In this chapter, we have used the model in Cropp & Gabric (2002) to demonstrate that this was a misinterpretation of the results. Maximising the traditional goal functions does not necessarily optimise resilience. However, we did find that ‘maximise resilience’ offers a compromise between the traditional goal functions. The mechanisms by which ecosystems would structure themselves in order to maximise resilience sensu Cropp & Gabric (2002) are not explained by the observation that ‘maximise resilience’ offers a compromise between the traditional goal functions. Hypothetically, if it had been found that resilience was maximised in the model when the other goal functions were maximised, it may be hypothesised that real ecosystems maximise resilience as a consequence of maximising the traditional goal functions. However, with the exception of ‘maximise flux’, evidence for this was not found. Rather, it has been found that resilience offers one possible way to simultaneously optimise the traditional goal functions. Now, in addition to the question of why ecosystems would maximise any of the traditional goal functions, we ask, assuming that they do, why ecosystems would ‘choose’ maximise resilience over any other compromise between the goal functions? This could only be justified if we assume that ‘maximise resilience’ is an independent goal function (e.g. on the basis of the Laws et al. (2000) result).

4.5 Conclusion

83

We will return to the question of ‘maximise resilience’ as an independent goal function in Chapter 7, when we discuss the results in Laws et al. (2000). For now, we will investigate the result that ‘maximise resilience’ offers a compromise between the traditional goal functions. ‘Maximise resilience’ may be recommended as a useful heuristic if the ability of ‘maximise resilience’ to compromise between the goal functions is shown to be a more general result. We are interested in determining if this observation is dependent upon this particular model, or if this represents a more general principle.

Chapter 5

The resilience heuristic 5.1

Introduction

5.1.1

Motivation

In Chapter 4, we investigated the resilience hypothesis: that ecosystems maximise resilience as a consequence of maximising the traditional goal functions. It was found that the resilience hypothesis was not supported by the CG model. We found that maximising resilience comes close to maximising the traditional goal functions, however we have no mechanism by which resilience will be maximised. While resilience offers a compromise between the traditional goal functions, there is no reason to suggest that ecosystems will ‘choose’ resilience over any other possible compromise. Even if one were to accept the validity of the traditional goal functions to begin with (but see criticisms in Chapter 2), if one will not accept the resilience hypothesis a priori, the result in Cropp & Gabric (2002) is not a sufficient reason to believe that ecosystems maximise resilience. Recall that there are four statements that we will consider in this thesis. 1. Resilience hypothesis: Ecosystems maximise resilience. Therefore, resilience is a goal function. There are two resilience hypotheses being considered. The first is taken from Cropp & Gabric (2002), and the second is based upon the Laws et al. (2000) result. (a) Resilience hypothesis by relationship with traditional goal functions: Maximising the traditional goal functions optimises resilience. Therefore, resilience is a goal function because of its relationship with the traditional goal functions. (b) Resilience hypothesis independent of traditional goal functions: Ecosystems maximise resilience for reasons that are independent of the relationship between 84

5.2 Methodology

85

resilience and the traditional goal functions. Therefore, resilience is a goal function by this (unspecified) mechanism. 2. Resilience heuristic: Resilience generally offers a good compromise between the traditional goal functions. There are two resilience heuristics, which arise from the acceptance or rejection of the resilience hypothesis. (a) Resilience heuristic without assuming the resilience hypothesis: Resilience generally offers a good compromise between the traditional goal functions. Therefore, if one is willing to assume that the traditional goal functions are valid, one could use resilience as a way of simultaneously optimising the traditional goal functions. However, there is no reason to suggest that ecosystems would ‘choose’ this compromise over any other. (b) Resilience heuristic assuming the resilience hypothesis: Resilience generally offers a good compromise between the traditional goal functions. Therefore, if one is willing to assume that the traditional goal functions are valid, and that the resilience hypothesis is true, one could use resilience as a way to satisfy the resilience hypothesis and simultaneously satisfy the traditional goal functions. In Chapter 2, we found no support for Statement 1(a) in the form proposed in Cropp & Gabric (2002). We will address Statement 1(b) in Chapter 7. In this chapter, we investigate the resilience heuristic: that maximising resilience in model ecosystems offers a compromise between the traditional goal functions. We ask, if one is to accept the traditional goal functions to begin with, does maximal resilience provide a sensible heuristic for simultaneously maximising all of the traditional goal functions? Also, if one has reason to believe that ‘maximise resilience’ is a valid independent goal function (e.g. on the basis of the Laws et al. (2000) result), will maximising resilience still offer a reasonable compromise with the traditional goal functions?

5.2

Methodology

We make structural adjustments to the CG model and explore the relationships between maximising resilience and maximising the traditional goal functions. First, we create 24 variations of the CG Model. For each of the model variations, we use a new measure, concordance (defined below), to quantify how well each of the goal functions compromises between the others. We are interested in determining if ‘maximise resilience’ offers a better compromise between the goal functions than any of the other goal functions. We are also interested in determining if this ability is robust to the variations in the model.

86

The resilience heuristic

5.2.1

Variations in the CG Model

The CG model described by Equation 4.1 has a Lotka-Volterra interaction between the zooplankton and phytoplankton compartments, and a Holling Type II interaction between the nutrient compartment and phytoplankton compartment (DeAngelis 1992). In this section, we generalise the CG Model by including a variety of phytoplankton-zooplankton (P-Z) and nutrient-phytoplankton (N-P) interactions, as shown in Table 5.1. A full description of differential equations governing all of the models may be obtained from Appendix C.1. The variations on the CG model are: • All possible combinations of Lotka-Volterra and Holling Type II interactions for N − P (column 2) and P − Z (column 3) were tested (denoted LL, LH, HL and HH); • An efficiency term for the assimilation of phytoplankton biomass by zooplankton was included or excluded (column 4); • Where applicable, the path of inefficiently assimilated phytoplankton grazed by zooplankton was passed directly to the nutrient compartment (e.g. DeAngelis 1992) or passed through the zooplankton compartment before the nutrient compartment (e.g. CG Model) (column 5); • A death term for phytoplankton was either excluded (e.g. Cropp & Gabric 2002) or included (e.g. Druon & Le Fèvre 1999) (column 6). Parameter ranges were chosen such that all models possessed a feasible and stable equilibrium. The base parameter values are shown in Table 5.2. The parameter value range is ±50% of the base values. This is the same as the methodology used in Cropp & Gabric (2002). HL4 is the rerun of the CG Model from Chapter 4.

5.2.2

Concordance - A measure of agreement between goal functions

Here we create a new metric, concordance, to quantify the agreement between goal functions. Concordance measures the extent to which maximising a given goal function also optimises the value of the other goal functions. Let the value of the goal function i be denoted Γi . The six goal functions are: Maximise Γ1 . Maximise phytoplankton biomass at equilibrium, Peq ; Maximise Γ2 . Maximise zooplankton biomass at equilibrium, Zeq ; Maximise Γ3 . Maximise flux of nutrients through the system at equilibrium, Feq ; Maximise Γ4 . Maximise flux to biomass ratio at equilibrium, (F/B)eq ; Maximise Γ5 . Minimise flux to biomass ratio at equilibrium, (F/B)eq ; and

87

5.2 Methodology

Model Name LL1 LL2 LL3 LL4 LL5 LL6 LH1 LH2 LH3 LH4 LH5 LH6 HL1 HL2 HL3 HL4 HL5 HL6 HH1 HH2 HH3 HH4 HH5 HH6

Table N-P Interaction Lotka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Holling II Holling II Holling II Holling II Holling II Holling II Holling II Holling II Holling II Holling II Holling II Holling II

5.1: A summary of attributes of models ran. P-Z Z Efficiency All P Pass Interaction Term? Through Z? Lotka-Volterra No N.A. Lotka-Volterra No N.A. Lotka-Volterra Yes No Lotka-Volterra Yes Yes Lotka-Volterra Yes No Lotka-Volterra Yes Yes Holling II No N.A. Holling II No N.A. Holling II Yes No Holling II Yes Yes Holling II Yes No Holling II Yes Yes Lotka-Volterra No N.A. Lotka-Volterra No N.A. Lotka-Volterra Yes No Lotka-Volterra Yes Yes Lotka-Volterra Yes No Lotka-Volterra Yes Yes Holling II No N.A. Holling II No N.A. Holling II Yes No Holling II Yes Yes Holling II Yes No Holling II Yes Yes

P Mortality Term? No Yes No No Yes Yes No Yes No No Yes Yes No Yes No No Yes Yes No Yes No No Yes Yes

Maximise Γ6 . Maximise resilience, Req . Let the parameter set (see Table 5.2) be denoted by a parameter vector, α. Each goal function is a function of the parameter set, Γi (α). Let the parameter set that maximises goal function j be denoted αmax j , and the parameter set that minimises goal function j be denoted αmin j . The normalised values of goal function i using the parameter values that maximise goal function j, ˆ i (αmax j ), is Γ ˆ i (αmax j ) = Γi (αmax j ) − Γi (αmin i ) , Γ Γi (αmax i ) − Γi (αmin i )

ˆ i (αmax j ) ≤ 1 ∀i. 0≤Γ

(5.1)

The concordance Cj of goal function j is a sum of these normalised values. It is a measure of the agreement between goal functions. Two types of concordance are considered. First, the concordance of a goal function with respect to all of the other goal functions, including resilience, is calculated. This is found by Cj =

i=6 X i=1 i6=j

ˆ i (αmax j ), Γ

0 ≤ Cj ≤ 4,

(5.2)

where Cj is the concordance of goal function j. Second, the concordance of a goal function with respect

88


Table 5.2: Base parameter values. Range used was ±50%. No was taken to be a constant 500 mg/m2 . Model µP µZ kP kZ dP dZ ηZ eP2 eZ2 mgN mgN m m 1 1 1 1 Name mgN d mgN d d d m2 m2 d d LL1 0.006 0.006 N.A. N.A. N.A. N.A. LL2 0.006 0.006 N.A. N.A. N.A. N.A. LL3 & LL4 0.006 0.006 N.A. N.A. N.A. N.A. LL5 & LL6 0.006 0.006 N.A. N.A. N.A. N.A. LH1 0.006 N.A. N.A. 3 N.A. 2000 LH2 0.006 N.A. N.A. 3 N.A. 2000 LH3 & LH4 0.006 N.A. N.A. 3 N.A. 2000 LH5 & LH6 0.006 N.A. N.A. 3 N.A. 2000 HL1 N.A. 0.006 3 N.A. 277 N.A. HL2 N.A. 0.006 3 N.A. 277 N.A. HL3 & HL4 N.A. 0.006 3 N.A. 277 N.A. HL5 & HL6 N.A. 0.006 3 N.A. 277 N.A. HH1 N.A. N.A. 3 3 277 2000 HH2 N.A. N.A. 3 3 277 2000 HH3 & HH4 N.A. N.A. 3 3 277 2000 HH5 & HH6 N.A. N.A. 3 3 277 2000 Note: eP is the consumption per day of nutrient per mass of phytoplankton µZ is the maximum zooplankton nutrient uptake rate kZ is the nutrient half saturation concentration for phytoplankton dP is the phytoplankton mortality

N.A. 0.005 N.A. 0.005 N.A. 0.005 N.A. 0.005 N.A. 0.005 N.A. 0.005 N.A. 0.005 N.A. 0.005

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

N.A. N.A. 0.4 0.4 N.A. N.A. 0.4 0.4 N.A. N.A. 0.4 0.4 N.A. N.A. 0.4 0.4

to the traditional goal functions only is calculated. This is found by excluding the normalised resilience values, Γ6 , from the sum, Cj =

i=5 X

ˆ i (αmax j ), Γ

i=1

0 ≤ Cj ≤ 4.

(5.3)

Note that i cannot be both 4 and 5 (F/B and −F/B) in Equation 5.2 or 5.3, as those goal functions are mutually exclusive.

5.2.3

Hypothesis tested

The two concordance measures, Equation 5.2 and Equation 5.3, may be interpreted as representing two assumptions about the status of ‘maximise resilience’ as a goal function. 1. Equation 5.2 – concordance with respect to all goal functions – assumes that ‘maximise resilience’ is a valid independent goal function. If one were to accept the traditional goal functions, and also accept ‘maximise resilience’ as a valid goal function (e.g. on the basis of the Laws et al. (2000) result), one may be interested in using the goal function which offers the best compromise between all of the goal functions. If it is known that ‘maximise resilience’ generally has high concordance of this type, then it would be a sensible choice.

5.2 Methodology

89

2. Equation 5.3 – concordance with respect to the traditional goal functions only – does not assume that ‘maximise resilience’ is a valid independent goal function. If one were to accept the traditional goal functions only, one may be interested in finding some function which offers a good compromise between the traditional goal functions. If it is known that ‘maximise resilience’ generally has high concordance of this type, then it would be a sensible choice. Although positive results for either of the above measures would support the use of ‘maximise resilience’ as a goal function, neither of them are equivalent to the resilience hypothesis of Cropp & Gabric (2002). Therefore, we refer to these as resilience heuristics. Some goal functions were not a function of every parameter in the model (e.g. in Model LH4, Peq did not depend upon the parameter µP ). The genetic algorithm would vary these parameter values, misrepresenting the effect of maximising each goal function on the selection of the parameter. Therefore, these parameters were set to their base values before being used in concordance calculations.

5.2.4

Results and discussion

‘Maximise resilience’ as a compromise Concordance values for each goal function are shown in Tables 5.3 to 5.6. Tables 5.3 and 5.4 are concordance values relative to all goal functions, including resilience. Tables 5.5 and 5.6 are concordance values relative to the traditional goal functions only (i.e. not including resilience). The results when concordance is measured relative to all goal functions are relatively consistent. Table 5.3 shows the results when (F/B)eq is maximised and Table 5.4 for when the (F/B)eq is minimised. It can be seen that, for most models, resilience has the highest (or equal-highest) concordance, and hence the maximisation of resilience leads to near compromise with the other goal functions. Inspection of the raw data reveals that the the converse is not true, that is, the maximisation of the traditional goal functions does not lead to the optimisation of resilience. For minimising (F/B)eq , all models give resilience with the highest or equal-highest (with flux) concordance. This suggests that the resilience heuristic assuming the resilience hypothesis is robust, and not specific to the CG Model type. It also supports the observed positive relationship between flux and resilience reported in the literature (DeAngelis et al. 1978). For maximising (F/B)eq , most models give resilience with the highest or equal-highest (with flux) concordance. The results when concordance is measured relative to only the traditional goal functions gives less consistent results. Table 5.5 shows the results when (F/B)eq is maximised and Table 5.6 for when the (F/B)eq is minimised. For minimising (F/B)eq , either ‘maximise flux’ or ‘maximise flux to biomass ratio’ has the highest concordance. Resilience never has the highest concordance. For maximising (F/B)eq , most

90


Table 5.3: Concordance (Equation 5.2) for each goal function in each model, relative to all goal functions including resilience. F/B is maximised. Model Maximise Maximise Maximise Maximise Maximise Support resilience Name Peq Zeq Feq F/Beq Req heuristic 1? LL1 3.25 0.82 3.90 1.71 3.90 Yes LL2 3.25 0.82 3.90 1.71 3.90 Yes LL3 2.95 0.56 2.84 1.43 3.58 Yes LL4 3.11 0.40 3.74 1.81 3.74 Yes LL5 2.95 0.56 2.84 1.43 3.58 Yes LL6 3.11 0.40 3.74 1.81 3.74 Yes LH1 2.25 0.75 2.53 1.56 3.04 Yes LH2 2.27 0.76 2.54 1.55 3.03 Yes LH3 0.04 0.89 2.44 1.73 2.68 Yes LH4 0.14 0.57 2.57 2.03 2.35 No LH5 0.05 0.89 2.45 1.73 2.62 Yes LH6 0.15 0.58 2.58 2.01 2.61 Yes HL1 2.72 0.88 3.92 1.63 3.92 Yes HL2 2.72 0.88 3.92 1.63 3.92 Yes HL3 2.47 0.61 2.84 1.45 3.65 Yes HL4 2.61 0.43 3.78 1.72 3.78 Yes HL5 2.47 0.61 2.84 1.45 3.65 Yes HL6 2.60 0.43 3.78 1.72 3.78 Yes HH1 2.08 0.67 2.42 1.88 3.08 Yes HH2 2.07 0.55 2.50 1.88 3.07 Yes HH3 0.09 0.91 2.39 1.93 2.70 Yes HH4 0.16 0.56 2.60 2.16 1.85 No HH5 0.09 1.16 2.58 1.98 2.83 Yes HH6 0.17 0.84 2.26 2.21 2.07 No

models give resilience with the highest or equal-highest (with flux) concordance. The exceptions to this are LH3 to LH6, and HH3 to HH6.

Does the parameter range affect concordance? Resilience is negative of the real part of the the most positive eigenvalue of the Jacobian Matrix (DeAngelis 1992). The eigenvalues can be real or complex conjugate. Real eigenvalues indicate that the steady state is a simple sink, whereas complex conjugate eigenvalues indicate that there is an oscillatory return to the steady state (Section 3.2.2). The parameter range chosen was such that all models with a Holling Type II interaction between the phytoplankton and zooplankton compartments (prefixed LH or HH) had the potential to possess either real or complex eigenvalues, whereas those with a Lotka-Volterra interaction (prefixed LL or HL) only possessed complex conjugate eigenvalues. For all models with potential for both types of eigenvalues, resilience was maximised at the discontinuity between real and complex conjugate eigenvalues. This interior point represents a local maximum in

5.2 Methodology

91

Table 5.4: Concordance (Equation 5.2) for each goal function in each model, relative to all goal functions including resilience. F/B is minimised. Model Maximise Maximise Maximise Minimise Maximise Supports resilience Name Peq Zeq Feq F/Beq Req heuristic 1? LL1 2.38 1.81 3.00 0.96 3.00 Yes LL2 2.38 1.81 3.00 0.95 3.00 Yes LL3 2.37 1.50 2.00 1.00 2.95 Yes LL4 2.36 1.34 2.95 0.73 2.95 Yes LL5 2.37 1.51 1.96 0.99 2.95 Yes LL6 2.36 1.40 2.95 0.72 2.95 Yes LH1 2.12 1.71 1.77 1.44 2.61 Yes LH2 2.13 1.71 1.78 1.44 2.61 Yes LH3 1.02 1.45 1.63 1.01 2.14 Yes LH4 1.12 1.51 1.78 1.11 2.18 Yes LH5 1.03 1.45 1.65 1.01 2.10 Yes LH6 1.12 1.50 1.80 1.10 2.16 Yes HL1 1.84 1.86 3.00 0.72 3.00 Yes HL2 1.84 1.86 3.00 0.72 3.00 Yes HL3 1.87 1.53 1.95 0.76 2.96 Yes HL4 1.86 1.42 2.97 0.53 2.97 Yes HL5 1.87 1.53 1.95 0.75 2.96 Yes HL6 1.86 1.42 2.97 0.52 2.97 Yes HH1 1.93 1.59 1.52 1.35 2.40 Yes HH2 1.93 1.47 1.62 1.35 2.41 Yes HH3 1.06 2.45 1.47 2.05 2.12 Yes HH4 1.13 1.51 1.70 1.12 2.15 Yes HH5 1.07 1.72 1.85 1.04 2.15 Yes HH6 1.14 1.76 1.55 1.12 2.17 Yes

resilience. An example of this is shown in Figure 5.1. Resilience is plotted against against eZ (zooplankton per capita per day nutrient consumption) for the simplest model, Model LL1. All other parameters are held constant. Figure 5.1 shows a typical relationship between parameter value and resilience. When the eigenvalues are complex conjugate, they both have the same real part, which is equal to resilience. In this case, resilience has a positive relationship with the parameter. When the eigenvalues are real, the eigenvalue closest to zero dominates the system, and is used to quantify resilience. In this case, resilience has a negative relationship with the parameter. From inspection of the raw data, it can be seen that critical points also exist for the goal functions ‘maximise zooplankton biomass’, and ‘maximise flux’. The position of the parameter range relative to the interior points affects where the goal function is maximised, and hence affects the concordance. When concordance is measured relative to all goal functions, including resilience, three model formulations (LH4, HH4, HH6) out of 24 do not give resilience the highest concordance (Table 5.3). For these models, flux has the highest concordance. These exceptions are all models with a zooplankton grazing inefficiency term where all inefficiently grazed phytoplankton biomass passes through the zooplankton compartment

92


Table 5.5: Concordance (Equation 5.3) for each goal function in each model, relative to traditional goal functions only. F/B is maximised. Model Maximise Maximise Maximise Maximise Maximise Supports resilience Name Peq Zeq Feq F/Beq Req heuristic 2? LL1 2.60 0.52 2.90 1.40 2.90 Yes LL2 2.60 0.52 2.90 1.40 2.90 Yes LL3 2.29 0.41 2.35 1.29 2.58 Yes LL4 2.44 0.25 2.73 1.48 2.73 Yes LL5 2.29 0.41 2.35 1.29 2.55 Yes LL6 2.44 0.25 2.73 1.81 2.73 Yes LH1 1.87 0.45 2.03 1.44 2.03 Yes LH2 1.87 0.45 2.02 1.42 2.03 Yes LH3 0.03 0.55 1.91 1.58 1.68 No LH4 0.13 0.24 1.82 1.69 1.35 No LH5 0.05 0.56 1.90 1.58 1.62 No LH6 0.14 0.25 1.83 1.69 1.62 No HL1 2.44 0.56 2.92 1.53 2.92 Yes HL2 2.44 0.56 2.92 1.53 2.92 Yes HL3 2.18 0.46 2.37 1.40 2.64 Yes HL4 2.31 0.29 2.78 1.61 2.78 Yes HL5 2.18 0.46 2.37 1.40 2.64 Yes HL6 2.31 0.29 2.78 1.61 2.78 Yes HH1 1.87 0.40 2.04 1.77 2.08 Yes HH2 1.88 0.37 2.06 1.78 2.07 Yes HH3 0.07 0.54 1.95 1.79 1.70 No HH4 0.14 0.22 1.92 1.86 0.85 No HH5 0.07 0.57 1.93 1.84 1.83 No HH6 0.15 0.25 1.88 1.91 1.07 No

(a Cropp & Gabric (2002) flux formulation), and a Holling Type II interaction exists between zooplankton and phytoplankton. For HH and LH models, the goal functions ‘maximise resilience’, ‘maximise flux’, and ‘maximise zooplankton biomass’ are maximised at an interior point. This has a general tendency to decrease the concordance of each of the goal functions, compared with their LL and HL counterparts, which are maximised at a vertex in the parameter space. The decrease in concordance is higher for resilience than flux because of the advantage ‘maximise flux’ has in optimising ‘maximise flux to biomass ratio’. Reversing this goal function to ‘minimise flux to biomass ratio’ gives resilience the advantage, as can be seen by comparing Table 5.3 and Table 5.4. When concordance is measured relative to the traditional goal functions only, the HH and LH models are less likely to predict resilience with a high concordance. The reason for this is the same as discussed above: it is an effect of the parameter range imposed upon the parameters. When compared to concordance measured relative to all goal functions, the frequency with which resilience has the highest concordance relative to the traditional goal functions is reduced. Resilience does not offer the best compromise between goal functions, unless it is also considered a goal function. This further implies that the resilience

5.3 Conclusion

93

Table 5.6: Concordance (Equation 5.3) for each goal function in each model, relative to traditional goal functions only. F/B is minimised. Model Maximise Maximise Maximise Minimise Maximise Supports resilience Name Peq Zeq Feq F/Beq Req heuristic 2? LL1 1.72 1.50 2.00 2.00 0.88 No LL2 1.72 1.50 2.00 2.00 0.88 No LL3 1.71 1.35 1.47 1.94 0.89 No LL4 1.70 1.25 1.95 1.95 0.70 No LL5 1.71 1.35 1.47 1.94 0.89 No LL6 1.70 1.25 1.95 1.95 0.69 No LH1 1.73 1.41 1.27 1.61 1.31 No LH2 1.73 1.41 1.26 1.61 1.29 No LH3 1.01 1.11 1.09 1.14 1.00 No LH4 1.11 1.18 1.04 1.17 1.10 No LH5 1.03 1.12 1.10 1.10 1.00 No LH6 1.12 1.17 1.05 1.16 1.09 No HL1 1.56 1.54 2.00 2.00 0.70 No HL2 1.56 1.54 2.00 2.00 0.69 No HL3 1.58 1.38 1.47 1.96 0.73 No HL4 1.57 1.27 1.96 1.96 0.52 No HL5 1.58 1.38 1.47 1.96 0.73 No HL6 1.57 1.27 1.96 1.96 0.51 No HH1 1.73 1.32 1.14 1.40 1.28 No HH2 1.74 1.29 1.18 1.41 1.28 No HH3 1.05 1.08 1.03 1.12 1.04 No HH4 1.12 1.14 1.02 1.15 1.11 No HH5 1.06 1.13 1.19 1.15 1.03 No HH6 1.13 1.17 1.18 1.17 1.11 No hypothesis described in Cropp & Gabric (2002) is not robust.

5.3

Conclusion

We found that, when concordance is measured relative to all goal functions, resilience has the highest concordance of all goal functions for most model types. Therefore, if one wanted to choose a goal function that offered a reasonable compromise with all goal functions, including ‘maximise resilience’, using ‘maximise resilience’ would be a sensible choice. However we ask: is the ability of ‘maximise resilience’ to provide a compromise between the traditional goal functions a sufficient reason to propose ‘maximise resilience’ as a heuristic? If one did not accept ‘maximise resilience’ as a valid independent goal function, yet still wanted to choose a function that offered a reasonable compromise with the traditional goal functions, would ‘maximise resilience’ be a sensible choice? It was found that, when concordance was measured relative to the traditional goal functions only, ‘maximise resilience’ only had the highest concordance for some model types. The mixed result for the second resilience heuristic (i.e. without assuming the resilience hypothesis) makes it difficult to answer

94


Resilience verses ez 3.5 Complex eigenvalues Real eigenvalues 3

Resilience

2.5

2

1.5

1

0.5

0 0

0.002

0.004

0.006

0.008 ez

0.01

0.012

0.014

0.016

Figure 5.1: Resilience verses eZ for Model LL1. All parameters except eZ are held constant. Note the maxima of resilience is at the transition from real to complex conjugate eigenvalues.

the question set out at the start of this chapter. We must find a way to get a clearer answer regarding our second resilience heuristic. We note that the strength of concordance between ‘maximise resilience’ and the traditional goal functions is sensitive to the parameter range used, and the position of the goal function maxima within that range. Removing this effect may give us a more consistent answer to our question of resilience as a heuristic. Therefore, we continue our investigation in the next chapter with a new model: one for which the parameter range is not so restricted.

Chapter 6

Resilience and the generalised Lotka-Volterra 6.1 6.1.1


The resilience heuristic proposes that ‘maximise resilience’ generally offers a good compromise between the goal functions. In Chapter 5, we tested the robustness of the resilience heuristic to variations in the CG Model. It was noted that the concordance between ‘maximise resilience’ and the traditional goal functions was effected by the parameter range used. In this chapter, we attempt to remove this effect, to further test the resilience heuristic. We are interested in the concordance of resilience when the size, parameter, structure, and formulation of the model is varied. In addition, we are interested in the discrepancies between the results of the resilience hypothesis and what is expected in natural systems. Cropp & Gabric (2002) finds that the resilience hypothesis predicts ecosystem attributes that did not agree with what would be expected from natural selection. We test the generality of this observation.

6.1.2

Overview

In Section 6.2, we develop a generalised Lotka-Volterra Model. We will use this to test the generality of the relationship between ‘maximise resilience’ and the traditional goal functions. Section 6.3 gives expressions for each of the goal functions, for use in Section 6.5.4. In Section 6.5.4, we use a similar methodology to 95

96

Resilience and the generalised Lotka-Volterra

that developed in Chapter 5 to test the resilience heuristic. We calculate the concordance of each goal function for each of the Lotka-Volterra Models developed. In Section 6.4, we use some analytic techniques to investigate the attributes of the model. We are particularly interested in stability measures, including permanence and sector stability. In Section 6.5, we test the analytic predictions of Section 6.4 by generating Lotka-Volterra models with random coefficients. We also make note of any trends that appear between the goal functions.

6.2

The generalised Lotka-Volterra model

6.2.1

Formalisation of the generalised Lotka-Volterra model

We are interested in testing the generality of the concordance result (Section 6.1.1), which is that ‘maximise resilience’ offers a compromise between the traditional goal functions. In order to do this, we consider a family of models: the generalised Lotka-Volterra (GLV) models. The change in nutrient concentration in a compartment may be described as some function of the concentration in that compartment, and the other compartments in the system (Pielou 1969, pp. 19) x˙i = xi fi (x)

(6.1)

where fi (x) is some autonomous function of the nutrient concentrations in each compartment, x. The function fi can take various forms. In the CG model, the forms used were Lotka-Volterra and Holling Type II. It is noted that the Lotka-Volterra form is easier to work with, so we begin our discussions with this (Chapter 7 will continue investigations with more complex interaction forms). The Generalised Lotka-Volterra (GLV) system is given by   n X dxi ai,j xj  , = xi di + dt j=1

(6.2)

where di represents the intrinsic rate of increase, and ai,j the interactions between compartment xi and compartment xj . While many authors object to the simplicity of this form, the Lotka-Volterra form characterises the stability characteristics of a much wider class of formulations when they are approximated around the equilibrium and taken to the first order (May 1973, pp. 41–42). We make the following assumptions for consistency with the CG model. The model is a nutrient cycling model, with the state variables describing the nutrient concentration of each each compartment. The model has n biotic compartments, and has one abiotic ‘nutrient compartment’, denoted n + 1. The biotic compartments interact with one another by predator-prey interactions. The nutrient compartment accepts the nutrient that is lost via excretion, inefficiency and death. The system is assumed to be

97

6.2 The generalised Lotka-Volterra model

closed, so there is no loss or gain of nutrient in the system. For simplicity, it is assumed that there is no nutrient cycling between biotic compartments, and no omnivory across trophic levels (Pimm 1982)1. Such a system is depicted in Figure 6.1.

1

2

3

5

4

6 n+1

Figure 6.1: A simple trophic system with a nutrient compartment. Numbered nodes are compartments, n + 1 is the nutrient compartment. Solid arrows indicate a flow of nutrients due to a predator-prey relationship, and dotted arrows indicate a flow of nutrients from biotic compartments to the nutrient compartment due to waste or mortality.

Let xi denote the nutrient content of compartment i. Assuming a Lotka-Volterra interaction between compartments, the change in nutrient concentration of compartment i can be described by dxi = ai,1 x1 xi + ai,2 x2 xi + . . . + ai,i xi xi + . . . + ai,n xn xi + ai,n+1 xn+1 xi + di xi , dt   n+1 X dxi ai,j xj  . = xi di + dt j=1

(6.3a)

(6.3b)

where ai,j is a coefficient describing nutrient flow from compartment j to compartment i, di is the mortality coefficient of compartment i (−1 ≤ di ≤ 0), and n + 1 represents the nutrient compartment. The use of di implies that species death (by mechanisms other than predation) is linearly proportional to the size of the compartment in terms of nutrient concentration. This does not allow for such nonlinear effects as exclusion (where as the population increases, resource competition, and territorialism increases the death rate). Two constraints are applied to to the values that ai,j may take: 1. Matter is conserved; ai,i = 0 In this model only the flow of nutrients is considered, so an ai,i 6= 0 would violate conservation of matter. 2. If compartment i preys on or is preyed upon by compartment j, the effect on compartment j is equal and opposite; 1 Note

that these assumptions are removed in later chapters.

98


ai,j = −aj,i A shortcoming of this assumption is that it ignores the effects of inefficiencies in nutrient transfer (DeAngelis 1975). A similar equation to Equation 6.3 can be derived for the nutrient content of the nutrient compartment   n X dxn+1 = xj (an+1,j xn+1 − dj ) . (6.4) dt j=1

The nutrient compartment differs from the biotic compartments in that it receives all nutrient from mortality (see Figure 6.1). If assimilation inefficiencies had been considered, there would have also been a term to account for the portion of matter that is wasted in the feeding process. For the nutrient compartment, dn+1 = 0.

The nutrient compartment in this model is treated as a group, rather than dealing with the complexities of decomposition. This includes all detrital matter, and the decomposers that act to make this matter available for recirculation. A shortcoming of this is that all detritus becomes available without the time lag required for its decomposition. Also, it does not differentiate between different forms of the nutrient, and the different rates at which they are re-entrained into the nutrient cycle. To solve for the steady state, Equation 6.3 and Equation 6.4 are set to zero. This gives n + 1 equations and n + 1 unknowns. Because the system is closed to material exchange, then n+1 X

xi = No ,

(6.5)

i=1

where No is the constant amount of nutrients in the system. This equation can be used to reduce the number of variables that Equation 6.3 must be solved for at steady state. At the non-trivial (x⋆i 6= 0 ∀i) steady state (dxi /dt = 0), Equation 6.3 reduces to 0 = ai,1 x1 +ai,2 x2 +. . .+ai,i−1 xi−1 +0+ai,i+1 xi+1 +. . .+ai,n+1 (No − x1 − x2 . . . − xi . . . − xn )+di , (6.6a)   n X (6.6b) 0 =  (ai,j − ai,n+1 )xj  + di + No ai,n+1 . j=1

Note that the di + No ai,n+1 term is the intrinsic rate of increase or decrease for compartment i.

6.2.2

Comments about building the model from first-principles

In general, early works on stability go directly to the creation of a Jacobian matrix, and randomly assign values to the elements of the Jacobian (e.g. May 1972). However, this would limit the insights available into the attributes of the species in the system. The absence of feasibility testing in stability investigations using the Jacobian form only has been criticised in several studies (Roberts 1974, DeAngelis

6.2 The generalised Lotka-Volterra model

99

1975, Lawlor 1978, Yodzis 1981, Paine 1992). Instead, we have specified the nonlinear system first before investigating stability. This is necessary for the calculation of the traditional goal functions’ values, and allows feasibility to be determined. Pimm & Lawton (1978) states that direct selection of Jacobian entries is preferable as it avoids the problem of specifying intrinsic rates of increase. While this may be true for models using population number rather than nutrient concentrations as their state variables, building the model from first principles, and closing the system, allows the intrinsic rates of increase specify themselves in Equation 6.6b. For autotrophic compartments, the large value for No ai,n+1 typically exceeds di , leading to a positive intrinsic rate of increase for producers. Conversely, ai,n+1 = 0 for consumers, so di is negative and represents mortality. Contrast this with Roberts, who arbitrarily set di to unity (Roberts 1974, Roberts & Tregonning 1980). Several authors state that it would be better to make di > 0 for autotrophs and di < 0 for heterotrophs (DeAngelis 1992, Tregonning & Roberts 1979), for example, assuming di = −1 for heterotrophs (Gilpin 1975). However, we did not need to make this assumption. It arises naturally from our model because we have assumed finite total amount of nutrient. Similarly, the intra-compartment interaction terms conform with theorists’ requirements. In models, ai,i is often set to −1, making all compartments self regulating. This means that there is some factor (e.g. competition for territory) independent of food supply and predation that lowers the population. Pimm (1982, p. 71) observes that this biases the models toward stability. He asserts that self-regulation can be achieved through other means. We have demonstrated that this is the case for our model, with Equation 6.6b. For the consumers, the diagonal term, ai,i − ai,n+1 , equals zero. However producers are self-regulating due to the non-zero −ai,n+1 term, which appears as a consequence of limiting total nutrient availability to No .

100


6.3

Expressions for goal functions

In this section, we give expressions for each of the goal functions in the GLV models. These will be used in Section 6.5.4, when we determine if ‘maximise resilience’ offers a compromise between the traditional goal functions, and if this ability is robust to variations in the size and structure of the food web.

6.3.1

Biomass

The biomass is the sum of nutrients in the biotic compartments at steady state, Equation 6.6 in matrix form (for i = 1, 2, . . . , n) gives an n × n   a1,1 − a1,n+1 a1,2 − a1,n+1 · · · a1,n − a1,n+1      a2,1 − a2,n+1 a2,2 − a2,n+1 · · · a2,n − a2,n+1    0= .. .. ..  ..   . . . .   an,1 − an,n+1

an,2 − an,n+1

···

an,n − an,n+1

matrix  x1    x2   ..  . 

n×n

xn

system ⋆ 

Pn

i=1

x⋆i . Expressing

d1 + a1,n+1 No

      d2 + a2,n+1 No  + ..     .   dn + an,n+1 No



    . (6.7)   

For convenience, we shall refer to the n × n matrix above as B. The form of the system in Equation 6.7

can be solved for the steady state solution x⋆i , i = 1, 2, . . . , n, and Equation 6.5 can be used to solve for the steady state nutrient compartment concentration, x⋆n+1 .

6.3.2

Flux

It is easiest to evaluate the flux of nutrients through the system at the interface between the nutrient compartment and the biota. This is a sum of the products of the per unit nutrient death rates, di , and the equilibrium nutrient concentrations in each compartment. The flux is the sum F =

n X

di x⋆i .

(6.8)

i=1

We shall see that this form is also convenient for predicting the attributes of a maximal flux system (Section 6.5.3 and 6.6.3).

6.3.3

Resilience

Resilience is defined as the negative real part of the maximum (i.e. most positive) eigenvalue of the Jacobian Matrix. The Jacobian evaluated at the steady state, x⋆ , is   dx1 dx1 dx1 ∂ ∂ ∂ · · · ∂xn dt   ∂x1 dt ∂x2 dt   ∂ dx2 dx2 ∂ 2 · · · ∂x∂ n dx   ∂x1 dt ∂x2 dt dt   J=  .. .. ..  . ..   . . . .   dxn dxn ∂ ∂ n · · · ∂x∂ n dx ∂x1 dt ∂x2 dt dt ⋆ x

(6.9)

6.4 Analytic results for the generalised Lotka-Volterra model

The diagonal elements of the Jacobian Matrix are (for more details, see Appendix D.1)   n+1 X dxi ∂ = ai,j x⋆j  − ai,n+1 x⋆i + di , ∂xi dt ⋆ x

101

(6.10a)

j=1

and the off-diagonal elements are ∂ dxi = x⋆i (ai,j − ai,n+1 ). ∂xj dt x⋆

(6.10b)

Using Equations 6.10a to 6.10b gives the Jacobian Matrix of size n × n,  P n+1 ⋆ ⋆ . . . x⋆1 (a1,j − a1,n+1 ) . . . x⋆1 (a1,n − a1,n+1 ) k=1 a1,k xk − a1,n+1 x1 + d1     .. .. ..   . . .     .. . ⋆ ⋆ J = . ... xi (ai,n − ai,n+1 ) xi (ai,1 − ai,n+1 ) ...     .. ..   ..   . . .   P n+1 ⋆ ⋆ ⋆ ⋆ xn (an,1 − an,n+1 ) . . . xn (an,j − an,n+1 ) . . . − a x + d a x n,n+1 n n k=1 n,k k (6.11)

This can be used to find the eigenvalues of the system at the steady state numerically. The steady state of interest is the non-trivial steady state, x⋆i > 0 ∀i = 1 . . . n + 1. Resilience is defined for a locally stable as the negative of the real part of the eigenvalue with the maximum (most positive) real part R = −max{Re(λi )},

(6.12)

where λi are the eigenvalues of the Jacobian Matrix J.

6.4

Analytic results for the generalised Lotka-Volterra model

This section is an aside, in which we use various analytic methods to gain insight into the GLV system. The reader may proceed directly to Section 6.6 without loss of continuity. We will investigate the diagonal elements of the Jacobian Matrix (Section 6.4.1), the Routh-Hurwitz criteria (Section 6.4.2), analytic approximations of the resilience for small systems (Section 6.4.3), Lyapunov functions (Section 6.4.4), and analytic results involving permanence (Section 6.4.5). We will verify some of these results numerically in Section 6.5.

6.4.1

The diagonal elements of the Jacobian Matrix

The sign of the diagonal elements of the Jacobian matrix Remark 1 For all diagonal elements of the Jacobian Matrix, ji,i , such that ai,n+1 = 0, we have ji,i = 0.

102


If ai,n+1 = 0, compartment i is a heterotroph. For these compartments, a particular row of Equation 6.7 is 0=

n+1 X k=1

(ai,k −

ai,n+1 )x⋆k

!

+ di + ai,n+1 No ,

(6.13)

which becomes 0=

n+1 X

ai,k x⋆k

k=1

!

+ di .

(6.14)

The diagonal of the Jacobian Matrix is ! n+1 X ⋆ ji,i = ai,k xk − ai,n+1 x⋆i + di ,

(6.15)

k=1

which becomes n+1 X

ji,i =

ai,k x⋆k

k=1

!

+ di ,

(6.16)

ji,i = 0.

(6.17)

Remark 2 For all diagonal elements of the Jacobian Matrix, ji,i , such that ai,n+1 > 0, ji,i < 0. If ai,n+1 > 0, compartment i is an autotroph. For these systems, a particular row of Equation 6.7 is ! n+1 X ⋆ (6.18) 0= (ai,k − ai,n+1 )xk + di + ai,n+1 No , 0=

0=

k=1 n+1 X

(ai,k −

k=1 n+1 X

ai,k x⋆k

k=1

ai,n+1 )x⋆k

!

!

+ di + ai,n+1

n+1 X k=1

x⋆k

!

,

+ di .

(6.19)

(6.20)

Substituting into Equation 6.15 gives ji,i = −ai,n+1 x⋆i .

(6.21)

Because nutrients always flow away from the nutrient compartment, ai,n+1 > 0, therefore ji,i < 0. Corollary 1 All diagonal elements of the Jacobian matrix in Equation 6.11 are less than or equal to zero.

Discussion As trace(J) =

Pn

k=1

λk , Corollary 1 has a stabilising affect upon the system. It does not, however,

guarantee stability, because most of the Gerschgorin Disks (Gerald & Wheatley 1997, pp. 545) are


103

centred about zero. The Gerschgorin Disks corresponding to the heterotroph compartments (ai,n+1 = 0) are centred about 0, and the Gerschgorin Disks corresponding to the autotroph compartments (ai,n+1 > 0) are centred about some negative real value, as found above (Figure 6.2). Unfortunately neither of these observations provide any useful predictions.

Imaginary axis

Real axis

−an,n+1 x* n

Figure 6.2: Gerschgorin Disks from a hypothetical three heterotroph, one autotroph system. The eigenvalues of the Jacobian Matrix will lie within these disks. Although all Gershgorin Disks are centred around non-positive values, there is no guarantee that all eigenvalues will be negative, and therefore stable.

6.4.2

Routh-Hurwitz criteria

The Routh-Hurwitz criteria are commonly used to determine if a system is locally stable (e.g. May 1973, p. 196). They are described in detail in Appendix D.2. In this section, we explore what insights these criteria can provide on the GLV model developed in Section 6.2.

Applying the Routh-Hurwitz criterion to the GLV

Remark 3 By the Routh-Hurwitz Criteria, generalised Lotka-Volterra systems that 1. are feasible (x⋆i > 0, ∀i = 1, . . . n + 1); 2. are chains; and 3. have 2 or 3 biotic compartments, are locally stable.

104


The Jacobian Matrix for a chain is J=              

0

x⋆1 a1,2

0

x⋆2 a2,1

0

x⋆2 a2,3

0 .. .

x⋆3 a3,2

0

0

···

−x⋆n an,n+1

··· ··· ..

···

0 −x⋆n an,n+1

··· . 0 x⋆n (an,n−1

− an,n+1 )

0



     0   . (6.22)   0   ⋆ xn−1 an−1,n   −x⋆n an,n+1 0

Two biotic compartment chain For a chain with two biotic compartments, the characteristic equation is 0 = λ2 − λ(j1,1 + j2,2 ) + j1,1 j2,2 − j1,2 j2,1 .

(6.23)

The Routh-Hurwitz Criterion are −(j1,1 + j2,2 ) > 0,

(6.24a)

j1,1 j2,2 − j1,2 j2,1 > 0.

(6.24b)

Equation 6.24a becomes x⋆2 a2,n+1 > 0,

(6.25)

which is true. Equation 6.24b becomes a21,2 x⋆1 x⋆2 > 0,

(6.26)

which is true. Three biotic compartment chain For a chain with three biotic compartments, the characteristic equation is 0 = λ3 + λ2 (−j3,3 ) + λ (−j3,2 j2,3 − j1,2 j2,1 ) + j1,2 j2,1 j3,3 − j1,2 j3,1 j2,3 .

(6.27)

The Routh-Hurwitz Criterion are −j3,3 > 0,

(6.28a)

j1,2 j2,1 j3,3 − j1,2 j3,1 j2,3 > 0,

(6.28b)

105


−j3,3 (−j3,2 j2,3 − j1,2 j2,1 ) > j1,2 j2,1 j3,3 − j1,2 j3,1 j2,3 .

(6.28c)

Equation 6.28a becomes a3,n+1 x⋆3 > 0

(6.29)

which is true. Equation 6.28b becomes j1,2 |{z}

j2,1 |{z}

j3,3 |{z}

(+)

(−)

(−)

−

(x⋆1 a1,2 ) (x⋆2 a2,1 ) (−x⋆3 a3,n+1 ) | {z } | {z } | {z }

j1,2 |{z}

− (x⋆1 a1,2 ) | {z }

−

(+)

j3,1 |{z}

j2,3 |{z}

>

0

(−x⋆3 a3,n+1 ) (x⋆2 a2,3 ) > | {z } | {z }

0

(−)

(+)

>

(6.30)

0

By the signs of the terms alone, it can be seen that this is true. After expanding and cancelling, Equation 6.28c becomes j3,3 |{z}

−x⋆3 a3,n+1

j3,2 |{z}

>

x⋆3 (a3,2 − a2,n+1 ) >

x⋆3 (a3,2 − a2,n+1 )
1, may be approximated by p(R > M ) =

1 . 8M

(6.34)

106


1 2 N

Figure 6.3: Diagrammatic representation of Ecosystem 3-1. Solid arrows indicate the direction of nutrient transfer due to predator-prey interactions. Dotted arrows show nutrient transfer from each biotic compartment to the nutrient compartment via death.

A 3-compartment chain, is shown in Figure 6.3. At steady state, the biomass of compartment 1 and compartment 2 is a2,3 d1 1 + d2 , a2,3 No + x⋆1 = a1,2 + a2,3 a1,2 x⋆2 =

−d1 . a1,2

(6.35) (6.36)

Because the system gives a Jacobian matrix of size 2 × 2, the resilience may also be found analytically. The eigenvalues, λ, are found by the equation s 2 −d1 a2,3 a2,3 d1 d1 a2,3 ± 0.5 + d2 . λ= + 4d1 a2,3 No + 2a1,2 a1,2 a1,2

(6.37)

Equation 6.37 gives two eigenvalues, which may be real or complex conjugate. In order for the eigenvalues to be real, the following condition must be satisfied −

d1 a22,3 4a2,3 d1 − 4d2 − > 4a2,3 No 2 a1,2 a1,2

(6.38)

which is unlikely for large No unless a1,2 is very small (in these experiments, No = 500). This simplifies the resilience to the first term in Equation 6.37 d1 a2,3 2a1,2 a2,n . = x⋆2 2

R=−

(6.39) (6.40)

From the analytical description, it can be seen that x⋆2 and R share a linear relationship. This is reflected in results in Appendix D.3. The probability that the resilience, R, is greater than some number, M , where M > 1, can be found as follows. First, we find the probability that x⋆2 > N −d1 . p (x⋆2 > N ) = p a1,2 < N

(6.41)

Recall that the coefficients are chosen from a random uniform distribution, where −1 ≤ d1 ≤ 0, and 0 ≤ a1,2 ≤ 1. We let y = −d1 , so that our notation is easier to follow. Then, the total probability is


found by the integral Z ⋆ p (x2 > N ) =

0

1

y .dy N

1 . 2N

=

107

(6.42) (6.43)

We can now substitute Equation 6.42 into our expression for R (Equation 6.39). p(R > M ) = p(

a2,3 x⋆2 > M) 2

= p(x⋆2 > N ). As N =

2M a2,3

p(R > M ) =

(6.44) (6.45) (6.46)

a2,3 . 4M

Now, we let y = −a2,3 Z 1 y .dy p(R > M ) = 4M 0 1 = . 8M

(6.47)

(6.48) (6.49)

Discussion This result implies that high values of resilience are difficult to obtain at random for three-compartment chains. Unfortunately, the result is only applicable to the particular size and structure of the food web investigated.

6.4.4

Sector stability

Details Lyapunov functions were discussed in Section 3.2.3. We employ some of Lyapunov functions developed in the literature to our GLV system. Remark 5 A sufficient condition for Goh-Lyapunov Stability, and therefore global asymptotic stability, is if a cn×1 can be found such that c−1 i si,j = bi,j , where bi,j are elements of the matrix B in Equation 6.7, and si,j = −sj,i , so S = (si,j ) is antisymmetric.

108


Refer to Rouche et al. (1977, p. 260-264) and Porati & Granero (2000) for a derivation. Note that it is assumed that bi,i ≤ 0 for all i (which is apparent from Equation 6.7). This condition can be demonstrated with the original Goh-Lyapunov function. Let xi − x⋆i = δi , then V˙ =

n X n X

ci δi δj bi,j < 0,

(6.50)

i=1 j=1

=

n X

ci bi,i δi2

i=1

!



 + 

n X

i,j=1 i6=j



 ci δi δj bi,j  .

(6.51)

If ci bi,j = −cj bj,i , then the second term is zero and V˙ becomes ! n X 2 ci bi,i δi , V˙ = i=1

which is negative semidefinite for bi,i ≤ 0.

Remark 6 The feasible 3-compartment-chain LV Network is Goh-Lyapunov Stable (Remark 5), and therefore globally asymptotically stable. B is of the form 

0

a1,2



, B= −(a1,2 + a2,3 ) −a2,3

so c of the form cT = a1,2a+a2,3 1,2

1

(6.52)

(6.53)

will satisfy the requirements. Remark 7 If bi,i = 0 for more than one i, then the sufficient condition for Goh-Lyapunov stability described in Remark 5 can be strengthened to a necessary one. For our GLV, bi,i 6= 0 implies that compartment i is an autotroph. The derivation relies upon finding a

situation for which the only guarantee that V˙ ≯ 0 is that the second term in Equation 6.51 described above is equal to 0. That is, that c−1 i si,j = bi,j can be found. Consider first a system such that for only one i = k, bk,k < 0, else bi,i = 0.   0    0      ..   .   , B=     bk,k < 0     . ..     0


then, V˙ of Equation 6.51 becomes 



n  X . c δ δ b V˙ = ck bk,k δk2 +  i i j i,j  

109

(6.54)

i,j=1 i6=j

Now consider the possibility that the perturbation vector δ is such that all δi 6= 0 except δk = 0. Applying this to the Goh-Lyapunov function leaves only the second term in Equation 6.54. As there is no guarantee that this term will be ≤ 0, c must be chosen such that this second term is zero, which can only be done by the method described in Remark 5. A similar type of argument can be used for all cases up to that in which two of the bi,i are zero. Corollary 2 All generalised Lotka-Volterra models in which there are two or more heterotrophs are GohLyapunov unstable. It should be noted that this is not to say that they are unstable or Lyapunov unstable in the general sense. Remark 8 Sufficient conditions for a single autotroph compartment system with more than 1 heterotrophic compartment to be Goh-Lyapunov stable are that 1. every heterotrophic compartment must feed upon the autotrophic compartment; and 2. the feeding rates are to be equal. The condition that c exists as described above applies to this system by Remark 7. The first requirement, that every biotic compartment must feed upon the autotroph, can be observed by noting that in order for c−1 i si,j = bi,j where si,j = −sj,i to hold for j = n, bi,n cannot equal zero. In other words, each compartment i feeds upon the autotroph in compartment n. The second requirement, that all feeding rates are equal, can be derived from consideration of the need to create antisymmetry between b:,n and bn,: , where : denotes the row or column of the matrix. Let all heterotroph grazing (upon the single autotroph) rates be equal to some constant, h1 . Then the last row of the interaction matrix is bn,: = −(h1 + an,n+1 ) −(h1 + an,n+1 ) . . . −(h1 + an,n+1 ) −an,n+1 , and the last column is b:,n = h1

h1

. . . h1

so cn =

h1 , h1 + an,n+1

−an,n+1 ,

110


satisfies the condition for Goh-Lyapunov stability. As all other (i, j = 1, . . . , n − 1) elements of B are of the form bi,j = sgn(j − i)ai,j , ci = 1 for those rows. Remark 9 Remark 8 is also a necessary condition. It is noted that, in order to preserve the antisymmetry of the leading minor B1...n−1,1...n−1 , ci = 1 for each row i = 1 . . . n − 1. This implies that the antisymmetry between the last row and last column of B must be achieved with cn alone. Consider first the case in which all heterotroph grazing rates are equal, h1 , except one, h2 . Then cn must simultaneously satisfy h1 , and h1 + an,n+1 h2 , cn = h2 + an,n+1 cn =

which is not possible with h1 6= h2 . Similarly for hn .

Discussion

The previous section employs Lyapunov functions developed in the literature. It is found that, even with many simplifying assumptions, the results are not particularly useful. One of the problems with Lyapunov functions is that if one does not find Lyapunov stability by the function employed, it may simply mean that the wrong function was employed. For example, Corollary 2 shows that the particular Lyapunov function, the Goh-Lyapunov function, gives no information about the stability of GLV models. However, this does not mean that some other unknown Lyapunov function could not be found that would perform well. The result from Remark 8 is uninteresting because it describes an ecologically implausible event: the coexistence of many compartments upon a single resource, where the coexistence relies upon the compartments being exactly equivalent. This further demonstrates the difficulties inherent in demonstrating sector stability for ecological models.


6.4.5

111

Permanence of chains

The meaning of permanence used in this section has already been introduced (Section 3.2.4). A model ecosystem is permanent if it is bounded within the feasible region in the phase space, and below some finite value. This section describes analytic results concerning the permanence of chains. Remark 10 A generalised Lotka-Volterra model that is a chain is permanent provided that the removal of a predator increases the steady state nutrient concentration of its prey. If it can be shown that at least one transversal eigenvalue (Section 3.2.4) is positive for each subsystem of an n-chain on bd(Sn ), then the system is permanent (Hofbauer & Sigmund 1988, pp. 166–169).

m−1

m

m+1

n

n+1

Figure 6.4: A diagram showing subsystems of a chain food web, and the notation used in the text.

First note that only boundary steady states, p, of the form pTm = 0 0

. . . 0 x⋆m

x⋆m+1

. . . x⋆n ,

(where m denotes the highest trophic level of the subsystem, see Figure 6.4) are considered, as these are the only subsystems on bd(Sn ) that exist for chains. We denote each of the transversal eigenvalues γi such that n X x˙ i bi,j xj . = l + γi = i xi i=1

112


Consider pm , which is the only subsystem of pm−1 (see Figure 6.4). The transversal eigenvalue associated with the system γm−1 = am−1,m x⋆m + dm−1 , is greater than 0 (implying that the system is permanent with respect to that subsystem) if x⋆m >

−dm−1 . am−1,m

(6.55)

Note that if Equation 6.55 was an equality, it would describe the steady state value of xm in the original system pm−1 . This implies that the transversal eigenvalue will increase if x⋆m increases when the subsystem is formed. If m = n + 1, the constraint is x⋆n = No >

−dn−1 , an−1,n

which is always true. Remark 11 If the top predator of a feasible chain is removed, the steady state value of its prey will increase. Consider the first two steps of the Gaussian elimination performed upon the last two rows of the matrix in Equation 6.7, where the system is a chain.    .. .. . .       gn−1,:  =  0 . . . −an−1,n−2    gn,: −an,n+1 . . . −an,n+1

| 0

an−1,n

−(an−1,n−2 + an,n+1 )

−an,n+1

|



.. . −dn−1

| −dn − an,n+1 No

   

where gi,j is the ith row and jth column of the Gaussian elimination matrix, G. To get G into (lower) triangular form, each of the ak,k+1 terms need to be set to zero. This is done by operations of the form gk−1,: = gk−1,: − gk,:

gk−1,k . gk,k

(6.56)

As all of the elements of gn,1...n are negative, and assuming that gn,n+1 is negative (which is necessary for the system to be feasible), the operation described in Equation 6.56 will result in all elements of gn−1,: being negative. It follows that each iteration of the Gaussian Elimination will cause the corresponding resultant gk−1,: to have all negative elements. Consider the completed Gaussian    g1,1 0 ... g1,:       g2,:  = g2,1 g2,2 0    .. .

elimination upon the chain system. The top two entries will be  0 | g1,n+1   ... 0 | g2,n+1  , (6.57)  .. .. . . |

6.5 Investigation of random-coefficient generalised Lotka-Volterra models

113

such that g2,1 x⋆1 + g2,2 x⋆2 = g2,n+1 ,

(6.58)

where all gi,j have the same sign. Therefore setting x⋆1 = 0 increases the value of x⋆2 . Corollary 3 All feasible Lotka-Volterra chains described are permanent. This follows from a combination of Remarks 10 and 11.

Discussion It is well documented that local stability analysis can fail to capture the intuitive meaning of stability and give ‘false positives’. For example, local stability analysis may result in a system being classified as stable when the basin of attraction is infinitesimally small (see Section 3.2.2). However, the GLV chains above are an example of the converse situation: that local stability analysis will give ‘false negatives’. We know, by a small proof (Section 6.4.5), that all GLV chains are permanent. Therefore, all chains are stable, in the sense of permanent, even if they are not stable by local stability analysis. They may tend toward limit cycles (e.g. Appendix D.4), or chaotic attractors. Permanence is perhaps a more suitable measure of stability, if stability is being used as a proxy for survival of the system (Grimm et al. 1992). If this is the case, then the observation that the probability of local stability decreases with system size (Section 3.3.1) is irrelevant.

6.5

Investigation of random-coefficient generalised Lotka-Volterra models

6.5.1

Introduction

This section is an aside, in which the GLV is explored by generating models with random coefficient values. Several analytic predictions about the GLV systems have been made (Section 6.4). The purpose of this section is to explore the GLV, and to verify these analytic results. In addition, any relationships observed between the goal functions are noted. The reader may proceed directly to Section 6.6 without loss of continuity.

6.5.2

Methods

For every food web structure with up to five biotic compartments (Figure 6.5), and for every food chain with up to ten biotic compartments, GLV systems were generated randomly. First, coefficient values were

114


chosen from a random uniform distribution, subject to the constraints described in Section 6.2. Then the rank of the B matrix in Equation 6.7 was found numerically. Those systems that consistently had rank(B) < n are singular due to their structure, and are not investigated further. They are noted in Figure 6.5. For the remaining food web structures, random coefficient values were generated until 500 feasible-stable systems were obtained. The algorithm is as follows: 1. A predefined template (Figure 6.5) is used to define the structure of the ecosystem. The template specifies which compartments will interact, and the direction of the flow of nutrients from that interaction. 2. For each interaction, a random values are assigned to the parameters subject to constraints described in Section 6.2. 3. The values of each of the goal functions (biomass, flux, flux to biomass ratio, resilience) are found and used to calculate: (a) the mean goal function values; and (b) the ‘concordance’ of the mean values. Any relationships observed between the goal functions were noted. 4. The values of goal functions and other information about each system are stored, and the process is repeated from step 2, until 500 feasible-stable model ecosystems are generated. We also investigated various attributes of the systems generated, according to our goal of comparing the results to analytic predictions. Attributes investigated are: 1. the proportion of infeasible and unstable systems in each run (cf. predictions in Section 6.4.2); 2. the probability of resilience greater than some value (cf. predictions in Section 6.4.3); and 3. the implications of the permanence of chains (Section 6.4.5).

6.5.3

Results

Ecosystem structures Figure 6.5 shows each of the models investigated. Biotic compartments are denoted by a circle, and the nutrient compartment by a square. For clarity, the return of biomass to the nutrient compartment via death is not shown. For each model in Figure 6.5, if the rank of B is less than n, Rank(B) is displayed (as found numerically).


3−1 2−1

115

3−2 Rank(B)=1

4−1

4−2

4−3

Rank(B)=2

5−1 4−4

4−5 Rank(B)=2

5−2

5−3 5−4

Rank(B)=2

5−5 5−6

5−7

Rank(B)=3

5−8

5−9

Rank(B)=3

Rank(B)=3

5−10 Rank(B)=2

Figure 6.5: All possible arrangements of compartments (excluding omnivory) for n = 1 . . . 4. Rank(B) < n are displayed.

116


Frequency of feasible-stable systems The number of infeasible and feasible-but-unstable systems generated were counted, and the results summarised in Table 6.1. Sections containing relevant analytic results are indicated.

Model 3-1 4-1 4-2 4-4 5-1 5-2 5-3 5-6 5-7

Table 6.1: Random runs and failures. Infeasible Feasible Reference but unstable Sections 1 0 6.4.2, 6.4.5, 6.4.4 2 0 6.4.2, 6.4.5 5 × 105 0 5 × 102 0 7 20 6.4.5, A. D.5 5 × 102 0 1 × 105 0 2 × 103 3 × 102 7 1 × 102

Expected value of goal functions In Section 6.4.3, for ecosystem structure 3-1, the probability of resilience greater than some value was predicted to be p(R > M ) =

1 . 8M

(6.59)

Figure 6.6 compares the prediction of Equation 6.59 with data generated. Table 6.2 shows the mean goal function values found from the 500 data points generated. The concordance of the mean values is calculated with respect to the traditional goal functions only. Both the ‘maximise flux to biomass ratio’ and ‘minimise flux to biomass ratio’ scenarios are considered. Table 6.2: Mean goal function values for randomly chosen points in parameter space. Concordance of random points calculated with respect to traditional goal functions only. Maximum possible concordance is 3. Maximise and minimise F/B scenarios shown. Mean value of Mean value’s concordance (Max F/B) (Min F/B) Model B F F/B R 3-1 248 123 0.49 0.613 1.23 1.25 4-1 491 252 0.51 0.431 2.00 1.97 4-2 492 299 0.61 0.049 2.19 1.98 4-4 495 303 0.61 0.062 2.21 1.98 5-1 342 166 0.49 0.225 1.51 1.53 5-2 282 163 0.59 0.034 1.48 1.30 1.11 0.88 5-3 152 97 0.62 0.014 5-6 287 152 0.54 0.006 1.42 1.34 1.57 1.53 5-7 345 179 0.52 0.020


117

p(R>M) verses M 1 Predicted Empirical 0.9 0.8 0.7

p(R>M)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

2

4

6

8

10

12

14

16

18

M

Figure 6.6: The probability that resilience, R, is greater than some value, M , found empirically and compared to predicted value 1/(8M ).

Relationships between goal functions The upper limit on flux had a negative relationship with biomass in the nutrient compartment. Figure 6.7 shows a representative example. This relationship may be predicted from the form of Equation 6.8. Equation 6.8 describes flux as the weighted sum of the biomass in each of the biotic compartments (where the weightings are the di terms). No clear relationship between flux and resilience could be determined. The relationship between resilience and the biomass in the biotic and abiotic compartments provided the most consistent results over the models trialled. It was found that resilience and the biomass in the nutrient compartment shared a negative relationship. This pattern is particularly pronounced in chain systems, where an asymptotic relationship may be discerned. An example is shown in Figure 6.8. Results for all models are available in Appendix D.3.

6.5.4

Discussion

Verifying analytic results Figure 6.6 verifies the analytic prediction that high resilience has low probability for a 3-biotic-compartment chain. However, if resilience is being used as a proxy for survival, the result that all feasible chains are

118


Biomass in nutrient compartment verses Flux 450 400 350 300

Flux

250 200 150 100 50 0 0

50

100

150

200 250 300 350 Biomass in nutrient compartment

400

450

500

Figure 6.7: Flux versus biomass in the nutrient compartment. Five hundred randomly chosen points.

permanent (Section 6.4.5) makes this finding less relevant.

Table 6.1 supports the prediction that all feasible 2- and 3-biotic-compartment chains are locally stable, and provides information about the probability of feasibility and local stability for other structures. The numerical results suggest that this is also true for several of the other systems (cf. Roberts 1974).

In contrast with Nisbet & Gurney (1976), we did not find that all feasible chains were locally stable. For 5-compartment chains (ecosystem 5-1 in Table 6.1), 20 feasible systems were found that were not locally stable. The proof given in Nisbet & Gurney (1976) assumes that successive submatrices of the Jacobian matrix have all negative eigenvalues. In Appendix D.5, we show that this assumption does not hold.

Trends in relationships between goal functions

It was found that resilience and the nutrient concentration in the nutrient compartment shared a negative relationship. The implication of this is that maximising resilience is compatible with the maximisation of biomass, for the range of models described so far. Flux also had a negative relationship with biomass, however this did not translate to a relationship between flux and resilience.

119

6.6 Investigation of Lotka-Volterra chains using a genetic algorithm

Ecoysstem 5-1: Resilience verses Biomass in nutrient compartment 3

2.5

Resilience

2

1.5

1

0.5

0 0

50

100

150

200 250 300 350 Biomass in nutrient compartment

400

450

500

Figure 6.8: Comparison between Resilience versus biomass in the nutrient compartment for the 5 biotic compartment chain.

6.6

Investigation of Lotka-Volterra chains using a genetic algorithm

6.6.1

Introduction

The resilience heuristic states that ‘maximise resilience’ generally offers a good compromise between the goal functions. In Chapter 5, we tested the robustness of the resilience heuristic to variations in the CG Model. It was found that the relationship was affected by the parameter range used. In Section 6.2, we develop a GLV Model upon which to further test the resilience heuristic. Section 6.3 gives expressions for each of the goal functions in the GLV. In this section, we further test the robustness of the resilience heuristic using the GLV model. The model was developed to reduce the effect of the parameter range upon concordance calculations. Therefore, we are interested in determining if the concordance of ‘maximise resilience’ will be reduced in the GLV compared to the results of Chapter 5. The CG Model used in Chapter 5 is a food chain. It has been noted in the literature that food chain size affects stability, and resilience (Chapter 3). Here, we will also examine the robustness of the resilience heuristic to an increase in the length of the food chain.

120


6.6.2

Methodology

In this section, we describe the relationships between the goal functions in the GLV. Similar to Chapter 4, a genetic algorithm is used to find the parameter set that maximises each of the goal functions. This parameter set is then used to find the value of the other goal functions, and the concordance is calculated. The goal functions investigated were: 1. Biomass. Pn B = i=1 x⋆i .

The total steady state biomass in the biotic compartments.

2. Flux. F =−

Pn

i=1

di x⋆i .

The flux of nutrients through the system at steady state. 3. Flux to Biomass ratio. F/B The ratio of the flux of nutrients through the system to the total biomass in the biotic compartments at steady state. 4. Resilience. R = −max{Re(λ)}. The negative real part of the eigenvalue of the Jacobian Matrix with the most positive real part. Chains of size n = 3 to 10 are investigated. A 0 to 1 range on the parameters could not be used, as the matrix A would often become singular. Therefore, each interaction term ai,j is given the range 1 × 10−3 ≤ ai,j ≤ 9.99 × 10−1 . The death rate terms are −1 × 10−3 ≥ ai,j ≥ −9.99 × 10−1 . The total amount of nutrient in the system was chosen for consistency with Chapter 4; No = 500. The case No = 3000 was used to investigate the effect of total nutrient on maximal resilience. Concordance was found in the same manner as described in Chapter 4. Concordance was found relative to all goal functions according to the equation Cj =

i=5 X

ˆ i (αmax j ), Γ

i=1 i6=j

0 ≤ Cj ≤ 3,

(6.60)

and was found relative to the traditional goal functions only according to the equation Cj =

i=4 X i=1

ˆ i (αmax j ), Γ

0 ≤ Cj ≤ 3,

(6.61)

ˆ i (αmax j ) is the normalised value of goal function where Cj is the concordance of goal function j, and Γ i when the parameter values that maximise goal function j are applied. Goal functions are numbered as


121

follows: 1. ‘maximise biomass’, 2. ‘maximise flux’, 3. ‘maximise flux to biomass ratio’, 4. ‘minimise flux to biomass ratio’, and 5. ‘maximise resilience’.

6.6.3

Results

Observations regarding the maximal goal function values Table 6.3 shows the value of each of the goal functions when the parameter set that maximises the goal function, as indicated, is applied. The maximum values for each of the traditional goal functions are approximately equal across chain sizes. Raw data collected during the runs shows that, for both ‘maximise flux’ and ‘maximise flux to biomass ratio’, death rate terms selected are approximately equal to each other, and equal to one. Table 6.3 demonstrates that maximising each of the goal functions does not optimise resilience. Further, as the length of the chain increases, the maximisation of the goal functions leads to lower values of resilience and normalised resilience. For example, for a 3-compartment chain, ‘maximise biomass’ gives a resilience value of 7.79 × 10−3 , which is 3.8 × 10−4 when normalised against the maximum resilience

value. However, for a 10-compartment chain, ‘maximise biomass’ gives a resilience value of 2.07 × 10−6 ,

which is 1.5 × 10−6 when normalised.

Trends in the relationship between system size and maximal resilience Figure 6.9 shows a plot of the number of species versus the maximum resilience (found with the genetic algorithm) for two different total nutrient values. A high total nutrient gives a higher maximum resilience. Both curves show a decrease in maximum resilience with system size.

122


Goal Maximised Size = 3 B F F/B -F/B R Size = 4 B F F/B -F/B R Size = 5 B F F/B -F/B R Size = 6 B F F/B -F/B R Size = 7 B F F/B -F/B R Size = 8 B F F/B -F/B R Size = 9 B F F/B -F/B R Size = 10 B F F/B -F/B R

Table 6.3: Values for goal functions (chains sized 3-10). Goal Function’s Value B F F/B -F/B

R

4.99 × 102 4.95 × 102 6.44 × 101 1.6 × 101 4.88 × 102

4.8 × 100 4.95 × 102 6.44 × 101 1.6 × 10−2 4.73 × 102

9.61 × 10−3 1 × 100 1 × 100 0.99 × 10−3 9.68 × 10−1

−9.62 × 10−3 −1 × 100 −1 × 100 −1 × 10−3 −9.69 × 10−1

7.79 × 10−3 3.5 × 100 7.39 × 10−2 1.65 × 10−5 2.04 × 101

4.99 × 102 4.98 × 102 4.11 × 102 4.99 × 102 4.85 × 102

5 × 10−1 4.98 × 102 4.11 × 102 4.99 × 10−1 4.69 × 102

1 × 10−3 9.99 × 10−1 9.99 × 10−1 0.99 × 10−3 9.65 × 10−1

−1.01 × 10−3 −10 × 10−1 −10 × 10−1 −1 × 10−3 −9.66 × 10−1

1.65 × 10−5 5.76 × 10−3 1.69 × 10−1 9.65 × 10−4 7.95 × 100

4.99 × 102 4.98 × 102 2.31 × 102 3.25 × 102 4.82 × 102

8.38 × 10−1 4.98 × 102 2.31 × 102 3.25 × 10−1 4.26 × 102

1.67 × 10−3 9.99 × 10−1 9.99 × 10−1 0.99 × 10−3 8.83 × 10−1

−1.68 × 10−3 −10 × 10−1 −10 × 10−1 −1 × 10−3 −8.84 × 10−1

2.54 × 10−4 1.16 × 10−2 1.07 × 10−3 1.39 × 10−5 6.08 × 100

4.99 × 102 4.98 × 102 4.98 × 102 3.72 × 102 4.83 × 102

6.67 × 10−1 4.98 × 102 4.98 × 102 3.72 × 10−1 4.55 × 102

1.33 × 10−3 9.99 × 10−1 9.99 × 10−1 0.99 × 10−3 9.41 × 10−1

−1.34 × 10−3 −10 × 10−1 −10 × 10−1 −1 × 10−3 −9.42 × 10−1

6.02 × 10−5 3.21 × 10−6 2.84 × 10−7 1.48 × 10−5 4.02 × 100

4.99 × 102 4.98 × 102 1.65 × 102 4.76 × 102 4.79 × 102

8.97 × 10−1 4.96 × 102 1.65 × 102 4.77 × 10−1 3.99 × 102

1.79 × 10−3 9.96 × 10−1 9.99 × 10−1 1 × 10−3 8.33 × 10−1

−1.8 × 10−3 −9.97 × 10−1 −10 × 10−1 −1.01 × 10−3 −8.34 × 10−1

8.05 × 10−5 5.44 × 10−4 1.22 × 10−5 7.15 × 10−4 2.77 × 100

4.99 × 102 4.98 × 102 4.98 × 102 4.92 × 102 4.83 × 102

5.49 × 10−1 4.96 × 102 4.98 × 102 4.92 × 10−1 3.81 × 102

1.09 × 10−3 9.96 × 10−1 9.98 × 10−1 1 × 10−3 7.88 × 10−1

−1.1 × 10−3 −9.97 × 10−1 −9.99 × 10−1 −1.01 × 10−3 −7.89 × 10−1

1.05 × 10−5 3.59 × 10−5 1.66 × 10−3 4.22 × 10−6 2.24 × 100

4.99 × 102 4.97 × 102 4.2 × 102 2.96 × 102 4.83 × 102

1.77 × 100 4.95 × 102 4.2 × 102 2.96 × 10−1 3.49 × 102

3.54 × 10−3 9.95 × 10−1 9.99 × 10−1 1 × 10−3 7.22 × 10−1

−3.55 × 10−3 −9.96 × 10−1 −10 × 10−1 −1.01 × 10−3 −7.23 × 10−1

−2.45 × 10−5 4.28 × 10−3 2.81 × 10−4 1.97 × 10−7 2.17 × 100

4.99 × 102 4.98 × 102 4.98 × 102 4.99 × 102 4.85 × 102

5.08 × 10−1 4.98 × 102 4.97 × 102 5.64 × 10−1 3.77 × 102

1.01 × 10−3 9.99 × 10−1 9.97 × 10−1 1.12 × 10−3 7.78 × 10−1

−1.02 × 10−3 −10 × 10−1 −9.98 × 10−1 −1.13 × 10−3 −7.79 × 10−1

2.07 × 10−6 4.01 × 10−6 1.39 × 10−5 2.91 × 10−6 1.8 × 100

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Maximum Resilience verses System Size 55 No = 500 No = 3000

50 45

Maximum Resilience

40 35 30 25 20 15 10 5 0 3

4

5

6 7 System Size (n+1)

8

9

10

Figure 6.9: The number of compartments in the system versus the maximum resilience (as found with the genetic algorithm).

Compartment attributes associated with high resilience Inspection of the raw data suggests that flux, biomass and resilience are maximised when the interaction strengths at the lower trophic levels are high, and the interaction strengths at the higher trophic levels are low. This tendency is most apparent when the values of a1,2 and an,n+1 are investigated. Table 6.4 demonstrates this for the maximally resilient system. This trend was not observed for the maximisation of flux to biomass ratio, which implies that it is not merely a dynamical constraint necessary for feasibility and/or stability. This was the only discernible trend in the raw data. Table 6.4: Summarising statistical data for the attributes of maximally resilient chains for system sizes n = 2 . . . 22. Standard Mean Deviation a1,2 0.321 0.36 an,n+1 0.945 0.0788

Concordance Table 6.5 and 6.6 gives the concordance of each of the goal functions relative to all goal functions (Equation 6.60), when the flux to biomass ratio is maximised and minimised respectively. The normalised goal function values may be found in Appendix D.6.

124


For all chains, and for both the maximisation and minimisation of the flux to biomass ratio, resilience has the highest concordance. Flux generally has the second highest concordance. Table 6.7 and 6.8 gives the concordance of each of the goal functions relative to the traditional goal functions only (Equation 6.61), when the flux to biomass ratio is maximised and minimised respectively. Resilience never has the highest concordance. When F/B is maximised, ‘maximise flux’ generally has the highest concordance, followed by ‘maximise flux to biomass ratio’. When F/B is minimised, ‘maximise biomass’ and ‘maximise flux’ generally have the highest concordance. Table 6.5: Concordance of chains, relative to all goal functions. The maximum concordance possible is 3. F/B is maximised. Highest concordance indicated in bold. Chain Maximise Maximise Maximise Maximise Size B F F/B R 3 0.97 2.16 0.26 2.90 4 0.00 2.00 1.66 2.87 0.01 2.00 0.92 2.69 5 0.00 2.00 2.00 2.82 6 7 0.01 1.99 0.66 2.59 8 0.00 1.99 2.00 2.52 0.00 1.98 1.68 2.39 9 10 0.00 2.00 1.99 2.50

Table 6.6: Concordance of chains, relative to all goal functions. The maximum concordance possible is 3. F/B is minimised. Highest concordance indicated in bold. Chain Maximise Maximise Minimise Maximise Size B F -F/B R 3 1.95 2.16 1.03 2.96 1.00 1.00 2.00 2.95 4 5 0.99 1.00 0.65 1.93 1.00 1.00 1.74 1.94 6 7 0.99 0.99 0.95 1.93 8 1.00 0.99 0.98 1.94 1.00 0.98 0.59 1.95 9 10 1.00 1.00 1.00 1.94


125

Table 6.7: Concordance of chains, relative to the traditional goal functions only. The maximum concordance possible is 3. F/B is maximised. Highest concordance indicated in bold. Chain Maximise Maximise Maximise Maximise B F F/B R Size 3 1.97 2.99 1.26 2.90 4 1.00 3.00 2.64 2.87 5 1.01 3.00 1.92 2.69 1.00 3.00 3.00 2.82 6 7 1.01 2.99 1.66 2.59 1.00 2.99 3.00 2.52 8 9 1.00 2.98 2.68 2.39 10 1.00 3.00 2.99 2.50

Table 6.8: Concordance of chains, relative to the traditional goal functions only. The maximum concordance possible is 3. F/B is minimised. Highest concordance indicated in bold. Chain Maximise Maximise Minimise Maximise Size B F -F/B R 3 2.95 1.99 1.03 1.96 2.00 2.00 3.00 1.95 4 5 1.99 2.00 1.65 1.93 6 2.00 2.00 1.74 1.94 1.99 1.99 1.95 1.93 7 8 2.00 1.99 1.98 1.94 2.00 1.98 1.59 1.95 9 10 2.00 2.00 2.00 1.94

6.6.4

Discussion

These experiments test the robustness of the concordance hypothesis to changes in the size of the system, and remove the parameter restrictions imposed upon the system by Cropp & Gabric (2002). While the parameter range used is unrealistic, the benefit of this method is that it has removed the question of parameter range interfering with the concordance values, as discussed in Chapter 5. Table 6.3 shows that the maximum value of resilience decreases as the size of the system increases. This result has been reported in the literature (Pimm 1982, Pimm & Lawton 1977, pp. 62, 118), and is consistent with the prediction in Section 6.4.3, which was verified in Section 6.5.3. Cropp & Gabric (2002, Fig. 3) observes that the resilience decreases with increasing total nutrient, No . Cropp & Gabric (2002) only varies ks , the autotroph half-saturation nutrient concentration, to adapt the systems to the nutrient load. Half-saturation concentration terms do not appear in the generalised Lotka-Volterra model. We have shown the opposite is true for GLV chains where all parameters are varied: resilience increases with increasing No . This suggests that the result in Cropp & Gabric (2002) is not robust to changes in the interaction type and the parameter range.

126


The most notable aspect of Table 6.3 is the consistency of the maximum values for each of the traditional goal functions. To maximise flux and flux to biomass ratio, the genetic algorithm selects death rate terms such that they are approximately equal to each other, and equal to one, Thus the maximum values for each of the traditional goal functions can be derived from the expressions given for them in Section 6.3: • Biomass. 0 < B < No . • Flux. 0 < F < No . • Flux to biomass ratio. min(−di ) ≤ F ≤ max(−di ). 0.001 ≤ F ≤ 0.999.

In general, systems which maximised the nutrient flux and biomass had a high interaction strength for lower trophic level interactions, and a low interaction strength for higher trophic levels. This is expected, as a high throughput through lower levels is required to make up for losses back to the nutrient pool as nutrient makes its way up the food chain. The lower interaction strengths at the higher end of the food chain is necessary to keep the solution feasible. The highly resilient system shares these characteristics, partially explaining the concordance between resilience and these goal functions. The maximisation of the traditional goal functions does not optimise resilience (Table 6.3 and Table D.2). Rather, the opposite is the case: the traditional goal functions tend to minimise resilience. This is consistent with the investigations of variations on the CG model (Chapter 4). Resilience offers the best compromise between all goal functions (when resilience is included in the concordance calculation). This is consistent with the CG Model variations described in Chapter 4. However, when concordance is measured relative to the traditional goal functions only, resilience has a relatively low concordance. This implies that, if resilience is not considered a valid goal function a priori, and one wanted to find a compromise between the traditional goal functions only, maximisation of one of the traditional goal functions would be a better strategy than using ‘maximise resilience’.

6.6.5

Concluding remarks

In this section, we explored the relationship between goal functions in variations on the GLV. We found that longer GLV food chains have lower maximal resilience, in agreement with the literature (Pimm 1982, Pimm & Lawton 1977, pp. 62, 118).

6.7 Investigation of non-chain Lotka-Volterra models

127

Cropp & Gabric (2002) observes a low grazing rate for herbivores when the goal functions are maximised, stating that “the herbivore attributes result in organisms that are less fit to compete for limiting resources at the individual level”, which is “contrary to attributes that might be expected from individual-based evolution”. Our results for GLV chains are similar. We also found that ‘maximise flux’, ‘maximise biomass’, and ‘maximise resilience’ led to a low interaction strength for higher trophic levels. In this section, we used GLV chains to further test the robustness of the resilience heuristic. There are two resilience heuristics being tested. The first assumes the resilience hypothesis, and is tested using a concordance measure that includes all goal functions, including ‘maximise resilience’. The second does not assume the resilience hypothesis, and tests the ability of goal functions to compromise between the traditional goal functions only. In agreement with Chapter 5, strong evidence was found for the first resilience heuristic. This is a consequence of the difficulty of achieving high resilience. In general, for long chains, the traditional goal functions resulted in resilience values of the order of 10−5 . As a consequence, while maximising resilience gives moderate values for the traditional goal functions, maximising the traditional goal functions gives moderate values for the traditional goal functions, but a low value for resilience, giving them a disadvantage in concordance comparisons. In contrast, when concordance is measured relative to the traditional goal functions only, ‘maximise resilience’ loses this advantage. As a consequence, no model resulted in ‘maximise resilience’ having the highest concordance of this type. So no evidence for the second resilience heuristic was found. Therefore, we conclude that if the resilience hypothesis is not assumed, maximisation of one of the traditional goal functions would be a better strategy for achieving a compromise between the goal functions than using ‘maximise resilience’.

6.7

6.7.1

Investigation of non-chain Lotka-Volterra models

Introduction

These investigations follow on from the investigation of random Lotka-Volterra chains in Section 6.6. In Chapter 5, found that the relationship between goal functions was affected by the parameter range used. As before, we are interested in determining if the concordance of ‘maximise resilience’ will be reduced in the GLV compared to the results of Chapter 5. We are also interested in the robustness of the resilience heuristic to changes in the food web size and structure. In Chapter 5 and Section 6.6, food chains were investigated. In this section, we investigate the effects of changes to the food web structure.

128

6.7.2


Methodology

The genetic algorithm used is the same as that described in Section 6.6. The experiment is the same in all aspects, except that the food web structures investigated are those shown in Figure 6.5. To reiterate, we investigate the following goal functions: ‘maximise biomass’, ‘maximise flux’, ‘maximise flux to biomass ratio’, and ‘maximise resilience’. For each food web structure, a genetic algorithm is used to find the set of parameter values that maximises each of the goal functions. These parameter values are then used to calculate the concordance of each goal function according to Equations 6.60 and 6.61. Equation 6.60 is concordance relative to all goal functions, including resilience, and is used to test the robustness of the first resilience heuristic (i.e. assuming the resilience hypothesis). Equation 6.61 is concordance relative to the traditional goal functions only, and is used to test the robustness of the second resilience heuristic (i.e. not assuming the resilience hypothesis).

6.7.3

Results

It was again observed that the maximum values for each of the traditional goal functions did not vary much between food web structures (cf. Section 6.6.3). The details are in Appendix D.7. Table 6.9 and 6.10 show the concordance of each of the goal functions, for each of the ecosystem structures, where the concordance is measured relative to all goal functions. Resilience has the highest concordance of all of the goal functions for both the maximisation and the minimisation of the flux to biomass ratio (Table 6.9 and 6.10). The exception to this is ecosystem 4-2 when ‘maximise flux to biomass ratio’ is a goal function, for which the ‘maximise flux’ and ‘maximise flux to biomass ratio’ have the highest concordance. Tables 6.11 and 6.12 show the concordance of each of the goal functions, for each of the ecosystem structures, where the concordance is measured relative to the traditional goal functions only. Resilience never has the highest concordance. When F/B is maximised, ‘maximise flux’ generally has the highest concordance, followed by ‘maximise flux to biomass ratio’. When F/B is minimised, ‘maximise biomass’ has the highest concordance. For all models except Ecosystems 4-2 and 4-4, ‘maximise resilience’ provides a higher concordance than the mean of randomly chosen points in Table 6.2.

6.7 Investigation of non-chain Lotka-Volterra models

129

Table 6.9: Concordance of models, relative to all goal functions including resilience. Maximum concordance possible is 3. F/B is maximised. Highest concordance indicated in bold. Ecosystem Maximise Maximise Maximise Maximise B F F/B R 4-2 0.12 2.00 2.00 1.22 4-4 0.00 1.99 1.99 2.69 5-2 0.00 2.10 1.33 2.27 5-3 0.43 1.98 1.31 2.92 5-6 0.00 1.99 1.54 2.82 5-7 0.07 1.99 1.84 2.56

Table 6.10: Concordance of models, relative to all goal functions including resilience. Maximum concordance possible is 3. F/B is minimised. Highest concordance indicated in bold. Ecosystem Maximise Maximise Minimise Maximise B F F/B R 4-2 1.00 1.00 0.89 2.05 4-4 1.00 0.99 1.00 1.90 5-2 1.00 1.10 0.99 1.96 5-3 1.00 0.98 0.06 1.94 5-6 1.00 0.99 0.65 1.82 5-7 1.00 0.99 0.84 1.88

Table 6.11: Concordance of models, relative to the traditional goal functions only. Maximum concordance possible is 3. F/B is maximised. Highest concordance indicated in bold. Ecosystem Maximise Maximise Maximise Maximise B F F/B R 4-2 1.12 3.00 2.99 1.22 4-4 1.00 3.00 2.99 2.69 5-2 1.00 2.99 2.33 2.27 5-3 1.43 2.98 2.32 2.92 5-6 1.00 2.99 2.55 2.82 5-7 1.08 2.99 2.84 2.56

Table 6.12: Concordance of models, relative to the traditional goal functions only. Maximum concordance possible is 3. F/B is minimised. Highest concordance indicated in bold. Ecosystem Maximise Maximise Minimise Maximise B F F/B R 4-2 2.00 1.99 1.76 1.87 4-4 2.00 1.99 2.00 1.91 5-2 2.00 2.00 2.00 1.97 5-3 2.00 1.98 1.06 1.95 5-6 2.00 1.99 1.66 1.83 5-7 2.00 1.99 1.85 1.88

130


6.7.4

Discussion

It was found that, with the exception of Ecosystems 4-2 and 4-4, the maximally resilient point in parameter space had a higher concordance than the mean of randomly chosen points (Table 6.2). This result implies that resilience does act as a compromise between the goal functions. However, similar to Section 6.6, if ‘maximise resilience’ is not assumed to be a valid goal function, and the only purpose is to find a goal function that compromises between the traditional goal functions, maximising one of the traditional goal functions would be the best strategy.

6.8

Chapter discussion

A significant analytic result is that feasible GLV chains are always permanent. If one is willing to accept permanence as a valid measure of stability, this implies that local stability analysis may be too strict a criteria for stability in food chains of this type. We will explore permanence as a stability criteria in Chapter 11. We have tested the robustness of the results from Chapter 4 to variations in the system size, structure, and parameter range. Reducing the restriction upon parameter range was particularly important, as it was found in Chapter 4 that the concordance of resilience was reduced when the traditional goal functions found their maxima in the interior of the parameter space. In this chapter, we continued the investigation of the resilience hypothesis and resilience heuristic, as described in Section 5.1.1. Recall that we are interested in the following statements: 1. Resilience hypothesis: Ecosystems maximise resilience. Therefore, resilience is a goal function. (a) Resilience hypothesis by relationship with traditional goal functions. (b) Resilience hypothesis independent of traditional goal functions. 2. Resilience heuristic: Resilience generally offers a good compromise between the traditional goal functions. (a) Resilience heuristic without assuming the resilience hypothesis. (b) Resilience heuristic assuming the resilience hypothesis. In Chapter 4, we found no evidence supporting the resilience hypothesis sensu Cropp & Gabric (2002). However, as discussed in Chapter 4, ‘maximise resilience’ may still provide a valuable heuristic for modelling efforts (Item 2 above). If one were to assume that the traditional goal functions are valid, one might be interested in finding some way to compromise between them. If it could be shown that ‘maximise

6.8 Chapter discussion

131

resilience’ provides a compromise between these goal functions, then one could apply ‘maximise resilience’ as a heuristic. In Chapter 4, we developed a measure of the ability of a goal function to compromise between other goal functions, which was called concordance. Which goal functions are included in the concordance measure should reflect which goal functions are thought to be valid. If we measure concordance relative to the traditional goal functions only, we are interested in ‘maximise resilience’ for its relationship to the traditional goal functions, and nothing else (Item 2a above). We are not assuming that resilience is maximised, but we are interested in the possibility that it offers one possible way to compromise between the traditional goal functions. If we measure concordance relative to all of the goal functions, we are interested in ‘maximise resilience’ for its relationship to the traditional goal functions, but also as an independent goal function (Item 2b). In this chapter, and Chapter 5, we make the following findings: 1a Resilience hypothesis by relationship with traditional goal functions. We have verified the result of Chapter 4, that the resilience hypothesis sensu Cropp & Gabric (2002) is incorrect. Maximising the traditional goal functions does not optimise resilience. We have found no support for the resilience hypothesis in the GLV model developed, particularly for larger systems. We have consistently found that the maximisation of the traditional goal functions leads to low values for resilience, both for GLV chains (Table 6.3), and for other structures that we investigated (Table D.2). If a general statement must be made about the effect of the traditional goal functions upon resilience, I would assert that maximisation of the traditional goal functions minimises resilience. 1b Resilience hypothesis independent of traditional goal functions. (To be addressed in Chapter 7). 2a Resilience heuristic without assuming the resilience hypothesis. When concordance was measured relative to the traditional goal functions only, ‘maximise resilience’ never had the highest concordance. Therefore, if we do not assume that ‘maximise resilience’ is a valid independent goal function, we are better off maximising one of the traditional goal functions to achieve a compromise between goal functions. 2b Resilience heuristic assuming the resilience hypothesis. For GLV chains and the structures investigated, with only one exception (Ecosystem 4-2), when concordance is measured relative to all of the goal functions, including resilience, ‘maximise resilience’ has the highest concordance. This implies that, if we were to assume that all of the goal functions were valid, including resilience, and would like to compromise between them by maximising one of the goal functions, maximising resilience would be a good choice.

132

6.9


Conclusion

In this chapter, we tested the robustness of results from Chapter 4. We were interested in testing the resilience heuristic, which is that ‘maximise resilience’ offers a good compromise between goal functions. We calculated the concordance of resilience with the traditional goal functions when the following attributes of the ecosystem model were changed: size, structure, parameter range, and model formulation. We calculated 1. Concordance with respect to all goal functions, including resilience. 2. Concordance with respect to the traditional goal functions only. The former assumes that ‘maximise resilience’ is a valid goal function in its own right. The latter assumes that only the traditional goal functions are valid goal functions, and that resilience is only interesting as a goal function because of its ability to compromise between them. In this chapter, we found support for the resilience heuristic when all goal functions, including resilience, were included in the compromise. However, when ‘maximise resilience’ was not included in the compromise, no support for the resilience heuristic was found. This implies that the use of the goal function ‘maximise resilience’ cannot be justified by its relationship with other goal functions, neither in the way originally hypothesised in Cropp & Gabric (2002), nor by viewing it as a heuristic as described in Chapter 4. However, if ‘maximise resilience’ is an independent goal function, and can be justified by some other theory (e.g. the results of Laws et al. (2000)), the results in this chapter and Chapter 4 strongly support its use. If a modeller was interested in choosing a goal function that would offer some compromise with the traditional goal functions and maximal resilience, the results of this chapter suggest that ‘maximise resilience’ would be a good choice (at least for the model types that we have investigated). As no support for the use of the goal function ‘maximise resilience’ can be found in relationships between it and traditional goal functions, neither by a resilience hypothesis sensu Cropp & Gabric (2002), nor by the resilience heuristic of Chapter 4, we must now turn our attention to the question of ‘maximise resilience’ as an independent goal function. The remaining chapters will investigate this question.

Chapter 7

Laws Model: Independence and robustness of resilience 7.1 7.1.1


In Chapter 4 it was argued that, while resilience offers a compromise between the traditional goal functions, there is no reason to presuppose that ecosystems will ‘choose’ maximal resilience over any other possible compromise. In Chapter 5, it was suggested that ‘maximise resilience’ offered a compromise between the traditional goal functions, and hence, could be used as a heuristic. However, this result was not robust to variations in the model. In Chapter 6, it was found that unless one had reason to believe that ‘maximise resilience’ was a valid independent goal function, choosing one of the traditional goal functions was a better way to achieve a compromise between the traditional goal functions than choosing ‘maximise resilience’. Thus, the relationship between ‘maximise resilience’ and the traditional goal function is not a sufficient reason to recommend ‘maximise resilience’ as a goal function. With no further evidence in support of the resilience hypothesis, it would be reasonable to conclude our investigations. However, one study does exist in which ‘maximise resilience’ was used as a goal function to predict field data, which may yet provide support for the ‘maximise resilience’ goal function. Laws et al. (2000) used ‘maximise resilience’ to select two free parameters in a 10 compartment marine pelagic model (henceforth the Laws Model ), and were able to predict the ef-ratio measured in the field. However, this result does not necessarily mean that resilience is a good goal function. As was demonstrated in previous chapters, resilience can offer a compromise between the traditional goal functions for some model types. 133

134

Laws Model: Independence and robustness of resilience

Therefore, it is possible that ‘maximise resilience’ was a successful goal function in the Laws et al. (2000) study only because of its relationship with the other goal functions. If this is the case, Laws et al.’s (2000) result provides support for the resilience heuristic for this specific model (Chapter 5) and nothing more. Our objectives in this chapter are as follows: 1. To become acquainted with the Laws Model; 2. To test if ‘maximise resilience’ offers a compromise between the goal functions in the Laws Model; 3. To determine if the success of ‘maximise resilience’ in the Laws Model was merely a result of the goal function’s ability to compromise between the traditional goal functions, or if ‘maximise resilience’ acted independently of the other goal functions; and 4. To test the robustness of the ‘maximise resilience’ goal function to changes in the Laws Model.

7.2

The ef-ratio

Primary production in the marine pelagic environment is the rate of photosynthetic carbon fixation by phytoplankton (Berger, Smetacek & Wefer 1989b). Primary production is dependent upon nitrate supply, which comes from two sources: nutrient loading from outside of the system (the fertile upper layer of water), termed new production, and nutrient being cycled within the system (Berger et al. 1989b). Eppley & Peterson (1979) defined the f-ratio as the ratio of new production to total primary production, and estimated the following relationship for global oceans f=

Pnew = 0.0025Ptotal, Ptotal

(7.1)

where Pnew is new production, and Ptotal is total primary production. Since that time, much work has been done estimating and predicting the f-ratio (see Eppley (1989) for a review). Export production is that portion of primary production that leaves the system (Eppley 1989), mainly through sinking of organic matter and diffusion from the fertile zone to the deep ocean (Fasham et al. 1990). At steady state, the export production and the new production should be equal. In recognition of this, Laws et al. (2000) defined the ef-ratio, which is the ratio of export (or new production) to total production, as ef =

Pexport Pnew = . Ptotal Ptotal

(7.2)

The ef-ratio is essentially the same as the f-ratio, except that export production may be used in the place of new production in the equation. The new production is equal to the loading rate, L, given in the field data examined by Laws et al. (2000), so the ef-ratio is calculated by ef =

L . Ptotal

(7.3)

135

7.3 Method

Laws et al. (2000) was interested in using a model described by a system of differential equations to predict this quantity in real systems. Table 7.1 summarises the ef-ratios observed in the literature, against which the Laws Model was tested. The ef-ratios are calculated from data taken from 10 different papers published over the period 1987 to 1999, which are cited in Table 3 of Laws et al. (2000). Details of the data set are presented in Appendix E.1. Table 7.1: Ocean regions and observed ef-ratio. Adapted from Table 3 of Laws et al. (2000), with citations therein. ID Abbreviation ef-ratio Region description 1 BATS 0.095 Bermuda Atlantic Time-series Study 2 HOT 0.147 Hawaiian Ocean Time-series 3 NABE 0.505 North Atlantic Bloom Experiment 4 EqPac-normal 0.123 Pacific equatorial upwelling system 5 EqPac-El Nino 0.073 Pacific equatorial upwelling system during El Nino 6 Arabian Sea 0.150 Arabian Sea 7 Ross Sea 0.679 Ross Sea 8 Subarctic Pacific Station P 0.424 North Pacific subarctic gyre (Station P) 9 Peru-normal 0.421 Peruvian upwelling system 10 Peru-El Nino 0.295 Peruvian upwelling system during El Nino 11 Greenland polynya 0.563 Greenland polynya

7.3

Method

7.3.1

Laws Model details

The marine pelagic food web model depicted in Figure 7.1 is taken from Laws et al. (2000). It is a 10 compartment nutrient cycling model, which is assumed to be at steady-state. The 10 compartments of the Laws model are governed by the following differential equations: X˙ 1 = L − (1 − r2l )F2l − (1 − r2s )F2s + r3 F3 + r4 F4 + r5 F5 + r6 F6 + rb Fb , X˙ 2s = q2s F2s − F3 X˙ 2l = q2l F2l − F5

X2s , X2s + Xb

X2l , X2l + X4

X˙ 3 = q3 F3 − F4 , X˙ 4 = q4 F4 − F5

X4 , X2l + X4

(7.4a) (7.4b) (7.4c) (7.4d) (7.4e)

X˙ 5 = q5 F5 − F6 ,

(7.4f)

X˙ 6 = q6 F6 − M6 X6 ,

(7.4g)

X˙ c = s2s F2s + s3 F3 + s4 F4 − Fb ,

(7.4h)

136


External Nutrient

Filter feeders (5)

Large phytoplankton (2l)

Carnivores (6)

Detrital Particulate Organic Matter (d)

Inorganic nutrient (1)

Small phytoplankton (2s)

Flagellates (3)

Ciliates (4) Export

Dissolved Organic Matter (c)

Bacteria (b)

Figure 7.1: The Laws Model.

X˙ d = s2l F2l + s5 F5 + s6 F6 − DXd , X˙ b = qb Fb − F3

Xb , X2s + Xb

where notation is defined in Table 7.2. Fi is defined as P Xi−1 − Pi Fi = Ai Xi P , Xi−1 Fi = Ai Xi fi ,

where

P

(7.4i) (7.4j)

(7.5a) (7.5b)

Xi−1 is the sum of the nutrient concentrations of all prey compartments of Xi . Note that

fi = (0, 1).

7.3 Method

137

Table 7.2: Notation used in the Laws Model. Adapted from Laws et al. (2000). Symbol Definition V Sinking rate M Mixed layer depth L External loading rate of limiting nutrient Fi Feeding rate of compartment i Xi Concentration of nutrient in compartment i fi Fraction of maximal uptake by compartment i qi Fraction of ingested nutrient converted to biomass by compartment i ri Fraction of ingested biomass respired by compartment i si Fraction of ingested nutrient converted to DOM or detritus D Fractional loss rate of detritus due to sinking S Sinking rate of detritus M6 Mortality of carnivores Ai Maximal prey/nutrient saturated grazing/uptake rate by compartment i Pi Threshold concentration for grazing/uptake by compartment i Compartment subscripts 1 Inorganic nutrient 2s Small phytoplankton 2l Large phytoplankton 3 Flagellates 4 Ciliates 5 Filter feeders 6 Carnivores c Dissolved organic matter d Detrital particulate organic matter b Bacteria

Maximal resilience to determine free parameters

Most parameter values were sourced from the literature (Laws et al. 2000, citations therein), which are reproduced in Table 7.3. As detailed in Appendix E.2, Equation 7.4 can be solved at the steady state, leaving five parameters undefined. These free parameters can be chosen as f2L , f3 , f5 , f6 , and X5⋆ . Laws et al. (2000) arbitrarily chose the free parameter values such that resilience was maximised. “We assumed the system to adjust to the external loading rate L in a manner that produced a stable steady state as defined by May [1974]. May’s definition of stability implies that the system is stable to small perturbations of the Xm from their equilibrium values. This is a reasonable requirement, since there is little point in discussing the characteristics of a steady state if the steady state is not stable to small perturbations. In order to obtain a unique solution to the equation, we further required that the steady state solution be more stable to small perturbations than any other steady state solution. This is an arbitrary assumption, and there is no a priori reason to believe that pelagic ecosystems evolve toward a condition of maximum stability. ... Requiring that the system has maximum stability implies that the least negative eigenvalue associated with a given steady state be more negative than the least

138


negative eigenvalue of any other steady state.” Table 7.3: Parameter values Parameter q2s q2L q3 q4 q5 q6 qB

used in the Laws Model (Laws et al. 2000). Value 0.7 0.7 0.35 0.35 0.3 0.35 1.00

r2s r2L r3 r4 r5 r6 rB

0 0 0.3 0.3 0.3 0.3 1 − qB

A2s A2L A3 A4 A5 A6 AB

(1.2/q2s ) exp(0.0633(T − 25)) (1.2/q2L) exp(0.0633(T − 25)) (2.4/q3 ) exp(0.1(T − 25)) (2.4/q4 ) exp(0.1(T − 25)) 0.5 exp(0.1(T − 25)) 0.5 exp(0.1(T − 25)) (1.2/qB ) exp(0.0633(T − 25))

P2s P2L P3 P4 P5 P6 PB

14 × 7.5M/1000 14 × 75M/1000 (x⋆B + x⋆2s )(1 − f3 ) x⋆3 (1 − f4 ) (x⋆4 + x⋆2L )(1 − f5 ) x⋆5 (1 − f6 ) 14 × 7.5M/1000

V

10

Laws et al. (2000) states that “after exploring a wide range of conditions, we discovered that the optimum [maximal resilience] values of f2l , f3 , f5 and f6 were highly correlated with one another and that there was little to be gained by treating them as independent variables.” Consequently, f3 , f5 , and f6 were given as functions of f2l as follows (Equations 18, 19 and 20 in Laws et al. (2000)) f2l , 3 0.13 + 1.52f2l f5 = 0.65 + f2l f6 = 0.6f5 . f3 =

(7.6a) (7.6b) (7.6c)

We will discuss the importance of these parameter relationships in Section 7.6. By including Equations 7.6 in the model, the Laws Model has two free parameters that can be selected by a grid search. They are

139

7.3 Method

chosen as: 1. f2L : The fraction of maximal uptake by the large phytoplankton compartment; and 2. X5⋆ : The steady state biomass of the filter feeder compartment. For each ocean region, the f2l and X5⋆ ranges are divided into a grid of 500 increments, and the value of the ef-ratio and resilience is evaluated at each grid point. The ef-ratio associated with the highest resilience for each ocean region is noted and compared to the field data.

7.3.2


Figure 7.2 shows the model predictions against the field data. Similar to Laws et al. (2000), the model shows excellent agreement with the field data. Table 7.4 shows the parameter values (f2L and X5⋆ ) that maximise the resilience. The rate of uptake by large phytoplankton, f2L , that maximises resilience tends to be low relative to its range (0, 1). The low feeding rate implies that a species may have to act altruistically in order to maximise the resilience of the entire system (c.f. Section 6.6.3). The range of X5⋆ is unconstrained, so a similar assessment is difficult (but see Section 7.6). Table 7.4: The parameter set that maximises resilience for the Laws et al. (2000) model, and the ef-ratio predicted by the ‘maximise resilience’ goal function. ID 1 2 3 4 5 6 7 8 9 10 11

Region BATS HOT NABE EqPac-normal EqPac-El Nino Arabian Sea Ross Sea Subarctic Station Peru-normal Peru-El Nino Greenland polynya

f2L 1.2327 × 10−01 1.2819 × 10−01 1.9714 × 10−01 1.7251 × 10−01 1.3804 × 10−01 1.9221 × 10−01 1.5774 × 10−01 1.3312 × 10−01 2.3653 × 10−01 2.3161 × 10−01 1.3804 × 10−01

X5⋆ 8.8146 × 10+01 8.9648 × 10+01 1.4648 × 10+03 2.3844 × 10+02 7.1618 × 10+01 1.8762 × 10+02 9.0653 × 10+03 1.5082 × 10+03 3.1616 × 10+03 2.3141 × 10+03 2.1583 × 10+03

ef-ratio 1.7482 × 10−01 1.8005 × 10−01 5.0701 × 10−01 1.6382 × 10−01 1.8014 × 10−01 1.6215 × 10−01 6.4823 × 10−01 3.5886 × 10−01 4.2424 × 10−01 3.9521 × 10−01 5.3518 × 10−01

Resilience 4.4356 × 10−02 6.3094 × 10−02 3.1290 × 10−02 8.3333 × 10−02 7.7986 × 10−02 1.0951 × 10−01 7.6023 × 10−03 1.2726 × 10−02 5.5224 × 10−02 5.9594 × 10−02 7.2965 × 10−03

140


Maximise Resilience 1

0.8

Model ef-ratio

7 0.6 3 0.4

10

11

9 8

0.2

5

1

4 2 6

0 0

0.2

0.4 0.6 Observed ef-ratio

0.8

1

Figure 7.2: Model ef-ratio versus observed ef-ratio for the Laws Model using ‘maximise resilience’ as a goal function. Numbers correspond to ocean regions given in Table 7.4. The diagonal line indicates the position of a perfect prediction.

7.3.3

Concluding remarks

We have reproduced the Laws et al. (2000) result and verified that combining the Laws Model with the ‘maximise resilience’ goal function gives excellent predictions of the field data. The question remains – why does this work?

7.4 ‘Maximise resilience’ as a compromise between the traditional goal functions

7.4

141

‘Maximise resilience’ as a compromise between the traditional goal functions

7.4.1

Introduction

As stated in Section 7.1.1, we are exploring the possibility that ‘maximise resilience’ offered good predictions in the Laws Model simply because of its relationship with the traditional goal functions. However, this presupposes that: 1. The traditional goal functions are valid; and 2. The goal function ‘maximise resilience’ does indeed offer a compromise between the traditional goal functions. We discussed Item 1 in Chapter 2. In this section, we assess the validity of Item 2.

7.4.2

Method

The f2l and X5⋆ range is divided into a grid of 500 increments, and free parameter values are found such that the value of each of the goal functions used by Cropp & Gabric (2002) is optimised. The goal functions are: • ‘maximise biomass’, B; Biomass is simply the sum of the nutrient concentrations in each of the biotic compartments at steady state B = Large phytoplankton + Small phytoplankton + Filter feeders + Carnivores + Flagellates + Ciliates + Bacteria.

(7.7)

• ‘maximise production’, P ; New production is found directly P = Production of large phytoplankton + Production of small phytoplankton,

(7.8)

• ‘maximise production to biomass ratio’, P/B; and • ‘maximise resilience’, R. The resilience is found by R = −max{Re(λ)}, where λ are the eigenvalues of the linearised system.

(7.9)

142


The ef-ratios predicted by each goal function, for each ocean, are compared with field data. While the grid-search method is inefficient compared to the genetic algorithm of Cropp & Gabric (2002), it allows the simultaneous generation of data necessary to perform a graphical comparison between goal functions. Previously, the high dimensionality of the parameter space necessitated the use of concordance (a normalisation technique) to compare goal functions (Chapters 5 and 6). In this study, however, the dimensionality of both models is two, which renders the data amenable to visualisation. Data returned by the grid-search method is used to generate a contour plot demarcating the feasible-stable region in parameter space. For each ocean region, the point in parameter space that optimises each of the goal functions is indicated within the feasible-stable region.

7.4.3


Figures 7.3 to 7.6 show the regions for which the system is both feasible and stable. The parameter range searched is restricted to this region. The parameter set that maximises each of the goal functions is indicated. It was found that, for all regions, each of the traditional goal functions varies smoothly and monotonically in the feasible and stable region. This causes the traditional goal functions to be maximised at the edges of the feasible-stable region. In contrast, resilience is maximised in the interior of the feasible-stable region. It was observed that resilience would swap from one eigenvalue to another, depending upon which eigenvalue had the most positive real part, and resilience was maximised on the cusp of these changes. These results explain intuitively why resilience acts as a compromise between the goal functions (Kristensen, Gabric, Braddock & Cropp 2003), but the traditional goal functions do not necessarily optimise resilience. Consider, for example, the Bermuda Atlantic Time-series Study result (Figure 7.3). It can be seen that, similar to the other regions, biomass is maximised at the upper-left of the feasible and stable region, production is maximised at the furthest upper-right, and production to biomass ratio at the lower right. The maximisation of resilience, however, is found in the interior of the feasible and stable region. Resilience necessarily compromises between the goal functions because ‘maximise resilience’ tends to be an interior point, whereas the other goal functions are maximised at opposing extremes of the feasible-stable parameter range.


143

Feasible and stable region in parameter space for Bermuda AtlanticTime-series Study

300 250 200

oB

150 100

oR oP o P/B

0

0.05

0.1

0.15

x5

0.2

0.25

0.3

0.35

0.4

0.45

50 0 0.5

f2l

Feasible and stable region in parameter space for Hawaiian OceanTime-series

300 250 oB

200 150

oP

100

oR o P/B

0

0.05

0.1

0.15

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Feasible and stable region in parameter space for North AtlanticBloom Experiment

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Figure 7.3: The location of the parameter set that maximises each of the goal functions in the Laws Model. The boundary of the feasible and stable region is marked as a dotted line. The key is B: ‘maximise biomass’, P: ‘maximise production’, P/B: ‘maximise production to biomass ratio’, and R: ‘maximise resilience’.

144


Feasible and stable region in parameter space for Pacific equatorialupwelling - normal

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Feasible and stable region in parameter space for Pacific equatorialupwelling - El Nino

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Feasible and stable region in parameter space for Arabian Sea

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145

Feasible and stable region in parameter space for Ross Sea

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Feasible and stable region in parameter space for Subarctic Pacific Station P

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Feasible and stable region in parameter space for Peru upwelling - normal

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146


Feasible and stable region in parameter space for Peru upwelling - El Nino

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Feasible and stable region in parameter space for Greenland polynya

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7.5 Is the success of the Laws Model independent of the other goal functions?

7.5

147

Is the success of the Laws Model independent of the other goal functions?

7.5.1

Approach

In the previous section (Section 7.4), it was found that ‘maximise resilience’ offered a compromise between the traditional goal functions. The question becomes: was the success of ‘maximise resilience’ merely a consequence of this ability to compromise between the traditional goal functions, or would any point in the feasible and stable region suffice? In this section, we test the predictive ability of the feasible-stable region. For each ocean region, free parameter values are chosen from a random, uniform distribution until 500 points in the feasible-stable region were found. For each of these points, the ef-ratio is calculated, and box-plots of the distribution of ef-ratios generated. Presumably, some traditional goal functions are more important, or better predictors of the data, then others. We are interested in identifying these. A genetic algorithm is used to find the parameter set that maximises each of the goal functions used in Cropp & Gabric (2002): ‘maximise biomass’, ‘maximise production’, and ‘maximise production to biomass ratio’. The parameter set that maximises each of the goal functions is used to calculate the ef-ratio. The ef-ratio predicted by ‘maximise resilience’ and the traditional goal functions are plotted for comparison.

7.5.2


Figure 7.7 shows box-plots of the randomly generated ef-ratios in the feasible and stable region compared with the observed ef-ratios reported in Table 3 of Laws et al. (2000). The median is shown as a ×, the 25th and 75th percentiles are used as the upper and lower hinges, 1.5 times the interquartile range is used as the upper and lower fences of the whiskers, and outliers are shown as points beyond the whiskers. Compared with Figure 3a of Laws et al. (2000) (results replicated in Figure 7.2), the feasible-stable region does not have the predictive power of ‘maximise resilience’. While the feasible-stable region of the ocean regions with low ef-ratios (Bermuda Atlantic Time-series Study, Hawaiian Ocean Time-series, Pacific equatorial upwelling during normal conditions, Pacific equatorial upwelling during El Nino, and Arabian Sea) are biased toward lower ef-ratios, the bias is very slight, and shows much less agreement with the field data than the point of maximal resilience. Figure 7.8 compares the ef-ratio predicted by each of the goal functions with that observed in the literature (see Laws et al. (2000) and references therein). By comparison with Figure 7.7, ‘maximise biomass’ and ‘maximise production’ do not appear to improve upon the predictive capabilities of the feasible-stable region as a whole. These goal functions show no qualitative agreement with the data.

148


Boxplots of feasible-stable region for Laws model 0.7

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Figure 7.7: A box-plot of the ef-ratios in the feasible and stable region for each of the oceans versus the observed ef-ratios. Laws Model. The diagonal line indicates the position of a perfect prediction.

Comparing ‘maximise resilience’ to the other goal functions, it is apparent that it is the most successful goal function in terms of both qualitative and quantitative agreement. The goal function ‘maximise production to biomass ratio’ also shows strong agreement with the data, but not to the extent that ‘maximise resilience’ does.

149

7.5 Is the success of the Laws Model independent of the other goal functions?

Comparison between model prediction for different goal functions. Laws Model. 1

Model ef-ratio

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1

Figure 7.8: Observed versus model ef-ratio for each of the goal functions, indicated as follows: ‘maximise biomass’ ×, ‘maximise new production’ ◦, ‘maximise production to biomass ratio’ +, and ‘maximise resilience’ . The diagonal line indicates the position of a perfect prediction.

7.5.3

Concluding remarks

In this section, we have demonstrated that, while ‘maximise resilience’ offered a compromise between the traditional goal functions, its predictive ability was independent of this relationship. Indeed, two of the three traditional goal functions had no predictive ability for the Laws Model, and the final goal function (‘maximise production to biomass ratio’) performed poorly in comparison. The question remains as to whether or not the success of ‘maximise resilience’ in the Laws Model is indicative of some general principle, or if it is peculiar to the Laws Model formulation.

150

7.6 7.6.1


Variations on the Laws Model Introduction

‘Maximise resilience’ performs successfully as an independent goal function for the Laws Model. In this section, we will test the robustness of the predictive ability of ‘maximise resilience’ to variations in the model formulation. By augmenting the original Laws Model, we identify which aspects of the model are necessary for ‘maximise resilience’ to make good predictions of the field ef-ratios. For brevity, only the successful variation is presented here. A summary of the methods used and the variations created is presented in Appendix E.4.

7.6.2

The successful variation

Approach Laws et al. (2000) states that, when resilience is maximised, there is a simple relationship between some fi values. Correspondence with E.A. Laws suggests that these parameter restrictions may have been fitted for high temperature scenarios (see also Appendix E.3 and Chapter 9 for details on the effect of temperature on the maximal resilience surface). They reduced dimensionality of the parameter space to two dimensions by imposing this relationship upon the fi parameters, as described in Equation 7.6. To verify this, the Laws Model is augmented such that those relationships described by Equation 7.6 are removed, and all five unknown parameters are free parameters, selected by a genetic algorithm to maximise resilience. The relationship between parameters and the predictive ability of ‘maximise resilience’ is investigated.

Results and discussion Figure 7.9 shows plots of the relationship between parameter values, compared to the relationships imposed upon them in Equation 7.6. It can be seen that the relationship between f2l and f3 in particular is weak. Removing Equations 7.6 from the model, thus allowing each of the f values to be free parameters, results in the ef-ratio predictions in Figure 7.10. The removal of the relationships between parameters in Equation 7.6 decreases the predictive ability of both the ‘maximise production to biomass ratio’ and ‘maximise resilience’ goal functions. It was found that the relationship between f2l and f3 in the original Laws Model (Equation 7.6) was significant to the predictive ability of ‘maximise resilience’ for ocean regions that possessed two local

151

7.6 Variations on the Laws Model

Maximally resilient points compared with Equation 18 of Laws et al. (2000) 0.4 5 11

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Figure 7.9: A comparison between the parameter values that maximised resilience in the Laws Model when parameter relationships were removed, and the equations used by Laws et al. (2000) to describe those relationships. Numbers indicate the ocean region. The point in parameter space that maximises resilience is indicated by a ‘◦’, and the equation by a dotted line.

152


Laws Model predictions when all f values are free parameters. 1

Model ef-ratio

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Oberved ef-ratio

Figure 7.10: Observed versus model ef-ratio for each of the goal functions, indicated as follows: ‘maximise biomass’ ×, ‘maximise new production’ ◦, ‘maximise production to biomass ratio’ +, and ‘maximise resilience’ . Modelled such that all f values were free parameters. Details in Appendix E.4 ‘Laws Model 2’. The diagonal line indicates the position of a perfect prediction.

resilience optima in parameter space. These ocean regions were the North Atlantic Bloom Experiment, Ross Sea, Subarctic Pacific Station P, and Greenland Polynya (Appendix E.4, Figures E.2, E.3, E.4, and E.7). Although the ef-ratio is relatively insensitive to the f3 value, restricting the f3 value effects where the point of maximal resilience is found with respect to other, more sensitive, parameter values. For those ocean regions with two optima, when f3 is restricted to low values, the optima found by the genetic algorithm describes a system with a large ef-ratio due to the predominance of large phytoplankton. However, when the f3 value is allowed to vary as a free parameter, it is revealed that the afore-mentioned optima is only a local optima, not a global one. The genetic algorithm then finds maximal resilience for a lower ef-ratio, with correspondingly fewer large phytoplankton. Details may be found in Appendix E.4. Although the predictive ability of the ‘maximise resilience’ goal function is greatly reduced, Figure 7.10 shows that the model and the goal function combination still possess some qualitative agreement with the field data. Further, this qualitative agreement is not reflected in any of the traditional goal functions, suggesting that the success of ‘maximise production to biomass ratio’ was due to the parameter restrictions in Equation 7.6.

7.7 Conclusion

7.7

153

Conclusion

Laws et al. (2000) provides the only empirical support in the literature for the goal function ‘maximise resilience’. In previous chapters, it was found that the relationship between ‘maximise resilience’ and the traditional goal functions was not sufficient reason to recommend ‘maximise resilience’ as a goal function. Therefore, in this chapter, we were interested determining whether ‘maximise resilience’ was successful in the Laws Model simply because of its relationship with the traditional goal functions, or if it showed independent predictive ability. It was found that the predictive ability of ‘maximise resilience’ acted independently of the traditional goal functions. This independence was most evident when the relationships imposed upon the free parameters (Equation 7.6), which were assumed in the original model, were removed. For this case, none of the traditional goal functions had any predictive ability, yet ‘maximise resilience’ showed qualitative agreement with the field data. Having established that ‘maximise resilience’ is an independent goal function in the Laws Model, we are left with the question of why ‘maximise resilience’ works. Cropp & Gabric (2002) proposes that maximisation of resilience is reasonable given that “highly resilient ecosystems are more likely to remain extant than ecosystems with low resilience”. Earlier authors have made similar statements (e.g. Lenton 1998, Dunbar 1960, Jørgensen & Mejer 1981). In the following chapters, we will explore this hypothesis, and test if it explains the predictive ability of the Laws Model.

Chapter 8

Quantitative stability and system-level selection 8.1

Introduction

8.1.1

Motivation

The results of Laws et al. (2000), discussed in Chapter 7, motivate the search for a theoretical justification for the use of maximal resilience as a goal function. Chapters 4 to 7 suggest that this justification must be made independent of a relationship between resilience and the traditional goal functions. In this chapter, we address the following statements in Cropp & Gabric (2002). First: Although there is little thermodynamic or ecological evidence to suggest [resilience] is a legitimate selection pressure, ecological networks that develop stabilizing feedbacks are considered to be more likely to remain extant than those that do not (Lenton 1998). And also: There is ... no a priori indication that the optimization of an ecosystem’s response to thermodynamic or ecological imperatives should result in a maximally resilient ecosystem. This outcome, however, is reasonable, given that all ecosystems exist within the constraints of thermodynamic laws and that highly resilient ecosystems are more likely to remain extant than ecosystems with low resilience [emphasis added]. This style of reasoning has a history in the stability literature, as discussed in Section 2.5 and Section 3.3. In this chapter, we construct a system-level feedback mechanism based upon the assumption that more 154

8.2 Overview

155

stable systems are more likely to remain extant than less stable systems. We explore if this mechanism can be used to justify the use of ‘maximise resilience’ as a goal function. We: 1. identify the assumptions necessary for the mechanism to affect the stability of the system; and 2. make predictions based upon the hypothesis, so that they can be tested in subsequent chapters.

8.2

Overview

Taking a simple definition of stability — the ability to return to a given state after a disturbance — it seems likely that ecosystems that do not possess stability will inevitably disappear in a variable environment. This line of reasoning is a motivation for the long-running complexity-stability debate (MacArthur 1955, Hairston et al. 1968, Gardner & Ashby 1970, May 1972, Yodzis 1981, Paine 1992, Tilman 1996, Doak et al. 1998, Tilman et al. 1998), and has been used as a constraint upon ecological systems, to predict what other qualities they should possess. For example, May (1973) used stability considerations to argue for a limit to similarity between competing species, Pimm & Lawton (1978) proposed that food web length may be limited by resilience (but see Sterner et al. 1997), and McMurtrie (1975) suggested that food webs would be compartmentalised due to stability constraints (see also May 1979, Solé & Montoya 2001). When using stability as a prerequisite for ecosystem survival, various qualifications may be made regarding how stringently the condition for stability is defined, and what constitutes the continuation of an ecosystem. Stringency often concerns the size of perturbations the ecosystem is expected to withstand. For example, if we only consider small perturbations, local asymptotic stability (Lewontin 1968, May 1973) is a necessary constraint for ecosystem survival. However, if we expect ecosystems to withstand much larger perturbations, measures such as sector stability (Goh & Jennings 1977, Goh 1977, Goh 1978) will be needed. The continuity of an ecosystem often concerns the state to which we expect the system to return. For example, both local and sector stability analysis restrict the state of the ecosystem to the neighbourhood of one particular combination of species populations. However, if we are willing to define an ecosystem by its species composition, measures such as permanence may be used (Hofbauer & Sigmund 1988, Grimm et al. 1992, Jansen & Sigmund 1998). While the above discussions highlight the substantial differences between definitions of stability, stability as described above always gives a binary ‘stable’ or ‘unstable’ result, corresponding to the interpretations ‘this system may remain extant’ or ‘this system cannot remain extant’. In contrast, in this chapter, we are interested in the question of the degree of ecosystem stability. We ask, would it be reasonable to

156

Quantitative stability and system-level selection

state that quantitatively more stable systems are more likely to remain extant than those that are less so (e.g. Pimm & Lawton 1977)? Could we extend this idea to say that ecosystems significantly increase, or even maximise, their stability? Such statements exist in the literature (MacArthur 1955, Dunbar 1960, Margalef 1968, Odum 1969, Patten 1975, Jørgensen et al. 1995, Cropp & Gabric 2002), and are the subject of this study. Cropp & Gabric’s (2002) statements quoted in Section 8.1.1 is reminiscent of theories that involve a feedback mechanism from system level survival to the survival of particular system compositions (c.f. Dunbar 1960, Jørgensen et al. 1995). They hypothesise that systems composed of species with stabilitydecreasing attributes will be less likely to persist. So no matter how much those attributes benefit the species themselves, those species will be replaced by other species, due to their deleterious effect upon the system as a whole. Potentially, they may be replaced by stability-enhancing species, which promote the survival of the system. As such, in the long-term, stability of the system should increase, or perhaps be maximised. In this chapter, we address two issues. First, we we seek to create a minimalist mechanism by which highly resilient ecosystems might be more likely to remain extant than ecosystems with low resilience. Secondly, we address whether or not this will provide the basis for a feedback mechanism by which system resilience is maximised.

8.3

The Models

8.3.1

Introduction

We present a very simple ecosystem model, with a single compartment, upon which to perform our experiments. The compartment consists of several functionally similar coexisting species (e.g. phytoplankton (Grenney, Bella & Curl 1973) or zooplankton (Pejler 1962)), similar to that often described in aquatic ecosystems, and modelled as such (e.g. Fasham et al. 1990). For simplicity, the number of species, n, is kept constant. Each species has its own attributes. The attributes of each species is not specified nor described in detail, except to say that these attributes have some affect upon the stability properties of the whole system. The stability of the compartment is the average of the contributions of the species. Note that this is not to say that the species themselves possess stability. Stability is a system level property. However, we find it convenient to refer to the stability associated with a particular species, which is to say, the system stabilising properties of that species. We will approach our study questions from two different areas. First, we temporarily set aside the

8.3 The Models

157

question of resilience and assume that whichever measure of stability we are using, it is directly related to the resistance of the system. We define resistance as the ability to withstand species loss, perhaps by the buffering of perturbations, or perhaps by the ability to recover from a perturbation. Either way, we will take for granted that ecosystems that possess more of this stability property are more likely to remain extant than those that do not, sensu Cropp & Gabric (2002). We apply successive perturbations to the system, which cause some proportion of the species to be replaced, where that proportion reflects the resistance of the system. We are interested in determining if this process provides a feedback mechanism by which the resistance of a system increases, or is maximised. Note that which particular species are removed is independent of their contribution to resistance. This is in recognition of the fact that stability is a system-level attribute, so a link between resistance and individual species’ survival cannot be assumed. Our second approach will involve resilience specifically. We are interested in the following three questions. First, can we plausibly argue that more resilient systems more likely to remain extant than less resilient systems? Second, if the answer to the first question is yes, does this imply that ecosystem resilience will increase, or be maximised? And finally, can resilience be used as a goal function to predict the attributes of the ecosystem? We emphasis that we will not be using the traditional differential equation approach to modelling the system responses to perturbation. While such an approach is more realistic, it will also introduce complicating factors into the model, potentially masking the feedback mechanism. We argue that, if the feedback mechanism does not perform convincingly for this simplified system, there is little motivation for increasing the complexity of the model.

8.3.2

Model for exploring resistance

Formulation Let us assume that environmental fluctuations perturb the model ecosystem, causing some species to become (locally) extinct at each time step. It is the effective strength of these perturbations that will be of interest in this experiment. The effective strength of the perturbation is emphasised because, while the strength of a perturbation is dependent upon complex external factors, its effect upon the system in terms of local species extinctions may be dampened, in part, by the system’s biotic attributes. Therefore, we assume that the effective strength of the perturbations will be negatively correlated with the compartment’s resistance. A simplification of the way in which each species’ attributes contributes to resistance is made by defining a species’ contribution to resistance, ri , for each species in the system, where the i are the indices denoting

158


the species in question, and i = 1 . . . n, where n is the total number of species in the compartment. Noting that modelling efforts concerning resilience (Laws et al. 2000) implicitly average species attributes to quantify compartment-level behaviour, the survival attribute of the entire compartment, r¯, is simplified as the average of the ri . Initially, the contribution to system resistance from each species, ri , is taken from a uniform random distribution, 0 ≤ ri ≤ 1. It is assumed that the mean ri value, r¯, is taken as the resistance of the entire system. For each time step, a perturbation is applied to the system such that a proportion of the species go extinct, and are replaced by new species, to keep the total number of species constant. The proportion removed is negatively related to resistance, as a low resistance implies a high effective strength of the perturbation. We let the proportion of species removed be equal to 1 − r¯, such that r¯ = 1 is perfect resistance to a perturbation such that no species are removed, and r¯ = 0 implies that all species are removed. After a proportion of species are removed from the system, lost species are replaced by species that have a resistance, ri , chosen from a random, uniform distribution, until the system once again contains the original n species. It should be noted that the assumption of randomness is a kind of null model; no assumption is made about the mapping from a species’ attribute to resistance, and no assumption is made regarding the relationship between invasion ability and contribution to resistance.

Results Figure 8.1 shows a trajectory of the mean resistance of the system over time. The control in Figure 8.1 is for the case where the proportion of species replaced at each time step is independent of r¯, that is, a random variable drawn between 0 and 1. When the ecosystem is in a state of higher resistance, losses of species will be few, allowing this state to persist for longer. Conversely, ecosystems in states of low resistance will lose large numbers of species when perturbed. This high loss of species will continue until a system with a higher mean resistance is achieved. This intuition is confirmed by Figure 8.1, which shows plateaus at high resistance, corresponding to the higher survival of these states. To determine the typical impact of this mechanism on the resistance of the entire system we calculated the mean resistance over time (Figure 8.2). The mean resistance of the system over time is higher than the mean resistance of a system without this mechanism (expected mean resistance, E(¯ r ) = 0.5, for the control), however, while statistically significant, the difference is small. Further, Figure 8.2 shows that the strength of this tendency decreases when the fixed number of species, n, in the system increases.

159

8.3 The Models

Compartment resistance versus Time 1 0.9

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Figure 8.1: A typical trajectory of the resistance (solid line), r¯ of the model compared to a control (dotted line). Trajectory shown is for a 3 species compartment.

Mean resistance versus Compartment size 1 0.95 0.9 Mean resistance

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Figure 8.2: The effect of the algorithm upon the mean resistance versus the (fixed) size of the system. Each data point is for 10000 time steps, or invasions.

160


Discussion The effect of the mechanism upon mean resistance is weak enough to discount the model as a plausible mechanism for the maximisation of resistance, particularly for large systems. The dilution of the mechanisms’ effect with increased system size is due to the increased probability of ‘misdirection’ of the feedback from system level to species. For example, for a single species system, low resistance leads to the replacement of that single species. However, for a many species system, it may be some time before the low-resistance species are removed. In that time, by chance, the system may lose species that confer a higher resistance to the system. This is a consequence of our assumption that species resistance contributes to system stability, but not to the likelihood of that species itself persisting. We see qualitative similarities between our result and the empirical results of Swenson, Wilson & Elias (2000). Swenson et al. (2000) investigated system-level selection on two types of ecosystems. The first consisted of 85 g pots of soil growing Arabidopsis thaliana plants. The second consisted of 2 ml of sediment and 28 ml of water, both from a local pond. Subsequent generations of pot-plant ecosystems were created by inoculating sterile soil with soil from three parents from the previous generation, and the pond ecosystems by inoculating pond sediment and water with water from one parent from the previous generation. The attributes selected for were maximum (minimum) plant weight, and pH, respectively. They found that the system level attribute selected for, while at some points increasing substantially, passed through short periods of collapse. Their results for plant weight resemble our model results, in that the period of near mean weight is short compared to the period of above (below) mean weight. While they relied upon the dynamics of the system, rather than invasion, for randomisation of species composition, we suggest that the qualitative similarities reflect similarities in the underlying principles at work. We speculate that Swenson et al.’s (2000) empirical results bear resemblance to our own because the system level feedback in their real system is subject to the same types of ‘misdirections’ as our model system.

8.3.3

Model for exploring resilience

Linking resilience to survival Up until this point, we have assumed that our stability measure is directly related to resistance. However, our stability measure of interest, resilience, is logically independent from resistance. A system may have high resistance and low resilience, and visa versa. Unless a positive relationship between the magnitude of resilience and the magnitude of the other components of stability is assumed, a connection between return time and the effective strength of a perturbation must be elaborated upon.

161

8.3 The Models

We might begin by postulating that a system which remains near the axis of the state space for extended periods of time risks the extinction of its species (Jansen & Sigmund 1998). For example, if a species is subject to low resilience such that its population is maintained for prolonged periods of time at only a few individuals, it is more likely that some small catastrophe will send the entire population extinct. However, the mathematical formulation of resilience used by Cropp & Gabric (2002) and Laws et al. (2000) is only applicable to the region in the state space about which the linearisation of the system holds (Pimm 1982, pp. 21). An entirely different approach is required to determine this aspect of an ecosystem’s return to equilibrium (see Kirlinger 1986, Kirlinger 1988). Is there some other way to link small-scale disturbances and system change? In Lozon & MacIsaac’s (1997) literature review, the authors observe that “one of the most frequently cited features of communities thought vulnerable to invasion is that they tend to be disturbed”. Shea & Chesson (2002) identify two stages of ecosystem invasion: transport and establishment. Similarly, Davis, Grime & Thompson (2000) note three preeminent factors affecting invasion success: propagule pressure, invasibility, and the characteristics of the new species. Working within this framework, the relationship between resilience and the probability of species replacement may be formulated. It is assumed that there is a window of opportunity after a perturbation, during which the system is particularly vulnerable to invasion. Therefore, invasion relies upon the invader being ‘in the right place at the right time’ (cf. Hubbell & Foster 1986). If ∆t is the proportion of time spent far enough from equilibrium to be vulnerable to invasion, and r now denotes the resilience aspect of stability, the probability of arriving at the right time is p(arrive at right time) ∝ ∆t ∝

1 r¯

(8.1)

We further assume that the probability that a new species may invade and establish itself is proportional to level of disturbance of the system, or the distance of the system away from equilibrium. Approximating an asymptotic approach to equilibrium of the form e−¯rt , where t is the time elapsed after a perturbation, the probability of a successful establishment by an arrived species at time t1 is proportional to e−¯rt1 . Therefore, the probability of invasion taken for the duration of the system’s return time ∆t is proportional to the integral of the return curve from perturbation time to ∆t p(successful establishment|arrive at right time) ∝

1 . r¯

(8.2)

Finally, the ability of the invading species to replace the resident is independent of resilience, therefore p(arrive at right time and successful establishment and replacement) ∝

1 . r¯2

(8.3)

This means that we can use our previous model to consider the way in which resilience changes, by changing the probability of species replacement from 1 − r¯ to 1/¯ r2 .

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Addressing increased versus maximal resilience The new resilience model is essentially the same as the old resistance model, except that the probability of species replacement is now 1/¯ r2 instead of 1 − r¯. However, this change does not avoid the problems of the original model – namely the weakness of the tendency to increased stability due to ‘misdirections’ of the feedback. To devise a model in which higher resilience is more likely than lower resilience is not the same as devising a mechanism by which resilience is maximised. Even if one were to accept the model as a valid, albeit simplified, reflection of mechanisms underlying invasion and replacement in planktonic systems, the model does not support the hypothesis that ecosystems maximise resistance, or resilience. Does this mean, however, that maximal resilience cannot be used as a goal function? Recall that one purpose of a goal function is to provide some heuristic by which attributes of the system may be predicted (Chapter 2). In the case of Laws et al. (2000), maximal resilience was used to predict the steady state biomass of the filter feeder, and the uptake of the large phytoplankton. In turn, these two parameters were used to predict the system attribute the ef-ratio, and the ef-ratio was verified against empirical data. The use of resilience as a goal function involves a mapping from resilience to a species’ attribute, and then from a species’ attribute to system-level attribute. This implies that if the species’ attribute, and system level attribute, are in the neighbourhood of the point in attribute space corresponding to maximal resilience, maximising resilience in a model will approximate those attributes, even if resilience itself is not maximised. The shape of the mapping from attribute space to resilience is dependent upon the structure of the system, and the attribute in question. As such, it is difficult to generalise about this shape and include it in the model. An alternative approach is to assume the best shape for achieving our end, and investigating how well this best-case scenario performs. Intuitively, resilience will be most effective as a goal function if the shape of the resilience function is one that has a very localised peak. In this way, the tendency of the system to oscillate about the point of maximal resilience will imply that it is also oscillating in the neighbourhood of the point in attribute space that maximises resilience. We define this function, shown in Figure 8.3, as ri (ai ) =

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point a = 0. If the mechanism has no effect, a frequency distribution of a will be centred about the centre of the a range. Putting the point of maximal resilience off-centre allows us to differentiate between the mechanisms’ effects and this statistical effect when histograms of the system resilience are plotted.

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We use p(species i is replaced by another) = f /ri (ai )2 , where ri (ai ) is the resilience as a function of the species’ attribute ai , described by Equation 8.4, and f is a coefficient of proportionality. We have chosen f = 1/1000, on the grounds that this ensures 0 < ri < 1 for −30 < a < 10 (see Figure 8.4).

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Result For a single species system, a typical resilience verses time plot is shown in Figure 8.5. The resilience is larger than that expected by random replacement, but for similar reasons to our previous result, not the maximum. However, a histogram of the attribute space (Figure 8.6) shows that the attributes tend to cluster near a = 0, which is the point in attribute space that confers maximal resilience. This implies that, if one were to take such a system and choose the point in species attribute space that maximised resilience, the point chosen will be close to the likely state of the system, and in the neighbourhood about which the system fluctuates. Further, if one were to map from this species-attribute value to a system-level attribute (e.g. ef-ratio (Laws et al. 2000)), and if the system-level attribute is less sensitive to the species-attribute space than resilience, then the maximally resilient point may still provide a good approximation for this system-level attribute. It is noted that, similar to our result in Figure 8.2, the agreement between the likely state of the system and the point of maximal resilience decreases with increasing system size. Figure 8.7 shows this decrease in terms of the histograms of the attribute value. Because we have chosen a = 0 as the maximal point, but allowed ai = [−30, 10], the expected value of the attribute, a ¯, moves away from the point of maximal resilience toward the median of the a range.

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8.4 How reasonable are the model assumptions?

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Discussion We have addressed the issue of increased versus maximal resilience by drawing attention to one way in which goal functions are employed; namely the mapping from goal function to attribute space, and attribute space to system-level attribute of interest. By specifically choosing a centred and peaked mapping from attribute to resilience, we are able to say that the point in attribute space that maximises resilience is in the neighbourhood about which the system fluctuates, for systems with small numbers of species. This is not to say that these systems maximise resilience, but that a modeller who chooses the point in attribute space which maximises resilience will approximate the attributes of such a system. However, once again, the mechanism is weakened with increased system size.

8.4

How reasonable are the model assumptions?

We find that, given the right circumstances, choosing the point in attribute space that maximises resilience will approximate the point in attribute space about which the system fluctuates. What we are interested in, however, is what constitutes ‘the right circumstances’. We have made assumptions specifically contrived to make the hypothesis work, the object being to clarify what is sometimes taken for granted in the literature (Cropp & Gabric 2002). We note, however, that these assumptions are difficult to justify, and their removal can render the feedback mechanism ineffectual. We have already discussed how the number of species is related to the probability of feedback misdirection. Similar arguments suggest that other dimensionality-increasing alterations to the model will reduce the effectiveness of the feedback mechanism. For example, it is assumed that there is only one compartment in the ecosystem, to which all species modelled belong. However, if one were to model several compartments, misdirection of feedback would occur not only between species, but between compartments. We could make a similar argument with respect to the number of species’ attributes that contribute to resilience. Both of these are reasonable extensions to the model. This lack of robustness to increased dimensionality is a serious issue if one is to apply this mechanism to large and complex ecosystems. The shape of the mapping from attribute space to resilience was specifically contrived to make resilience an effective goal function. Without a localised peak, variation about the peak will cover a wider neighbourhood in attribute space. If several peaks exist, the variability will become less centred about the point of maximal resilience. Interestingly, the discontinuous shape of the function peak is often observed (e.g. Section 5.2.4, Figure 5.1), as maximisation of resilience can occur at transitions from real to complex eigenvalues, or transitions in dominance of one complex conjugate eigenvalue pair to another. However, we are not aware of any justification for using a single, highly localised peak. It is not possible to make any sensible generalisations about how an arbitrarily selected attribute influences resilience.

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It is assumed that the perturbation strength and frequency is ‘just right’ to balance the stochastic and selection forces. We imply, by directly relating the time spent in the disturbed state, ∆t, and resilience, r¯, that the perturbations are small enough to allow the system sufficient time to return to equilibrium between perturbations. This removes the possibility that large or successive perturbations will negate the feedback mechanism. Conversely, we assume that perturbations are strong enough to affect invasibility of the system. In effect, we require perturbations of a specifically intermediate strength and frequency. There is no a priori reason to suggest that the ecological system will conform to these particular conditions, and their specificity makes them difficult to justify.

8.5

Conclusion

In an effort to explore the maximisation of resilience as an ecological goal function, we investigate the oftused premise that more stable systems are more likely to remain extant than less stable systems. We ask if this can justify the hypothesis of increased, or maximal, stability generally, and resilience specifically. We describe a feedback mechanism by which ecosystems will have high resilience. We find that, in specifying the logic used in a step-wise manner, we uncover assumptions implicit in the formulation which we find difficult to justify in the context of ecological systems. In order for the feedback mechanism to have an appreciable effect on the resilience of a system, it must be assumed that: 1. the system is simple; 2. the system has few members; and 3. the perturbation strength and frequency are of a specifically moderate value (Section 8.4). In order for ‘maximise resilience’ to be an effective goal function in such a system, it must be further assumed that there is a single-peaked mapping from attribute space to resilience. Our result is not restricted to resilience and resistance, but any hypothesis in which the quantitative degree of stability of a system is maximised by a feedback from the system level to some probabilistic species-replacement mechanism. It is possible that similar reasoning may be used to revisit goal functions created on the basis of their relationship with stability (e.g. exergy of Jørgensen & Mejer 1977). While models using qualitative stability as a constraint are often successful in constructing large, improbably stable systems (Tregonning & Roberts 1979, Post & Pimm 1983, Taylor 1988, Drake 1990, Law & Blackford 1992, Law & Morton 1996), such a methodology relies upon a definitive feedback from system level instability to the lower levels. That is, if the system is unstable it is immediately reordered. We will investigate this in Chapter 11. In contrast, quantitative stability is not definitive. We must allow a low stability system to persist, even if a highly stable system will persist longer. This prevents highly stable

8.5 Conclusion

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systems from being common, except in certain exceptional circumstances. In general, one cannot assume that a species’ contribution to system stability affects that particular species’ persistence. Therefore, we believe that the diluting affect of ‘misdirection’ of feedbacks from the system level to the species level should be addressed in any theory of system level selection. While high stability has intuitive appeal, we suggest that reasoning about high stability, based upon its relationship with ecosystem persistence in the manner prescribed, should only be done when the principle of feedback ‘misdirection’ is also addressed.

Chapter 9

Laws Model and peaks in the resilience surface 9.1 9.1.1


In Chapter 7, it was found that using the goal function ‘maximise resilience’ in the Laws Model lost much of its predictive power when a relationship between the parameter describing the fraction of maximal uptake, fi , used by Laws et al. (2000), was removed. However, after this was done, some qualitative agreement remained between the field data and the model predictions when resilience was maximised. The purpose of this chapter is determine why this qualitative agreement occurred. In Chapter 8, we discussed the hypothesis that ‘maximise resilience’ is the result of a system-level feedback mechanism. We found that assuming that highly resilient systems were more likely to remain extant than less resilient systems was not sufficient to imply that ecosystems maximise resilience. In simulated ecosystems, resilience will only be higher than expected when the feedback mechanism is absent, not maximised. In Chapter 8, it was proposed that ‘maximise resilience’ can still be used to predict the likely attributes of the system, provided that the mapping from attribute space to resilience contains a single steep peak. In this chapter, we will investigate the shape of the mapping from the ef-ratio to resilience in the Laws Model. First, we are interested in determining if the mapping contains a single steep peak, as required by the theory from Chapter 8. Second, we are interested in testing a prediction that arises from the theory of Chapter 8: that the efficacy of ‘maximise resilience’ as a goal function should be correlated with how 170

9.2 Peaks in the resilience surface

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well the ef-ratio to resilience mapping conforms to the assumed single steep peak.

9.2 9.2.1

Peaks in the resilience surface Introduction

In Chapter 8, a system-level feedback mechanism was proposed by which ecosystems would tend toward high resilience. The feedback mechanism only works well for particular types of ecosystems: those that are small, simple, and subject to perturbations of the particular frequency and strength described in Chapter 8. In addition, in order for the high resilience resulting from the mechanism to translate into an ability to use ‘maximise resilience’ as a goal function, it must be further assumed that the mapping from the attribute of interest to resilience has a simple peaked shape. The assumptions restricting the complexity of the system, and the nature of the perturbations to which it is subject, are difficult to justify for the marine pelagic ecosystem. However, we will set aside criticisms of these assumptions for now, and focus upon the assumption regarding the shape of the mapping from attribute space to resilience. A peaked mapping from attribute space to resilience is a necessary condition for the effectiveness of the goal function ‘maximise resilience’. In the Laws Model, the attribute of interest is the ef-ratio. We know from previous work that the point of maximal resilience shows qualitative agreement with the field data (Chapter 7), however we do not know if this point of maximal resilience is at a single localised peak, as required by the theory. Potentially, there may be several local maxima in the mapping from ef-ratio to resilience, which may predict quite different ef-ratios. If that is the case, then the mechanism developed in Chapter 8 should not be effective, as the feedback only increases the probability of the system having high resilience, and does not guarantee maximal resilience. In this section, we investigate the shape of the mapping from ef-ratio to resilience in the Laws Model. We determine if the mapping from ef-ratio to resilience contains a single distinct peak, as this is required for ‘maximise resilience’ to be an effective goal function.

9.2.2

Method

We are interested in determining if the field data is consistent with a peaked mapping from ef-ratio to resilience. In Section 7.6, we allowed the following parameters to be selected by a grid search: f2l , f3 , f5 , f6 , and X5⋆ . As before, the fi parameters are the fraction of maximal uptake for large phytoplankton, flagellates, filter feeders, and carnivores respectively, and X5⋆ is the steady-state filter feeder compartment’s nutrient concentration. In this section, we replace the free parameter X5⋆ with the ef-ratio. Appendix F.1

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Laws Model and peaks in the resilience surface

gives details on how this is achieved. This makes it easier to collect information about how the maximum resilience changes with respect to the ef-ratio values. Recall in Equation 7.3 that the ef-ratio is a ratio of two attributes measured in the field: the loading rate, and the total production rate (ef = L/Ptot ), so specifying the ef-ratio is a proxy for either the loading rate or the total production. Therefore, specifying the ef-ratio as the free parameter enables us to fully parameterise the model in two ways. Either: 1. Laws Model L: one can specify the ef-ratio, measure the loading rate, and use the model to predict the total production; or 2. Laws Model TP: one can specify the ef-ratio, measure the total production, and use the model to predict the loading rate. These two versions of the Laws Model search a different parameter space to one another (Section 9.2.3). For each ocean region, the ef-ratio is divided into a grid of 20 increments along the total production or loading rate axis, and the parameters fi in a grid of 10 increments. The point of maximal resilience at each ef-ratio grid point is found by a grid search, and the contours of the maximal resilience are plotted.

9.2.3

Results

Comparison between variations in the model Figure 9.1 compares the predictions of the model and ‘maximise resilience’ for both versions of the Laws Model: Laws Model L and Laws Model TP. Table 9.1 gives the values for each ocean region. For ocean regions where the observed ef-ratio was low (Bermuda Atlantic Time-series Study, Hawaiian Ocean Time-series, Pacific equatorial upwelling system in normal and El Nino, and Arabian Sea), there is little difference between the two versions of the model. The most significant differences occur for high ef-ratios. For example, the North Atlantic Bloom Experiment and the Peru upwelling during normal conditions (ocean IDs 3 and 9 respectively) have only slightly higher maximal resilience when total production is the free parameter (Table 9.1). However, this small difference in resilience results in a much lower ef-ratio than the other Laws Model version (Laws Model L) and the field data.

Resilience surface Figure 9.2 and Figure 9.3 show contour plots of the maximal resilience in loading rate versus total production space. Diagonal lines are ef-ratio isoclines (ef-ratio equals 0.1, 0.2, ... 0.6). Three circles are placed on the contours to indicate the following points: the observed data, the data predicted by maximal resilience in Laws Model L, and data predicted by maximal resilience in Laws Model TP. The depth of


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Table 9.1: The ef-ratios predicted by the goal function maximise resilience for two different free-parameter sets: total production as a free parameter, and loading rate as a free parameter. Most maximal resilience indicated in bold. Laws Model TP Laws Model L Ocean ID Ocean region ef-ratio max resilience ef-ratio max resilience 1 BATS 1.8166e-01 5.4437e-02 1.7338e-01 6.0272e-02 2 HOT 1.8459e-01 6.6667e-02 1.8447e-01 7.4432e-02 3 NABE 2.5099e-01 3.9118e-02 4.2299e-01 3.9010e-02 4 EqPac-normal 1.7287e-01 8.3333e-02 1.7338e-01 8.6284e-02 5 EqPac-El Nino 1.8654e-01 8.3333e-02 1.6228e-01 1.2326e-01 6 Arabian Sea 1.5732e-01 1.0832e-01 1.4010e-01 9.5352e-02 7 Ross Sea 4.5047e-01 1.3615e-02 5.6166e-01 1.3525e-02 8 Subarctic Station P 2.6662e-01 2.0788e-02 1.7338e-01 1.8425e-02 9 Peru-normal 2.4526e-01 6.9080e-02 5.0065e-01 6.4710e-02 10 Peru-El Nino 2.4526e-01 7.1038e-02 3.8971e-01 7.7103e-02 11 Greenland Polynya 3.3321e-01 1.6491e-02 4.0635e-01 1.1875e-02

colour on the contours indicates the relative value of the maximal resilience surface (cf. Figure F.1). Values may be found in Table 9.1. Some three-dimensional examples of the maximal resilience surface have been included in Figure 9.4 and 9.5 for comparison. The maximal resilience surface is fairly irregular and jagged. This is consistent with the peaked shape of the resilience profile in parameter space described in Figure 5.1 in Section 5.2.4. However, most of the ocean regions possess one or several ridges of high maximal resilience running along the ef-ratio isoclines. For example, the Bermuda Atlantic Time-series Study’s maximal resilience surface possesses a single ridge running close to the lower ef-ratio bound, shown in deep red in Figure 9.3. As a contrasting example, the North Atlantic Bloom Experiment has three ridges in the interior of the surface, which also run parallel to the ef-ratio isoclines. They can be seen as bands of deep red in Figure 9.3, and can be discerned in the three-dimensional plot in Figure 9.4. Generally speaking, the ocean regions can be split into two groups based upon the shape and number of ridges in the maximal resilience surface. The first group are typified by a gradually increasing surface culminating in a single distinct ridge, where this ridge is located at low ef-ratios, corresponding to the observed ef-ratio. The second group are typified by a very sudden increase in the maximal resilience surface to a plateau with several local ridges upon it. Ocean regions in the first group are: Bermuda Atlantic Time-series Study, Hawaiian Ocean Time-series, Pacific equatorial upwelling system during normal conditions, Pacific equatorial upwelling system during El Nino, and the Arabian Sea. Ocean regions in the second group are: North Atlantic Bloom Experiment, Ross Sea, Peru upwelling during normal conditions, Peru upwelling during El Nino, and Greenland Polynya. The exception to the division into two groups is the ocean region Subarctic Pacific Station P. A threedimensional plot of its maximal resilience surface is shown in Figure 9.5. Like the ocean regions of the first

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group, the maximal resilience surface of Subarctic Pacific Station P increases gradually as we decrease the ef-ratio. However, like the ocean regions in the second group, the surface ends in a plateau possessing several local ridges. It has two peaks in the relative probability surface corresponding to a low ef-ratios (approximately 0.25, and 0.15), however the observed ef-ratio for this region is high (0.424). With the exception of Subarctic Pacific Station P, the split into two groups of ocean regions based on the shape of the maximal surface corresponds to a split in the ef-ratio values of those regions. All of the ocean regions in the first group described above have low ef-ratio values, corresponding to the single peak at low ef-ratios. Conversely, those with a plateau-shaped maximal resilience surface have higher ef-ratios. For the two Peru upwelling ocean regions, the intermediate ef-ratio reported in the field data corresponds very closely to the largest ridge on the plateau. For the other ocean regions in the second group (North Atlantic Bloom Experiment, Ross Sea, and Greenland Polynya), the field data reported an ef-ratio that is in the upper ef-ratio edge of the feasible-stable region. For these ocean regions, there are ridges in the maximal resilience surface toward the supper edge of the feasible-stable region, however they are not always on the very edge of the region, nor are the ridges on the edge of the region always the highest. For example, the North Atlantic Bloom Experiment, the field data records an ef-ratio of 0.505. From Figure 9.3 and Figure 9.4, it can be seen that there is a small ridge near the field data point, however the most prominent ridges are near an ef-ratio of 0.25.

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9.2.4

Discussion

With the exception of Subarctic Pacific Station P, the result supports the hypothesis that ecosystems will have high resilience. For ocean regions that have a peaked mapping from the ef-ratio to resilience, when the resilience is high, the ef-ratio is constrained to the low end of the ef-ratio range. This agrees with the field data. For ocean regions that do not have a peaked mapping from the ef-ratio to resilience, there is a wide range of ef-ratios that have approximately equal resilience, which are close to the maximum value. Therefore, the field data which predicts a high ef-ratio for these ocean regions is also consistent with the hypothesis that ecosystems will have high resilience. Although all of the ocean regions except Subarctic Pacific Station P conform to high resilience, as expected from the theory developed in Chapter 8, not all of the maximal resilience surfaces contain a distinct peak. Recall that a distinct peak is required before the high resilience predicted by Chapter 8 allows the goal function ‘maximise resilience’ to be used effectively. Only those ocean regions with low ef-ratios possess a distinct peak in the maximal resilience surface. For those ocean regions, ‘maximise resilience’ shows qualitative agreement with the field data and predicts a low ef-ratio. However, ‘maximise resilience’ also shows qualitative agreement with the field data for ocean regions that do not possess a distinctly peaked maximal resilience surface.

9.3 Predicting general trends

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We would expect that, as the maximal resilience surface becomes more plateaued, and the number of ridges on the plateau increases, the feedback mechanism would be less likely to select the region of high resilience associated with the maximum resilience peak. Then why is it that ‘maximise resilience’ continues to give qualitative agreement with the field data for ocean regions that have maximal resilience surfaces with these properties? Either the feedback mechanism is working more effectively than we would expect, and it is able to differentiate between the small differences in resilience for those maximal resilience surfaces, or the feedback mechanism is not the real reason why ‘maximise resilience’ shows qualitative agreement with the field data. The problem with interpreting these results in this way is that we do not know for certain that the mechanism cannot differentiate between small differences in maximal resilience that occur for high efratio ocean regions. Perhaps the mechanism can sometimes find the particular ridge associated with the maximum resilience, however it is just less likely to do so. A better test would be see if the accuracy of the model predictions decreases with a flattening of the maximal resilience surface. In the next section, we will perform this test.

9.3 9.3.1

Predicting general trends Introduction

In Section 9.2.3, we found that the field data was consistent with high resilience as predicted in Chapter 8. For low ef-ratios, resilience was maximised at a single peak at the lower edge of the Laws Model’s ef-ratio range. However, for high ef-ratios, the resilience surface was relatively flat, with a large range of ef-ratios with approximately equal maximal resilience values. We argued that, for the high ef-ratio group of ocean regions, the feedback mechanism would be less likely to allow ‘maximise resilience’ to provide a good approximation of the field data. However, ‘maximise resilience’ does show some qualitative agreement for this group. Conditions that do not lead to a distinct peak in the maximal resilience surface should, according to the theory in Chapter 8, lead to less agreement between the peaks in the resilience surface and the field data. Can we detect a trend in the field data which is indicative of this? In this section, we use the resilience surface to make predictions about the response of the ef-ratio to changes in the three input variables: temperature, mixed layer depth, and loading rate. Using the theory of Chapter 8, we make predictions about how the field data should behave with respect to these three input variables. We are particularly interested in situations in which trends between the input variables and the ef-ratio should become weaker as a result of the maximal resilience surface becoming less peaked.

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If these predictions are successful, then the Laws Model is consistent with the hypothesis of high resilience via a system level feedback mechanism, as described in Chapter 8. If not, then we will need to investigate other explanations for the observations (Chapter 10).

9.3.2

Method

In Section 9.2, the temperatures and mixed layer depths measured in the field for each ocean region were used as input parameters into the Laws Model. In this section, we systematically explore the behaviour of the maximal resilience surface for a range of temperatures and mixed layer depths. We generate maximal resilience contours for varying temperatures and mixed layer depths. First, the mixed layer depth is kept at a constant 80m, and contours are generated for temperatures of: 0◦ C, 5◦ C, 10◦ C, 15◦ C, 20◦ C, and 25◦ C. Second, the temperature is kept at a constant 15◦ C, and contours are generated for mixed layer depths of: 30m, 50m, 80m, 110m, and 150m. For each temperature or mixed layer depth value, the ef-ratio is divided into a grid of 20 increments along the total production or loading rate axis, and the parameters fi in a grid of 10 increments. The point of maximal resilience at each ef-ratio grid point is found by a grid search. Recall that the steady-state of the Laws Model is solved by using the mixed layer depth for the ocean region as an input parameter into Equation E.1 (Appendix E.2). The steady-state nutrient concentration of the detritus compartment is found by Xd⋆ =

LZ , S

(9.1)

where L is the loading rate, Z is the mixed layer depth, and S is the detritus sinking rate. In this section, the only difference is that the mixed layer depth is set to the six values specified above. Recall from Table 7.3 that the temperature of the ocean region determines the maximal prey/nutrient saturated grazing/uptake rate for each compartment, Ai . The temperature dependence of nutrient uptake by the biotic compartments reflects growth rates reported in the literature (and citations within Laws et al. 2000). For example, the nutrient saturated uptake rate for small phytoplankton is A2s = (1.2/q2s ) exp(0.0633(T − 25)),

(9.2)

where T is the temperature, and other parameters are defined in Table 7.2. Again, in this section, the only difference is that instead of taking the temperature associated with each ocean region, we set the temperature to each of the values specified above. For the maximum and minimum temperature and mixed layer depth investigated, three-dimensional surface plots of the maximal resilience surface are also generated. Using the theory from Chapter 8, and


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the plots generated, testable predictions are made regarding trends in the field data with respect to the input parameters. They will be tested in Section 9.4.

9.3.3

Results

Variation with temperature Figure 9.6 shows contour maps of the maximum resilience in total production versus loading rate space for different temperatures. Diagonal lines on the contour maps show the lines of equal ef-ratio, beginning with ef = 0.1, ending with ef = 0.7 (see Appendix F.2 for a key). For low temperatures, there are several ridges of high maximum resilience running parallel to the ef-ratio isoclines. Where these ridges appear with respect to changes in the temperature does not follow any clear pattern. However, the general shape of the contour maps does show a distinct relationship with temperature: as temperature is increased, the maximum resilience surface becomes more strongly spiked toward low ef-ratios. For example, at 0◦ C, starting at high ef-ratios, the plateau of high resilience is reached by ef = 0.6. However, at 25◦ C, the plateau of high resilience is not reached until approximately ef = 0.4. The contours in Figure 9.6 only show the relative values of the maximal resiliences. Figure 9.7 compares the maximal resilience surface in three dimensions for temperatures of 0◦ C and 25◦ C. It can be seen that maximal resilience at high temperatures is much higher than at low temperatures. Figure 9.7 again shows the spike in the maximal resilience surface near low ef-ratio values at high temperature.

Variation with mixed layer depth Figure 9.8 shows contour maps of the maximum resilience in total production versus loading rate space for different mixed layer depths. For low mixed layer depths, the resilience surface is quite complex, however there are generally two ridges in the resilience surface. As the mixed layer depth is increased, the ridge at the lower ef-ratio becomes more dominant, and the ridge corresponding to the higher ef-ratio shrinks up toward the high loading rate and total production region (that is, the top right hand corner of the contour plots). Figure 9.9 compares the resilience surface in three dimensions for a mixed layer depth of 30m and 150m. Unlike the change in temperature, the two surfaces have comparable maximal resilience values.

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For high temperature and high mixed layer depth, there is a dominant ridge of high resilience corresponding to low ef-ratios. However, as the loading rate and total production is increased (moving diagonally up to the top right hand corner of the contour plots), the range of ef-ratios for which resilience is high is increased, and a second suboptimal peak in the resilience surface is formed at higher ef-ratios.

9.3.4

Predictions

We can make predictions about the trends in the ef-ratio with respect to temperature, mixed layer depth and loading rate. Using the maximal resilience surfaces of the previous sections, and the prediction that the system will correspond to high resilience (Chapter 8), we predict that the ef-ratio will: 1. decrease with increasing temperature; 2. decrease with increasing mixed layer depth; and 3. increase with increasing loading rate. Verification of the above predictions would support our observation that high resilience in the model shows qualitative agreement with the field data. This is not much different to our observation that using ‘maximise resilience’ in the model shows qualitative agreement with the field data (Chapter 7). What we are particularly interested in is evidence for the feedback mechanism. We expect that the relationship between the ef-ratios predicted by the peaks in the maximal resilience surface and the ef-ratio reported in the field data should become less distinct as the peaks in the maximal resilience surface become less distinct. By less distinct, we mean that the maximal resilience surface becomes more plateau-shaped, and the number of ridges in the surface more numerous. Therefore, as secondary predictions: 1. For both the relationship between ef-ratio and temperature and the relationship between ef-ratio and mixed layer depth, the relationship should be clearer for high temperatures and mixed layer depths, where the maximal resilience surface has a distinct single peak. 2. The relationship between mixed layer depth and ef-ratio should be stronger than the relationship between temperature and ef-ratio, because the difference between the plateau and peaks in the maximal resilience surface for low mixed layer depths is larger than for low temperatures. Verification of the above two predictions would support our high resilience hypothesis and support the hypothesis that high resilience is achieved by a system-level feedback mechanism similar to that described in Chapter 8.

9.4 Field data

9.4

Field data

9.4.1

Approach

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In Section 9.3.4, we made some predictions about the relationship between the ef-ratio and the three input parameters: temperature, mixed layer depth, and loading rate. These predictions were based upon trends that we would expect if the system-level feedback mechanism hypothesised in Chapter 8 was the central determinant of the ef-ratio. In this section, we search the field data for evidence that may refute or support these predictions. Plots of each input parameter versus the ef-ratio from the field data are generated. The trends predicted in Section 9.3.4 are qualitative statements about whether the ef-ratio will have a positive or negative relationship with the input parameter. Therefore, for each plot, a linear regression is fitted, as it is the simplest regression that can describe positive or negative relationships between variables. Trends in the relationships between the input parameters and the ef-ratio are then compared to the predictions in Section 9.3.4.

9.4.2


Figures 9.10, 9.11, and 9.12 show the the ef-ratio from the field data plotted against temperature, mixed layer depth, and loading rate. The negative relationship reported by Laws et al. (2000) between temperature and ef-ratio is verified in Figure 9.10. For all three input variables, the slope of the linear regression agrees with the three predictions from the resilience contours. However, the secondary predictions made in Section 9.3.4, which are those related to the requirement that the mapping from attribute space to resilience is peaked, were not successful. It was predicted that the relationship between temperature and ef-ratio should be strongest when the peak in the surface is the most distinct, which is when the temperature and the mixed layer depth is high. This is not the case. It was predicted that the relationship between the mixed layer depth and the ef-ratio should be stronger than the relationship between the temperature and the ef-ratio. The opposite is the case. The temperature relationship gives r2 = 0.87, whereas the mixed layer depth has r2 = 0.44. It was also predicted that the ef-ratio would increase with increasing loading rate, but that this relationship would be most apparent for low temperature and mixed layer depth. Unfortunately, the field data does not include points of high loading rate with low temperature and mixed layer depth, so the area for which the relationship would be most apparent is missing from the data. The evidence for a relationship between loading rate and ef-ratio is very weak (r2 = 0.19).

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9.5 Conclusion

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9.5

Conclusion

In this chapter, we tested the system level feedback hypothesis developed in Chapter 8. It was hypothesised that, given certain conditions, ecosystems will have high resilience. We found that the field data was consistent with high resilience in the model for all ocean regions except one (Subarctic Pacific Station P). According to the theory developed in Chapter 8, ‘maximise resilience’ should have predictive ability when the mapping from ef-ratio to resilience is peaked. This was true for ocean regions with low ef-ratio, where a distinct peak in the maximal resilience surface at the low ef-ratios corresponded to the low efratio in the field data. However, for ocean regions with high ef-ratio, the maximal resilience surface was relatively flat. For these areas, the theory predicted that ‘maximise resilience’ should not be as effective. However, ‘maximise resilience’ was still able to predict trends in the ef-ratio that ‘high resilience’ could not. Similarly, the hypothesis predicted that the relationship between the ef-ratio and the three input variables (temperature, mixed layer depth, and loading rate) will be strongest when the mapping from attribute space to maximal resilience is peaked. This was not the case. The assumptions required for the system-level feedback hypothesis are stringent. The major assumptions made in Chapter 8 were that the system had few species, and that it had few compartments. Also, the perturbations were assumed to be of the correct frequency and strength to give highly resilient systems a relative survival advantage compared to systems with low resilience, as described in Chapter 8. These assumptions are difficult to justify in this context.

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In addition, the predictions based upon the system-level feedback mechanism failed. We were not able to detect a weakening of the relationship between the ef-ratio and the input parameters when the maximal resilience surface became flatter, and the ridges it possessed more numerous. This suggests that the hypothesis does not explain why qualitative agreement exists between ‘maximise resilience’ and the field data. In the next chapter, we will investigate another model of the marine pelagic ecosystem, the Fasham Model. We aim to gain some insight into why the Laws Model and ‘maximise resilience’ showed qualitative agreement with the field data using Fasham et al. (1990) model.

Chapter 10

The Fasham Model and the feasible-stable region 10.1

Introduction

10.1.1

Motivation

The Laws Model is the only model known to successfully use the goal function ‘maximise resilience’ to predict field observations. In Chapter 7, we found that the predictive ability of ‘maximise resilience’ was independent of the traditional goal functions. This implied that an explanation for its predictive ability should also be formulated independent of the relationship between ‘maximise resilience’ and the traditional goal functions. In Chapter 8, we used a suggestion from Cropp & Gabric (2002) as to why ‘maximise resilience’ was an effective goal function. Cropp & Gabric (2002) and others (May 1973, Pimm & Lawton 1977) have related the resilience of a system to the probability that it may remain extant (Section 3.3.2). We investigate a system-level feedback mechanism, where it was assumed that highly resilient systems were more likely to remain extant than those that were less so. We found that, at best, the mechanism would lead to high, but not maximal, resilience. The assumptions required for the mechanism to make an appreciable difference to the resilience of the system were difficult to justify. Further, in order for the goal function ‘maximise resilience’ to be able to predict the likely attributes of the system, the mapping from the attribute space to resilience had to be peaked. In Chapter 9, we tested the Laws Model for evidence of the mechanism proposed in Chapter 8. It was found that the mapping from the ef-ratio to the maximal resilience surface was indeed peaked for low efratios, as required by the theory, but not for high ef-ratios. Evidence for other predictions based upon the 191

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mechanism were not found. The strength of the relationship between the input parameters (temperature and mixed layer depth) and the ef-ratio did not decrease as the peak in the resilience surface flattened, as predicted by the theory in Chapter 8. There is no reason to believe that the Laws Model is the definitive model describing the behaviour of marine pelagic ecosystems. Other models, such as Fasham et al.’s (1990) model, have been developed and have proved successful in predicting the behaviour of the marine pelagic ecosystem. Having reached the apparent limit of insights into the resilience hypothesis that can be gained from the Laws Model, we explore the Fasham et al. (1990) model. In this chpater, we are are interested in determining if it behaves similarly to the Laws Model. Are we able to predict the ef-ratio using the goal function ‘maximise resilience’ ? If it does, what explanations are there for this predictive ability? Does it show evidence of a system level feedback mechanism, as hypothesised in Chapter 8, or can another mechanism be used to explain the agreement between the model and the field data?

10.2

Fasham Model

10.2.1

Method

We investigate Fasham et al.’s (1990) model, which has been augmented to include a temperature dependent response of the plankton’s nutrient uptake, as described below (subsection entitled Temperature dependence of parameters). We investigate the response of the Fasham Model to each of the goal functions, including ‘maximise resilience’. The investigations are similar to those performed in Chapter 7. We compare the ef-ratio predicted when each of the goal functions are maximised to that observed in the field data. We also explore the predictive ability of the feasible-stable region, as was done with the Laws Model in Section 7.5. In Section 10.3 we explore the shape of the maximal resilience surface as we did in Chapter 9 for the Laws Model.

Fasham Model details The model depicted in Figure 10.1 (the Fasham Model ) is adapted from Fasham et al. (1990). The major differences between the original model published by Fasham et al. (1990) and our Fasham Model are: 1. The original model allows for diffusion from each compartment to below the mixed layer depth. Our model assumes that the only export is from the sinking of detritus. This simplifies the model, and makes it more consistent with the Laws Model. 2. In our model, temperature dependence of feeding and excretion, similar to that used in the Laws Model, is included. This was done for consistency with the Laws Model. Details may be found in

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Appendix G.1 and G.2. Differences between the Fasham Model and the Laws Model are detailed in Appendix G.1. The major differences between the Fasham Model and the Laws Model are: 1. There is no differentiation between large and small phytoplankton in the Fasham Model; 2. There is no carnivore compartment in the Fasham Model; 3. The flagellate, ciliate and filter feeder compartments in the Laws Model are replaced with a single zooplankton compartment; 4. The dissolved organic matter compartment in the Laws Model is differentiated into the ammonium and dissolved organic nitrogen compartments in the Fasham Model; 5. There are nutrient flows from detritus in the Fasham Model that are not included in the Laws Model: a flow from detritus to zooplankton, and a flow from detritus to dissolved organic nitrogen; and 6. The Fasham Model has two free-parameters, in contrast with the Laws Model’s five free-parameters.

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where γ1 is the fraction of total net primary production that is exuded by phytoplankton as dissolved organic nitrogen, σ is the average daily phytoplankton specific growth rate, G1 is the loss of phytoplankton to the zooplankton compartment, and µ1 is the mortality of phytoplankton to the detritus compartment. The expression for σ is σ = J(Q1 + Q2 ),

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where J is the light limited growth rate of phytoplankton, and Q1 and Q2 are the nutrient limitation factors associated with nitrate uptake and ammonium uptake respectively. The rate of change of the nutrient concentration in the zooplankton compartment, Z, is dZ = β1 G1 + β2 G2 + β3 G3 − (µ2 + µ5 )Z, dt

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where β1 , β2 , and β3 are assimilation efficiencies for phytoplankton, bacteria, and detritus respectively, G1 , G2 , and G3 are grazing rates for phytoplankton, bacteria, and detritus respectively, and µ2 and µ5 are zooplankton excretion and mortality respectively. The rate of change of the bacteria nutrient concentration, B, is dB = U1 + U2 − G2 − µ3 B, dt

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Fixed parameter values are taken from Table 1 of the Fasham et al. (1990) paper. These fixed parameter values were chosen by Fasham et al. (1990) to test the model against data from Bermuda Station ‘S’

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(collected between 1958 and 1960). Fasham et al. (1990) provides a detailed discussion of the parameters, which the interested reader may refer to. The parameter values chosen for Bermuda Station ‘S’ will not be applicable to the wide range of ocean regions that we are comparing the model to. Nevertheless, we choose to use the original Fasham et al. (1990) parameter set to provide contrast with the Laws Model and parameters. We are no longer purely interested in predicting the ef-ratio, but we are also interested in exploring model behaviour, and the response of the predictions to variations in the model used. This is to test the robustness of ‘maximise resilience’ as a goal function, and it is also done to avoid the charge that our parameters have been specifically selected to make ‘maximise resilience’ successful.

Temperature dependence of parameters The Fasham et al. (1990) model is augmented to include the temperature dependence of the nutrient uptake rate of the biotic compartments. Recall that in the Laws Model (Chapter 7), the nutrient-saturated uptake rate by biotic compartment i was given by a temperature dependent function Ai . For example, the nutrient-saturated uptake rate of the small phytoplankton compartment is given by A2s = 1.2/q2s exp(0.0633(T − 25)),

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where q2s is the fraction of ingested nutrient converted to biomass by small phytoplankton, and T is the temperature in degrees centigrade. In the Fasham Model, the maximal growth rate J is formulated analogously J = 1.2/(1 − γ1 ) exp(0.0633(T − 25)),

(10.3)

where γ1 is the fraction of ingested nutrient that is exuded to the dissolved organic nutrient compartment. Details for other temperature dependent uptake rates may be found in Appendix G.2.

Prediction the ef-ratio and the resilience profile Details of the Fasham Model, including Octave code, are presented in the Literate Programming style (Knuth 1984) in Appendix G.2. The free parameters in the Fasham Model are: 1. Q2 : the phytoplankton nutrient limitation factor with respect to ammonium uptake; and 2. ef-ratio: the ratio of nutrient loading rate to total production, ef = L/Ptot . For consistency, investigations performed for the Laws Model (described in Chapter 7) are also performed for the Fasham Model. Detailed results may be found in the Appendices. For brevity, we only cover the three key investigations here:

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1. For each ocean region, and for each goal function (Section 7.4.2), we divide the Q2 and ef-ratio range into 500 increments, and a grid-search is used to find the free parameter values that maximises each of the goal functions. The ef-ratios predicted by the model and goal functions are compared to the field data. 2. For each ocean region, we chose free parameter values from a random, uniform distribution until 500 points in the feasible-stable region are found. Box-plots of the distribution of ef-ratios predicted are generated. 3. For each ocean region, the Q2 and ef-ratio range is divided into 100 increments, and the maximum resilience for each ef-ratio is plotted. The shape of the maximal resilience profile in parameter space is compared to the resilience surface for the Laws Model (Chapter 9).

10.2.2

Results

Predicting the ef-ratio The Fasham Model possesses similar qualities to the Laws Model. The relative sizes of the feasible stable region in parameter space are comparable (Appendix G.3), and it is possible to create a variation on the Fasham Model (named Fasham 5 and described in Appendix G.4) that shows predictive ability comparable to that of the Laws Model. Figure 10.2 compares the observed and model ef-ratio for each of the goal functions. Figure 10.2 shows that all of the goal functions give some qualitative agreement with the observed data. Figure 10.3 shows box-plots of the ef-ratios in the feasible-stable region. Unlike the Laws Model, the ef-ratio range predicted by the feasible stable region shows qualitative agreement with the observed data. Comparing Figure 10.2 and Figure 10.3, none of the goal functions have predictive power beyond that inherent in the feasible-stable region.

The resilience profile versus the ef-ratio Figures 10.4 and 10.5 show the maximum resilience found against the ef-ratio values. The following ocean regions all possess a distinct peak in the maximum resilience with respect to the ef-ratio: Bermuda Atlantic Time-series Study, Hawaiian Ocean Time-series, Pacific equatorial upwelling during normal conditions, Pacific equatorial upwelling during El Nino, and Arabian Sea. This peak corresponds to low ef-ratios. In contrast, the following ocean regions possess a broader resilience profile, generally possessing two peaks: North Atlantic Bloom Experiment, Ross Sea, Subarctic Station P, Peru upwelling during normal conditions, Peru upwelling during El Nino, and Greenland Polynya.

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Comparing Figures 10.4 and 10.5 with Figures 9.2 and 9.3 in Section 9.2.3, there is a correlation between the maximal resilience surface in the Laws Model and the maximum resilience profile for the Fasham Model. Those ocean regions that possess a distinct peak near low ef-ratio values in Figures 10.4 and 10.5 also possess a distinct peak in Figures 9.2 and 9.3. Those that possess multiple peaks Figures 10.4 and 10.5 also possess multiple peaks in Figures 9.2 and 9.3.

10.2.3

Discussion

There are several similarities between the Laws Model and the Fasham Model. For both models, the sizes of the ocean regions’ feasible-stable regions relative to one another are comparable (compare Appendix G.3 with Figures 7.3 and 7.4). Also, the relative values of the ef-ratios predicted by the feasible-stable region of the Fasham Model are similar to the relative values predicted by maximal resilience in the Laws Model (compare Figure 10.3 with Figure 9.1). Finally, the peaks in the maximal resilience profile against the ef-ratio in the Fasham Model have a similar shape to those observed in the Laws Model (compare Figures 10.4 and 10.5 of the Fasham Model with Figures 9.2 and 9.3 of the Laws Model). Despite the similarity between the two models, the Fasham and Laws Model differ in one important regard: the feasible-stable region of the Laws Model does not possess the same level of predictive ability as the feasible-stable region of the Fasham Model. The parameter values of the whole feasible stable region in the Fasham Model give qualitative agreement between the model ef-ratio and the field data. As a consequence, all of the goal functions investigated (including ‘maximise resilience’) also show qualitative agreement with the field data. This is in contrast with the Laws Model, for which there is only a slight agreement between the predicted feasible-stable region and the field data (Section 7.5.2). For the Laws Model, the use of maximal resilience led to much better agreement with the field data than the feasiblestable region. The close relationship between the two models in all other aspects motivates us to ask: is the predictive ability of ‘maximise resilience’ in the Laws Model because of its relationship with the feasible-stable region in other models such as the Fasham Model? There is no strong reason to suppose that the behaviour of the feasible-stable region in the Laws Model is more representative of the ecological reality than the Fasham Model, or its behaviour more robust to the structural and parameter value differences between the ocean regions modelled. The Fasham Model has fewer free-parameters than the Laws Model (there are two free-parameters in the Fasham model compared to the Laws Model’s five free-parameters), which restricts the size of its feasible-stable region. Although the Fasham Model uses parameter values that were devised specifically for the ocean region Fasham et al. (1990) were attempting to model, the size of the feasible-stable region of the model with these parameters fixed may be closer to the size of the feasible-stable region of the real systems, if real systems are indeed restricted in the range of values

10.3 Fasham-Laws Model

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available to these parameters according to the local conditions of the region. Additionally, one may consider the structural differences between the Laws and Fasham Models equally valid. For some conditions the ecosystem may behave more like the Fasham Model, and for other conditions more like the Laws Model. Thus, the only attributes of the system that could be reliably estimated are those that are robust to these changes in conditions, and common to the models that represent some of this variety. If a continuity with respect to variations in the model can be demonstrated for the shape of the resilience profiles and feasible-stable regions, then the shape of the resilience profile is one such robust attribute. If this could be demonstrated, then the agreement between the point of maximal resilience in the Laws Model and the position of the feasible-stable region is less likely to be mere coincidence. This would allow the predictive ability of ‘maximise resilience’ in the Laws Model to be explained in terms of its relationship with the feasible-stable region in other models, which in turn, describe some of the variety of behaviours that real systems possess. One way to test for this continuity is to create a hybrid Fasham-Laws Model, for which the structure of the food-web is intermediate between the Fasham and Laws Models, and the fixed-parameter values are a combination between the two. We will do this in the following section.

10.3

Fasham-Laws Model

10.3.1

Introduction

In Section 10.2, we found that the ef-ratio predicted by the parameter values of the entire feasible-stable region of the Fasham Model showed qualitative agreement with the field data. This was in contrast to the Laws Model, for which the predictive ability of the feasible-stable region was slight, and much lower than the predictive ability of ‘maximise resilience’. We have observed a similarity between the Laws and Fasham Models, and speculated that the predictive ability of ‘maximise resilience’ in the Laws Model is because of its relationship with the feasible-stable region in other models. In this section, we investigate the behaviour of a model intermediate to the Fasham and Laws Models, which we will call the Fasham-Laws Model. If a continuity in the shape of the resilience profile can be demonstrated between the Fasham and Laws Models, then the relationship between the point of maximal resilience in the Laws Model and the position of the feasible-stable region in the Fasham Model is less likely to be coincidental. A relationship between ‘maximise resilience’ in the Laws Model and the feasible-stable region in other models like the Fasham Model would explain why using ‘maximise resilience’ in the Laws Model leads to qualitative agreement with the field data.

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10.3.2

Method

We make three investigations of the Fasham-Laws Model. First, we investigate its response to the goal functions, including resilience. This is done by the same method as used in Section 10.2.1. Next, we determine if the ef-ratio predicted by the feasible-stable region of the model shows agreement with the field data. Finally, we compare the maximal resilience profile of the model with that of the Laws Model and Fasham Model.

Fasham-Laws Model details The model depicted in Figure 10.6 (the Fasham-Laws Model ) is a hybrid between the Fasham Model and the Laws Model. The major differences between the Fasham Model and the Fasham-Laws Model are: 1. The DON and ammonium compartments in the Fasham Model are merged into a single combined nutrient compartment; 2. Phytoplankton and zooplankton are divided into large and small by the use of ratios gp and gz (defined below); 3. The flow of nutrients from the detritus compartment back to the DON compartment in the Fasham Model is removed; and 4. The interaction terms between compartments are of the same form as in the Laws Model (i.e. terms of the form Ai fi Xi ). Therefore, fixed parameter values sourced from Fasham et al. (1990) are removed, increasing the number of free-parameters from two to five. The major differences between the Laws Model and the Fasham-Laws Model are: 1. Although gp and gz allow some differentiation between small and large phytoplankton and zooplankton, the small and large plankton remain combined into the same compartment, and so the distinct streams of nutrient flow in the Laws Model do not exist; and 2. Carnivores are not included in the Fasham-Laws Model. The model is described by the following equations. Notation and fixed parameter values are given in Table 10.1 and Table 10.2. The solution to the steady state is given in Appendix G.5.2. X˙ p = qp Fp − Fz

Xp − µp Xp , Xp + Xb + Xd

(10.4a)

X˙ z = qz Fz − µz Xz ,

(10.4b)

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(10.4c)

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Xd V Xd − + µz Xz + µp Xp , Xp + Xb + Xd M

(10.4d)

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Figure 10.6: Fasham-Laws Model.

X˙ n = L − Fp + rz Fz ,

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(10.6)

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Table 10.1: Notation used in the Fasham-Laws Model. Definition Mixed layer depth Sinking rate of detritus External loading rate of limiting nutrient Feeding rate of compartment i Concentration of nutrient in compartment i Fraction of maximal uptake by compartment i Fraction of ingested nutrient converted to biomass by compartment i Fraction of ingested biomass respired by compartment i Fraction of ingested nutrient converted to DOM or detritus si = 1 − (qi + ri ) D Fractional loss rate of detritus due to sinking Ai Maximal prey/nutrient saturated grazing/uptake rate by compartment i Pi Threshold concentration for grazing/uptake by compartment i µi Mortality from compartment i to the combined nutrient compartment gi Fraction of large to total zooplankton (i = z) or phytoplankton (i = p) Compartment subscripts n Nitrate compartment p Phytoplankton pl Large phytoplankton ps Small phytoplankton z Zooplankton zs Small zooplankton zl Large zooplankton b Bacteria d Detrital particulate organic matter c Combined nutrient compartment Symbol M V L Fi Xi fi qi ri si

Prediction the ef-ratio and the resilience profile The free-parameters in the Fasham-Laws Model are: 1. fp : the fraction of maximal uptake by phytoplankton; 2. fb : the fraction of maximal uptake by bacteria; 3. fz : the fraction of maximal uptake by zooplankton; 4. gz : the ratio of large zooplankton to total zooplankton; and 5. ef : the ratio of nutrient loading rate to total production, ef = L/Ptot . For consistency, investigations performed for the Laws Model (described in Chapter 7) and the Fasham Model (Section 10.2) are also performed for the Fasham-Laws Model. 1. For each ocean region, a genetic algorithm is used to find the parameter values that maximise each of the goal functions. The ef-ratios predicted by the model and goal functions are compared to the field data.

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Table 10.2: Parameter values used in the Fasham-Laws Model. Parameter Value qp 0.7 qz 0.35 qb 1.00 rp rz rb

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S

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10.3.3

Results

Predicting the ef-ratio

Figure 10.7 compares the observed and model ef-ratio for each of the goal functions. The figure demonstrates that all of the goal functions show some qualitative agreement with the observed data, however not to the extent that the Fasham Model displayed (Figure 10.2). Figure 10.3 shows a box-plot of the ef-ratios in the feasible-stable region. Unlike the Laws Model, the range of ef-ratios predicted by the feasible-stable region shows qualitative agreement with the observed data. However, the spread of the feasible-stable region is much larger in the Fasham-Laws Model (Figure 10.8) than in the Fasham Model (Figure 10.3).

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Comparing resilience profiles versus the ef-ratio Figures 10.9 and 10.10 compare the maximal resilience profile versus the ef-ratio for the Fasham Model, the Fasham-Laws Model, and the Laws Model. Two ocean regions are chosen: Bermuda Atlantic Time-series Study, and Greenland Polynya. The full collection of maximal resilience profiles for the Fasham-Laws Model is available in Appendix G.5.

10.3.4

Discussion

The differences between the Laws and Fasham Models are presumably due to the different experiences of the researchers that developed them. They reflect the particulars of the ecosystems with which they are most familiar, and the which aspects of the system the researchers believe are the most interesting and the most important. No single model can be applicable for all of the regions to which the models were applied. Further, just as the biological parameters are acknowledged to not be fixed (e.g. Fasham et al. 1990), the best model formulation for a region may change as the seasons and other conditions change. For example, the zooplankton ensemble is known to change over the year with climactic and associated current-pattern changes (Mackas, Thomson & Galbraith 2001). It is possible that, for any particular ocean region investigated, for some part of the year the ecosystem might behave more like the Laws Model, and for other parts of the year more like the Fasham Model. Therefore, we are interested in those attributes of the models that are robust to plausible variations in the model, as these attributes should provide the best prediction of the real system. Both models agree on the general shape of the resilience profile in ef-ratio space. For the Fasham Model (and to a lesser extent, the Fasham-Laws Model), this shape correlates to a bias in the feasible-stable region toward the ef-ratio observed in the field data. The agreement is qualitative; both models suggest that the following ocean regions will have a low ef-ratio: Bermuda Atlantic Time-series Study, Hawaiian Ocean Time-series, Pacific equatorial upwelling during normal conditions, Pacific equatorial upwelling during El Nino, and the Arabian Sea. Both models also suggest that the following ocean regions will have high ef-ratios: the North Atlantic Bloom Experiment, Ross Sea, Peru upwelling during normal conditions, Peru upwelling during El Nino, and Greenland Polynya. Also, both models place Subarctic Station P somewhere between the dichotomy of low and high ef-ratio ocean region groups. In contrast with the Fasham Model and Fasham-Laws Model, the bias of the feasible-region toward the ef-ratio reported in the field is absent in the Laws Model. However, because of the similarity in the general shape of the maximal resilience profile, the bias is preserved in the position of the peaks of the maximal resilience profile. We propose that this is the reason that using ‘maximise resilience’ in the Laws Model leads to qualitative agreement between the model and the field data; because the point of

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maximal resilience in the Laws Model follows the trends in the feasible-stable region of other models like the Fasham Model, even though its feasible-stable region is too broad to possess those trends itself. The hypothesis that feasibility and stability are constraints upon ecosystems is simpler and easier to justify than the hypothesis that ecosystems will have high or maximal resilience. First, there is a long-running precedent for assuming that ecosystems must be locally stable (as discussed in Chapter 3). However, even if one does not accept that local stability is a constraint upon real ecosystems (e.g. criticisms in Section 3.2.1, and alternative measures such as permanence in Section 3.2.4), feasibility and stability are prerequisites for high or maximal resilience, making them at least simpler than resilience constraints. Our previous efforts to formulate a resilience hypothesis by its relationship with the traditional goal functions have failed (Chapters 4 and 8). We also have not been able to demonstrate that ecosystems maximise resilience, or even tend toward significantly high resilience independent of the traditional goal functions (Chapter 9). Further, it was difficult to justify the stringent assumptions required for the system level feedback mechanism to make an appreciable difference to resilience in the Laws Model. Although it was found that the mapping for ef-ratio to resilience was peaked for low ef-ratio values (Section 9.2), simulations in Chapter 8 suggest that the mechanism will not work well for systems as complex as the marine pelagic system modelled. For example, in the simplified model in Chapter 8, the system-level feedback was ineffective for systems with more than 5 species. However, it is known that there can be large numbers of phytoplankton and zooplankton coexisting in a compartment (Hutchinson 1961, Grenney et al. 1973). As another example, in Chapter 8 it was assumed that there was only a single compartment. However, the marine pelagic ecosystem as described by Laws et al. (2000) has 10 compartments. Further, the hypothesis developed to explain high resilience in the Laws Model could not make predictions beyond that which it was created to explain (Section 9.3.4 and 9.4). For example, the feedback mechanism implies that the strength of the relationship between high resilience and the attribute should decrease as the peak in the mapping becomes less distinct. This implied that the relationship between temperature and ef-ratio, and mixed-layer depth and ef-ratio, should have weakened with decreasing temperature and mixed-layer depth. This was not the case. There are two reasons why the behaviour of the Fasham Model should be considered relevant to marine pelagic food web. The first reason is that the Fasham Model is just as plausible as the Laws Model. The second reason is that, although the lower number of free-parameters that Fasham Model possesses (two compared to the Laws Model’s five) may restrict its applicability to a particular ocean region (Bermuda Station ‘S’) and ocean regions very similar to it, having those parameters fixed may be more representative of the qualitative behaviour of a particular ocean region. These two reasons are discussed below. We have no strong reason to suggest that the ecosystem will behave exactly like the Laws Model in all situations. If we assume that the marine pelagic food web will behave in different ways depending upon


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the particulars of the environment and assemblage present, we could argue that the Fasham Model and others like it are plausible representations of this behaviour. Thus, the predictive ability of ‘maximise resilience’ in the Laws Model may be explained in terms of its relationship with the feasible-stable region of the other models such as the Fasham Model, where these other models represent the variety of behaviours that the ecosystem could conceivably possess. The Fasham-Laws Model was created as an example of a model that is intermediate between the two models investigated. The significant thing about the Fasham-Laws Model is that the shape of the resilience profile is also intermediate as a consequence. Comparing the box-plots of the feasible-stable region for the Laws Model (Figure 7.7), the Fasham-Laws Model (Figure 10.8), and the Fasham Model (Figure 10.3), it can be seen that the Fasham-Laws Model, being intermediate between the Laws and Fasham Models, has an intermediate spread in its feasible-stable region. This suggests that there is likely to be a continuity between the behaviours represented by the two models, for which for which the feasible-stable region and the peaks in the resilience profile maintain their predictive ability. This is further supported by Figures 10.9 and 10.10, where the shape of the maximal resilience profile for the Fasham-Laws Model is also intermediate between the Laws and Fasham Models. Fasham-Laws Model preserves some of the structural attributes of the Fasham Model, yet increases its generality by freeing some of the fixed parameters that were chosen by Fasham et al. (1990). Recall that Fasham et al. (1990) chose the parameter values that would be suitable for the Bermuda Station ‘S’. As a consequence, the Fasham Model had only two free-parameters. In contrast, the Laws Model has five free parameters, thus allowing the goal functions to select the analogues of those parameters which Fasham et al. (1990) had fixed. Increasing the number of free-parameters can only increase the range of ef-ratios for which the system is feasible and stable. While allowing these extra parameters to be free makes the Laws Model flexible enough to be fitted for quantitative agreement with a wider range of ocean regions, within each region, the parameter values may be restricted like Fasham et al. (1990) model for Bermuda Station ‘S’. Therefore, the ‘true’ model may have a restricted feasible-stable region similar to that of the Fasham Model, because the parameters that were free in the Laws Model are constrained by that particular region’s conditions. As discussed in Chapter 9, the use of the peaks in the maximal resilience surface to predict the ef-ratio does not perform as well as a simple linear-regression on the temperature. However, the food web model has an advantage over the linear regression, in that it can be used to predict other attributes of the system, which may not possess a simple relationship with one of the input parameters. For example, Laws (2003) has used the model and ‘maximise resilience’ to predict the bacterial biomass in the system. However, we do not recommend the use of ‘maximise resilience’ as a goal function on the basis of this result.

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We have only investigated the marine pelagic food web. An open question is whether or not the presence of a relationship between resilience profiles in related models is a general quality for other ecosystems. We have investigated a variety of models and found that the general shape of maximal resilience in parameter space concur (variations in the Laws Model in Appendix E.4 and Fasham Model in Appendix G.4), however the continuity between the models may simply have been a consequence of a serendipitous choice of parameter and structural changes. It is possible that the similarity in resilience profile is peculiar to the models investigated, and that other ecosystem models do not possess this continuity. Dynamical systems are well-known for their ability to undergo dramatic changes in behaviour after relatively small changes in parameter values and model formulation. For example, in systems like the Brusselator (Appendix A.1), parameter changes can lead to bifurcations, such that the stability properties of the system are drastically changed after a small change in parameter values (Nicolis & Prigogine 1977). While the maximally resilient point in the Laws Model may be a good proxy for the position of the feasible-stable region in variations on the Fasham Model, there may exist some other model, differing only slightly from those investigated, for which the maximal resilience profile bears no resemblance to the models investigated. Perhaps the reason that we did not come across a model with vastly different behaviour is because the model variations were all closely related to published models of this well-studied system. However, not all modellers will have this luxury. Therefore, we are reluctant to recommend the use of ‘maximise resilience’ in the manner described here. For less well-known ecosystems, a modeller would need to explore a number of different plausible models, for the purpose of testing for the sensitivities described above. However pragmatically speaking, even if such investigations led to a reasonable assurance that the point of maximal resilience was a good proxy for the feasible-stable region of the model variations, doing this testing would yield sufficient information about the position of the feasible-stable region in various models to render unnecessary the use of proxies like ‘maximise resilience’. If ‘maximise resilience’ is to be recommended merely on the basis of its relationship with the feasible-stable region, then one is better off simply using the feasible-stable region to predict the behaviour of the system. Therefore, while we propose that the reason that ‘maximise resilience’ has predictive ability in the Laws Model is because of its relationship with the feasible stable region of the real system, we do not recommend the use of ‘maximise resilience’ as a goal function on the basis of this result.

10.4 Conclusion

10.4

213

Conclusion

In this chapter, we have compared the behaviour of the Laws Model to that of the Fasham Model, and that of a hybrid of the Fasham and Laws Models, named the Fasham-Laws Model. The Fasham Model shows qualitative agreement with the field data when ‘maximise resilience’ is used to choose the free parameters. This reflects the qualitative agreement between the field data and the feasible-stable region in the Fasham Model. It is found that the behaviour of maximal resilience in ef-ratio space in the Fasham Model has very similar properties to the maximal resilience in the Laws Model (Appendix G.3). The hybrid model, Fasham-Laws, also has a similar maximal resilience shape to the Fasham and Laws Models. The spread of the feasible-stable region in the Fasham-Laws Model is intermediate between the two models. As a consequence, the point of maximal resilience in the Laws Model is correlated with the position of the feasible-stable region in the Fasham and Fasham-Laws Models. We hypothesise that the reason that ‘maximise resilience’ showed qualitative agreement with the field data in the Laws Model is because the point of maximal resilience in the Laws Model is correlated with the feasible-stable region in other models, where these other models describe the variety of behaviours available to the system. First, we have no reason to suppose that the Laws Model is a better description of the behaviour of the ecosystems than the Fasham Model. Just as the biological parameters are not fixed, the structure of the food web will not be fixed either. As the planktonic ensembles change their composition, differences between the two models, such as the level of differentiation between large and small planktonic streams, will also change. Second, there is more evidence for the hypothesis that ecosystems are constrained to be feasible and stable than the hypothesis that ecosystems will have (high or) maximal resilience. The assumptions that need to be satisfied in order for the system-level feedback mechanism described in Chapter 8 to have an appreciable effect on resilience are stringent, and difficult to justify for the marine pelagic ecosystem. Further, in Chapter 9, we found no evidence for the system-level feedback in the field data. On one hand, we have created a hypothesis for why ‘maximise resilience’ in the Laws Model predicts the ef-ratio in the field data. However, we do not suggest its use as a goal function. It is much more efficacious to use the feasible-stable region of a model (or family of models) if the sole justification of the use ‘maximise resilience’ is its relationship with the feasible-stable regions. A large volume of work already exists on the use of local stability as a constraint on model food webs, which we briefly reviewed in Chapter 3. Rather than adding to this, we will close the thesis with a more recent line of investigation. We will choose permanence as the stability measure, and apply it as a constraint to model ecosystems.

Chapter 11

Permanence as a food web building algorithm constraint 11.1

Introduction

11.1.1

Motivation

In previous chapters, we have discussed the role of a quantitative measure of stability, resilience, as a constraint upon ecosystems. We have found that the most likely explanation for high resilience as a constraint upon ecosystems is its relationship with local stability. In this chapter, we take a final look at qualitative stability as an ecosystem constraint. Back in Chapter 3, we discussed work that had been done using local stability as an ecosystem constraint. Underlying this work is an assumption that local stability is a necessary condition for the survival of a system. As a consequence, many predictions have been made about the attributes of real systems based upon the attributes of locally stable model ecosystems. This work has been ongoing since May’s (1972) landmark paper, and a voluminous body of literature exists as a result. However, a major criticism of this work has been that local stability is a inadequate measure of ecosystem survival. As discussed in Chapter 3, local stability neglects the transient dynamics of ecosystems, and excludes persistent systems, such as limit cycles, that do not possess a locally stable steady state. Ultimately all measures of stability are poor measures of what it means for a system to be able to survive in the face of environmental perturbations. However, as discussed in Section 3.2.4, permanence addresses the shortcomings of local stability listed above. 214

11.1 Introduction

215

In this chapter, we are interested in testing the predictive ability of permanence as an ecosystem constraint.

11.1.2

Overview

A large body of empirical literature exists describing the structural attributes of food webs (Schoener 1989, Lawton 1988, Pimm & Kitching 1988, Hall & Raffaelli 1993). Aspects investigated include the proportions of species fulfilling different trophic roles (e.g. basal, intermediate, and top fractions, and predator-prey ratios), and the distribution of interactions between these species (e.g. proportion of basalintermediate link-types, basal-top link-types, and the ratio of links to species). Early authors examining published food webs reported that some food web attributes did not change with respect to the size of the food web, that is, they were scale-invariant (Briand & Cohen 1984, Briand & Cohen 1987, Cohen 1989b). However, more recent work has criticised these scale-invariance results, with respect to the quality of the data (Goldwasser & Roughgarden 1997), and the way in which it was analysed (Martinez et al. 1999). Further, several authors have observed systematic changes in values of attributes as food web sizes increase – scale-variance – for attributes that were previously described as scale-invariant (Sugihara et al. 1989, Winemiller 1990, Schoenly, Beaver & Heumier 1991, Havens 1992, Martinez & Lawton 1995). In this chapter, we discuss the empirical literature concerning food web attributes. By identifying conflicts in the literature, we will be better placed to measure how well our theoretical predictions compare with ecological reality. We identify studies that are highly regarded in the literature, and use this to compile a selected food webs data-set. In closing their review of food web attributes, Hall & Raffaelli (1993) state that simulated food web assembly is a promising avenue for investigating the cause of patterns observed in real food webs. Food web assembly simulations typically involve an iterative algorithm, by which invading species are sequentially added to a model ecosystem, and a stability constraint is used to determine if the new assemblage persists until the next invasion, or some species become extinct. In the past, food web assembly algorithms have used local stability as a constraint (Drake 1990, Post & Pimm 1983, Taylor 1988). Local stability implies that the populations will return back to some steady-state after an arbitrarily small perturbation. However, Hall & Raffaelli (1993) favour the use of permanence as a model constraint (Law & Blackford 1992, Law & Morton 1996). An ecosystem is permanent if, in the face of small perturbations, the populations of its species remain positive, and below some finite upper bound (Hofbauer & Sigmund 1988). In a permanent ecosystem, if the population of any set of species is brought close to zero, they recover. Unlike local stability, permanence does not presuppose the existence of a single steady-state. It allows for complex trajectories within the phase space like limit cycles. This is more

216

Permanence as a food web building algorithm constraint

consistent with ecologists’ experience of real food webs (Paine 1988, Hastings 1988). In this chapter, we follow Hall & Raffaelli’s (1993) suggestion. We create our own food web building algorithm, using permanence instead of local stability as its constraint, and use it to make predictions about food web attributes. We compare the results to the data-sets compiled from the literature, and generate hypotheses about food web attributes for future testing. Chen & Cohen (2001) found that the probability that a randomly assembled food web is permanent decreases with two measures of ecosystem complexity: size, and connectance. This is reminiscent of a similar finding by May (1972), where the probability of local stability decreased with the same two measures of ecosystem complexity. May’s (1973) observation triggered the long-running complexitystability debate (MacArthur 1955, Hairston et al. 1968, Gardner & Ashby 1970, May 1972, Yodzis 1981, Paine 1992, Tilman 1996, Doak et al. 1998, Tilman et al. 1998), in which theorists and empiricists struggled to reconcile May’s (1973) observation with the complexity observed in real ecological systems. The result was hotly debated, as it was the prevailing view at the time that complexity facilitated ecosystem stability (MacArthur 1955). In response to the ensuing debate, May (1973, pp. 3–4) stated that

“...theoretical work should not try to prove any general theorem that “complexity implies stability”, but instead should focus on elucidating the very special sorts of complexity, the singular strategies, which may promote such mathematically atypical stability.” One candidate for these “singular strategies” was the use of food web assembly algorithms. Authors showed that, if a food web was built via a food web building algorithm that used local stability as its constraint, atypically large and complex systems could result (Drake 1990, Post & Pimm 1983, Taylor 1988). In this chapter, we are also interested in testing if food web assembly can, once again, provide an explanation for how stable yet complex food webs are formed. Further, the food web attributes predicted by the algorithm, particularly those that find strong agreement with the empirical literature, will elucidate those “singular strategies” that allow ecosystems to maintain both complexity and permanence. This chapter is organised as follows. In the remainder of this section, we explore the scale-variant and scale-invariant literature. This exploration motivates the compilation of a data-set of selected food webs. In Section 11.3, we present the selected food webs, and the algorithm predictions. The algorithm predictions are compared to each of our empirical data-sets: the scale-invariant literature, the scalevariant literature, and our observations from selected food webs.

11.1 Introduction

11.1.3

217

Definitions

We are interested in generalisations about certain food web properties. We define these properties as follows. Cycle. A cycle is formed in a food web when nutrient leaving a given compartment returns back to that compartment by some path through the heterotrophic compartments. That is, compartment 1, which feeds upon compartment 2, ..., which feeds upon compartment k, feeds upon compartment 1, and none of the species 1, . . . , k are autotrophs. Maximum chain length. The maximum chain length of a food web is the longest path from an autotrophic compartment to a top predator, measured by the number of intermediate compartments (cf. Briand & Cohen 1987). Trophic fraction. The trophic fraction is the fraction of compartments that are classified as being of the specified trophic type. The trophic types are top, intermediate, and basal. A compartment is a basal compartment if it has no prey, an intermediate compartment if it has both prey and predators, and a top compartment if it has no predators (Cohen 1989b). Link-type fraction. The link-type fraction is the fraction of links between compartments that are classified as being of the specified link type. The link types are basal-intermediate, basal-top, intermediateintermediate, and intermediate-top (Cohen 1989b). Predator prey ratio. The fraction (number of prey)/(number of predators), which is equivalent to (number of basal + number of intermediate)/(number of intermediate + number of top) (Briand & Cohen 1984). Niche-overlap graph. A food web can be turned into a niche-overlap graph by placing an undirected edge between all predators who have a prey compartment in common (e.g. Cohen 1978, pp. 8–11). See Figures 11.1 and 11.2. Interval niche-overlap graph. If the niche-overlap graph can be represented by overlapping regions in a one-dimensional space, it is an interval graph. See Figure 11.1 for an example (adapted from May (1983)). A food web with an interval niche-overlap graph will be described as having the interval property. Having the interval property is a necessary condition for the presence of a one-dimensional niche-space (Cohen 1989b, pp. 19). Rigid-circuit niche-overlap graph. If the niche-overlap graph contains no cycles greater than length three, it possesses the rigid-circuit property. All interval graphs are rigid-circuit graphs, but the converse is not necessarily true (Figure 11.2). A food web with a rigid-circuit niche-overlap graph will be described as having the rigid-circuit property (Sugihara et al. 1989).

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(a)

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2 1

3 Food web (b)

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4 3

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2

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4? 2

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Figure 11.1: An example of an interval and non-interval niche overlap graph. Adapted from May (1983).

(a)

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Figure 11.2: (a) An example of a rigid-circuit niche-overlap graph that also has the interval property, (b) and a rigid-circuit niche-overlap graph that does not have the interval property.

11.1 Introduction

11.1.4

219

Generalisations concerning food web structure

Cohen (1989b) provided a summary of five food web generalisations from empirical studies (Briand & Cohen 1984, Briand & Cohen 1987): 1. Excluding cannibalism, cycles are rare. Of 113 webs, only 3 contained a cycle, which was of length 2. 2. Chains are short. The median value of the maximum chain length in the 113 webs was 4. The maximum 10. 3. Scale invariance of trophic type. Three trophic types were included: (1) top, (2) intermediate, (3) and basal. The proportions were 0.29, 0.53, and 0.19, respectively, which Cohen (1989b) described as scale-invariant. 4. Scale invariance in proportions of link-types. Four link-types where included: (1) Basal-intermediate, (2) Basal-top, (3) Intermediate-intermediate, and (4) Intermediate-top. The proportions were 0.274, 0.077, 0.301, and 0.348, respectively, which Cohen (1989b) described as scale-invariant. 5. Scale invariance in the links to species ratio. The ratio of links versus species in the 113 webs was approximately two. We also investigate the following two observations, which were made in related works: 1. Predator prey ratio. Briand & Cohen (1984) reported a scale-invariant predator-prey ratio of 0.88. 2. Interval niche-overlap graphs. Several authors have found that a large proportion of the niche-overlap graphs of food webs observed have either the interval property (Cohen 1978, pp. 40), or the rigid-circuit property (Sugihara et al. 1989). Sugihara et al. (1989) report that 95 % of food webs sampled had rigid-circuit nicheoverlap graphs. In this paper, we investigate each of these generalisations, and the hypothesis of scale invariance of trophic and link-type fractions. We will refer to the literature quoted above as the non-selective literature, to differentiate it from the selected literature in Section 11.3.1.

11.1.5

Criticisms of generalisations

The values reported by Cohen (1989b) and others for the scale-invariant attributes of food webs have received some criticism. In this section, we detail this criticism. First, we discuss methodological criticisms,

220


which are primarily concerned with methodological shortcomings in the literature. These criticisms highlight uncertainty regarding the values for attributes reported in the literature. Second, we discuss what we call interpretation criticisms. These criticisms highlight uncertainty about the validity of describing attributes as scale-invariant.

Methodological criticisms Definition of trophic types The categorisation of trophic species into basal, intermediate, and top species is dependent upon the extent of the food web captured by the study, rather than some intrinsic quality of the species considered. For example, Cohen (1989b) defined basal species as those that have no prey in the food web described. As a result, the basal fraction of food webs used by Cohen (1989b) included species that are not autotrophs, such as ducks, fish and grasshoppers (Hall & Raffaelli 1993, Hall & Raffaelli 1991). A similar comment may be made about the categorisation of top species. Several authors have questioned whether any food web has true top species, especially when cannibalism is included in the web (Havens 1992, Polis 1991). Aggregation A frequently cited criticism of food web data is the aggregation of species in food webs (Paine 1988, Pimm & Kitching 1988, Pimm, Lawton & Cohen 1991, Lawton & Warren 1988). There are two kinds of aggregation. First, equitable aggregation, in which species are lumped based upon trophic similarities. Second, inequitable aggregation, which recognises that the observer has a bias toward aggregating certain types of species more than others. Several studies have been performed to investigate the effects of equitable aggregation upon food web measures. Sugihara et al. (1989) lumped food webs on the basis of a similarity index (where similarity was measured as the quotient of the numbers of predators and prey shared in common, over the total number of predator and prey in their union). Of the attributes investigated, only two attributes showed a systematic shift in value with aggregation: the product of the number of species and connectance, and the basal fraction (observed again in Sugihara et al. (1997)). While the predator-prey ratio, and top and intermediate fractions were sensitive to aggregation, they did not show a systematic increase or decrease. This early result, however, is disputed by subsequent studies into the effects of aggregation (Warren 1989, Hall & Raffaelli 1991, Martinez 1993, Martinez et al. 1999). Using the same source web as Sugihara et al. (1989) and Schoenly et al. (1991), but a different aggregation methodology, Martinez (1993) found that aggregation decreased the number of links per species, the intermediate fraction, and the intermediate-intermediate link-type fraction, and increased the top fraction, basal fraction, and topbasal link-type fraction. Similar results by other authors are summarised in Table 11.1.

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221

The question of which study is more valid depends upon which aggregation method is judged to be more representative of the methodology used by the field ecologists who developed the webs. Web authors possess cultural differences depending upon their area of expertise (Paine 1988, May 1983), as well as individual idiosyncrasies (Paine 1988), so there is a tendency to aggregate trophic levels of less interest to the author than others. It is generally accepted that basal species are more strongly aggregated than other species in published webs (May 1983, Pimm & Kitching 1988, Polis 1991, Hall & Raffaelli 1991, Schoenly et al. 1991, Sugihara et al. 1997). To account for this, Hall & Raffaelli (1991) simulated food web aggregation such that lower trophic levels were aggregated more strongly than higher trophic levels. The results are included in Table 11.1. They found a decrease in the intermediate-intermediate link-type fraction and links per species, and an increase in the basal fraction, and basal-intermediate link type fraction, with aggregation of species at the basal level. However, it should be noted that the increase in the basal fraction is due to the fact that the basal fraction was highly aggregated to begin with. Therefore, we expect that aggregation in the literature will in fact have the opposite effect, decreasing the basal fraction and the basal-intermediate link-type fraction. Martinez’s (1991) recent work is the first food web in which an explicit effort was made to describe each trophic level equitably. Therefore, most food web data is subject to the effects of aggregation (Table 11.1). Sampling effort The effects of sampling effort have been studied to a lesser extent. A few authors have noted that the boundaries of documented food webs are often arbitrary, and that food web members feed outside of the catalogued web (Pimm et al. 1991, Pimm & Kitching 1988). Notable exceptions are webs involving ‘closed’ communities such as within pitcher plants and galls (Beaver 1985, Hawkins & Goeden 1984), however even they are subject to interactions with the wider ecosystem. Goldwasser & Roughgarden (1997) used a 44 species food web to simulate a researcher’s ability to identify members of a web. A sampling direction was used to simulate a researcher starting with key species and, in either prey or predator direction, or both, identifying interactions between those species and other species in the community for inclusion in the web. Detectability of interactions between species was taken as proportional to the the frequency of the acts of predation. Only two web attributes (number of species, and standard deviation in chain length) were insensitive to the level of sampling effort. Trophic fractions and link-type fractions were found to be moderately sensitive to sampling effort. Noteworthy was the observation that, for many of the web attributes investigated, the values derived from low sampling effort corresponded to the values reported in the ECOWeB database (Cohen 1989a), irrespective of the sampling direction. This was true for attributes including the number of links per species, the rigid-circuit property, and the maximum chain length, but not for trophic and link-type fractions, or the predator-prey

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ratio. This observation supports the hypothesis that sampling effort has had an effect upon food webs catalogued in the literature.

Interpretation criticisms The criticisms discussed in the previous section lead us to question the accuracy of the values given for various scale-invariant attributes. However, they do not address the underlying hypothesis that these attribute values are scale-invariant. Alternatives are that the attributes are scale-variant, or that there is no relationship between attribute values and food web size. We discuss the evidence for each of these possibilities in turn. Interpretation of scatter as no relationship There is a large amount of scatter in graphs of the trophic fraction and link-type fractions of Cohen (1989b). It has been suggested that, rather than being interpreted as evidence for a scale-invariant relationship between these attributes and the size of the web, the data should be interpreted as an absence of relationship (Hall & Raffaelli 1991, Hall & Raffaelli 1993). Hall & Raffaelli (1993) stated that the invariance of the predator-prey ratio was the only relationship which appeared meaningful, due to its low scatter. However, because the intermediate fraction is double-counted (as it is both predator and prey), its inclusion spreads the scatter-plot along the 1:1 line, giving a stronger impression of compactness compared to plots in which the intermediate fraction is omitted (but see Briand & Cohen (1984)). Scale-variant relationships Briand & Cohen (1984) mentioned “a slight tendency for the fraction of top species to increase [with web size] and for the fraction of basal species to decrease”. This observation has been repeated in recent work, along with other scale-variant tendencies. In Table 11.1, the changes in the attribute values with respect to web size are shown for two cases: where the web size differed across sampled webs (above the line), and where the web size was changed artificially by aggregating the trophic species (below the line). There is complete consistency between webs above the line where scale-variance has been observed. This replicability of the results suggests that the scale-variant relationships do exist. Interestingly, there is also a similarity between trends derived from collations of differently sized food webs, and those trends observed when web sizes are manipulated artificially. The significance of this is uncertain.

Concluding remark Clearly, there is a need for clarification of the empirical literature. Ambiguity regarding the attributes of food webs makes testing our permanence food web building algorithm difficult, as one could cynically predict that just about any result will agree with at least some part of the literature. We shall therefore

11.2 Methods

223

Table 11.1: Scale-variance literature. Increases and decreases only reported if P < 0.05, or monotonic trends in the case of nonlinear regressions, else marked 0. Slope given where regression was used, else simply Inc or Dec for increase and decrease respectively. No data indicated by ‘-’. Range of Trophic fractions Link type fractions Ref. web sizes Top Int Bas Bas-int Bas-top Int-int Int-top L/n Change in attribute with observed increase in web size (1) [10, 20] Dec Inc (2) [10, 74] Inc (3) [2, 87] 0 0 -0.002 (4) [3, 90] 0 0 -0.003 (5) [19, 104] 0 0 Dec 0.16 (6) [1, 6500000] Dec Inc Dec Change in attribute with simulated increase in web size (7)a [35, 44] Dec Inc 0 0 Dec Inc 0 Inc (8)s [35, 44] 0 Inc Dec Dec Dec Inc Inc Inc (9)a [2, 87] Both Inc Dec Inc (10)a [12, 89] 0 0 Dec Dec Inc 0 Inc (1): Schoener (1989). (3): Sugihara et al. (1989). (4): Schoenly et al. (1991). (5): Winemiller (1990). (2): Havens (1992). (6): Martinez & Lawton (1995). (7): Goldwasser & Roughgarden (1997) – prey directed. (8): Ibid. – predator directed. (9): Martinez et al. (1999). (10): Hall & Raffaelli (1991). s Studies where web size range was an effect of simulated sampling effort. a Studies where web size range was an effect of aggregation. compare our algorithm’s results with each literature group in turn. First, we seek some scale-invariant values for each of the attributes, which shall be compared with the values given in Cohen (1989b). Second, we shall test those same attribute values against the selected food webs, determined in Section 11.3.1. Lastly, we shall compare any scale-variant trends produced by the model with the trends summarised in Table 11.1.

11.2

Methods

11.2.1

Selected food web data-set

Several authors have suggested that, in order to rectify ambiguities and address criticisms of the existing food web catalogue, rather than simply adding webs to the pool, one should seek to replace poor data with better-described data (Hall & Raffaelli 1993, Lawton 1988). This has motivated us to tabulate results from recent, high-quality studies. The selection of studies in Table 11.2 is based upon favourable recommendations in the literature, where the recommended paper was not central to the recommending paper’s investigations, and the two papers did not share any authors. Some of the values reported in Table 11.2 may not have been reported by the authors directly, but calculated by us from the results presented.

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11.2.2


Food web building algorithm

Overview We base our food web building algorithm on that used by Law & Morton (1996). The algorithm consists of series of repeating steps, which are detailed in the following subsections, and shown diagrammatically in Figure 11.3.

START Recorded systems

Initial system

i=1

Generate new compartment

NO

Select system i

YES

NO

No. of subsystems exceeds max?

Can it invade?

Select largest food web

YES

Identify subsystems

New system permanent?

Calculate attribute values

NO

Allow invasion.

No. of recorded systems=500?

i=i+1

YES Record attribute values

YES NO i=500?

NO

Record resulting system

Randomly choose a permanent subsystem

Food Web Building Algorithm

YES STOP

Post−processing

Figure 11.3: A flow-chart showing the major steps in the food web building algorithm, and the post-processing of the results of the food web building algorithm.

First, we begin with an initial food web (Section 11.2.2). A potential invader is generated, such that its trophic type and interaction coefficient values are selected randomly (Section 11.2.2). Next, its ability to invade the system is tested using a transversal eigenvalue test (Section 11.2.2). If it can invade the system, it is permitted to. If it cannot, another invader is generated, and the process repeated. All subsystems of the new system are found (Section 11.2.2), and the permanence of the new system with respect to these subsystems is tested (Section 11.2.2). If the new system is permanent, it is recorded. If the new system is not permanent, a permanent subsystem of the new system is chosen randomly, and

11.2 Methods

225

the resulting system is recorded. This process is repeated until 500 systems have been recorded, at which point, the algorithm is stopped and post-processing of the results occurs. Full details of the algorithm may be found in the documentation of the code, which is available in Appendix H.1.

Initial system For all results in this paper, the initial system used is a food chain of size 2, with one autotroph, and one heterotroph predating upon that autotroph. All coefficient values are determined randomly, within the constraints described below. We have chosen to use an open generalised Lotka-Volterra system (GLV), similar to that used by Law & Morton (1996). The GLV system is defined by a system of equations   n X dxi ai,j xj  , = xi di + dt j=1

(11.1)

where xi is the biomass of compartment i, di represents the intrinsic rate of increase of compartment i, and ai,j represents the interactions between a compartment i and another compartment j. We use the term ‘compartment’ rather than ‘(trophic) species’ to emphasise that these are abstract entities. Compartments differ only in their growth rates, death rates, and interactions with other compartments. In this way, a compartment may consist of several species, or one species may be split into several compartments. Following from Law & Morton (1996), the following constraints are imposed. If compartment i is an autotroph, di > 0, and ai,j ≤ 0 ∀j. As such, the autotrophs are always the basal species, and visa versa. We set ai,i < 0, so that the autotrophs are self-limiting in the absence of predators. If compartment i is a heterotroph, di < 0, ai,i = 0, and ai,j may take either sign. A heterotroph with ai,j ≥ 0 ∀j is a top predator. All other heterotrophs are intermediate compartments. We set |di |, |ai,j | < 1. Law & Morton (1996) constrained predator-prey relationships based upon the respective body sizes of the interactors, such that predators could not predate upon prey that were larger than themselves. While we do not disagree with this generalisation, and observe that it is supported in the literature to varying degrees (Neubert & Blumenshine 2000, Lawton 1988), we have decided not to impose this restriction in our own work, to increase the generality of our result. Instead, the relative sizes of interaction coefficient pairs are only constrained by our interpretation of the state variable xi as the biomass (unit mass per unit space) of compartment i. If compartment i is the predator, and j the prey, (i.e. sgn(ai,j ) = + and sgn(aj,i ) = −), the flow out of the prey compartment

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must be greater than or equal to the flow into the predator compartment because of the conservation of matter (i.e. |ai,j | ≤ |aj,i |). For the initial system, the autotroph i is determined by choosing the di from a random uniform distribution in (0, 1). For the heterotroph j, its dj is determined by choosing from a random uniform distribution in (−1, 0). For the autotroph, ai,i is chosen from a random uniform distribution in (−1, 0). The heterotroph has aj,j = 0. aj,i is chosen from a random uniform distribution in (−1, 0), and to ensure conservation of mass, ai,j = −f aj,i , where f is chosen from a random uniform distribution in (0, 1).

Generating a new compartment

At each time step, a new compartment is created, to attempt an invasion into the existing system. The first decision is to determine if the invader will be an autotroph or an heterotroph. An autotroph is created with probability 0.2, and a heterotroph with probability 0.8. The second decision is to determine which compartments in the system the invader will interact with. If the new compartment is an heterotroph, one prey is selected for it at random. If the new compartment is a autotroph, and the number of independent (non-interacting) autotrophs exceeds a user-specified value, the new compartment is forced to have at least one predator, chosen at random. For every other compartment in the system, the invader is given a 0.2 probability of interacting with that compartment. There is a 0.5 probability of the invader predating upon the compartment, and a 0.5 probability of it being a prey to the compartment, with the following exceptions: 1. If the invader is an autotroph, in may not ‘predate’ upon an existing compartment. 2. If the compartment in the system is an autotroph, it may not ‘predate’ upon the invader. The constant probability of interaction implies that, as the number of compartments in the system increases, the expected number of species that an invader will interact with increases. This contrasts with Cohen, Newman & Briand’s (1985) Cascade Model, where the probability of an interaction between any two compartments is some constant divided by the number of species. It should be noted, however, that the two views may be reconciled. While the permanence algorithm permits a potential invader to have a number of interactions proportional to the number of species present, there is nothing to say that the potential invader will succeed in invading the system (see Section 11.3.2). We are interested in seeing if Cohen et al.’s (1985) assumption will arise spontaneously from the permanence constraint. All of the coefficients associated with the new compartment are chosen from a uniform random distribution, as described in Section 11.2.2.

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11.2 Methods

Can the new compartment invade? The test of whether or not a compartment may invade is based upon the same principle as the test for permanence (Section 11.2.2). Details about the test for invasion success may be found in the adaptive dynamics literature (Dieckmann 1996, Metz, Geritz, Meszéna, Jacobs & van Heerwaarden 1995, Dieckmann, Marrow & Law 1995, Dieckmann & Law 1996). For the GLV, a new compartment xi can invade if it has positive growth at low concentrations, that is, if its transversal eigenvalue, γi , satisfies γi = fi (x⋆ ) > 0, where x⋆ is the vector steady state solution to the system when the biomass of the new compartment is low (taken at zero), and fi is the function describing the rate of increase of biomass in compartment i. In the GLVs, fi (x) = di +

n X

ai,j x⋆j ,

j=1

where

⋆

again denotes that the state-variable is evaluated at the steady state.

Note that we will accept an invader into the system even if it violates feasibility constraints upon the system, provided that it can initially invade. Such violations are rectified when the system collapses back to some permanent subsystem.

Identify subsystems For each system there exists a set of subsystems to which the full system may collapse via the extinction of some of its compartments. To test for permanence is to test that the system can recover from each of these subsystems back to its original state. Therefore, these subsystems must be identified. Subsystems are identified by a recursive depth-first-search through the system, subject to two constraints. A subsystem is only considered for further analysis if it is non-singular, and not a floating subsystem, which is defined below. As an example of a singular system, consider the case of a simple competitive exclusion, as shown in Figure 11.4. The matrix of coefficients, A, is singular, which in this case implies that there is no steady state solution to the system. As such, we do not need to test the permanence of the full system with respect to this subsystem. If the full system were to assume this structure (perhaps by the removal of a shared top predator), or contain this structure, the structure will quickly collapse into either {1, 2} or {1, 3}, depending upon the particular attributes of the compartments. Similar arguments can be made regarding other singular systems. We define a floating subsystem as any system for which there exists at least one floating compartment. A floating compartment is a compartment for which no path exists from the autotrophic compartment(s)

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2

3

1

Figure 11.4: A singular system: competitive exclusion such that the system will either collapse to {1, 2} or {1, 3}.

to the compartment in question. It is obvious that, for such systems, the floating compartments will have a steady state value of 0. In effect, they will be absent. Therefore, testing the permanence with respect to such subsystems is redundant, as it is equivalent to testing the permanence with respect to the same subsystem with its floating compartment(s) removed. It can be shown analytically that all feasible food chains are permanent (Appendix H.2). This provides the algorithm with a termination point. Once a subsystem has been found that consists only of feasible chains (and independent autotrophs), we need not test the permanence of the system against further sub-subsystems. We have used the analytic result above and various other strategies described in the documentation of the code to reduce the size of the search considerably. However, the number of subsystems increases exponentially with the size of the system. As such, the user is given the option of specifying a maximum size for the set of subsystems. Once this size is exceeded, the algorithm is restarted. In results presented in this paper, the maximum number of subsystems is constrained to 1000. Once this value is exceeded, the system is re-initialised.

Is the new system permanent?

A permanent ecosystem is one which retains all of its constituent compartments after small perturbations. Alternately, it means that, when sets of compartments are brought arbitrarily close to extinction, they will recover by moving away from the boundaries of the feasible phase space. This is described formally below.

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229

Permanence. A dynamical system with state variables x is permanent if there exists a δ > 0 such that δ < lim inf xi (t) ∀i, t→∞

whenever xi (t = 0) > 0 ∀i and lim xi (t) ≤ M < ∞.

t→∞

(Hofbauer & Sigmund 1988). This means that an initially feasible ecosystem will stay within some bound between its limit inferior, δ and limit superior M for all time. No species will stay close to zero population (extinction) or grow unimpeded to some infinitely large population (as prescribed by M ). Permanence has two advantages over local stability analysis. First, it describes the behaviour of the dynamical system beyond some arbitrarily small region around which a linearisation holds (Pimm 1982, p. 57). Second, it places no restrictions upon the dynamics of the feasible ecosystem, allowing for complex limit cycles and chaotic dynamics, which are excluded from steady-state definitions of stability (Grimm et al. 1992, Jansen & Sigmund 1998). A convenient artifact of realistic ecological models is that the limit superior imposed upon the population may often be assumed. This is true for Lotka-Volterra systems where all autotrophs are self-limiting (ai,i < 0), and interactions are of the predator-prey type (Law & Blackford 1992). Such systems are termed dissipative. Our GLV is dissipative. It can be shown (Hofbauer & Sigmund 1988, pp. 166–169) that a sufficient (but not necessary) condition for permanence of a dissipative Lotka-Volterra system is that all of the equilibrium points upon the boundaries, p⋆ , are repellors, or not saturated. Saturated. A rest point is saturated iff all of its transversal eigenvalues are non-positive (Hofbauer & Sigmund 1988, p. 166). In this way, one need not test every point upon the boundary for permanence; just the steady states. This allows a convenient test, which we will refer to as the transversal eigenvalue test, for ensuring that a system is permanent. It is also a way to test if a new compartment can invade the system. Consider first the question of whether or not a new compartment can invade the system. The transversal eigenvalue is the eigenvalue associated with eigenvector that is perpendicular to the boundary face of the system, and parallel to the axis of the invader. For our GLV (Equation 11.1), the transversal eigenvalues Pn are the (di + j=1 ai,j xj ) values, where i is the compartment of the invader. If the value of the transversal

eigenvalue is greater than zero, the population of the invader increases when small, and it can invade the system (Dieckmann 1996, Metz et al. 1995, Dieckmann et al. 1995, Dieckmann & Law 1996).

230


Similarly, in the test for permanence, one is testing whether or not some set of compartments can reinvade after a reduction of their populations to near-extinction. If at least one transversal eigenvalue for the boundary equilibrium is positive, that boundary equilibrium is a repellor, and the system will move away from that boundary and its associated extinctions (Kirlinger 1986, Kirlinger 1988). If this is true for every boundary equilibrium, that is, every boundary equilibrium is a repellor or not saturated, then the system is permanent.

Randomly choose a permanent subsystem If the new system, including the invader, is permanent, the system is recorded, and the algorithm continues to the next invasion, or termination. However, if the system is not permanent, it must lose compartments until permanence is achieved. It is possible that such a system may have several distinct permanent subsystems that it can collapse to. How would one determine which subsystem to choose? In the food web assembly literature, various methods have been used to determine which subsystem results after the system becomes unstable. The most common method used was to remove the invader that ‘triggered’ instability (Drake 1990, Post & Pimm 1983). However, this method implies that stability, a system-level attribute, feeds back to a particular species, when theoretically all species suffer from the instability of a system. In contrast, Taylor (1988) used numerical simulations to determine the outcome of invasions. The present population was initialised at steady state, and the invader at some small value. After integrating the system numerically, those compartments whose biomass fell below a specified threshold value were removed. While this approach is an improvement over simply removing the invader, it has shortcomings in the context of permanence. Permanent systems are not necessarily at steady state. It is possible that the settled dynamics of the system are close to the boundaries of the feasible space, or anywhere within the positive phase-space. Further, the settled dynamics are not necessarily a single steady state, but may be limit cycles, or more complicated trajectories. As such, initialising the system requires that a decision be made about which point on the settled trajectory one should choose. Because this arbitrary decision will affect which permanent subsystem the full system collapses to after invasion, the decision as to which subsystem is chosen is, in an indirect way, also arbitrary1. Because of the indirect arbitrariness in which subsystem the impermanent system collapses to, we have decided to choose the resulting subsystem randomly. This is achieved as follows. The list of subsystems obtained in the previous section is randomised. Then, the system collapses to the first subsystem in the list that is permanent. 1 However, it is possible that a larger proportion of initial points correspond to one subsystem than another, in effect giving certain subsystems higher likelihoods than others.

231

11.2 Methods

11.2.3

The control

Underlying the food web building algorithm are rules governing the generation of invaders’ attributes, such as their trophic type, the number of species that they will interact with, and which species these will be. These underlying rules of interaction will influence the values of the food web attributes we are measuring will take, and in effect are also a model of food web assembly. However, unlike other models based on species interaction rules, such as the Cascade Model (Cohen & Newman 1985), these rules were not created with particular ecological principles in mind. As such, we must separate the effects of the underlying interaction-rules model from the effects of the permanence constraint itself. Throughout this paper, we will be comparing the predictions of the algorithm in its entirety with a ‘control’. This control is basically the food web building algorithm without the permanence constraint, as shown in Figure 11.5.

START

YES Generate new compartment NO Allow invasion.

No. of compartments exceeds max?

To Post−processing

Initial system

NO YES No. of recorded systems=500?

Record resulting system

Control Figure 11.5: A flow-chart showing the major steps in the control food web building algorithm.

As before, an invader is generated with a randomly chosen trophic type, and a randomly chosen set of compartments with which it will interact. However, unlike before, there is no restriction upon which invaders may successfully invade the system, and no test as to which parts of the new system will persist to the next time step. All invaders generated will successfully invade. The system grows unimpeded until it reaches a user-specified size, at which point the process is restarted.

232

11.2.4


Post-processing

After the completion of the food web building algorithm or the control, the recorded systems are passed to a post-processing algorithm. The purpose of this algorithm is to calculate attribute values, such as trophic fraction, and link-type fraction. However, the systems recorded may be composed of several food webs, as shown in Figure 11.6. Therefore, we must decide which food web(s) the attributes are calculated for. We have chosen to measure the properties of the largest food web in each system. This is done for two reasons. First, the independent autotrophs and small chains that exist in most systems are uninteresting from the perspective of comparing the model predictions to empirical literature. Particularly in recent literature, empiricists have become interested in large and complex food webs. Second, we expect that large food webs are more likely to be studied by field ecologists, by merit of being easier to locate, and of greater interest.

"FOOD WEBS"

"SYSTEM"

Figure 11.6: Systems generated by the GLV permanence algorithm may consist of several independent food webs.

11.3

Results

11.3.1

Selected food web data-set

Table 11.2 records the high-quality food webs selected from the literature. We have indicated the aggregation level for each of the webs, as described in the key. This set of food webs shall be used to test our algorithm’s predictions.

Reference 1 5 1 6♯♯ 2 7 3 8 2 9 1 10♯ 4 11♯♯ 3 3 3

4 12♯ 13♯♯

Web size [5, 14] [4, 15] [12, 23] 30 [19, 32] 50 [10, 74] 89 93 182

Trophic fractions Top Int. Basal - [0.25,0.43] - [0.20,0.45] - [0.27,0.4] 0.16 0 0.12 0.34 0.06 [0.01,0.41] 0.26 0.01 0.01

0.78 0.9 0.86 0.46 0.44 [0,0.67] 0.65 0.86 0.65

0.06 0.10 0.04 0.2 0.5 [0.23,0.74] 0.04 0.13 0.34

Basal-int. - [0.22,0.40] 0.18 0.08 0.38 0.53 [0,0.93] 0.1 0.09 0.27

Link type fractions Basal-top Int-int. - [0.14,0.33] - [0,0.33] 0.02 0.01 0 0 [0.05,1] 0 0 0

0.52 0.70 0.26 0.32 [0,0.65] 0.6 0.91 0.73

11.3 Results

Table 11.2: Selected literature. References are presented as Recommender and Recommended. Web sizes are presented as a range “[min, max]”. Attribute are presented as “mean[min, max]”. A ‘-’ indicates that a value is not meaningful (e.g. a mean is omitted because of a relationship between that attribute and web size), or not reported. The fraction of webs containing cycles does not include cannibalistic cycles. Int-top - [0.30,0.38] 0.28 0.20 0.37 0.05 [0,0.45] 0.3 0.001 0

233

Table 11.2 continued Web Contains Max chain size Prey/pred Link density cycles length Reference 1 5 [5, 14] 0.96 [0.8,1.2] 1.35 [1.14,1.60] 0 4 [2,5] 1 6♯♯ [4, 15] 2.59 [1.6,2.6] 0.73 [12, 23] 1.89 [1.72,2.63] - [2.3,4.8] 0 5.8 [5,7] 2 7 3 8 30 1.11 9.6 1 13 2 9 [19, 32] 1.27 [1,1.67] - [4.2,7.8] 0 8 [7,9] 50 1.1 1.8 0 4 1 10♯ [10, 74] - [1.4,7.9] 4 11♯♯ 3 4 89 0.72 4.6 0 10 93 1.14 11 1 12 3 12♯ 3 13♯♯ 182 1.52 13.6 1 12 (1): Lawton (1988), review. (2): Hall & Raffaelli (1993), review. (3): Goldwasser & Roughgarden (1997), study of effect of sampling effort. (4): Hall & Raffaelli (1991), estuary web (5): Beaver (1985), pitcher plant communities. (6): Sprules & Bowerman (1988), freshwater plankton. (7): Warren (1989), open water pond webs. (8): Polis (1991), desert web. (9): Warren (1989), pond margin webs. (10): Hawkins & Goeden (1984), parasitoid Atriplex spp. gall webs. (11): Havens (1992), freshwater pelagic webs. (12): Martinez (1991), lake web - trophic species. (13): Martinez (1991), lake web - raw web. ♯ : Indicates that author did not preferentially aggregated basal species. ♯♯ : Indicates that author did not aggregate species at all.

234


11.3.2

Algorithm predictions

The following sections summarise results from 10 runs of 500 invasions, initialised with a two-compartment chain. The algorithm permits webs if they have less than 1000 subsystems. All results are compared to a 500 time step control, restarted when the system size exceeds 15.

Food web size and structure Figure 11.7 shows the size (number of compartments) of the largest food-web in the system against time. Note that the collapse to size two after attaining a large system size, as indicated by the arrows, is an intentional consequence of the code employed (Section 11.2.2), rather than an attribute of the permanence algorithm itself. However, the permanence constraint does cause the system to occasionally collapse, as described in Section 11.2.2. Largest food web size versus time. Permanence algorithm. Biomass 18 16

Number of compartments

14 12 10 8 6 4 2 0 0

50

100

150

200 250 300 Time (invasion-steps)

350

400

450

500

Figure 11.7: Total number of compartments versus Time. Arrows indicate places where the algorithm was restarted.

Figure 11.8 shows the size of the food webs in terms of the maximum chain length present, compared with the same for the control. The permanence algorithm appears to reduce the maximum chain length compared to the control. Both the algorithm and the control show an increase in maximum chain length with increasing food web size.

235

11.3 Results

Maximum chain length versus Number of compartments. Permanence algorithm. 10

Maximum chain length

8

6

4

2

0 0

5

10 Number of compartments

15

20

Maximum chain length versus Number of compartments. Control. 10


8

6

4

2

0 0

5


15

20

Figure 11.8: Maximum chain length versus number of compartments. Permanence algorithm and control. Means reported in Table 11.2 indicated by the range shown on the left-hand side of the graph, where upper limit of range at Maximum Chain Length = 12. Note that the median value reported by Cohen (1989b) is 2.

236


Figure 11.9 shows the largest example system from the algorithm. Predation is indicated by an arrow from the prey compartment to the predator compartment. Autotrophs, shown on the bottom row, are given an arrow back to themselves to indicate that they are self-limiting. Numbers given to each of the compartments approximate their order of arrival, with the lowest number being the most recent arrival. The permanence food-web building algorithm performed on the GLV results in a wide variety of complex structures such as these, beyond the simple chain structure predicted analytically (Section H.2). However, the proportion of food webs produced by the algorithm that are chains (29%) is much greater than that produced by the control (4%), suggesting that the permanence constraint does restrict the complexity of food web structures attained.

11

8

14

1

19

10

20

18

13

12

4

3

6

15

5

17

7

16

2

9

Figure 11.9: Example of a system generated by the permanence algorithm. Numbers approximate order of arrival, with lowest numbers being the most recent additions.

As discussed in Section 11.2.2, the probability with which a potential invader interacts with a species present is constant. This implies that, all else being equal, the number of compartments with which each compartment interacts should increase with time. This can be verified by inspecting the control in Figure 11.10. However, the algorithm result in Figure 11.10 shows that the total number of links per compartment versus the total number of compartments does not increase as steeply with increasing food web size as would occur without the permanence constraint. This implies that the permanence constraint also restricts the connectance of food webs.

237

11.3 Results

Links per compartment. Permanence algorithm.

Total links/Total compartments

2.5

2

1.5

1

0.5 0

5


15

20

15

20

Links per compartment. Control.


2.5

2

1.5

1

0.5 0

5


Figure 11.10: Total number of links per compartment versus the total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. Means reported in Table 11.2 indicated by the range shown on the left-hand side of the graph, where upper limit of range at Total Links/Total Compartments = 13.6.

238


Table 11.3 compares selected structural attributes predicted by the algorithm and the control with those reported in the literature. The fraction of webs with rigid-circuit niche-overlap graphs for both the algorithm and the control is close to that reported in the literature. The fraction of webs with cycles predicted by the algorithm is closer to the literature than the control. Table 11.3: Comparison between algorithm and literature results. The fraction of cycles is taken from (Cohen 1989b). The proportion of food-webs with rigid-circuit niche-overlap graphs is taken from (Sugihara et al. 1989). Simulations Attribute Lit. Permanence Control Fraction with cycles 0.03 0.047 0.344 Fraction rigid-circ. 0.95 0.9838 0.868

Trophic relationships Figure 11.11 shows the number of predators versus the number of prey. The linear regression slope of 1.6 disagrees with Hall & Raffaelli’s (1993) observation that most ratios reported in the literature fall below 1. However it is not inconsistent with the range of values given in the selected literature in Table 11.2, which is represented in Figure 11.11 by a dotted-line cone. Figures 11.12 to 11.14 present the trophic fractions, and Figures 11.15 to 11.18 the link-type fractions, as predicted by the algorithm, compared with the control. For ease of comparison, the format of the figures is similar to that used by Cohen (1989b), with several features added to assist interpretation. First, the length of the horizontal lines intersecting each discrete data point describes the proportion of webs in that size class with an attribute-fraction of that value. Second, the mean attribute-fraction value for each web-size class is shown by a solid line passing through the points. Third, the Cohen (1989b) value is marked by a diamond on the left-hand side of the graph. Finally, the range of means values from the selected literature in Table 11.2 is shown by a vertical line on the left of the graph. The upper and lower arrow-heads are the highest and lowest mean reported in Table 11.2, respectively. The basal fraction, and to a lesser extent the intermediate fraction, both appear to be scale invariant, in agreement with Cohen (1989b). Comparing the results of the permanence algorithm to the control suggests that this flattening is a direct result of the permanence constraint. The permanence algorithm appears to lower the top fraction compared to the control, however the relationship is not scale invariant. For all trophic fractions, the mean values predicted by the control show greater agreement with the Cohen (1989b) result than the mean values predicted by the algorithm. For the algorithm, the mean basal and top fractions are in the upper limits, and the intermediate fraction is in the lower limits of the of the means reported in the selected literature (Table 11.2). For the basal-top link-type fraction, and the intermediate-intermediate link-type fraction, the control

11.3 Results

shows better agreement with the Cohen (1989b) result than the algorithm.

239

For the intermediate-

intermediate link-type fraction, the algorithm flattens the control’s scale variant relationship, but conversely, makes the scale relationship of the basal-intermediate link-type fraction become positive. The mean basal-intermediate and the intermediate-top link-type fractions, for both the algorithm and the control, show reasonable agreement with the selected literature (Table 11.2). Table 11.4 summarises the results of the permanence algorithm where scale-invariance was reported. We have determined scale variance or invariance on the basis of the relationship between the mean and the food web size beyond size 5 food webs. For those attributes where the relationship predicted by the control and those reported in the literature differ (basal fraction, basal-intermediate link-type fraction, and intermediate-top link-type fraction), the permanence constraint does not correct the trends of underlying algorithm of food web assembly. In contrast with food web literature predicting scaleinvariance, the permanence algorithm does not demonstrate any predictive ability regarding scale-variant trends. Table 11.4: Comparing scale-variance and invariance between the control, the permanence algorithm, and the literature reporting scale-variance. Trophic fractions Link type fractions Source Top Int. Basal Basal-int. Basal-top Int-int. Int-top L/n Control Dec Inc 0 0 Dec Inc Dec Inc Lit. Dec Inc Dec Dec Dec Inc Inc Inc Algorithm Dec Inc 0 Inc Dec Inc Dec Inc

240


Predator prey ratio. Permanence algorithm. 20 Data point 1:1 Line Regression Line

Number of prey compartments

15

10

5

0 0

5

10 Number of predator compartments

15

20

Predator prey ratio. Control. 20 Data points 1:1 Line Regresssion Line

Number of prey compartments

15

10

5

0 0

5

10 Number of predator compartments

15

20

Figure 11.11: Number of prey versus number of predators. For the permanence algorithm, the regression for the predator vs prey relationship is prey= −0.2+1.6 pred, which has a coefficient of determination of r 2 = 0.8. For the control, the regression for the predator vs prey relationship is prey= −0.9 + 1.1 pred, which has a coefficient of determination of r 2 = 0.9. Range of mean slopes reported in Table 11.2 indicated by dotted-line cone.

241

11.3 Results

Basal fraction. Permanence algorithm.

Basal compartments/Total compartments

1

0.8

0.6

0.4

0.2

0 0

5


15

20

15

20

Basal fraction. Control.


1

0.8

0.6

0.4

0.2

0 0

5


Figure 11.12: Basal fraction versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph.

242


Intermediate fraction. Permanence algorithm.

Intermediate compartments/Total compartments

1

0.8

0.6

0.4

0.2

0 0

5


15

20

15

20

Intermediate fraction. Control.


1

0.8

0.6

0.4

0.2

0 0

5


Figure 11.13: Intermediate fraction versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph.

243

11.3 Results

Top fraction. Permanence algorithm.

Top compartments/Total compartments

1

0.8

0.6

0.4

0.2

0 0

5


15

20

15

20

Top fraction. Control.

Top compartments/Total compartments

1

0.8

0.6

0.4

0.2

0 0

5


Figure 11.14: Top fraction versus Total number of compartments Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph.

244


Basal-intermediate link type fraction. Permanence algorithm.

Basal-intermediate links/Total links

1

0.8

0.6

0.4

0.2

0 0

5


15

20

15

20

Basal-intermediate link type fraction. Control.


1

0.8

0.6

0.4

0.2

0 0

5


Figure 11.15: Fraction of basal-intermediate links versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph.

245

11.3 Results

Basal-top link type fraction. Permanence algorithm. 1

Basal-top links/Total links

0.8

0.6

0.4

0.2

0 0

5


15

20

15

20

Basal-top link type fraction. Control. 1


0.8

0.6

0.4

0.2

0 0

5


Figure 11.16: Fraction of basal-top links versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph.

246


Intermediate-intermediate link type fraction. Permanence algorithm.

Intermediate-intermediate links/Total links

1

0.8

0.6

0.4

0.2

0 0

5


15

20

Intermediate-intermediate link type fraction. Control.


1

0.8

0.6

0.4

0.2

0 0

5


15

20

Figure 11.17: Fraction of intermediate-intermediate links versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph.

247

11.3 Results

Intermediate-top link type fraction. Permanence algorithm. 1

Intermediate-top links/Total links

0.8

0.6

0.4

0.2

0 0

5


15

20

15

20

Intermediate-top link type fraction. Control. 1


0.8

0.6

0.4

0.2

0 0

5


Figure 11.18: Fraction of intermediate-top links versus Total number of compartments. Permanence algorithm with state variable interpreted as biomass, and control. The Cohen (1989b) value is indicated by a diamond, and the means reported in Table 11.2 are indicated by the range, both on the left-hand side of the graph.

248


11.4

Discussion

11.4.1

Food web size and structure

Similar to work with local stability (Drake 1990, Post & Pimm 1983, Taylor 1988, Tregonning & Roberts 1979), we have shown that large and complex permanent systems can be created using food web assembly (Figures 11.7 and 11.9). This result strengthens the argument that, while stability imposes severe constraints upon the complexity of a food web (Chen & Cohen 2001), those constraints can be circumvented when the food web is built incrementally.

The food webs created by the algorithm are fragile to collapse with respect to invasions. However, it should be noted that sudden changes in the food web size shown in Figure 11.7 are not necessarily due to a proportional removal of compartments from the system. These may be the result of the removal or addition of a key compartment joining adjacent webs, thus suddenly increasing or decreasing the size of the largest web.

The algorithm employed assumes that there is an inexhaustible pool of invaders that can enter the system. Clearly this not the case. As such, one should be wary of interpreting the fragility of the food web in the algorithm as being representative of real system (cf. Law & Morton 1996).

The maximum chain length increases with increasing web size, however, the rate of increase is lower than that of the control. This implies that the permanence algorithm reduces the maximum chain length. This result provides an alternative to the hypotheses that food web length is constrained by energetics (Lindeman 1942) or resilience (Pimm & Lawton 1977). Our own preliminary results on systems with higher productivity suggests that maximum chain length does eventually asymptote, as the algorithm has not yet reached the largest maximum chain length observed in empirical studies (Appendix H.3).

Empiricists have observed that a large proportion of documented food webs possess niche-overlap graphs with the rigid-circuit property or the interval property. While authors have been careful to stress that the interval property does not necessarily imply that the underlying structure of the system (e.g. the niche space) is one-dimensional, the frequency with which it occurs is suggestive (Cohen 1978, pp. 19, 81). Our algorithm predicts high rigid-circuitry despite the algorithm having no underlying nice-space. Therefore, we have provided an example of a process by which food webs can possess high frequency of rigid-circuit niche-overlap graphs, without this being indicative of an underlying one-dimensional structure, in niche space or otherwise.

11.4 Discussion

11.4.2

249

Link density

The link density predicted by the algorithm is half the value quoted by Cohen (1989b) (2), and Schoenly et al. (1991) (2.2), but much less than that quoted in most of the recent literature (Table 11.2). Others (Winemiller 1990, Havens 1992) report a link density that increases with the number of compartments to values well above the algorithm’s predictions (i.e. a power relationship between L and n), however these studies were conducted on food webs with a much larger size than the range used for the algorithm. Goldwasser & Roughgarden (1997) reports that, for prey, predator, and undirected sampling, L/n increases with increasing sampling effort. Also, Sugihara et al. (1989) show that species aggregation slightly reduces the connectance measured in food webs. This implies that, if anything, the values reported in the literature should be an underestimate. Therefore, we cannot justify our result by suggesting that the sampling techniques used in the literature are at fault, neither regarding sampling effort, nor species aggregation. Although this implies that the algorithm does not correctly predict the link density of real webs, the algorithm does tend to flatten the relationship between link density and food web size. This agrees with Cohen (1989b) and the Cascade Model. The Cascade Model created by Cohen (1989b) assumed that link density was constant with food web size. We show that this constancy naturally arises from the permanence constraint. In this regard, our algorithm has higher predictive ability than the Cascade Model. It is perhaps unsurprising that the permanence algorithm results in a flatter relationship between link density and web size than the control. Previous theoretical work with local stability has suggested that connectance should decrease with increasing system size (May 1972), and one study has extended this generalisation to stability measured as permanence (Chen & Cohen 2001). Food webs that report a constant link density (or decreasing connectance) support the hypothesis that stability constrains food web complexity. The small slope retained (Figure 11.10) agrees with the more recent literature (Winemiller 1990, Havens 1992, Schoener 1989, Bengtsson 1994), where link density is reported to increase with system size (Table 11.1).

11.4.3

Trophic relationships

The predator-prey ratio of 1.6 for the permanence algorithm, shown in Figure 11.11, is inconsistent with Cohen’s (1989b) value of 0.88, and inconsistent with the observation that most ratios reported in the older literature are below 1 (Hall & Raffaelli 1993). This discrepancy may be due to species aggregation in the food web literature. This is supported by comparing the algorithm results to the ratio reported in recent, more detailed food web investigations (Table 11.2). For the selected food web data-set, the

250


algorithm predictions are at the higher end of the range of values reported. The starkest differences between the algorithm and the literature are that the algorithm predicts a higher fraction of basal compartments, a lower intermediate fraction, and a higher basal-top link-type. The higher basal fraction predicted by the algorithm disagrees with the predominantly low basal fractions reported in the literature. Again, this may be a result of aggregation of lower trophic levels, as suggested by Hall & Raffaelli (1993). Hall & Raffaelli’s (1993) hypothesis has not been verified in the literature. While species aggregation has been investigated, the results of these investigations are not applicable to this question. Either the intensity of aggregation is always independent of the trophic level considered (e.g. Sugihara et al. 1989), or else aggregation does not properly measure the effect upon the basal fraction, as the basal fraction was highly aggregated to begin with (Hall & Raffaelli 1991). Table 11.2 supports our contention that that aggregation has reduced the basal fraction reported in the literature. For the only two studies in which species were not aggregated, the basal fraction reported was much higher than in the rest of the literature (Martinez 1991, Havens 1992). However, both of the selected studies that reported a high basal fraction still reported a higher intermediate fraction and a lower basal-top link fraction than that of the algorithm. Havens (1992) observed a basal fraction of 0.5, yet the basal-top link fraction was only 0.06. Martinez (1991) observed a basal fraction of 0.34, yet observed no basal-top links. Clearly, aggregation of similar species is not sufficient to explain this discrepancy. The results imply that there is a fundamental difference between the algorithm and real webs. The algorithm predicts a predominance of ‘short but broad’ food webs, such as the system depicted in Figure 11.19. These are not common in real systems, as described by the literature.

1 6

2

3 10

4 9

8 5

7

Figure 11.19: Iteration 27 of 500. Numbers approximate order of arrival, with lowest numbers being the most recent additions. Note the absence of intermediate compartments.

One explanation for this difference is that the algorithm does not include the architectural constraints upon the digestive systems of herbivores and carnivores. In the algorithm, there is nothing to stop a

11.5 Conclusion

251

top predator from also feeding upon autotrophs, which allows for a larger basal-top link-type fraction. Comparing with the control, it is apparent that such links are stabilising, but they may be disallowed in real systems by more preeminent constraints.

11.5

Conclusion

In Chapter 3, we defined stability as a proxy for the survival of an ecosystem in the face of environmental perturbations. If the existence of an ecosystem is sufficient proof of its survival ability, and stability a reasonable measure of this survival, then model ecosystems should reflect this by also possessing stability. Further, there should exist some similarities between the attributes of the stable model ecosystem, and the real ecosystem, that reflects the stability constraint upon ecosystems. This study is the first to use permanence as a constraint in a food web building algorithm in an effort to explain the emergent and system level properties of real food webs. Similar to work with the local stability based algorithms, we have been able to demonstrate that complex permanent food webs can result if they are built incrementally (cf. Drake 1990, Post & Pimm 1983, Taylor 1988, Tregonning & Roberts 1979). In this chapter, we investigated the predictive ability of permanence as an ecosystem constraint. We were interested in identifying agreement between the model and real systems, as this would be evidence in support of the hypothesis that permanence constrains the attributes of real food webs. However, no measure of stability is a perfect proxy for survival, and no model a perfect representation of the real system. Therefore, we were also interested in disagreement between the model and real systems, as this would indicate which aspects of food web structure are outside of the influence of permanence constraints, or are influenced by other more preeminent constraints, to which permanence is subordinate. We will briefly recount the findings here, and then discuss the general points that can be made from them. We have shown that the permanence constraint reduces the maximum food chain length, when compared with the control. This implies that permanence offers an alternative explanation for why food webs are short, in addition to energetics (Yodzis 1984, Hutchinson 1959), ecosystem size (Spencer & Warren 1996a, Sterner et al. 1997, Post, Pace & Hairston 2000), and resilience (Pimm & Lawton 1977). The Cascade Model (Cohen et al. 1985) imposes the following constraint upon food webs: that the expected number of species that an invader will interact with is constant and independent of food web size. We did not need to impose this constraint, as we found that it naturally arose from the permanence constraint in the algorithm. The permanence constraint confines the number of links per species, flattening it compared to the control. While the mean values were well below those reported in the literature, the qualitative agreement implies that patterns of interaction between species can arise naturally as a result of stability constraints upon the whole system.

252


We have provided a mechanism by which food webs will possess rigid-circuit niche-overlap graphs at a high frequency, that does not rely upon an underlying one-dimensional structure in the niche space. Because the algorithm has no niche space imposed upon it, this result reinforces the fact that rigid-circuitry is a necessary, but not sufficient, condition for a one-dimensional niche space (Cohen 1989b, pp. 19). We have compared the predictions of the permanence food web building algorithm to the literature according to two separate hypothesis: scale-variance, and scale-invariance, of food web attributes with food web size. We have used both early works (non-selective literature including (Cohen 1989b)), and later detailed studies (Table 11.2) to test our algorithm’s predictions. With respect to the scale-invariant literature, the predictions of the permanence food web building algorithm showed more agreement with the values given in the selected literature than with the values given in the non-selective literature. For example, the permanence algorithm predicted a predator-prey ratio greater than one, which is inconsistent with the non-selective literature, but supported the selected literature (Table 11.2). These results are consistent with criticisms of the methodology used in the earlier empirical literature, such as Hall & Raffaelli’s (1993) hypothesis that coarser taxonomic classification at lower trophic levels has led to the underestimation of the prey fraction. However, some major differences did occur between the algorithm and the literature that could not be explained away by methodological shortcomings in the literature. The largest difference was that the algorithm predicted a higher basal fraction, lower intermediate fraction, and higher basal-top linktype fraction, than all of the literature. This represents a fundamental structural difference between the algorithm and the literature, in that the algorithm predicts a predominance of “short but broad” food webs. We propose that this discrepancy could be addressed by increasing the productivity of the autotrophic compartments, which may increase the maximal food chain length (Kaunzinger & Morin 1998, Townsend, Thompson, McIntosh, Kilroy, Edwards & Scarsbrook 1998, Spencer & Warren 1996b, Jenkins, Kitching & Pimm 1992, Yodzis 1984), thus bringing the algorithm results into closer agreement with the literature. Our preliminary work in this area supports this (Appendix H.3). Alternately, differences between the algorithm and the empirical work could be due to the simplifications in the model. For example, the algorithm does not include the architectural constraints upon omnivores that would restrict the taxonomic range upon which they may predate. In the algorithm, there is nothing to prevent a top predator from feeding on autotrophs. In the real world, restrictions in organism morphology and biochemistry will constrain this possibility. Some of the control’s predictions showed scale-variant trends, which were in agreement with the literature. These trends were preserved when the permanence constraint was added. However, for those instances where the control did not match the literature, the permanence constraint did not correct the trend. Therefore, we have not been able to demonstrate that the scale-variant trends in real food webs are

11.5 Conclusion

253

related to permanence. Examples of agreement between the model and the real system are: short food chain length, a restricted fraction of links per species, a large basal fraction, and a high frequency of rigid-circuit niche-overlap graphs. While each of these is evidence in support of the hypothesis that permanence constrains real systems, it is no guarantee that this is what has caused the similarities. For example, while it is true that the short food chain length in the field data is predicted by the model, there are other explanations for the predominance of short food chains in real ecosystems. Further, some of the similarities are more meaningful than others. For example, a high frequency of rigid-circuit niche-overlap graphs occurs in the model and the control, which suggests that this pattern is unrelated to the permanence constraint. It also suggests that the meaning attributed to it (i.e. a one dimensional niche structure) is not correct, and it may merely be a consequence of food web assembly in general. Examples of disagreement between the model and the real system are the basal-top fraction, and the scale-variant trends. The disagreement on the basal-top fraction reinforces the fact that, even if stability constrains ecosystems, there may exist more preeminent constraints that will shape their general attributes. In the case of the basal-top fraction, we speculated that the constraint was a morphological constraint upon the organisms. The reason for the disagreement on scale-variant trends is not as clear. The fact that many of the trends were predicted by the control suggests that, like the high frequency of rigid-circuit niche-overlap graphs, it may simply be a consequence of food web assembly. Therefore, this exercise suggests a philosophical framework for the use of stability as a constraint for predicting and explaining ecosystem generalisations. Agreement between stability models and field data, while supporting the hypothesis that the stability-measure used constrains real food webs, must be interpreted in light of other possible explanations for the patterns observed. Further, agreement is not always meaningful (e.g. rigid-circuit niche-overlap graphs). Yet conversely, disagreement between the model and the field data does not negate the hypothesis that stability constrains the ecosystem. In the case of the basal-top fraction, it forces us to acknowledge organising and constraining forces on levels lower in the hierarchy than stability.

Chapter 12

Conclusion The main findings from the thesis are now briefly restated, along with recommendations for future research. Evidence for the resilience hypothesis in the literature was taken from two sources: the theoretical work in Cropp & Gabric (2002), and the empirical modelling results in Laws et al. (2000). We addressed each of these works in turn. The response of a three compartment marine pelagic food web model (the CG Model ) to thermodynamic goal functions and the function ‘maximise resilience’ was investigated in Cropp & Gabric (2002). A similarity between the parameter set that maximised each of the functions used was observed, and it was hypothesised that “the biotic attributes that optimize the thermodynamic goal functions also maximize the system’s resilience” (Cropp & Gabric 2002). Hence the resilience hypothesis was proposed: “ecosystems evolve to maximize resilience” (Cropp & Gabric 2002). We investigated the resilience hypothesis from two directions. First, we examined the validity of the goal functions used (Chapter 2), and second, we investigated the relationship between the goal functions in the Cropp & Gabric (2002) model (Chapters 4 and 5). The goal function approach is an attempt to encapsulate the trends in ecosystems within simple principles. It is based in theories concerned with the organisation of ecosystems, and the functions are often based upon the trends that are observed during ecosystem succession. Thermodynamic goal functions are ecological goal functions inspired by the organised behaviour of dissipative systems. We have found that the assumptions underlying thermodynamic goal functions are often difficult to justify. These issues with the goal function theory aside, we also found that the way in which the goal functions were used in Cropp & Gabric (2002) was inconsistent with the theory. We have, therefore, described the goal functions used in Cropp & Gabric (2002) as heuristics. We refer to them as the traditional goal functions. 254

255

In Chapter 4, we verified the result in Cropp & Gabric (2002) regarding the similarity between the parameter set that maximised each of the traditional goal functions and the parameter set that maximised resilience. However, contrary to the interpretation in Cropp & Gabric (2002), the parameter set the maximised each of the traditional goal functions does not maximise resilience; this was only true for the goal function ‘maximise flux’. Therefore, even if one were to assume that the traditional goal functions were valid, there is no reason to suppose that resilience will be maximised as a consequence of the maximisation of the traditional goal functions. While we did not find any evidence for the resilience hypothesis in the CG Model, we did verify that there was a similarity between the parameter values that maximised each of the goal functions and the parameter values that maximised resilience. For the CG Model, the parameter values that maximised resilience came close to optimising the traditional goal functions. In response, we formulated the resilience heuristic: maximising resilience offers a compromise between the traditional goal functions. We further explored the relationship between the goal functions by investigating their response to variations in the CG Model. A new metric, concordance, was developed to measure the ability of each goal function to also simultaneously optimise the others. Two types of resilience heuristic were investigated, reflecting two assumptions regarding the resilience hypothesis. First, we investigated concordance relative to all goal functions, including resilience. This represents the assumption that all goal functions, including resilience, are equally valid, and that we are interested in a compromise between all of them. Second, we investigated concordance relative to the traditional goal functions only. This assumes that only the traditional goal functions are valid, and ‘maximise resilience’ is of interest only for its ability to offer a compromise between them. If ‘maximise resilience’ is found to have high concordance of this type, this would justify the use of ‘maximise resilience’ as a goal function. While it would not imply that ecosystems will maximise resilience as a consequence of maximising the traditional goal functions, it does mean that ‘maximise resilience’ will provide a compromise between the traditional goal functions. In Chapter 5, it was found that, in general, the variations in the CG Model supported the first resilience heuristic, where concordance was calculated relative to all goal functions. However, the variations of the CG Model gave mixed results for the second resilience hypothesis. In Chapter 5, it was noted that the parameter range effected the concordance calculations. In Chapter 6, the resilience heuristic was further tested by exploring the concordance of the goal functions in various generalised Lotka-Volterra models. It was found that the traditional goal functions generally predicted very low values of resilience. This partly explained why, when concordance was measured relative to all goal functions, ‘maximise resilience’ generally had a higher concordance than the other goal functions. This implies that, if ‘maximise resilience’ is a valid independent goal function, maximising resilience will still offer some compromise with the traditional goal functions. No support for the second resilience heuristic was found. This implies that the use of ‘maximise resilience’ cannot be justified purely on the

256

Conclusion

grounds that it maximises the traditional goal functions, as the other goal functions consistently provided a better compromise between the traditional goal functions than ‘maximise resilience’. The results from Chapters 5 and 6 motivated the search for a justification of the resilience hypothesis that was independent of the traditional goal functions. Laws et al. (2000) successfully used ‘maximise resilience’ to predict the ef-ratios of marine systems. In Chapter 7, we investigated whether the success of ‘maximise resilience’ in the Laws Model was merely because ‘maximise resilience’ provided a compromise between the traditional goal functions. This would imply that it was the traditional goal functions, and not ‘maximise resilience’, that held the predictive ability. It was found that ‘maximise resilience’ was the only goal function that gave good predictions of the ef-ratios. This left the question as to why this was the case. In Chapter 3, we discussed the use of stability as a proxy for the survival of an ecosystem in the face of environmental perturbations. Cropp & Gabric (2002) used this as a partial justification for proposing that ecosystems maximise resilience. They stated that “highly resilient ecosystems are more likely to remain extant than ecosystems with low resilience”. In Chapter 8, we investigated their reasoning in three steps. First, we assumed that highly resilient systems would be more likely to remain extant than systems that had low resilience. We described this in a model by giving the species of a system with low resilience a higher probability of going locally extinct and being replaced by another species. Next, we tested whether this system-level feedback mechanism was sufficient to cause ecosystems to maximise their resilience. We found that, although ecosystems of this type had a higher resilience than when the feedback mechanism was absent, the resilience of the model system was not maximised. The efficacy of the mechanism was strongly dependent upon assumptions regarding the size and simplicity of the system, and the perturbation strength and frequency. Finally, we contrived a situation in which ‘maximise resilience’ would still be an effective goal function when the mechanism was present. It was found that, if the mapping from the system attribute of interest to resilience was peaked, the system with high resilience would oscillate around the point of maximal resilience, and therefore, maximising resilience in an appropriate model would approximate the likely attributes of the real system. However, once again, the assumptions required for this mechanism to be effective were stringent, and difficult to justify. The work in Chapter 8 led to some testable predictions, which were investigated in Chapter 9. We investigated the shape of the maximal resilience surface in total production versus loading rate space in the Laws Model. The change in the shape of the surface with the two input parameters, temperature and mixed layer depth, was investigated. Combining information about the shape of the surfaces and the theory developed in Chapter 8, we made two predictions about the strength of the relationship between temperature and ef-ratio, and mixed layer depth and ef-ratio. Both of these predictions failed. The

257

failure of the predictions, combined with the difficulty in justifying the assumptions behind the systemlevel feedback mechanism hypothesised in Chapter 8, led to us rejecting the hypothesis as an explanation for why ‘maximise resilience’ in the Laws Model leads to qualitative agreement with the field data. In Chapter 10, we explore the predictive ability of the Fasham Model, which is adapted from a model in the literature (Fasham et al. 1990). We found that using all of the goal functions, including ‘maximise resilience’, led to qualitative agreement between the ef-ratios predicted by the model and the field data. The reason for this was that the ef-ratios predicted by the whole parameter set of the feasible-stable region also shows agreement with the field data. Therefore, as the points that maximise each of the goal functions are necessarily feasible and stable, they too agree with the field data. In Chapter 10, we noted similarities between the attributes of the Fasham and Laws Model. Significantly, both models have a similar shape to the maximal resilience profile in ef-ratio space. Although the Laws Model has a wide feasible-stable region in ef-ratio space, because the general shape is preserved, the point of maximal resilience in the Laws Model roughly corresponds to the position of the feasible-stable region in the Fasham Model. We used a hybrid model, the Fasham-Laws Model, to demonstrate that there is a continuity between the two models, such that an intermediate model will have an intermediately shaped maximal resilience profile. This implies that the similarity in resilience profile between the two models was not merely a coincidence. We have no reason to suppose that the Laws Model or the Fasham Model is the better representation of the behaviour of the ecosystem. We proposed that the ecosystems will behave as a combination of the two models. Different ocean regions will behave more like one model or the other. Further, for a particular ocean region, climactic and external forcings may change the behaviour of the particular system such that sometimes it behaves more like the Laws Model, and other times, more like the Fasham Model. We proposed that those attributes of the models that are robust to these variations will provide the best predictions of the real ecosystem. In Chapter 10, we found that the shape of the maximal resilience profile is one such attribute. For the Fasham models investigated, there was complete agreement as to which ocean regions would be constrained by the feasible-stable constraint to low ef-ratios, and which could have high ef-ratios. This also agreed with the ef-ratio trends predicted by ‘maximise resilience’ in the Laws Model. On one hand, while we have provided an explanation for why ‘maximise resilience’ worked in the Laws Model, we do not recommend its use. The generality of this result is unknown. Dynamical systems are well-known to produce drastically different behaviours for slight changes in parameter values, and this may be an interesting avenue for future research. If ‘maximise resilience’ is to be recommended as a goal function on the basis of its relationship with the feasible-stable region of the system’s behaviours, then it may be more efficient to simply use the feasible-stable region of a variety of models to predict behaviour,

258

Conclusion

rather than an proxy like ‘maximise resilience’. The hypothesis that feasibility and stability constrain ecosystem behaviour has been discussed in Chapter 3. There, we defined stability as a proxy for the survival of an ecosystem in the face of environmental perturbations. Chapter 10 provided supporting evidence of this definition for local stability. Rather than adding to the already large amount of work on ecosystems constructed using local stability rules (Chapter 3), in Chapter 11 we investigated another stability measure, permanence. We created a food web building algorithm using permanence as its constraint. We were interested in identifying agreement between model food webs and food web attributes reported in the literature, as these would be evidence for permanence as a constraint in real systems. We were also interested in disagreement, which would indicate which food web properties were likely to be outside of the influence of permanence, or subject to other constraints. We found that permanence constrained maximum food chain length, link density, increased the basal fraction, and led to a high frequency of rigid-circuit niche-overlap graphs. However, the similarity between these attributes and those in real systems does not necessarily mean that permanence constrains the real system. For example, the short maximum chain length in both the model and field data may be better explained as an energetic constraint (Kaunzinger & Morin 1998, Townsend et al. 1998, Spencer & Warren 1996b, Jenkins et al. 1992, Yodzis 1984). Further, some similarities between the model and the real system may not be meaningful. For example, the predominance of rigid-circuit niche-overlap graphs predicted by the model and in the field data was also observed in the control, which implies that it is more likely to be a consequence of the food web assembly than an indication of dimensionality of the underlying niche structure. We found that the model and the real system disagreed as to the scale-variant trends, and the fraction of basal-top link-types. These findings indicate where the limits of permanence’s influence upon real ecosystems are likely to be. We suggest that the lower basal-top link-type fraction in real systems compared to the model predictions may be explained by the morphological constraints upon organisms. This reminds us that, even if stability constrains ecosystems, it is not necessarily the dominant driver of system-level attributes. There are several open questions arising from the permanence food web building algorithm of Chapter 11. The effect of productivity upon the web, particularly on the length of the food chains, could be further explored. In addition, a separation between the effects of the feasibility constraint and the stability constraint upon the food webs is yet to be investigated. We make the following conclusions regarding the four central issues addressed by this thesis (Chapter 1): 1. The resilience hypothesis: Do ecosystems maximise resilience? If they do then ‘maximise resilience’ can be used as a goal function. We find little support for the resilience hypothesis sensu Cropp & Gabric (2002) in this thesis. We find that:

259

(a) The selection pressures used by Cropp & Gabric (2002) were controversial (Chapter 2); (b) The selection pressures used by Cropp & Gabric (2002) were inconsistent with the goal function theory (Chapter 2); and (c) There is no support for the resilience hypothesis sensu Cropp & Gabric (2002) in the CG Model (Chapter 4). We also find little support for the resilience hypothesis via a system-level feedback mechanism, as alluded to in Cropp & Gabric (2002), and developed in Chapter 8. We find that: (a) Assuming that more resilient systems are more likely to remain extant than less resilient systems is not sufficient reason to propose that ecosystems will maximise resilience (Chapter 8); (b) That, at best, a system-level feedback will give an ecosystem a higher probability of high resilience, however the assumptions required for this to work are stringent and difficult to justify (Chapter 8); and (c) That we do not find any evidence for such a feedback mechanism in the Laws Model (Chapter 9). 2. The resilience heuristic: Does maximising resilience offer a compromise between the traditional goal functions? We find that ‘maximise resilience’ does offer a compromise between the traditional goal functions in a wide variety of models (Chapters 4, 5, and 6), however, even if one were to accept the validity of the traditional goal functions, this alone is not sufficient reason to recommend the use of ‘maximise resilience’ as a goal function (Chapter 6). 3. Independence of resilience: Is ‘maximise resilience’ an effective goal function independent of its relationship with the traditional goal functions? We found that the predictive ability of ‘maximise resilience’ in the original Laws Model is greater than the predictive ability of any of the traditional goal functions. This implies that it’s effectiveness as a goal function is independent of its relationship with the traditional goal functions (Chapter 7). Work with the Fasham Model suggests that it is the relationship between the point of maximal resilience and the feasible-stable region of related models that gives ‘maximise resilience’ its predictive ability (Chapter 10). This is evidence in support of the stability/survival hypothesis below. 4. Stability/survival hypothesis: Are the attributes of real ecosystems are restricted by the constraint that they possess some level of stability?

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Conclusion

We do not find support for the hypothesis that, because more stable ecosystems are more likely to remain extant than less stable ecosystems, ecosystems will tend to have high stability. We find that: (a) The assumptions required for this system-level feedback mechanism to have an appreciable effect on the stability of the system are stringent and difficult to justify (Chapter 8); and (b) There is no evidence for this in the Laws Model. However, we do find some support for the hypothesis that stability constrains ecosystems. We find that: (a) The ef-ratio predicted by the parameter values of the feasible stable region of the Fasham Model show qualitative agreement with the field data (Chapter 10); and (b) Model food webs assembled using permanence as a constraint have some attributes in common with real food webs (Chapter 11). However, we make the following cautions regarding the hypothesis above. We note that: (a) No measure of stability is a perfect proxy for the survival of the ecosystem (Chapter 3); (b) Similarities between stable model food webs and real food webs may be due to reasons other than that food webs are constrained by stability (e.g. short food chains, a high basal fraction, and rigid-circuit niche-overlap graphs in Chapter 11); and (c) Even if the aspect of stability considered does constrain the ecosystem, its effects may be masked by more preeminent constraints (e.g. the basal-top link-type fraction in Chapter 11).

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Appendix A

Appendices relating to Chapter 2 A.1

Two necessary conditions for organised behaviour in dissipative systems

A.1.1

Item 1: Open systems

deS

di S

Figure A.1: Entropy flux and entropy production in an open system. Reproduced from Nicolis & Prigogine (1977).

In an isolated system (no matter or energy flux), the change in entropy, S, is (Nicolis & Prigogine 1977, pp. 22) dS ≥ 0, dt

(A.1)

so entropy is always constant or increasing. In an open system (Figure A.1), the total entropy change dS is the sum of two contributions; that internally generated, di S, and an exchange with the environment outside the system, de S. That is (Nicolis & Prigogine 1977, pp. 24) dS = de S + di S.

(A.2) 276

A.1 Two necessary conditions for organised behaviour in dissipative systems

277

In Equation A.2, we can imagine a situation in which a steady state has been achieved, dS = 0, and no more entropy is produced. One possibility is that both the internally generated entropy, and the entropy exchange terms, are zero (de S = 0, and di S = 0). Nicolis & Prigogine (1977, pp. 25) call this thermodynamic equilibrium. To reach this state with the exchange term omitted from Equation A.2, we find the entropy of the system must increase until it finds its equilibrium at some maximal entropy state, dS = di S ≥ 0.

(A.3)

However another route to equilibrium, and another steady state, is possible. If a continual negative entropy exchange exists, theoretically it may be possible to maintain a steady state such that de S = −di S < 0.

(A.4)

In other words, a continual addition of negative entropy to the system may allow the system to remain at steady state away from the maximal entropy state. This would increase the order of its configuration compared to thermodynamic equilibrium. Nicolis & Prigogine (1977) differentiate this steady state from the thermodynamic equilibrium discussed above by calling it non-equilibrium 1 . In summary, the an open system can decrease entropy within the system without a violation of the second law of thermodynamics by transferring entropy outside of the system. From a biological perspective, conceiving of ecosystems as open systems may provide a way to reconcile the universal tendency to disorder with the order apparent in natural systems.

A.1.2

Item 2: Far from equilibrium

Entropy production forms a Lyapunov function (see Chapter 3, Section 3.2.3), which implies that the state of minimal entropy production, the thermodynamic equilibrium, is stable to fluctuations (Nicolis & Prigogine 1977, pp. 57). However, if some parameter in the system is changed, it possible that the thermodynamic equilibrium will become unstable, and the system could be compelled to move away from the thermodynamic equilibrium to some other steady-state, far from equilibrium. Three key ingredients are identified for far-from-equilibrium behaviour: fluctuation, feedback and complexity. Prigogine referred to this as the principle of order through fluctuations, however I believe that this is misleading, as the latter two ingredients are just as important as the first. In contrast with a Boltzmann system, where slight fluctuations are dampened, the slight fluctuations in a self-organising system are autocatalytically amplified through some positive feedback mechanism (DeAngelis, Post & Travis 1986, pp. 39–43). This amplification moves the system away from the 1 Note that their usage of ‘equilibrium’ and ‘steady state’ differ from that typically used in the ecological context, in which both terms are interchangeable.

278

Appendices relating to Chapter 2

equilibrium situation, in which the system is disorganised, to an organised non-equilibrium steady state. In order for a system to have more than one steady state, and to have one of these steady states exhibit some organised behaviour, the system must possess some level of complexity. In the dynamical systems that will be the focus of this thesis, this complexity is manifest as nonlinearity in the differential equations governing the system, and a large number of state variables in the system. Below, we use a dynamical system called the Brusselator as an example of how this occurs. Brusselator Nicolis & Prigogine (1977, Chp. 7) presents a simplified version of the Brusselator. A chemical reaction is described by dX = A − (B + 1)X + X 2 Y + D1 ∇2 X, dt dY = BX − X 2 Y + D2 ∇2 Y, dt

(A.5a) (A.5b)

where each state variable is the concentration of a different chemical species. Local stability analysis (Nicolis & Prigogine 1977, p. 96 for working) provides a simple steady state solution X ⋆ = A, Y⋆ =

B , A

(A.6a) (A.6b)

and the conditions for its stability. The steady state described by Equation A.6 is what we would ‘expect’ of a chemical reaction: that after some time reaction rates reach equilibrium and both species are present in constant quantities. The interesting thing about the Brusselator is what occurs when this steady state becomes unstable. When the concentration of B exceeds a certain amount, the steady state described by Equation A.6 becomes unstable, and the dynamics of the system proceed toward a limit cycle. This means that there are oscillations in the concentrations of X and Y at regular intervals, such that, at one moment X is the dominant species, then Y , then X, and so on. Prigogine & Stengers (1984, pp. 147–148) describes the behaviour as follows. Let us pause for a moment to emphasise how unexpected such a phenomenon is. Suppose we have two kinds of molecules, “red” and “blue”. Because of the chaotic motion of the molecules ... [the] vessel would appear to us as “violet”, with occasional irregular flashes of red or blue. However, this is not what happens with a chemical clock; here the system is all blue, then it abruptly changes its color to red, then again to blue. Because all these changes occur at regular time intervals, we have a coherent process. This example demonstrates how the three ingredients for far-from-equilibrium behaviour work. First, the system is complex enough to possess an equilibrium steady state, and also a limit-cycle. Nicolis &

A.2 The general evolution criterion

279

Prigogine (1977, pp. 90–92) show that only trimolecular or higher system may have an unstable node or focus as a centre of a limit cycle. Slight fluctuations are provided by the random motions of the molecules in the mixture. However, because the equilibrium steady-state is unstable, rather than these fluctuations being dampened, they are amplified, until the system moves from the static equilibrium to a coherent, self-organised, behaviour.

A.2

The general evolution criterion

The restriction of the theorem of minimum entropy production to steady states in the immediate vicinity of thermodynamic equilibrium motivated the derivation of a theory for the nonlinear region (Nicolis & Prigogine 1977, pp. 37). Variously called the ‘general evolution criterion’ or ‘universal evolution criterion’, the theorem derived was dx P ≤ 0(= 0 at steady state) dt

(A.7)

where dx P is entropy production resulting from forces (as opposed to flows) in the system (Nicolis & Prigogine 1977, pp. 51). Unfortunately, away from local thermodynamic equilibrium, no assumptions can be made regarding the sign of the change in entropy production resulting from flows. Hence, no generalisation can be made about the sign of entropy production in its entirety, as is done in the minimum entropy production principle. Perhaps for this reason, I have not been able to find any references to application of the general evolution criterion in the ecology literature.

A.3

The assumptions behind the theorem of minimum entropy production

Reading through derivations of the minimum entropy production theorem ((Glandsdorff & Prigogine 1971, pp. 30–43),(Nicolis & Prigogine 1977, pp. 31–45)) and using commentaries as a guide (Kay 2002), I provide the following list of assumptions behind the theorem. 1. Local equilibrium thermodynamics (LET) LET implies that while the system is away from thermodynamic equilibrium, the behaviour of molecules on the micro-scale is a close enough approximation to thermodynamic equilibrium to allow simplifying assumptions to be made. This assumes: (a) Maxwell Boltzmann distribution of momenta and position of particles (Nicolis & Prigogine 1977, pp. 32) implies:

280


• Reactive collisions are sufficiently rare that the assumption of elastic collisions is not invalidated. • Macroscopic boundary gradients do not greatly perturb distribution. (b) The Onsager’s Reciprocity Relationship holds (Nicolis & Prigogine 1977, pp. 40). 2. Linearisation of phenomenological relations The phenomenological relations are linear relationships between flows and forces (e.g. Fick’s diffusion) (Glandsdorff & Prigogine 1971, p. 30). The linearisation of the relationship between flows (diffusion and reaction rates) and forces (chemical potentials and affinities) (Nicolis & Prigogine 1977, pp. 36, Eq 3.21–3.22), which implies: (a) A restatement of LET for transport forces and flows. (b) A strict restriction upon chemical forces and flows, equating to assuming either • the immediate vicinity of chemical equilibrium; or • reactions with extremely low activation energy. 3. Strict linearity The phenomenological coefficients are the derivatives of flows with respect to forces. Strict linearity is to assume that they are independent of time. 4. Boundary conditions imposed upon the system are time independent This is simply a consequence of the the assumption of steady state (Nicolis & Prigogine 1977, pp. 43).

A.4

The least specific dissipation principle

Least specific dissipation principle (LSD) is the name given by some authors (Fath et al. 2001, Choi et al. 1999) to the application of the minimum entropy production principle to ecological systems. The specific rate of entropy production is approximated by the respiration to biomass ratio. Choi et al. (1999) state: The LSD principle indicates that any system, in the face of a small but continued energy gradient, will tend toward a local minimum in the rate of energy dissipation per unit volume or mass (di s/dt ...). This ‘specific’ dissipation rate can be approximated by the ratio of respiration rate to biomass (or the ‘R/B’ ratio) when the mass of specific rate of entropy flow (de s/dt) is assumed to represent the dominant part of biological activity, relative to structural elaboration or anabolic processes ...

A.5 Lotka’s maximal flux hypothesis as a Prigoginean concept

A.5

281

Lotka’s maximal flux hypothesis as a Prigoginean concept

Choi et al. (1999) states: Lotka (1922) proposed that the macroscopic consequence of the microscopic struggle from free energy is to make the energy flux through the system a maximum, subject to the constraints of the system. The thermodynamic mechanism that describes this struggle for free energy is known as the ‘order through fluctuation’ (‘OTF’) principle proposed by Glansdorff and Prigogine (1971). The OTF principle is as follows: small statistical deviations in the flow of energy exist in any thermodynamic system. Such deviations are generally dampened out by dissipative processes. However, as the energy gradient increases, the amplification of these fluctuations becomes more and more probable. Any fluctuations that can better dissipate the energy gradient becomes positively selected and amplified over time. These positive feedback structures (e.g. autocatalytic cycles, food webs) are known as ‘dissipative structures’. A comparison between Lotka’s formulation and Prigogine’s later work (Prigogine & Stengers 1984, pp. 189–190) can be made: ... new constituents, introduced in small quantities, lead to a new set of reactions among the system’s components. This new set of reactions then enters into competition with the system’s previous mode of functioning. If the system is “structurally stable” as far as this intrusion is concerned, the new mode of functioning will be unable to establish itself and the “innovators” will not survive. If, however, the structural fluctuation successfully imposes itself – if, for example, the kinetics whereby the “innovators” multiply fast enough for the latter to invade the system instead of being destroyed – the whole system will adopt a new mode of functioning: its activity will be governed by a new “syntax”.

A.6

Patten’s evidence for linearity

Patten (1975) finds several pieces of evidence for linearity of ecosystems, which are discussed here: 1. Combining the statement that linear systems are regular and reliable, and the observation that ecosystems are also well-behaved, Patten (1975), whilst acknowledging other homoeostatic mechanisms, considers the behavioural regularity of ecosystems a product of their linearity. 2. Patten (1975) cites one study specifically, where precipitation was linearly related to cation output. 3. Combining the statement that nonlinear feedback “dithering” can be used to linearise a system, and the observation that ecosystems are rich in feedback, Patten (1975) proposes that nonlinear

282


feedback in ecosystems may be responsible for linearising them. 4. By assuming that the ecosystem is at or near steady state, which may be approximated by linearisation of nonlinear dynamics, the nonlinear ecosystem behaves in a linear fashion. Regarding Item 1, while a linear system may be well-behaved, a system does not have to be linear in order to be well-behaved (for example, permanence Chapter 11). Further, Patten (1975) own list of four homoeostatic mechanisms are not peculiar to linear systems, and two of them (density-dependent population regulation, and environmental control by feedback) explicitly require nonlinearity. As with Item 3, where a similar implication reversal is used, while the mechanism (linearity, nonlinear feedback respectively) will produce the observation, the observation does not imply the mechanism. Regarding Item 2, Dwyer & Perez (1983) provides a counterexample to linearity from the same study cited by Patten (1975). Dwyer & Perez (1983) note that “...the linear relationship does not hold for the export of particulate matter from the same watersheds, or for export of some of the same dissolved cations from watersheds subjected to a clearcutting perturbation”. Further, their own controlled study concluded nonlinearity in microcosms used. Regarding Item 3, I would first acknowledge that control theory is well outside of my realm of expertise. Nonetheless, it is interesting to note that the nonlinear feedback mechanism cited by Patten (1975) is a human operator (Graham & McRuer 1961, pp. 440), using human faculties such as internally generated criteria for linearity (Graham & McRuer 1961, pp. 442), and prediction (Graham & McRuer 1961, pp. 444). Further, Patten (1975) interprets “ecosystem linearization ... as an evolutionary self-design process wherein natural selection favors the reliability of linear dynamics”, for similar reasons that the human operator favours linear control systems (Graham & McRuer 1961, pp. 436). The ‘design’ element, however, is sometimes unnecessary, as suitable dither arises as consequence of the machinery in contactorcontrolled servomechanism, for example, “by backlash in the gears connecting the motor to the load” (Graham & McRuer 1961, pp. 448). Patten (1975) mentions dither inherent in ecosystems, such as stochastic variation and Brownian motion. Whether such dither will serendipitously suit linearising ecosystem behaviours becomes the question. Item 4 is discussed in the main body of the text. It is worth noting that Patten (1975) is motivated by a preconception of ecosystems as being well behaved. He states The bias herein is in favor of regularity and reliability as the rule and poor behaviour as the exception. The exception, then, becomes aligned with non-linear systems. Further, he interprets linearity as being beneficial to the ecosystem ... ecosystem linearization may be construed as an evolutionary self-design process wherein

A.7 The history of the exergy goal function

283

natural selection favors the reliability of linear dynamics and either eliminates the irregularities or nullifies the consequences of nonlinear behaviour. Accordingly, the ecosystem is seen as a holistic unit of coevolution ... wherein component-level nonlinear processes are constrained by the system organization to operate within linear dynamic ranges, or the output from such processes appears linear when integrated at whole-system levels. Such theories of system-level adaptation are discussed in the body of the text.

A.7

The history of the exergy goal function

Cropp & Gabric (2002) cite exergy as a justification for the maximisation of autotroph and heterotroph biomass. The purpose of this Appendix is to provide additional description of what exergy is, and summarise its historical development through the citation of key papers by its main proponent, Jørgensen. This information is summarised in the main body of the text. Jørgensen & Mejer first described the use of exergy in ecosystems in (1977). Jørgensen & Mejer (1977) were interested in relating complexity to stability, partly in response to May’s (1972) result. Jørgensen & Mejer (1977) began by noting that phosphorus concentration in an experimental lake (Lake Fure) rapidly increases after the ‘buffer capacity’ inherent in the sediment was exceeded. The buffer capacity was defined as β=

δL , δSV

(A.8)

where L is the loading, and SV is the state variable. Jørgensen & Mejer (1977) compared two measures of complexity to buffer capacity: Shannon’s index and exergy. Exergy was defined as E = RT

X

aj P˙j ln

j

Pj kJ/m3 Pjeq

(A.9)

where R is the gas constant, T is mean temperature, aj is the volume ratio of the compartment j to the lake, Pj is phosphorus in compartment j, and Pjeq is the concentration in compartment j expected at thermodynamic equilibrium. They stated that “exergy measures the mechanical energy equivalent of the distance from thermodynamic equilibrium”. The main result was that buffer capacity increased with Shannon index and exergy. One can see the beginning of ‘goal function’ thinking in this work. On the basis of the examination carried out it seems to be possible to select the state variables needed in a model built to answer certain questions. The selection should be based upon an examination of the actual state variables and their contribution to the exergy or buffer capacity.

284


Jørgensen & Mejer (1981) continued using exergy in similar terms. The meaning of exergy was also interpreted as measuring “the amount of information incorporated in the structure”. The formulation of exergy was changed to E = RT

X

aj

Pj ln

j

Pj Pjeq

!

− (Pj −

Pjeq )

kJ/m3

!

(A.10)

By 1986, Jørgensen had begun to think about exergy as a goal function in explicit terms. Using the term ‘control function’, exergy is described as a way for an ecosystem to meet environmental perturbations that were beyond the range of physiological homoeostatic mechanisms. If, for instance, the temperature changes, the species might also change their optimum temperature accordingly ... If the external factors are changed more radically, the ecosystem might even change its structure and species composition ... This particular work is interesting, in that it explicitly describes the exergy goal function in terms of Prigoginean concepts. The quotation continues ... the ecosystem might even change its structure and species composition to maintain the basic function of the ecosystem: growth of biological components on various trophic levels to maintain a state far from thermodynamic equilibrium (Jørgensen 1986a). Svirezhev (2000) similarly identifies exergy as a measure of the systems distance from thermodynamic equilibrium. Unfortunately, while dissipative structures exist away from thermodynamic equilibrium, there is no reason to suggest that they maximise this distance. As such, this does not explain its use as a goal function, or the mechanism by which exergy is maximised. During this time in exergy’s development, it is difficult for a reader to discern exactly why the maximisation occurs. On one hand, the above quotation implies, somewhat tautologically, that the ecosystem is impelled to stay as far from equilibrium as possible, where “the exergy measure is seen as the distance from thermodynamic equilibrium in energy terms.” This thermodynamic interpretation is emphasised by the reformulation of exergy in terms of information, and distance from thermodynamic equilibrium. Jørgensen (1986b) describe exergy as E = T I = T (Seq − S)

(A.11)

where T is temperature, I is thermodynamic information, Seq is the entropy of the considered system at thermodynamic equilibrium, and S the entropy of the system. However, the quotation below, taken from the same work, implies a species replacement process akin to Dunbar’s (1960) ecosystem adaptation as the mechanism, rather than appealing to Prigoginean concepts. A certain species composition prevails under one combination of external factors (forcing functions), but if the external factors are changed, a new combination of species will prevail.

A.7 The history of the exergy goal function

285

A selection takes place in the genetic pool and the species that are better fit to grow (produce biomass) under the external factors now dominate the system. Growth means increased biomass and increased organisation, which can be measured by use of exergy. Further, Jørgensen (1986b) state that What is really changed in the system is a shift from one species composition to another. This concept of species selection in accordance with system-level fitness objectives was present in much earlier work, albeit in a less clarified form. For example, Jørgensen & Mejer (1979) describe exergy as a Darwinian theory applied to the level of ecosystems. Jørgensen & Mejer (1979) state that A species which can adapt itself to changed environment will survive – this is a well accepted theory (compare with Darwin) – and, in accordance with the theory presented here, the same is equally valid for the ecosystem. The ecosystem structure which can adapt itself to changed circumstances will survive. They describe the mechanism as a species selection, using an ecosystem’s response to changes in temperature as an example, however it is not yet clear whether species are responding directly to their own fitness with respect to perturbations perturbations, or are responding to perturbations filtered through the system-level response If we change the temperature pattern some species will be eliminated, others better fitted to the new temperature pattern will be more dominant. It is even possible to demonstrate by use of models, that species able to change to the optimum temperature in the right direction will win ... In the later work, in becomes clear that it is the latter: that the deleterious effect of perturbations on the system-level is passed down to the species. For example, Jørgensen et al. (1995) state The core in the tentative approach can be considered an extension of Darwin’s theory ... to the level of the ecosystem and to the application in modelling. Darwin’s theory is based on the level of organisms and species: those with the properties best fitted to prevailing conditions, including the conditions determined by the presence of other species, will survive. If, however, we have to account for the simultaneous struggle of survival by all the species present in an ecosystem, we have to consider the level of the ecosystem. While energetic interpretations have persisted (Marques et al. 1997), similar statements may be found in Jørgensen (1999), and Jørgensen (2000). In (1995), Jørgensen et al. (1995) modify exergy to “account for the higher organisation of higher organisms”. This modification, to later become known as ‘specific exergy’, is best described by an example.

286


Consider a prokaryotic cell. Jørgensen et al. (1995) calculate exergy for the cell as follows. Ep /RT = 7.34 × 105 ci + ci ln 20329000 ,

(A.12)

where 7.34 × 105 is the exergy of detritus per litre of water, ci is the concentration in the ecosystem

of compartment i, and 20329000 represents the number of possible states the genetic code could have. In essence, Jørgensen et al. (1995) derive a way to give higher organisms, those with more genetic information, a higher exergy contribution. Exergy is generalised as E=T

X j

βi × c i

(A.13)

where βi compares quantity of information in the biomass to that of detritus, and ci is the concentration of compartment i. The reasoning behind this weighting of higher organisms is described in Jørgensen (1999) We know, that each species tries to get the best possible survival and fastest growth. Survival could be measured by the biomass, but an optimisation of the biomass of several species at the same time, requires that we sum up the biomass or perhaps make an addition of the weighted biomass. An addition of the biomass seems not to be an appropriate idea ... The possibility for a fish to find a new pathway for survival under a new and emergent circumstances [sic] is furthermore much better than for phytoplankton due to the more advanced properties of a fish. A fish can move to the corner of the ecosystem where the food resources are most abundant, it can see, smell and hear in which direction it is most beneficial to move. A fish carries more information in its genes, an information which is used to obtain a better survival. The reasoning, however, is clouded by the method in which it is applied. The example used by Jørgensen (1999) has two free parameters, maximal growth rate of phytoplankton, and settling rate of phytoplankton. The question then becomes, how or why should phytoplankton modify their attributes to suit the survival (measured as biomass) of other compartments such as fish? The justification is based upon heuristics, and wholism. The first justification is relatively easy to describe. Jørgensen (1999) sees parallels between the increasing complexity in the fossil record, and the changes that should take place in ecological time. Further, he observes that fixed parameters models do not adequately describe changes in the ecosystem. The rigid parameters of the various species make it difficult for the species to survive under changing circumstances. After some time only a few species will still be present in the model, opposite of what is the case in reality, where more species survive because they are able to adapt to the changing circumstances.

A.8 An example ascendency calculation

287

However, such heuristics do not explain the apparent altruism required of compartments in the models. The second assumption is that each compartment of the system will not only maximise its own biomass but that of other compartments. The mechanism is not stated explicitly, however, Jørgensen (1999) does refer to coevolution in the introductory sections Coevolution means that the evolution process cannot be described as reductionistic, but that the entire system is evolving. A holistic description of the evolution of the system is needed. and extends this to say ... the species composition and their adaptation processes have adapted to the new situation. Organisms develop and coevolve in the fitness landscape. When we analyse the ecosystem, we can presume that the organisms and their network have found if not the optimum solution then at least an excellent solution of combination of properties to obtain the highest possible survival and growth for all the organisms. This appears to be the main weakness in the exergy formulation. Even if one were to accept biomass and organisation as proxies for survival, there is no reason to suggest that one compartment will behave to promote the survival of another. That higher organisms will be able to exploit lower organisms into altruistic behaviour does not necessarily follow from the fact that they have higher information. Marrow et al. (1992) demonstrate the opposite in an adaptive dynamics scheme, with the ‘loser wins’ principle. Buffer capacity is rarely mentioned in later works, however a link between exergy and stability in general remains. For example, Jørgensen (2000) states Exergy measures the distance from thermodynamic equilibrium ... exergy measures the amount of energy needed to break down the ecosystem. Exergy is therefore a reasonably good measure of ... (1) absence of disease; (2) stability or resilience; and (3) vigour and scope for growth.

A.8

An example ascendency calculation

The following example of an ascendency calculation is taken from Ulanowicz (1980). I have clarified the example by elaborating upon the intermediate steps taken. Figure A.2 is reproduced from Ulanowicz’s (1980) Figure 1. The flows shown are flows of carbon in Lake Findley. All flows are in g Carbon m−2 year−1 . Four steps of the ascendency calculation are shown in Table A.1. Ulanowicz (1980) quantifies ascendency

288


0.3 ZOO 7.9 2.2

2.1 1.1

POC

16.3

BENTH 0.1 9.4 5.7

0.5 11.7 SED 5.0

3.7

4.8

Figure A.2: Figure 1 of Ulanowicz (1980), reproduced to provide an example of how ascendency is calculated.

as: A=T

XX

fkj Qk ln fkj /

j

k

X i

fij Qi

!!

,

(A.14)

where Qi = T i /

n X

Tj ,

(A.15)

j=1

and T =

n X

Tj ,

(A.16)

j=1

where fkj is the nutrient flow from compartment k contributing to compartment j, Ti is the throughput of compartment i, and T may be thought of as the total throughput.

289

A.8 An example ascendency calculation

Table A.1: A summary of the ascendency calculation in Ulanowicz (1980). Step 1: Find throughput and fraction of Step 3: Multiply each fij by corresponding throughput for each compartment. Qi and take sums. fij Qi for each j i Ti Qi Qi (ln(Qi )) Z POC SED B 0.005917 0 0.001972 0 Z 2.5 0.04931 -0.1484 POC 16.3 0.321499 -0.36482 0.043393 0 0.112426 0 SED 20.2 0.398422 -0.36665 0 0 0 0.230769 B 11.7 0.230769 -0.33839 0 0 0.185404 0 T =

50.7

C=

61.76575

Step 2: Find flow from i to j, and divide by total throughput of i. fij Z 0.12 0.134969 0 0

POC 0 0 0 0

SED 0.04 0.349693 0 0.803419

B 0 0 0.579208 0

P

i

fij Qi =

0.04931

0

0.299803

0.230769

Step 4: Substitute into the rest of the equation. Calculate ascendency.

j= Z POC SED B

P ln(fkj / i fij Qi ) Z POC SED 0.005263 0 -0.00397 0.043693 0 0.017306 0 0 0 0 0 0.182763

P P P ln(fkj / fij Qi ) = P kP j P A = T k j (ln(fkj / fij Qi )) =

B 0 0 0.212364 0 0.457415 23.19095

290

A.9


Response of P/B to perturbation

Figure A.3 shows the time-averaged response of P/B to a 99% reduction in the biomass for the CG Model. The time average is found by Z T P Time averaged P/B = dt 0 B

(A.17)

Figure A.3 shows both the maximally resilient and minimally resilient CG Models. Cropp & Gabric (2002) stated that the maximally resilient system “maximizes P/B over evolutionary time, but minimizes P/B over shorter time scales”. However Figure A.3 shows that both the maximally and minimally resilient systems behave in this way. P/B as system returns to steady state after a perturbation 0.8 Min R Max R

Time averaged production to biomass ratio

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

20

40

60

80 Time

100

120

140

160

Figure A.3: Time averaged production to biomass ratio in response to a reduction in the biomass. Both the maximum and minimum resilience models are shown.

Appendix B

Appendices relating to Chapter 3 B.1

Lyapunov function example

The following example is taken from Borrelli & Coleman (1996, pp. 415). We show that x˙ 1 = −x1

(B.1)

x˙ 2 = −2x2

(B.2)

is Lyapunov stable. Let the Lyapunov function be the square of the distance to the steady state (0, 0) V (x) = x21 + x22 ,

(B.3)

which satisfies V (x⋆ ) = 0, and V (x) > 0 ∀x 6= x⋆ . The rate of change of V is 2

X dV dxi dV = , dt dxi dt i=1

(B.4)

= −2x21 − 4x22 ,

(B.5)

which is always negative.

B.2

MacArthur stability

MacArthur (1955) defined stability follows Suppose, for some reason, that one species has an abnormal abundance. Then we shall say the community is unstable if the other species change markedly in abundance as a result 291

292


of the first. The less effect this abnormal abundance has on the other species, the more stable the community. MacArthur (1955) then reasoned about the attributes that would promote stability The amount of choice which the energy has in following the paths up through the food web is a measure of the stability of the community (Odum, 1953). To see this, consider first a community in which one species is abnormally common. For this to have a small effect upon the rest of the community there should be a large number of predators among which to distribute the excess energy, and there should be a large number of prey species of the given species in order that none should be reduced too much in population. MacArthur (1955) described this link between stability and energy pathways with an equation S=−

X

pi log pi

(B.6)

where pi is the product of indices qi from bottom to top of the food web for each path, and the qi the inverse of the number of prey for a given predator. This was consistent with the stability observed in complex tropical systems (Goodman 1975), and was generally supported by field studies (more recent studies include: Naeem & Li 1997, Tilman 1996). Several authors have noted that MacArthur’s (1955) definition of stability is very closely related to diversity (Hairston et al. 1968, Pimm 1979), and therefore it is somewhat tautological to explore the relationship between MacArthur stability and diversity.

B.3

An error in the work of Yodzis

It appears that Yodzis (1981) misunderstood Pimm & Lawton’s (1978) description of the community matrix, and this misunderstanding may have implications for the relative importance of the dynamical form chosen. Yodzis (1981) states that “element Ai,j of the community matrix gives the per capita effect of species j on species i in the neighbourhood of the presumed equilibrium”, whereas his source (Pimm & Lawton 1978) correctly states that the per capita affect of species j on i is the ai,j coefficient in the generalised LotkaVolterra equation. The elements of the community matrix, however, are “the products of the per capita effect of the predator on the prey multiplied by the prey’s equilibrium density” (Pimm & Lawton 1978), that is Ai,j = ai,j Xi⋆ . The fact that Yodzis’s (1981) relative ordering of the community matrix element magnitudes with respect to expected real interaction strength remained effective implies that the interaction strength has a stronger

B.3 An error in the work of Yodzis

293

determining effect upon stability than the equilibrium populations. Further, while Yodzis (1981) states that the results are completely independent of assumptions about global dynamics, logic suggests that weighting interaction strength more strongly than equilibrium population in community matrix entries assumes lower order dynamical systems (e.g. Lotka-Volterra) with corresponding smaller powers of X ⋆ in Ai,j .

Appendix C

Appendices relating to Chapter 5 C.1

Differential equations governing models and goal functions

C.1.1

Model LL

Differential Equations: dP = eP N P − eZ P Z − dP P, dt

(C.1a)

dZ = eZ (1 − ηZ )P Z − dZ Z, dt dN = dZ Z + dP P + eZ ηZ P Z − eP N P. dt

(C.1b) (C.1c)

Goal Functions: Phytoplankton biomass at equilibrium: Peq =

dZ . eZ (1 − ηZ )

(C.2a)

Zooplankton biomass at equilibrium: Zeq =

eP Neq − dP , eZ

(C.2b)

where nutrient at equilibrium is Neq =

dZ (No − Peq ) + eZ ηZ Peq (No − Peq ) + dP Peq . dZ + eZ ηZ Peq + eP Peq

Flux at equilibrium for the DeAngelis (1992) flux formulation: Feq = (1 − ηZ )eZ Peq Zeq = dZ Zeq = eP Neq Peq − ηZ eZ Peq Zeq − dP Peq . 294

(C.2c)

C.1 Differential equations governing models and goal functions

295

Flux at equilibrium for the Cropp & Gabric (2002) flux formulation: Feq = eZ Peq Zeq = dZ Zeq + ηZ eZ Peq Zeq = eP Neq Peq − dP Peq . Flux to biomass ratio at equilibrium: Feq F = . B eq Peq + Zeq

(C.2d)

(C.2e)

Resilience at equilibrium is found from (refer to Appendix: Finding resilience for details): A = eP (Neq − Peq ) − eZ Zeq − dP ,

(C.2f)

B = −Peq (eP + eZ ),

(C.2g)

C = eZ (1 − ηZ )Zeq ,

(C.2h)

D = 0.

(C.2i)

C.1.2

Model LH

Differential Equations: µZ P Z dP = eP N P − − dP P, dt kZ + P dZ µZ (1 − ηZ )P Z = − dZ Z, dt kZ + P dN µZ ηZ P Z = dZ Z + dP P + − eP N P. dt kZ + P

(C.3a) (C.3b) (C.3c)


dZ kZ . µZ (1 − ηZ ) − dZ

(C.4a)

Zooplankton biomass at equilibrium: Zeq =

(eP Neq − dP )(kZ + Peq ) , µZ

(C.4b)

where nutrient at equilibrium is Neq =

dZ (kZ + Peq )(No − Peq ) + µZ ηZ Peq (No − Peq ) + dP Peq (kZ + Peq ) . dZ (kZ + Peq ) + µZ ηZ Peq + eP Peq (kZ + Peq )

Flux at equilibrium for the DeAngelis (1992) flux formulation: Feq =

µZ ηZ Peq Zeq µZ (1 − ηZ )Peq Zeq = dZ Zeq = eP Neq Peq − − dP Peq . kZ + Peq kZ + Peq

(C.4c)

296


Flux at equilibrium for the Cropp & Gabric (2002) flux formulation: Feq =

µZ Peq Zeq µZ ηZ Peq Zeq = dZ Zeq + = eP Neq Peq − dP Peq kZ + Peq kZ + Peq

Flux to biomass ratio at equilibrium: Feq F = . B eq Peq + Zeq

(C.4d)

(C.4e)

Resilience at equilibrium is found from (refer to Appendix: Finding resilience for details): µZ kZ Zeq − dP , kZ + Peq µZ , B = −Peq eP + kZ + Peq A = eP (Neq − Peq ) −

C=

µZ kZ (1 − ηZ )Zeq , (kZ + Peq )2

D = 0.

C.1.3

(C.4f) (C.4g) (C.4h) (C.4i)

Model HL

Differential Equations: µP N P dP = − eZ P Z − dP P, dt kP + N dZ = eZ (1 − ηZ )P Z − dZ Z, dt µP N P dN = dZ Z + dP P + eZ ηZ P Z − . dt kP + N

(C.5a) (C.5b) (C.5c)


dZ . eZ (1 − ηZ )

Zooplankton biomass at equilibrium: 1 µP Neq Zeq = − dP , eZ kP + Neq

(C.6a)

(C.6b)

where nutrient at equilibrium is √ −b ± b2 − 4ac , Neq = 2a where: a = eZ ,

(C.6c)

C.1 Differential equations governing models and goal functions

297

b = −µP + dP + eZ (No − Peq − kP ),

(C.6d)

c = kP eZ (No − Peq ) + dP kP .

(C.6e)

Flux at equilibrium for the DeAngelis (1992) flux formulation: Feq = eZ (1 − ηZ )Peq Zeq = dZ Zeq =

µP Neq Peq − eZ ηZ Peq Zeq − dP Peq . kP + Neq

(C.6f)

Flux at equilibrium for the Cropp & Gabric (2002) flux formulation: Feq = eZ Peq Zeq = eZ ηZ Peq Zeq + dZ Zeq =

µP Neq Peq − dP Peq . kP + Neq

Flux to biomass ratio at equilibrium: Feq F = . B eq Peq + Zeq

(C.6g)

(C.6h)

Resilience at equilibrium is found from (refer to Appendix: Finding resilience for details): A=

µP (Neq − Peq )(kP + Neq ) + µP Peq Neq − eZ Zeq − dP , (kP + Neq )2

(C.6i)

B=

−µP Peq kP − eZ Peq , (kP + Neq )2

(C.6j)

C = eZ (1 − ηz )Zeq ,

(C.6k)

D = 0.

(C.6l)

C.1.4

Model HH

Differential Equations: dP P N − µZ Z , = µP P dt N + kP P + kZ (1 − ηZ )P dZ − dZ Z, = µZ Z dt kZ + P dN N ηZ P − µP P . = dZ Z + µZ Z dt kZ + P N + kP

(C.7a) (C.7b) (C.7c)


dZ kZ . (1 − ηZ )µZ − dZ

Zooplankton biomass at equilibrium: µP N kZ + P − dP , Zeq = µZ kP + N

(C.8a)

(C.8b)

298


where nutrient at equilibrium is √ −b ± b2 − 4ac , Neq = 2a

(C.8c)

where: a = −µZ ,

(C.8d)

b = µP (kZ + Peq ) + µZ (kP − No + Peq ) − dP (kZ + P ),

(C.8e)

c = −µZ kP (No − Peq ) − dP (kZ + P ).

(C.8f)

Flux at equilibrium for the DeAngelis (1992) flux formulation: (1 − ηZ )Peq Neq ηZ Peq Feq = µZ Zeq = µP Peq − µZ Zeq = dZ Zeq . kZ + Peq kP + Neq kZ + Peq Flux at equilibrium for the Cropp & Gabric (2002) flux formulation: Peq Neq ηZ Peq Feq = µZ Zeq = µP Peq = dZ Zeq + µZ Zeq . kZ + Peq kP + Neq kZ + Peq Flux to biomass ratio at equilibrium: Feq F = . B eq Peq + Zeq

(C.8g)

(C.8h)

(C.8i)

Resilience at equilibrium is found from (refer to C.2 for details): A=

µP (Neq − Peq )(kP + Neq ) + µP Peq Neq µZ Zeq kZ − − dP , 2 (kP + Neq ) (kZ + Peq )2

(C.8j)

B=

−µP Peq kP µZ Peq − , (kP + Neq )2 kZ + Peq

(C.8k)

C=

µZ Zeq kZ (1 − ηZ ) , (kZ + Peq )2

(C.8l)

D = 0.

C.2 C.2.1

(C.8m)

Finding resilience General Jacobian Matrix for CG Model variations

The Jacobian Matrix is used to find the resilience of the system. The Jacobian Matrix is described by:   ∂ dP   dP ∂ A B . J =  ∂P dt ∂Z dt  =  (C.9) dZ ∂ dZ ∂ C D ∂P dt ∂Z dt

C.2 Finding resilience

From Equation C.9 it can be said that the eigenvalue, λ, is found by ∂ dP ∂ dP ∂P dt − λ ∂Z dt . |J − λI| = 0 = dZ ∂ dZ ∂ − λ ∂P dt ∂Z dt

299

(C.10)

The quadratic equation to be solved from Equation C.10 is 0 = λ2 − (A + D)λ + (AD − BC),

(C.11)

for equilibrium values of P and Z. Note that for complex eigenvalues Equation C.12 must be satisfied (A + D)2 − 4(AD − BC) < 0,

(C.12)

for which the resilience, R, is the negative of the real part of the eigenvalues R=−

A+D . 2

(C.13)

For real eigenvalues, the larger (most positive) of the two eigenvalues found from the characteristic equation (Equation C.10) is used to find resilience.

eZ

Max Peq Max Zeq Max Feq Max (F/B)eq Max Req Min Peq Min Zeq Min Feq Min (F/B)eq Min Req

0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003

0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003

0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009

Zeq

Goal Function Values Feq (F/B)eq Req

Req R?

316.7 368.8 356.3 245.8 356.3 198.9 122.9 124.3 245.8 124.3

23.8 9.2 26.7 18.4 26.7 5.0 9.2 3.1 6.1 3.1

0.070 0.024 0.070 0.073 0.070 0.025 0.070 0.024 0.024 0.024

0.0750 0.0375 0.1125 0.0375 0.1125 0.0083 0.0125 0.0042 0.0125 0.0042

No No No No No No No No No No

316.1 368.5 356.0 245.7 355.8 198.6 122.3 123.7 244.6 123.9

23.7 9.2 26.7 18.4 26.7 5.0 9.2 3.1 6.1 3.1

0.070 0.024 0.070 0.073 0.070 0.025 0.070 0.024 0.024 0.024

0.0750 0.0375 0.1125 0.0375 0.1125 0.0083 0.0125 0.0042 0.0125 0.0042



eP

300

Goal

Table C.1: Model LL1 and LL2 Parameter Values µP µZ kP kZ dP dZ ηZ Peq Model LL1 0.075 25.0 0.025 8.3 0.075 25.0 0.075 8.3 0.075 25.0 0.025 2.8 0.075 8.3 0.025 2.8 0.025 8.3 0.025 2.8 Model LL2 0.0050 0.075 25.0 0.0025 0.025 8.3 0.0025 0.075 25.0 0.0025 0.075 8.3 0.0050 0.075 25.0 0.005 0.025 2.8 0.0075 0.075 8.3 0.0075 0.025 2.8 0.0075 0.025 8.3 0.005 0.025 2.8

eP

eZ


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003

0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003

0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009

LL4


Peq

Zeq

Req R?

62.5 10.4 31.3 10.4 62.5 3.5 20.8 6.9 20.8 3.5

291.7 367.2 351.6 244.8 328.1 198.6 119.8 123.3 239.6 124.1

21.9 9.2 26.4 18.4 24.6 5.0 9.0 3.1 6.0 3.1

0.062 0.024 0.069 0.072 0.063 0.025 0.064 0.024 0.023 0.024

0.1875 0.0469 0.1406 0.0469 0.2813 0.0104 0.0313 0.0104 0.0313 0.0052


62.5 10.4 62.5 20.8 62.5 3.5 20.8 3.5 10.4 3.5

291.7 367.2 328.1 239.6 328.1 198.6 119.8 124.1 244.8 124.1

54.7 11.5 61.5 44.9 61.5 6.2 22.5 3.9 7.6 3.9

0.154 0.030 0.158 0.173 0.158 0.031 0.160 0.030 0.030 0.030

0.1875 0.0469 0.2813 0.0938 0.2813 0.0104 0.0313 0.0052 0.0156 0.0052



Goal

Table C.2: Model LL3 and Parameter Values µP µZ kP kZ dP dZ ηZ Model LL3 0.075 0.6 0.025 0.2 0.075 0.2 0.075 0.2 0.075 0.6 0.025 0.2 0.075 0.6 0.025 0.6 0.025 0.6 0.025 0.2 Model LL4 0.075 0.6 0.025 0.2 0.075 0.6 0.075 0.6 0.075 0.6 0.025 0.2 0.075 0.6 0.025 0.2 0.025 0.2 0.025 0.2

301

eZ


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003

0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003

0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009

and LL6


ηZ

Peq

Zeq

Req R?

0.6 0.2 0.2 0.2 0.6 0.2 0.6 0.6 0.6 0.2

62.5 10.4 31.3 10.4 62.5 3.5 20.8 6.9 20.8 3.5

291.1 367.0 351.4 244.7 327.7 198.3 119.2 122.6 238.3 123.7

21.8 9.2 26.4 18.3 24.6 5.0 8.9 3.1 6.0 3.1

0.062 0.024 0.069 0.072 0.063 0.025 0.064 0.024 0.023 0.024

0.1875 0.0469 0.1406 0.0469 0.2813 0.0104 0.0313 0.0104 0.0313 0.0052


0.6 0.2 0.6 0.6 0.6 0.2 0.6 0.2 0.2 0.2

62.5 10.4 62.5 20.8 62.5 3.5 20.8 3.5 10.4 3.5

291.1 367.0 327.9 239.4 327.7 198.3 119.2 123.5 243.5 123.7

54.6 11.5 61.5 44.9 61.4 6.2 22.3 3.9 7.6 3.9

0.154 0.030 0.157 0.172 0.157 0.031 0.160 0.030 0.030 0.030

0.1875 0.0469 0.2813 0.0938 0.2813 0.0104 0.0313 0.0052 0.0156 0.0052



eP

302

Goal

Table C.3: Model LL5 Parameter Values µP µZ kP kZ dP dZ Model LL5 0.005 0.075 0.0025 0.025 0.0025 0.075 0.0025 0.075 0.005 0.075 0.005 0.025 0.0075 0.075 0.0075 0.025 0.0075 0.025 0.005 0.025 Model LL6 0.005 0.075 0.0025 0.025 0.0025 0.075 0.0025 0.075 0.005 0.075 0.005 0.025 0.0075 0.075 0.0075 0.025 0.0075 0.025 0.005 0.025

eP


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003

eZ

µP

Goal Function Values Zeq Feq (F/B)eq Req

Req R?

317.0 430.8 386.5 325.5 360.6 283.2 195.2 198.4 385.9 198.4

23.8 10.8 29.0 24.4 27.0 7.1 14.6 5.0 9.6 5.0

0.050 0.023 0.065 0.071 0.057 0.025 0.069 0.024 0.022 0.024

0.1968 0.1530 0.2517 0.0656 0.5026 0.0132 0.0182 0.0059 0.0747 0.0059

Yes No No No Yes No No No No No

316.2 430.4 386.2 323.5 260.4 282.7 194.2 197.4 383.8 198.1

23.7 10.8 29.0 24.3 27.0 7.1 14.6 4.9 9.6 5.0

0.050 0.023 0.065 0.071 0.057 0.025 0.069 0.024 0.022 0.024

0.1962 0.1501 0.2493 0.0643 0.4800 0.0132 0.0183 0.0059 0.0747 0.0059



Goal

Table C.4: Model LH1 and LH2 Parameter Values µZ kP kZ dP dZ ηZ Peq Model LH1 1.5 3000 0.075 157.9 1.5 2040 0.025 34.6 3.3 2460 0.075 57.2 4.4238 1000 0.075 17.2 1.78 2558.2 0.075 112.8 4.5 1000 0.025 5.6 4.5 1000 0.075 16.9 4.5 1000 0.025 5.6 1.5 3000 0.025 50.8 4.5 1000 0.025 5.6 Model LH2 1.5 3000 0.005 0.075 157.9 1.5 2002.9 0.0042 0.025 33.9 4.0601 3000 0.0028 0.075 56.5 4.5 1000 0.005 0.075 16.9 1.93 2792.8 0.005 0.075 112.36 4.5 1000 0.005 0.025 5.6 4.5 1000 0.0075 0.075 16.9 4.5 1000 0.0075 0.025 5.6 1.5 3000 0.0075 0.025 50.8 4.5 1000 0.0025 0.025 5.6

303


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003

eZ

µP

Zeq


Req R?

66.6 423.3 374.9 321.4 339.7 282.6 62.3 62.3 62.3 198.0

5.0 10.6 28.1 24.1 25.5 7.1 4.7 4.7 4.7 5.0

0.010 0.023 0.065 0.070 0.056 0.024 0.010 0.010 0.010 0.024

0.0110 0.1706 0.2660 0.0810 0.4883 0.0166 0.0111 0.0111 0.0111 0.0074

Yes No No No Yes No Yes Yes Yes No

66.6 423.3 333.3 308.6 339.7 282.6 62.3 198.0 62.3 198.0

5.0 13.2 62.5 57.9 25.5 8.8 11.7 6.2 11.7 6.2

0.010 0.029 0.149 0.164 0.056 0.030 0.024 0.030 0.024 0.030

0.0110 0.1706 0.3615 0.1679 0.4883 0.0166 0.0111 0.0074 0.0111 0.0074

Yes No No No Yes No Yes No Yes No


eP

304

Goal

Table C.5: Model LH3 and LH4 Parameter Values µZ kP kZ dP dZ ηZ Peq Model LH3 1.5 3000 0.075 0.6 428.6 1.5 1819.1593 0.025 0.2 38.7 4.5 2841.6422 0.075 0.2 60.5 4.5 1000 0.075 0.2 21.3 2.8196 2137.8299 0.075 0.5 113.1 4.5 1000 0.025 0.2 7.0 1.5 3000 0.075 0.6 428.6 1.5 3000 0.075 0.6 428.6 1.5 3000 0.075 0.6 428.6 4.5 1000 0.025 0.2 7.0 Model LH4 1.5 3000 0.075 0.6 428.6 1.5 1819.1593 0.025 0.2 38.7 3.4765 1486.8035 0.075 0.6 84.8 4.5 1000 0.075 0.6 43.5 2.8196 2137.8299 0.075 0.5 113.1 4.5 1000 0.025 0.2 7.0 1.5 3000 0.075 0.6 428.6 4.5 1000 0.025 0.2 7.0 1.5 3000 0.075 0.6 428.6 4.5 1000 0.025 0.2 7.0

eP


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003


0.006 0.009 0.009 0.009 0.009 0.006 0.003 0.003 0.003 0.003

eZ

µP

Peq


Req R?

428.6 39.2 63.4 21.3 113.3 7.0 428.6 428.6 428.6 7.0

65.8 423.0 374.7 321.2 331.2 282.1 60.2 60.2 60.2 197.7

4.9 10.6 28.1 24.1 24.8 7.1 4.5 4.5 4.5 4.9

0.010 0.023 0.064 0.070 0.056 0.024 0.009 0.009 0.009 0.024

0.0109 0.1730 0.2790 0.0810 0.4987 0.0166 0.0107 0.0107 0.0107 0.0074

Yes No No No Yes No Yes Yes Yes No

428.6 39.2 87.0 43.5 113.3 7.0 428.6 7.0 428.6 7.0

65.8 423.0 333.0 308.4 331.2 282.1 60.2 197.0 60.2 197.7

12.3 13.2 62.4 57.8 24.8 8.8 11.3 6.2 11.3 6.2

0.025 0.029 0.149 0.164 0.056 0.030 0.023 0.030 0.023 0.030

2.1855 0.1730 0.3763 0.1679 0.4987 0.0166 0.0107 0.0074 0.0107 0.0074



Goal

Table C.6: Model LH5 and LH6 Parameter Values µZ kP kZ dP dZ ηZ Model LH5 1.5 3000 0.005 0.075 0.6 1.544 1897.3607 0.0025 0.025 0.2 3.9751 2626.5885 0.0025 0.075 0.2 4.5 1000 0.0025 0.075 0.2 4.0337 2589.4428 0.005 0.075 0.56 4.5 1000 0.005 0.025 0.2 1.5 3000 0.0075 0.075 0.6 1.5 3000 0.0075 0.075 0.6 1.5 3000 0.0075 0.075 0.6 4.5 1000 0.0025 0.025 0.2 Model LH6 1.5 3000 0.005 0.075 0.6 1.544 1897.3607 0.0025 0.025 0.2 4.5 2000 0.0025 0.075 0.6 4.5 1000 0.0025 0.075 0.6 4.0337 2589.4428 0.005 0.075 0.56 4.5 1000 0.005 0.025 0.2 1.5 3000 0.0075 0.075 0.6 4.5 1000 0.0075 0.025 0.2 1.5 3000 0.0075 0.075 0.6 4.5 1000 0.0025 0.025 0.2

305

eP

µP


0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009

3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5


0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009

3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5

Zeq


Req R?

335.3 435.1 421.0 294.0 421.0 178.4 82.7 83.2 204.4 121.8

25.1 10.9 31.6 22.1 31.6 4.5 6.2 2.1 5.1 3.0

0.070 0.025 0.071 0.073 0.071 0.025 0.068 0.024 0.024 0.024

0.0598 0.0682 0.2102 0.0230 0.2102 0.0033 0.0038 0.0013 0.0053 0.0011


334.6 435.0 420.8 293.8 420.8 177.9 81.9 82.4 202.6 121.0

25.1 10.9 31.6 22.0 31.6 4.4 6.1 2.1 5.1 3.0

0.070 0.025 0.071 0.073 0.071 0.025 0.068 0.024 0.024 0.024

0.0596 0.0681 0.2099 0.0230 0.2099 0.0032 0.0038 0.0013 0.0052 0.0011



eZ

306

Goal

Table C.7: Model HL1 and HL2 Parameter Values µZ kP kZ dP dZ ηZ Peq Model HL1 277 0.075 25.0 138.5 0.025 8.3 138.5 0.075 25.0 138.5 0.075 8.3 138.5 0.075 25.0 277 0.025 2.8 415.5 0.075 8.3 415.5 0.025 2.8 415.5 0.025 8.3 138.5 0.025 2.8 Model HL2 277 0.005 0.075 25.0 138.5 0.0025 0.025 8.3 138.5 0.0025 0.075 25.0 138.5 0.0025 0.075 8.3 138.5 0.0025 0.075 25.0 277 0.005 0.025 2.8 415.5 0.0075 0.075 8.3 415.5 0.0075 0.025 2.8 415.5 0.0075 0.025 8.3 138.5 0.0075 0.025 2.8

eP

eZ


0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009


0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009

HL4


Peq

Zeq

Req R?

62.5 10.4 31.3 10.4 62.5 3.5 20.8 6.9 20.8 3.5

311.9 433.3 415.7 293.2 389.0 178.2 81.5 82.8 200.7 121.7

23.4 10.8 31.2 22.0 29.2 4.5 6.1 2.1 5.0 3.0

0.062 0.024 0.070 0.072 0.065 0.025 0.060 0.023 0.023 0.024

0.1602 0.0856 0.2653 0.0289 0.5570 0.0041 0.0098 0.0032 0.0135 0.0014


62.5 10.4 62.5 20.8 62.5 3.5 20.8 3.5 10.4 3.5

311.9 433.3 389.0 289.2 389.0 178.2 81.5 83.1 203.8 121.7

58.5 13.5 72.9 54.2 72.9 5.6 15.3 2.6 6.4 3.8

0.156 0.031 0.162 0.175 0.162 0.031 0.149 0.030 0.030 0.030

0.1602 0.0856 0.5570 0.0602 0.5570 0.0041 0.0098 0.0016 0.0066 0.0014



Goal

Table C.8: Model HL3 and Parameter Values µP µZ kP kZ dP dZ ηZ Model HL3 3 277 0.075 0.6 4.5 138.5 0.025 0.2 4.5 138.5 0.075 0.2 4.5 138.5 0.075 0.2 4.5 138.5 0.075 0.6 3 277 0.025 0.2 1.5 415.5 0.075 0.6 1.5 415.5 0.025 0.6 1.5 415.5 0.025 0.6 1.5 138.5 0.025 0.2 Model HL4 3 277 0.075 0.6 4.5 138.5 0.025 0.2 4.5 138.5 0.075 0.6 4.5 138.5 0.075 0.6 4.5 138.5 0.075 0.6 3 277 0.025 0.2 1.5 415.5 0.075 0.6 1.5 415.5 0.025 0.2 1.5 415.5 0.025 0.2 1.5 138.5 0.025 0.2

307

eP

µP


0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009

3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5


0.003 0.003 0.003 0.009 0.003 0.009 0.009 0.009 0.003 0.009

3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5

and HL6


ηZ

Peq

Zeq

Req R?

0.6 0.2 0.2 0.2 0.6 0.2 0.6 0.6 0.6 0.2

62.5 10.4 31.3 10.4 62.5 3.5 20.8 6.9 20.8 3.5

311.3 433.2 415.5 293.0 388.9 177.8 80.8 82.0 198.9 120.9

23.3 10.8 31.2 22.0 29.2 4.4 6.1 2.1 5.0 3.0

0.062 0.024 0.070 0.072 0.065 0.025 0.060 0.023 0.023 0.024

0.1597 0.0855 0.2650 0.0289 0.5563 0.0041 0.0098 0.0032 0.0134 0.0014


0.6 0.2 0.6 0.6 0.6 0.2 0.6 0.2 0.2 0.2

62.5 10.4 62.5 20.8 62.5 3.5 20.8 3.5 10.4 3.5

311.3 433.2 388.9 289.0 388.9 177.8 80.8 82.4 202.0 120.9

58.4 13.5 72.9 54.2 72.9 5.6 15.1 2.6 6.3 3.8

0.156 0.031 0.162 0.175 0.162 0.031 0.149 0.030 0.030 0.030

0.1597 0.0855 0.5563 0.0601 0.5563 0.0041 0.0098 0.0016 0.0066 0.0014



eZ

308

Goal

Table C.9: Model HL5 Parameter Values µZ kP kZ dP dZ Model HL5 277 0.005 0.075 138.5 0.0025 0.025 138.5 0.0025 0.075 138.5 0.0025 0.075 138.5 0.0025 0.075 277 0.005 0.025 415.5 0.0075 0.075 415.5 0.0075 0.025 415.5 0.0075 0.025 138.5 0.0075 0.025 Model HL6 277 0.005 0.075 138.5 0.0025 0.025 138.5 0.0025 0.075 138.5 0.0025 0.075 138.5 0.0025 0.075 277 0.005 0.025 415.5 0.0075 0.075 415.5 0.0075 0.025 415.5 0.0075 0.025 138.5 0.0075 0.025

eP

eZ

µP


3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5


3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5


Req R?

327.0 459.6 429.0 395.1 413.5 287.0 150.7 151.5 388.5 222.2

24.5 11.5 32.2 29.6 31.0 7.2 11.3 3.8 9.7 5.6

0.051 0.024 0.070 0.072 0.064 0.025 0.067 0.024 0.022 0.024

0.1757 0.23 0.33 0.08 0.855 0.0063 0.0039 0.0011 0.068 0.0007


326.5 456.4 429.7 394.9 412.4 286.4 149.3 150.2 386.2 220.9

24.5 11.4 32.2 29.6 30.8 7.2 11.2 3.8 9.7 5.5

0.051 0.024 0.069 0.072 0.064 0.025 0.067 0.024 0.022 0.024

0.1755 0.16 0.38 0.0883 0.8820 0.0063 0.0039 0.0011 0.067 0.0007



Goal

Table C.10: Model HH1 and HH2 Parameter Values µZ kP kZ dP dZ ηZ Peq Model HH1 1.5 277 3000 0.075 157.9 1.5 138.5 1142.71 0.025 19.4 3.2625 138.5 1437.92 0.075 33.8 4.5 138.5 1000 0.075 16.9 2.9164 144.2 2689.15 0.075 70.98 4.5 277 1000 0.025 5.6 4.5 415.5 1000 0.075 16.9 4.5 415.5 1000 0.025 5.6 1.5 415.5 3000 0.025 50.8 4.5 138.5 1000 0.025 5.6 Model HH2 1.5 277 3000 0.005 0.075 157.9 4.5 138.5 2638.31 0.0025 0.025 14.7 4.5 138.5 2200.39 0.0025 0.075 37.3 4.5 138.5 1000 0.0025 0.075 16.9 2.2 147.7 2040.08 0.005 0.075 71.7 4.5 277 1000 0.005 0.025 5.6 4.5 415.5 1000 0.0075 0.075 16.9 4.5 415.5 1000 0.0075 0.025 5.6 1.5 415.5 3000 0.0075 0.025 50.8 4.5 138.5 1000 0.0075 0.025 5.6

309

eP

eZ

µP 3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5


3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5


Peq

Zeq

Req R?

428.6 25.0 39.4 21.3 68.28 7.0 428.6 14.1 428.6 7.0

68.6 455.0 420.4 392.3 413.0 286.6 63.6 150.9 63.6 222.1

5.1 11.4 31.5 29.4 28.0 7.2 4.8 3.8 4.8 5.6

0.010 0.024 0.069 0.071 0.058 0.024 0.010 0.023 0.010 0.024

0.0110 0.3070 0.3721 0.1131 0.8409 0.0080 0.0110 0.0031 0.0110 0.0009


428.6 25.0 65.2 43.5 68.28 7.0 428.6 7.0 428.6 7.0

68.6 455.0 389.0 377.9 413.0 286.6 63.6 151.4 63.6 222.1

12.9 14.2 72.9 70.9 28.0 9.0 11.9 4.7 11.9 6.9

0.026 0.030 0.161 0.168 0.058 0.031 0.024 0.030 0.024 0.030

0.0110 0.3070 0.5729 0.2534 0.8409 0.0080 0.0111 0.0015 0.0111 0.0009




310

Goal

Table C.11: Model HH3 and HH4 Parameter Values µZ kP kZ dP dZ ηZ Model HH3 1.5 277 3000 0.075 0.6 2.9282 138.5 2317.6931 0.025 0.2 3.6554 138.5 1480.9384 0.075 0.209 4.5 138.5 1000 0.075 0.2 3.7791 138.5 2874.8878 0.0678 0.227 4.5 277 1000 0.025 0.2 1.5 415.5 3000 0.075 0.6 4.5 415.5 1000 0.025 0.6 1.5 415.5 3000 0.075 0.6 4.5 138.5 1000 0.025 0.2 Model HH4 1.5 277 3000 0.075 0.6 2.9282 138.5 2317.6931 0.025 0.2 4.1422 138.5 1375.3666 0.075 0.6 4.5 138.5 1000 0.075 0.6 3.7791 138.5 2874.8878 0.0678 0.227 4.5 277 1000 0.025 0.2 1.5 415.5 3000 0.075 0.6 4.5 415.5 1000 0.025 0.2 1.5 415.5 3000 0.075 0.6 4.5 138.5 1000 0.025 0.2

eP

eZ

µP


3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5


3 4.5 4.5 4.5 4.5 3 1.5 1.5 1.5 1.5

Peq


Req R?

428.6 37.0 77.1 21.3 71.0 7.0 428.6 14.1 428.6 7.0

68.2 450.1 405.7 391.9 407.5 286.0 61.7 149.5 61.7 221.7

5.1 11.3 30.4 29.4 29.2 7.2 4.6 3.7 4.6 5.5

0.010 0.023 0.063 0.071 0.061 0.024 0.009 0.023 0.009 0.024

0.0109 0.50 0.5647 0.11 0.8522 0.0080 0.0107 0.0032 0.0107 0.0009

Yes No Yes No Yes No Yes No Yes No

428.6 37.0 103.1 43.5 71.0 8.1 428.6 7.0 428.6 7.0

68.2 450.1 372.3 377.7 407.5 285.7 61.7 150.1 61.7 221.7

12.8 14.1 69.8 70.8 29.2 10.3 11.6 4.7 11.6 6.9

0.026 0.029 0.147 0.168 0.061 0.035 0.024 0.030 0.024 0.030

4.51 0.50 2.05 0.25 0.8522 0.009 1.46 0.0015 1.46 0.0009



Goal

Table C.12: Model HH5 and HH6 Parameter Values µZ kP kZ dP dZ ηZ Model HH5 1.5 277 3000 0.005 0.075 0.6 2.02 138.5 2364.61 0.0025 0.025 0.2 2.7522 138.5 2186.70 0.0025 0.075 0.2 4.5 138.5 1000 0.0075 0.075 0.2 3.7522 138.5 2491.69 0.0075 0.072 0.310 4.5 277 1000 0.005 0.025 0.2 1.5 415.5 3000 0.0075 0.075 0.6 4.5 415.5 1000 0.0075 0.025 0.6 1.5 415.5 3000 0.0075 0.075 0.6 4.5 138.5 1000 0.0025 0.025 0.2 Model HH6 1.5 277 3000 0.005 0.075 0.6 2.02 138.5 2364.61 0.0025 0.025 0.2 3.71 138.5 1938.41 0.0025 0.075 0.6 4.5 138.5 1000 0.0025 0.075 0.6 3.7522 138.5 2491.69 0.0075 0.072 0.310 4.5 277 1000 0.005 0.025 0.309 1.5 415.5 3000 0.0075 0.075 0.6 4.5 415.5 1000 0.0075 0.025 0.2 1.5 415.5 3000 0.0075 0.075 0.6 4.5 138.5 1000 0.0025 0.025 0.2

311

312


Table C.13: Normalized Goal Function Values. Model LL Goal Goal Normalized Maximized P Z F (F/B) R −(F/B) Model LL1 Peq 1.00 0.79 0.87 0.94 0.65 0.06 Zeq 0.25 1.00 0.26 0.01 0.31 0.99 Feq 1.00 0.95 1.00 0.95 1.00 0.05 (F/B)eq 0.25 0.50 0.65 1.00 0.31 0.00 Req 1.00 0.95 1.00 0.95 1.00 0.05 −(F/B)eq 0.25 0.50 0.13 0.00 0.08 1.00 Model LL2 Peq 1.00 0.79 0.87 0.94 0.65 0.06 Zeq 0.25 1.00 0.26 0.01 0.31 0.99 Feq 1.00 0.95 1.00 0.95 1.00 0.05 (F/B)eq 0.25 0.50 0.65 1.00 0.31 0.00 Req 1.00 0.95 1.00 0.95 1.00 0.05 −(F/B)eq 0.25 0.50 0.13 0.00 0.08 1.00 Model LL3 Peq 1.00 0.69 0.81 0.79 0.66 0.21 Zeq 0.12 1.00 0.26 0.03 0.15 0.97 Feq 0.47 0.94 1.00 0.94 0.49 0.06 (F/B)eq 0.12 0.51 0.66 1.00 0.15 0.00 Req 1.00 0.84 0.92 0.82 1.00 0.18 −(F/B)eq 0.29 0.48 0.12 0.00 0.09 1.00 Model LL4 Peq 1.00 0.69 0.88 0.87 0.66 0.13 Zeq 0.12 1.00 0.13 0.00 0.15 1.00 Feq 1.00 0.84 1.00 0.89 1.00 0.11 (F/B)eq 0.29 0.48 0.71 1.00 0.32 0.00 Req 1.00 0.84 1.00 0.89 1.00 0.11 −(F/B)eq 0.12 0.51 0.07 0.00 0.04 1.00 Model LL5 Peq 1.00 0.69 0.81 0.79 0.66 0.21 Zeq 0.12 1.00 0.26 0.03 0.15 0.97 Feq 0.47 0.94 1.00 0.94 0.49 0.06 (F/B)eq 0.12 0.51 0.66 1.00 0.15 0.00 Req 1.00 0.84 0.92 0.82 1.00 0.18 −(F/B)eq 0.29 0.48 0.12 0.00 0.09 1.00 Model LL6 Peq 1.00 0.69 0.88 0.87 0.66 0.13 Zeq 0.12 1.00 0.13 0.00 0.15 1.00 Feq 1.00 0.84 1.00 0.89 1.00 0.11 (F/B)eq 0.29 0.49 0.71 1.00 0.32 0.00 Req 1.00 0.84 1.00 0.89 1.00 0.11 −(F/B)eq 0.12 0.50 0.07 0.00 0.04 1.00


Table C.14: Normalized Goal Function Values. Model LH Goal Goal Normalized Maximized P Z F (F/B) R −(F/B) Model LH1 Peq 1.00 0.52 0.78 0.57 0.38 0.43 Zeq 0.19 1.00 0.24 0.02 0.30 0.98 Feq 0.34 0.81 1.00 0.88 0.49 0.12 (F/B)eq 0.08 0.55 0.81 1.00 0.12 0.00 Req 0.70 0.70 0.92 0.71 1.00 0.29 −(F/B)eq 0.30 0.81 0.20 0.00 0.14 1.00 Model LH2 Peq 1.00 0.52 0.78 0.57 0.40 0.43 Zeq 0.19 1.00 0.24 0.02 0.30 0.98 Feq 0.33 0.81 1.00 0.88 0.51 0.12 (F/B)eq 0.07 0.55 0.80 1.00 0.12 0.00 Req 0.70 0.70 0.92 0.71 1.00 0.29 −(F/B)eq 0.30 0.80 0.19 0.00 0.15 1.00 Model LH3 Peq 1.00 0.01 0.01 0.01 0.01 0.99 Zeq 0.08 1.00 0.25 0.22 0.34 0.78 Feq 0.13 0.87 1.00 0.91 0.54 0.09 (F/B)eq 0.03 0.72 0.83 1.00 0.15 0.00 Req 0.25 0.77 0.89 0.77 1.00 0.23 −(F/B)eq 1.00 0.00 0.00 0.00 0.01 1.00 Model LH4 Peq 1.00 0.01 0.11 0.01 0.01 0.99 Zeq 0.08 1.00 0.13 0.03 0.34 0.97 Feq 0.18 0.75 1.00 0.89 0.74 0.11 (F/B)eq 0.09 0.68 0.92 1.00 0.33 0.00 Req 0.25 0.77 0.74 0.59 1.00 0.41 −(F/B)eq 1.00 0.00 0.10 0.00 0.01 1.00 Model LH5 Peq 1.00 0.02 0.02 0.01 0.01 0.99 Zeq 0.08 1.00 0.26 0.22 0.34 0.78 Feq 0.13 0.87 1.00 0.90 0.55 0.10 (F/B)eq 0.03 0.72 0.83 1.00 0.15 0.00 Req 0.25 0.75 0.86 0.76 1.00 0.24 −(F/B)eq 1.00 0.00 0.00 0.00 0.01 1.00 Model LH6 Peq 1.00 0.02 0.11 0.01 0.01 0.99 Zeq 0.08 1.00 0.13 0.04 0.34 0.96 Feq 0.19 0.75 1.00 0.89 0.75 0.11 (F/B)eq 0.09 0.68 0.92 1.00 0.33 0.00 Req 0.25 0.75 0.89 0.73 1.00 0.27 −(F/B)eq 1.00 0.00 0.09 0.00 0.01 1.00

313

314


Table C.15: Normalized Goal Function Values. Model HL Goal Goal Normalized Maximized P Z F (F/B) R −(F/B) Model HL1 Peq 1.00 0.72 0.78 0.94 0.28 0.06 Zeq 0.25 1.00 0.30 0.01 0.32 0.99 Feq 1.00 0.96 1.00 0.96 1.00 0.04 (F/B)eq 0.25 0.60 0.68 1.00 0.10 0.00 Req 1.00 0.96 1.00 0.96 1.00 0.04 −(F/B)eq 0.25 0.35 0.10 0.00 0.02 1.00 Model HL2 Peq 1.00 0.72 0.78 0.94 0.28 0.06 Zeq 0.25 1.00 0.30 0.01 0.32 0.99 Feq 1.00 0.96 1.00 0.96 1.00 0.04 (F/B)eq 0.25 0.60 0.68 1.00 0.10 0.00 Req 1.00 0.96 1.00 0.96 1.00 0.04 −(F/B)eq 0.25 0.34 0.10 0.00 0.02 1.00 Model HL3 Peq 1.00 0.65 0.73 0.80 0.29 0.20 Zeq 0.12 1.00 0.30 0.04 0.15 0.96 Feq 0.47 0.95 1.00 0.95 0.48 0.05 (F/B)eq 0.12 0.60 0.68 1.00 0.05 0.00 Req 1.00 0.87 0.93 0.84 1.00 0.16 −(F/B)eq 0.29 0.34 0.10 0.00 0.02 1.00 Model HL4 Peq 1.00 0.65 0.79 0.87 0.29 0.13 Zeq 0.12 1.00 0.16 0.01 0.15 0.99 Feq 1.00 0.87 1.00 0.91 1.00 0.09 (F/B)eq 0.29 0.59 0.73 1.00 0.11 0.00 Req 1.00 0.87 1.00 0.91 1.00 0.09 −(F/B)eq 0.12 0.35 0.05 0.00 0.01 1.00 Model HL5 Peq 1.00 0.65 0.73 0.80 0.29 0.20 Zeq 0.12 1.00 0.30 0.04 0.15 0.96 Feq 0.47 0.95 1.00 0.95 0.47 0.05 (F/B)eq 0.12 0.60 0.68 1.00 0.05 0.00 Req 1.00 0.87 0.93 0.84 1.00 0.16 −(F/B)eq 0.29 0.34 0.10 0.00 0.02 1.00 Model HL6 Peq 1.00 0.65 0.79 0.87 0.29 0.13 Zeq 0.12 1.00 0.16 0.01 0.15 0.99 Feq 1.00 0.87 1.00 0.91 1.00 0.09 (F/B)eq 0.29 0.59 0.73 1.00 0.11 0.00 Req 1.00 0.87 1.00 0.91 1.00 0.09 −(F/B)eq 0.12 0.34 0.05 0.00 0.01 1.00


Table C.16: Normalized Goal Function Values. Model HH Goal Goal Normalized Maximized P Z F (F/B) R −(F/B) Model HH1 Peq 1.00 0.57 0.73 0.57 0.20 0.43 Zeq 0.09 1.00 0.27 0.04 0.27 0.96 Feq 0.19 0.90 1.00 0.95 0.39 0.05 (F/B)eq 0.07 0.79 0.91 1.00 0.10 0.00 Req 0.43 0.85 0.96 0.84 1.00 0.16 −(F/B)eq 0.30 0.77 0.21 0.00 0.08 1.00 Model HH2 Peq 1.00 0.58 0.73 0.57 0.20 0.43 Zeq 0.06 1.00 0.27 0.04 0.18 0.96 Feq 0.21 0.91 1.00 0.94 0.44 0.06 (F/B)eq 0.07 0.80 0.91 1.00 0.10 0.00 Req 0.43 0.86 0.95 0.83 1.00 0.17 −(F/B)eq 0.30 0.77 0.21 0.00 0.08 1.00 Model HH3 Peq 1.00 0.01 0.05 0.01 0.01 0.99 Zeq 0.04 1.00 0.27 0.23 0.36 0.77 Feq 0.08 0.91 1.00 0.96 0.44 0.04 (F/B)eq 0.03 0.84 0.92 1.00 0.13 0.00 Req 0.15 0.89 0.87 0.79 1.00 0.21 −(F/B)eq 1.00 0.00 0.04 0.00 0.01 1.00 Model HH4 Peq 1.00 0.01 0.12 0.01 0.01 0.99 Zeq 0.04 1.00 0.14 0.04 0.36 0.96 Feq 0.14 0.83 1.00 0.95 0.68 0.05 (F/B)eq 0.09 0.80 0.97 1.00 0.30 0.00 Req 0.15 0.89 0.46 0.35 1.00 0.65 −(F/B)eq 1.00 0.00 0.11 0.00 0.01 1.00 Model HH5 Peq 1.00 0.02 0.05 0.01 0.01 0.99 Zeq 0.07 1.00 0.28 0.22 0.59 0.78 Feq 0.17 0.89 1.00 0.87 0.66 0.13 (F/B)eq 0.03 0.85 0.96 1.00 0.13 0.00 Req 0.15 0.89 0.95 0.84 1.00 0.16 −(F/B)eq 1.00 0.00 0.03 0.00 0.01 1.00 Model HH6 Peq 1.00 0.02 0.12 0.01 0.01 0.99 Zeq 0.07 1.00 0.14 0.04 0.59 0.96 Feq 0.23 0.80 1.00 0.85 0.38 0.15 (F/B)eq 0.08 0.81 1.02 1.00 0.30 0.00 Req 0.15 0.89 0.58 0.45 1.00 0.55 −(F/B)eq 1.00 0.00 0.11 0.00 0.01 1.00

315

Appendix D


D.1

Detailed derivation of the Jacobian Matrix

Equation 6.3 gives the general form for the change in species’ i biomass with time t. Repeated, it is   n+1 X d(xi ) ai,j xj  . = xi di + dt j=1

(D.1)

where ai,j is a coefficient describing nutrient flow from species j to species i, di is the mortality coefficient of species i (−1 ≤ di ≤ 0), and n + 1 represents the nutrient compartment. Equation D.1 can combined with Equation 6.5 (the assumption that the system is closed to nutrient input and output)   n+1 d(xi )  X = ai,j xi xj  + di xi , dt j=1   n X = ai,j xi xj  + ai,n+1 xi xn+1 + di xi ,

(D.2)

(D.3)

j=1



=



 = 

n X j=1

n X j=1 j6=i









ai,j xi xj  + ai,n+1 xi No −    ai,j xi xj   + ai,n+1 xi No −

n X j=1

n X j=1 j6=i



xj  + di xi ,

(D.4)

 2 2 xj   + di xi + ai,i xi − ai,n+1 xi .

(D.5)

316



317

D.2 Routh-Hurwitz criteria

So the diagonal elements of the Jacobian Matrix are

∂ ∂xi

d(xi ) dt



 = 



 = 



 = 



 = 



=

D.2

n X j=1 j6=i

n X j=1 j6=i

n X j=1 j6=i

n X j=1 j6=i

n+1 X j=1





   ai,j xj   ai,n+1 No − ai,n+1  



   ai,j xj   ai,n+1 No − ai,n+1  

n X j=1 j6=i

n X j=1 j6=i



 xj   + di + 2ai,i xi − 2ai,n+1 xi .

(D.6)



 xj   + di − 2ai,n+1 xi ,

(D.7)

  n X   xj  + di − ai,n+1 xi , ai,j xj   + ai,n+1 No −

(D.8)

j=1



 ai,j xj   + ai,n+1 xn+1 + di − ai,n+1 xi , 

ai,j xj  − ai,n+1 xi + di .

(D.9)

(D.10)

Routh-Hurwitz criteria

If J is an n × n matrix, the characteristic equation for finding the eigenvalues of J is λn + c1 λn−1 + c2 λn−2 + . . . + cn = 0,

(D.11)

the following conditions, called Routh-Hurwitz Criteria, are necessary and sufficient to ensure that all eigenvalues are negative, and hence the system is locally stable. (May 1973, p 196). n=2

(D.12a)

c1 > 0;

(D.12b)

c2 > 0.

(D.12c)

n=3

(D.12d)

c1 > 0;

(D.12e)

c3 > 0;

(D.12f)

c 1 c 2 > c3 .

(D.12g)

318


n=4 c1 > 0;

(D.12i)

c3 > 0;

(D.12j)

c4 ≫ 0;

(D.12k)

c1 c2 c3 > c33 + c21 c4 .

(D.12l)

n=5

D.3

(D.12h)

(D.12m)

ci > 0, i = 1, 2, 3, 4, 5;

(D.12n)

c1 c2 c3 > c33 + c21 c4 ;

(D.12o)

(c1 c4 − c5 )(c1 c2 c3 − c23 − c21 c4 ) > c5 (c1 c2 − c3 )2 + c1 c2 .

(D.12p)

Trends between resilience and biomass in the nutrient compartment

Figure D.1 and Figure D.2 shows the resilience verses biomass in the nutrient compartment for a number of GLV models. Data points are randomly generated, as described in Section 6.5.2.

D.3 Trends between resilience and biomass in the nutrient compartment

319

Ecosystem 3-1: Resilience verses Biomass of nutrient compartment 5 4.5 4 3.5

Resilience

3 2.5 2 1.5 1 0.5 0 0

50

100

150

200

250

300

350

400

450

500

Biomass in nutrient compartment Ecosystem 4-1: Resilience verses Biomass in nutrient compartment 7

6

Resilience

5

4

3

2

1

0 0

50

100

150

200

250

300

350

400

Biomass in nutrient compartment Ecosystem 4-2: Resilience verses Biomass in nutrient compartment 1.8 1.6 1.4

Resilience

1.2 1 0.8 0.6 0.4 0.2 0 0

50

100

150

200

250

300

350

400

450

Biomass in nutrient compartment

Figure D.1: Comparison between Resilience verses biomass in the nutrient compartment for a number of models.

320


Ecosystem 4-4: Resilience verses Biomass in nutrient compartment 2 1.8 1.6 1.4

Resilience

1.2 1 0.8 0.6 0.4 0.2 0 0

20

40

60

80

100

120

140

160

Biomass in nutrient compartment Ecoysstem 5-1: Resilience verses Biomass in nutrient compartment 3

2.5

Resilience

2

1.5

1

0.5

0 0

50

100

150

200

250

300

350

400

450

500

450

500

Biomass in nutrient compartment Ecosystem 5-2: Resilience verses Biomass in nutrient compartment 1.8 1.6 1.4

Resilience

1.2 1 0.8 0.6 0.4 0.2 0 0

50

100

150

200

250

300

350

400

Biomass in nutrient compartment

Figure D.2: Comparison between Resilience verses biomass in the nutrient compartment for a number of models.

321

D.4 An example of a feasible, unstable chain

D.4

An example of a feasible, unstable chain

Consider an A matrix  0.00000 0.57536 0.00000 0.00000   −0.57536 0.00000 0.38565 0.00000   A =  0.00000 −0.38565 0.00000 0.23870    0.00000 0.00000 −0.23870 0.00000  0.00000 0.00000 0.00000 −0.01511

 0.00000   0.00000   0.00000 ,   0.01511  0.00000

with

DT = −0.87654 −0.00127 −0.32397 −0.59377 −0.00000 ,

(D.13)

(D.14)

which gives the feasible solution (x⋆ )T = 17.4726 1.5235 26.0711 3.8185 451.1144 ,

(D.15)

The eigenvalues of the Jacobian Matrix are λ = −0.0304 ± 1.7712i, 0.0015 ± 4.1967i,

(D.16)

which is a saddle. However, after some time it is found that the system settles into a continuous cycle in all four dimensions (Figure D.3). Example: Unstable 5 chain cycle

45 40

Compartment 3

35 30 25 20 15 10 10 8

30 6

25 20

4 15

2 Compartment 2

10 0

5

Compartment 1

Figure D.3: Nearing the cycle of the example 5-compartment chain described in the text.

322


D.5

Feasibility in closed chains does not imply stability

Nisbet & Gurney (1976) state that closed generalised Lotka-Volterra chains, with the same form and constraints as used in Chapter 6, will always be locally stable provided that they are feasible. However, results in this study contradict this (Table 6.1). The proof given by Nisbet & Gurney (1976) assumes that successive submatrices of the Jacobian matrix have all negative eigenvalues. Below, we show that this assumption does not hold. Nisbet & Gurney (1976) begin by defining a closed chain with a nutrient compartment such that the nutrient compartment is described by x˙ 0 =

n X i=1

ki xi − A01 x0 x1 ,

(D.17a)

the biotic compartments described by x˙ 1 = −k1 x1 − A10 x1 x0 − A12 x1 x2 ,

(D.17b)

and the top predator by x˙ n = −kn xn − An,n−1 xn xn−1 .

(D.17c)

We retain the notation used by Nisbet & Gurney (1976), where their ki is analogous to our di , their Aij is analogous to our ai,j , and our compartments are ordered in the opposite direction, that is, their top predator is subscript n, and the nutrient compartment subscript 0. Nisbet & Gurney (1976) limit the total amount of nutrient in the system, so the Jacobian matrix is described by 

−Sn

   Sn−1 Bn =    0  .. .

−Sn − tn−1

−Sn

−Sn

0

−tn−2

0

Sn−2 .. .

0 .. .

−tn−3 .. .

...



  ...  ,  ...  

(D.18)

where Si = −An−i+1,n−i x⋆n−i+1 ,

(D.19)

ti = An−i,n−i+1 x⋆n−i .

(D.20)

It can be shown, with some manipulation, that Equation D.18 is materially equivalent to Equation 6.9. Nisbet & Gurney (1976) go on to state

323

D.6 Normalised value of goal functions for chains sized 3-10

By a succession of column operations on the matrix Bn − λI (where I is the unit matrix

[sic]1 ) we find that

det(Bn − λI) = −Sn det(Bn−1 − λI) + det(An − λI) . . . . . . (4), where An is the matrix obtained by setting Sn = 0 in the matrix Bn . If the matrix Bn−1 is stable we can use Equation (4) to show that it is impossible for Bn to have any pure imaginary eigenvalues. Our task is now to determine what the undefined Bn−1 matrix is. We may take the determinant of Bn by starting the spaghetti method with the (1,1) and (2,1) elements of the matrix, so we find 0 − λ −tn−2 |Bn − λI| = (−Sn − λ) Sn−2 0 − λ .. .. . .

0 −tn−3 .. .

... −Sn − tn−1 . . . − Sn−1 Sn−2 .. .

−Sn

−Sn

0 − λ −tn−3 .. .. . .

... ... .

(D.21a)

Separating the −λ| · | and grouping it with the −Sn−1 | · |, we find that the combined terms give 0 − λ −tn−2 |Bn −λI| = −Sn Sn−2 0 − λ .. .. . .

0 −tn−3 .. .

0 − λ . . . Sn−1 . . . + 0 .. .

Thus, the first matrix must be their ‘Bn−1 − λI’.

−Sn − tn−1

−Sn

−Sn

0−λ

−tn−2

0

Sn−2 .. .

0−λ .. .

−tn−3 .. .

... ... . (D.21b) ...

The proof given by Nisbet & Gurney (1976) relies upon finding Bn−1 always locally stable, however, this is impossible. The trace of a matrix is equal to the sum of the eigenvalues. The expression trace(Bn−1 ) = 0, which implies that either all eigenvalues of Bn−1 have zero real parts, or some eigenvalues of Bn−1 are positive, and some are negative. Both of these possibilities imply that Bn−1 is locally unstable, and so their proof is incorrect.

D.6

Normalised value of goal functions for chains sized 3-10

Table D.1 shows the normalised effect of the maximisation of each of the goal functions upon the other goal functions for GLV chains. 1 This

should be “where I is the identity matrix”.

324


Table D.1: Normalised values for size Goal Maximised B 3 B 1 F 1 F/B 0.13 -F/B 0.03 R 0.98 4 B 1 F 1 F/B 0.82 -F/B 1 R 0.97 5 B 1 F 1 F/B 0.46 -F/B 0.65 R 0.96 6 B 1 F 1 F/B 1 -F/B 0.74 R 0.97 7 B 1 F 1 F/B 0.33 -F/B 0.95 R 0.96 8 B 1 F 1 F/B 1 -F/B 0.98 R 0.97 9 B 1 F 0.99 F/B 0.84 -F/B 0.59 R 0.97 10 B 1 F 1 F/B 1 -F/B 1 R 0.97

goal functions (chains sized 3-10). Goal Normalised F F/B -F/B R 0.96 0.01 0.99 0 0.99 1 0 0.17 0.13 1 0 0 0 0 1 0 0.95 0.97 0.03 1 0 0 1 0 1 1 0 0 0.82 1 0 0.02 1 0 1 0 0.94 0.96 0.04 1 0 0.01 0.99 0 1 1 0 0 0.46 1 0 0 0 0 1 0 0.85 0.88 0.12 1 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 0 0.91 0.94 0.06 1 0 0.01 0.99 0 0.99 1 0 0 0.33 1 0 0 0 0 1 0 0.80 0.83 0.17 1 0 0 1 0 0.99 1 0 0 1 1 0 0 0 0 1 0 0.76 0.79 0.21 1 0 0 1 −0 0.99 1 0 0 0.84 1 0 0 0 0 1 0 0.70 0.72 0.28 1 0 0 1 0 1 1 0 0 0.99 1 0 0 0 0 1 0 0.75 0.78 0.22 1

D.7 Values and normalised value of goal functions for non-chain structures

D.7

325

Values and normalised value of goal functions for non-chain structures

Table D.2 shows the value of each of the goal functions when a particular goal function is maximised. Consistency between the maximum values for each of the traditional goal functions is observed. Table D.3 shows the normalised effect of the maximisation of each of the goal functions upon the other goal functions. Entries indicated as ≈ 0 were very small values close to 1. Table D.2: Values for goal functions. Goal Goal Function’s Value Maximised B F F/B -F/B Ecosystem 4-2 B 4.99 × 102 2.98 × 101 5.96 × 10−2 −5.97 × 10−2 2 2 F 4.98 × 10 4.98 × 10 9.99 × 10−1 −10 × 10−1 2 2 −1 F/B 4.98 × 10 4.98 × 10 9.99 × 10 −10 × 10−1 2 −1 −3 -F/B 3.78 × 10 3.78 × 10 0.99 × 10 −1 × 10−3 2 1 −1 R 4.46 × 10 7.71 × 10 1.72 × 10 −1.73 × 10−1 Ecosystem 4-4 B 4.99 × 102 5 × 10−1 1 × 10−3 −1.01 × 10−3 2 2 −1 F 4.99 × 10 4.98 × 10 9.99 × 10 −10 × 10−1 2 2 −1 F/B 4.98 × 10 4.98 × 10 9.99 × 10 −10 × 10−1 2 −1 −3 -F/B 4.99 × 10 4.99 × 10 1 × 10 −1 × 10−3 2 2 −1 R 4.75 × 10 4.25 × 10 8.94 × 10 −8.95 × 10−1 Ecosystem 5-2 B 4.99 × 102 6.87 × 10−1 1.37 × 10−3 −1.38 × 10−3 F 4.97 × 102 4.97 × 102 1 × 100 −1 × 100 2 2 0 F/B 3.32 × 10 3.32 × 10 1 × 10 −1 × 100 2 −1 −3 -F/B 4.97 × 10 4.97 × 10 0.99 × 10 −1 × 10−3 2 2 −1 R 4.9 × 10 3.21 × 10 6.53 × 10 −6.54 × 10−1 Ecosystem 5-3 B 4.98 × 102 1.09 × 102 2.19 × 10−1 −2.2 × 10−1 2 2 F 4.95 × 10 4.95 × 10 9.99 × 10−1 −10 × 10−1 2 2 −1 F/B 3.29 × 10 3.29 × 10 9.99 × 10 −10 × 10−1 1 −2 −3 -F/B 3.13 × 10 5.9 × 10 1.88 × 10 −1.89 × 10−3 2 2 −1 R 4.86 × 10 4.8 × 10 9.86 × 10 −9.87 × 10−1 Ecosystem 5-6 B 4.99 × 102 9.79 × 10−1 1.95 × 10−3 −1.96 × 10−3 F 4.98 × 102 4.98 × 102 9.99 × 10−1 −10 × 10−1 F/B 3.87 × 102 3.87 × 102 1 × 100 −1 × 100 -F/B 3.28 × 102 3.28 × 10−1 0.99 × 10−3 −1 × 10−3 R 4.57 × 102 4.56 × 102 9.98 × 10−1 −9.99 × 10−1 Ecosystem 5-7 B 4.99 × 102 1.98 × 101 3.97 × 10−2 −3.98 × 10−2 2 2 F 4.98 × 10 4.98 × 10 9.99 × 10−1 −10 × 10−1 2 2 0 F/B 4.6 × 10 4.6 × 10 1 × 10 −1 × 100 2 −1 −3 -F/B 4.23 × 10 4.23 × 10 0.99 × 10 −1 × 10−3 2 2 −1 R 4.68 × 10 3.92 × 10 8.37 × 10 −8.38 × 10−1

R 2.93 × 10−5 3.44 × 10−8 4.89 × 10−7 1.8 × 10−1 1.31 × 100 3.78 × 10−7 4.9 × 10−10 2.66 × 10−12 1.2 × 10−7 5.9 × 100 3.76 × 10−4 4.78 × 10−1 6.44 × 10−6 9.05 × 10−4 4.03 × 100 1.54 × 10−2 1.17 × 10−2 9.88 × 10−7 8.24 × 10−8 3.9 × 100 ≈0 8.04 × 10−5 2.93 × 10−4 2.87 × 10−10 2.04 × 100 1.33 × 10−4 5.16 × 10−7 2.06 × 10−3 3.76 × 10−8 3.16 × 100

326


Table D.3: Normalised values for goal functions. Goal Goal Normalised Maximised B F F/B -F/B Ecosystem 4-2 B 9.99 × 10−1 5.96 × 10−2 5.87 × 10−2 9.41 × 10−1 −1 −1 0 F 9.97 × 10 9.97 × 10 1 × 10 ≈0 F/B 9.97 × 10−1 9.97 × 10−1 1 × 100 ≈0 -F/B 7.57 × 10−1 7.57 × 10−4 ≈0 1 × 100 −1 −1 −1 R 8.92 × 10 1.54 × 10 1.71 × 10 8.28 × 10−1 Ecosystem 4-4 B 9.99 × 10−1 1 × 10−3 5.8 × 10−7 9.99 × 10−1 −1 −1 0 F 9.98 × 10 9.97 × 10 1 × 10 −10 × 10−4 −1 −1 0 F/B 9.97 × 10 9.97 × 10 1 × 10 −1.01 × 10−3 −1 −4 -F/B 9.99 × 10 9.99 × 10 ≈0 1 × 100 −1 −1 −1 R 9.51 × 10 8.51 × 10 8.94 × 10 1.05 × 10−1 Ecosystem 5-2 B 9.99 × 10−1 1.37 × 10−3 3.76 × 10−4 9.99 × 10−1 −1 −1 0 F 9.95 × 10 9.95 × 10 1 × 10 −1.01 × 10−3 −1 −1 0 F/B 6.65 × 10 6.65 × 10 1 × 10 −1.01 × 10−3 −1 −4 -F/B 9.95 × 10 9.95 × 10 ≈0 1 × 100 −1 −1 −1 R 9.81 × 10 6.42 × 10 6.54 × 10 3.45 × 10−1 Ecosystem 5-3 B 9.97 × 10−1 2.18 × 10−1 2.18 × 10−1 7.81 × 10−1 −1 −1 0 F 9.91 × 10 9.91 × 10 1 × 10 −9.7 × 10−4 −1 −1 0 F/B 6.59 × 10 6.59 × 10 1 × 10 −1.01 × 10−3 −2 −4 −4 -F/B 6.26 × 10 1.18 × 10 8.86 × 10 9.99 × 10−1 −1 −1 −1 R 9.73 × 10 9.6 × 10 9.87 × 10 1.28 × 10−2 Ecosystem 5-6 B 9.99 × 10−1 1.95 × 10−3 9.6 × 10−4 9.99 × 10−1 −1 −1 0 F 9.97 × 10 9.97 × 10 1 × 10 −9.05 × 10−4 −1 −1 0 F/B 7.74 × 10 7.74 × 10 1 × 10 −1.01 × 10−3 −1 −4 -F/B 6.56 × 10 6.56 × 10 ≈0 1 × 100 −1 −1 −1 R 9.14 × 10 9.13 × 10 9.99 × 10 9.37 × 10−4 Ecosystem 5-7 B 9.99 × 10−1 3.97 × 10−2 3.88 × 10−2 9.61 × 10−1 −1 −1 0 F 9.96 × 10 9.96 × 10 1 × 10 −9.99 × 10−4 −1 −1 0 F/B 9.2 × 10 9.2 × 10 1 × 10 −1.01 × 10−3 −1 −4 -F/B 8.46 × 10 8.46 × 10 ≈0 1 × 100 R 9.37 × 10−1 7.85 × 10−1 8.38 × 10−1 1.61 × 10−1

R 1.91 × 10−4 ≈0 ≈0 1.37 × 10−1 1 × 100 6.4 × 10−8 8.3 × 10−11 4.51 × 10−13 2.04 × 10−8 1 × 100 9.32 × 10−5 1.18 × 10−1 1.59 × 10−6 2.24 × 10−4 1 × 100 3.95 × 10−3 3 × 10−3 2.53 × 10−7 2.11 × 10−8 1 × 100 ≈0 3.93 × 10−5 1.43 × 10−4 1.4 × 10−10 1 × 100 4.21 × 10−5 1.63 × 10−7 6.53 × 10−4 1.18 × 10−8 1 × 100

Appendix E

Appendices relating to Chapter 7 E.1

Field results used to determine the ef-ratio

Table E.1 summarises field results for the determination of the ef-ratio in various ocean regions. This data was taken from a variety of sources, cited in Laws et al.’s (2000) Table 3.

Ocean ID 1 2 3 4 5 6 7 8 9 10 11

E.2

Table E.1: Field data adapted from Table 3 of (Laws et al. 2000). Region Mixed layer Temperature Export or depth (m) (◦ C) New production BATS 140 21 7.8 HOT 150 25 12.2 NABE 35 12.5 98 EqPac-normal 120 24 32.1 EqPac-El Nino 120 27 12.3 Arabian Sea 65 25 29.2 Ross Sea 40 0 165 Subarctic P 120 6 40.3 Peru-normal 25.5 16.8 339 Peru-El Nino 17.8 17.4 256 Greenland polynya 50 0 35.6

Total production 82 83 194 260 169 195 243 95 806 867 63.2

Solving the Laws Model at steady state

The steady-state detritus concentration is found by a mass balance about the entire system Xd⋆ =

LM , V

(E.1)

where L is the loading rate, M is the mixed layer depth, and V is the detritus’ vertical sinking rate. 327

328


As X1 and Xc do not explicitly appear in the dynamical equations, the equations describing f2l and fb are rearranged to give X1⋆ =

P2l , 1 − f2l

(E.2)

Xc⋆ =

Pb , 1 − fb

(E.3)

and

respectively. Equation E.2 can be used to find f2s f2s = 1 −

P2s . X1⋆

(E.4)

Equations 1 to 10 of Laws et al. (2000) may be expressed at the steady state by equating the derivatives to zero. Equations 2 and 10 become ⋆ 0 = q2s f2s A2s (X2s + Xb⋆ ) − f3 A3 X3⋆ ,

(E.5a)

⋆ 0 = qb fb A2s (X2s + Xb⋆ ) − f3 A3 X3⋆ ,

(E.5b)

and

respectively at the steady state. Unless assumptions are made about the values of the parameters f , the system will find a solution to these equations by setting the state variables to zero. As we are interested in ⋆ non-trivial solutions, we must equate the f3 A3 /(X2s + Xb⋆ ) terms in both equations to find a relationship

between b and 2s parameters. Rearranging gives fb =

q2s f2s A2s . qb Ab

(E.6)

A similar situation exists between Equations 3 and 5, which gives f4 =

q2l f2l A2l . q4 A4

(E.7)

Equation 7 evaluated at the steady state becomes q6 f6 A6 X6⋆ = µ6 X6⋆ ,

(E.8)

which implies either a trivial solution or a relationship between the parameters. In this instance, we have used the equality to substitute for µ6 X6⋆ in Equation 9. It was thought convenient to reduce the number of equations used as far as possible. To this end, Equation 6 at steady state f6 A6 X6⋆ = q5 f5 A5 X5⋆ ,

(E.9)

E.3 Correspondence with Edward Laws regarding the parameter restrictions

329

was used to replace all instances of X6⋆ in the steady state equations. The remaining equations (1,2,4,5,8 and 9) may be used to attempt a solution to the steady state. However the rank of the 6 × 6 matrix to be solved is 5. Laws et al. (2000) addressed this by specifying the steady

state concentration of nutrients in the filter feeders compartment, X5⋆ .

E.3

Correspondence with Edward Laws regarding the parameter restrictions

Date: Wed, 14 May 2003 13:54:58 -1000 From: Ed Laws To: "’Nadiah Kristensen’"

Nadiah:

I don’t have the raw data anymore, but I think the numbers came from optimization studies at relatively high temperatures, probably 25 degrees C, where there would have been no chance for the system to go into a truly high export ratio mode.

On my recent trip to DC, I tried

the Fasham et al model that appears in Limnol. Oceanogr. 44: 80-94 (1999).

I am enclosing a matlab program if you want to follow up.

Interestingly, I could not find a stable steady state.

It bothers me that there are so many adjustable parameters and only one criterion - maximum resiliency. It seems that there are many combinations of these numerous parameters that produce steady states with comparable resiliency, and some of these combinations produce very different ef ratios.

I am thinking that there must be some other

constraints, or that the true constraint is some combination of ef ratio and resiliency.

Edward Laws University of Hawaii Department of Oceanography

330


E.4

Variations on the Laws Model

E.4.1

Introduction

The purpose of this appendix is to summarise the variations on the Laws Model that were investigated. The purpose of these investigations was to determine which aspects of the Laws Model were significant to its ability to predict the ef-ratio field data.

E.4.2

Method

Four models have been created, which are summarised in Table E.2. The column labelled ‘fi related?’ indicates whether each free fi parameter was related to other fi by the relationship described in Laws et al. (2000): ‘yes’ means that the fi parameters were constrained by Laws et al.’s (2000) Equations 18–20, and ‘no’ means that the fi parameters were treated as free parameters. The column labelled ‘final parameter’ indicates if X5⋆ was treated as a free parameter (Section E.2) as done by Laws et al. (2000), or if the ratio of small phytoplankton to total phytoplankton, gp was used in instead (as described in Section E.4.2). Note that the sinking rate was 1 m d−1 , as described in Appendix E.5. Table E.2: A summary of variations on the Laws Model investigated. Model Name fi related? final parameter Laws 2 no gp = X2s /(X2s + X2l ) Laws 3 yes gp = X2s /(X2s + X2l ) Laws 4 no X5 Laws 5 no g5 = X5 /(X5 + X3 ) Laws 6 yes g5 = X5 /(X5 + X3 )

Choosing the final parameter As described in Section E.2, Laws et al. (2000) used the steady state concentration in the filter feeders compartment, X5⋆ , as the final free parameter to solve the system of equations at steady state. However, this parameter is difficult to search, because its range is not restricted. A solution to this is to specify steady-state concentrations as fractions between zero and one. The equations describing these fractions can then be used to replace the redundant equation in the steady state expression. For Laws Model 2 and 4, the proportion of small phytoplankton is used gp =

⋆ X2s ⋆ + X ⋆ , where gp = (0, 1). X2s 2l

(E.10)

For Laws Model 5 and 6, the proportion of filter feeders is used g5 =

X5⋆ , where g5 = (0, 1). + X3⋆

X5⋆

(E.11)

E.4 Variations on the Laws Model

331

In summary, the steady state was described by the following six linear equations ⋆ ⋆ −L = −f2l A2l X2l − f2s A2s X2s + r3 f3 A3 X3⋆ + r4 f4 A4 X4⋆ + r5 f5 A5 X5⋆ + r6 q5 f5 A5 X5⋆ ,

(E.12a)

⋆ ⋆ 0 = (1 − gp )X2s − gp X2l , or

(E.12b)

0 = (1 − g5 )X5⋆ − gp X3⋆ ,

(E.12c)

0 = q3 f3 A3 X3⋆ − f4 A4 X4⋆ ,

(E.12d)

⋆ 0 = q2l f2l A2l (X2l + X4⋆ ) − f5 A5 X5⋆ ,

(E.12e)

⋆ 0 = s2s f2s A2s X2s + s3 f3 A3 X3⋆ + s4 f4 A4 X4⋆ − fb Ab Xb⋆ ,

(E.12f)

⋆ L = s2l f2l A2l X2l + s5 f5 A5 X5⋆ + s6 q5 f5 A5 X5⋆ + q6 q5 f5 A5 X5⋆ .

(E.12g)

E.4.3


Figure E.1 shows the predictive ability of each of the goal functions for each of the Laws Model variations. For Laws Models 3 and 6, which both had the fi parameter relationships of Equations 18–20, ‘maximise resilience’ shows predictive ability comparable to the original Laws Model. In contrast, the remaining models have reduced predictive ability. All models for which all fi parameters are free parameters underestimate the ef-ratio when ‘maximise resilience’ is used. Using a biomass fraction as the free parameter instead of X5⋆ has little effect upon the results. Boxplots of the feasible stable region showed no predictive ability, and are omitted for brevity. Tables E.3 and E.4 show the parameter values for the maximally resilient point for Laws 2 and Laws 3 respectively. For brevity, the oceans investigated have been restricted to those which had ef ratios within the range of the Laws models. Because Laws 2 has a larger parameter volume to search, the maximum resilience value found in Laws 2 is always higher than that in Laws 3. Table E.3: Parameter values that maximise resilience for Laws 2. Ocean ID R f2l f3 f5 f6 gp 3 0.0397 0.344 0.189 0.777 0.446 0.499 7 0.0094 0.621 0.397 0.419 0.407 0.625 8 0.0188 0.212 0.309 0.582 0.358 0.469 9 0.0721 0.274 0.081 0.715 0.399 0.399 10 0.0619 0.306 0.055 0.637 0.353 0.159 11 0.0106 0.189 0.313 0.586 0.360 0.625 Experimentation with the Laws model revealed that 1. Restricting f3 affected resilience, but the ef-ratio was insensitive to f3 ; and 2. the ef-ratio was sensitive to gp .

332


Table E.4: Parameter values that maximise Ocean ID R f2l 3 0.0321 0.196 7 0.0076 0.163 8 0.0128 0.134 9 0.0554 0.254 10 0.0613 0.218 11 0.0073 0.135

resilience for Laws 3. gp 0.083 0.011 0.157 0.176 0.227 0.0576

Figures E.2 to E.7 were generated to explore this relationship. These figures were generated such that all parameters were held constant, at the values that maximised resilience, except gp . Both Laws 2 and Laws 3 are shown, with the resulting ef values and resilience scaled for comparison. From Figures E.2 to E.7, it can be seen that, in general, Laws 2 results in maximal resilience found at lower values of gp than Laws 3. As the relationship between gp and ef is a consistent monotonic decrease, this implies that maximising resilience in Laws 2 gives higher ef estimates than Laws 3. Greenland Polynya provides an example of how restricting f3 affects the predicted ef-ratio. Keeping all other parameters constant (where the parameter values were taken as the average of the values that maximised resilience in Laws 2 and Laws 3), the resilience surface was plotted against f3 and gp (Figure E.8). Figure E.8 shows two resilience optima. One is located on the left of the figure near (gp , f3 ) = (0.058, 0.045), and the second is a ridge running almost parallel to gp = (0.35, 0.7) at about f3 ≈ 0.35. If f3 is restricted to low values, the local optima is located at the lower end of the gp range maximises resilience. However, if f3 is free, the local optima at the higher end of the gp range becomes the global optima, which corresponds to a lower ef-ratio.

333


Laws Model 2

Laws Model 3

0.7

0.7 1:1 Line Max B Max P Max P/B Max R

0.6

0.5

0.5

0.4

0.4

Model ef

Model ef

0.6

1:1 Line Max B Max P Max P/B Max R

0.3

0.3

0.2

0.2

0.1

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0

0 0

0.1

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0

0.1

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Laws Model 4

0.6

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0.7

0.7 1:1 Line Max B Max P Max P/B Max R

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1:1 Line Max B Max P Max P/B Max R

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0

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Laws Model 6 0.7 1:1 Line Max B Max P Max P/B Max R

0.6

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Model ef

0.4 0.5 Observed ef

0.4

0.3

0.2

0.1

0 0

0.1

0.2

0.3

0.4 0.5 Observed ef

0.6

0.7

0.8

Figure E.1: Model versus observed ef for Laws Model variations.

0.6

0.7

0.8

334


Resilience and ef ratio versus gp for Laws 2 optimal parameter set. Ocean 3.


0.7

0.7 line 1 line 2

0.6

0.6

0.5

0.5 10*Resilience and ef ratio

10*Resilience and ef ratio

line 1 line 2

0.4

0.3

0.4

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0.2

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0.1

0.1

0

0 0

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0.5

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0.7

0

0.1

0.2

0.3

gp

0.4

0.5

0.6

0.7

gp

Figure E.2: Ocean 3. Variation in resilience and ef with changes in the parameter gp , for models Laws 2 and Laws 3. All other parameters held constant at the values that maximised resilience.



1

0.8 line 1 line 2

line 1 line 2

0.9

0.7



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0.45 0.35 10*Resilience and ef ratio


0.4 0.35 0.3 0.25 0.2 0.15

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335




0.75

0.75 line 1 line 2

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line 1 line 2

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line 1 line 2 0.6

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gp

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gp


Resilience and ef ratio versus gp for Laws 2 optimal parameter set

Resilience and ef ratio versus gp for Laws 3 optimal parameter set

1.1

0.8 ef ratio 100*resilience

ef ratio 100*resilience

1 0.7

0.8



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0.5 gp

0.6

0.7

0.8

0.9

1


336


Resilience surface in parameter space

Resilience

0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

0

0.1

0.2

0.3 gp

0.4

0.5

0.6

0.7 0

0.1 0.05

0.2 0.15

0.3 0.25

0.4 0.35

0.45

f3

Figure E.8: Ocean 11. The resilience surface in f3 × gp parameter space.

E.5

A note on the sinking rate

In the literature, the sinking rate of detritus is generally quoted as being between 1 and 10m d−1 (e.g. Fasham et al. 1990). For the original Laws Model, Laws et al. (2000) used a value of 1m d−1 . In investigating variations upon the Laws Model, it was observed that, for all ocean regions except Ross Sea, and Greenland Polynya, when the sinking rate of 1m d−1 was used, there was a plateau in the resilience surface in the parameter space. Figure E.9 shows an example of this for the BATS ocean region. This plateau is due to the (9, 9) element of the Jacobian matrix, which is the only entry in the ninth column. J9,9 =

∂(X˙ d ) = −V /M = −Xd⋆ , ∂Xd

(E.13)

where V is the sinking rate, and M the mixed layer depth. It is associated with the unit vector in the Xd direction. When V = 1m d−1 , J9,9 is a relatively large (more positive) negative number, and this eigenvalue dominates the system. Therefore, the resilience becomes equal to this eigenvalue for a wide

337

E.5 A note on the sinking rate

Resilience versus attribute space for Laws 3

R

0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 1 0.8 0

0.6 0.2

0.4 f2l

0.4 0.6

gp

0.2 0.8

1 0

Figure E.9: Plateau in attribute space about which resilience is maximised. V = 1m d−1 . Ocean ID 1.

range of parameter values. I believe that the reason that this did not occur in the original Laws Model was because of a small error in the calculation of the steady-state detritus concentration, Xd⋆ , overestimating its value. When Xd⋆ is large, the eigenvalue resulting from the (9, 9) element of the Jacobian Matrix no longer dominates the system. A similar effect can be achieved by changing the sinking rate to 10m d−1 . Because this encapsulates more information about the system, and is consistent with Laws et al.’s (2000) earlier results, I have done this for all models used. This miscalculation does not affect the parameters that maximise the resilience goal function, nor the prediction of the ef-ratio, in the original Laws Model.

Appendix F

Appendices relating to Chapter 9 F.1

Making ef-ratio the free parameter

The purpose of this appendix is to detail how the steady state to the Laws Model was solved such that one of the free parameters was the ef-ratio. Recall in Appendix E.2 that the filter-feeder steady state nutrient concentration, X5⋆ , was set as a free parameter, which allowed us to solve the system at steady state. In Appendix E.4.2, it was shown that instead of using X5⋆ as the free parameter, a new parameter, gp , could be defined and used to replace the redundant equation in the steady state expression. Similarly, if one defines the ef-ratio, the following equation can be used to replace the redundant steady state equation L = f2s A2s + f2l A2l . ef

(F.1)

The field data offers measurements of both L and Ptotal , so specifying the ef-ratio is equivalent to specifying either L or Ptotal , and using the model to predict the other.

338

339

F.2 The ef-ratio isocline key

F.2

The ef-ratio isocline key

Section 9.3.3 of the text presents contours of the maximal resilience for each ocean region in total production versus loading rate space. They are Figures 9.6 and 9.8. These figures include ef-ratio isoclines showing the position of equal ef-ratio values in total production versus loading rate space. Figure F.1 below provides a reference for the position and values of the ef-ratio isoclines. Position of ef−ratios 500 ef=0.7

ef=0.6

ef=0.5

ef=0.4

450

ef=0.3

400

350

Loading rate

300

ef=0.2

250

200 ef=0.1

150

100

50

200

400

600

800 Total production

1000

1200

1400

Figure F.1: A graph showing the position of ef-ratio isoclines in total production versus loading rate space. For use with Figure 9.6 and Figure 9.8.

Appendix G

Appendices relating to Chapter 10 G.1

Comparing the Fasham Model and the Laws Model

This appendix was created as a quick reference for comparing the Fasham Model and the Laws Model. Table G.1 compares differential equations governing the system. Table G.2 compares fixed parameters. Table G.3 compares the flow of nutrients between compartments.

340

G.1 Comparing the Fasham Model and the Laws Model

Table G.1: Comparing the differential equations governing the Fasham Model and the Laws Model. Laws Model Fasham Model Name Equation Equation Name N˙n = −JQ1 P Nitrate Inorganic nutrient X˙ 1 = L − (1 − r2l )F2l − (1 − r2s )F2s + r3 F3 + r4 F4 + r5 F5 + r6 F6 + rb Fb 2l P˙ = (1 − γ1 )σP − G1 − µ1 P Phytoplankton Large phytoplankton X˙ 2l = q2l F2l − F5 X2lX+X 4 X2s ˙ Small phytoplankton X2s = q2s F2s − F3 X2s +Xb σ = J(Q1 + Q2 ) Z˙ = β1 G1 + β2 G2 + β3 G3 − (µ2 + µ5 )Z Zooplankton Flagellates X˙ 3 = q3 F3 − F4 X4 ˙ Ciliates X4 = q4 F4 − F5 X2l +X4 Filter feeders X˙ 5 = q5 F5 − F6 Carnivores X˙ 6 = q6 F6 − M X6 (not included) ˙ ˙ Nr = −JQ2 P − U2 + µ3 B Ammonium DOM Xc = s2s F2s + s3 F3 + s4 F4 − Fb N˙ d = γ1 J(Q1 + Q2 )P + µ4 D + (1 − ǫ)Z − U1 DON D˙ = (1 − β1 )G1 + (1 − β2 )G2 − β3 G3 − Detritus Detritus X˙ d = s2l F2l + s5 F5 + s6 F6 − DXd V D µ4 D + µ1 P − M X b ˙ = U1 + U2 − G2 − µ3 B B Bacteria Bacteria X˙ b = qb Fb − F3 X2s +X b

341

342

Table G.2: Comparing the parameter values in the Fasham Model and the Laws Model. Laws Model Fasham Model Description Notation Value Value Notation Description Small phytoplankton assimilation efficiency q2s 0.7 0.7 1 − γ1 1− (total net primary production Large phytoplankton assimilation efficiency q2L 0.7 exuded by phytoplankton) Flagellate assimilation efficiency q3 0.35 0.35 β Zooplankton assimilation efficiency Ciliate assimilation efficiency q4 0.35 Filter feeder assimilation efficiency q5 0.3 Carnivore assimilation efficiency q6 0.35 (not included) Bacteria assimilation efficiency qB 1.00 1.00 (implicit) Small phytoplankton respiration r2s 0 0 (not included) Small phytoplankton respiration r2L 0 0 Flagellate respiration r3 0.3 0 Ciliate respiration r4 0.3 0 Filter feeders respiration r5 0.3 0 Carnivore respiration r6 0.3 0 Bacteria respiration rB 1 − qB 0


flow to phytoplankton small phyto large phyto zooplankton filter feeders flagellates detritus detritus DON DOM ammonium DON DOM DOM detritus detritus detritus

type uptake uptake uptake grazing grazing grazing death exudation exudation exudation death & excretion excretions exudation exudation feeding inefficiency excretion excretion

Notation (1 − γ1 )JQ1 P q2s F2s q2l F2l βG1 q5 F5 (X2l /(X2l + X4 )) q3 F3 (X2s /(X2s + Xb )) µ1 P s2l F2l γ1 JQP s2s F2s (ǫµ2 + (1 − ω)µ5 )Z (1 − ǫ)µ2 Z s3 F3 s4 F4 (1 − β)(G1 + G2 ) s5 F5 s6 F6

Value 1.2Q1 exp(0.0633(T − 25))P 1.2f2s exp(0.0633(T − 25))X2s 1.2f2l exp(0.0633(T − 25))X2l 2.4fZ exp(0.1(T − 25))Z 0.15f5 exp(0.1(T − 25))(X2l /(X2l + X4 ))X5 2.4f3 exp(0.1(T − 25))(X2s /(X2s + Xb ))X3 0.045 exp(0.0633(T − 25))P 0.51f2l exp(0.0633(T − 25))X2l 0.51Q exp(0.0633(T − 25))P 0.51f2s exp(0.0633(T − 25))X2s 0.596 exp(0.1(T − 25))Z 0.1875 exp(0.1(T − 25))Z 2.4f3 exp(0.1(T − 25))X3 2.4f4 exp(0.1(T − 25))X4 1.7fZ exp(0.1(T − 25))Z 0.2f5 exp(0.1(T − 25))X5 0.175f6 exp(0.1(T − 25))X6

zooplankton flagellates ammonium

grazing grazing excretion

βG2 q3 F3 (Xb /(Xb + X2s )) µ3 B

2.4fZ exp(0.1(T − 25))Z 2.4f3 exp(0.1(T − 25))(Xb /(Xb + X2s ))X3 0.05 exp(0.0633(T − 25))B

bacteria bacteria bacteria

uptake uptake uptake

qb Fb U1 U2

Fasham Laws Fasham Laws Fasham Laws Laws

ammonium (not included) detritus (not included) detritus (not included) biota

phytoplankton

uptake

(1 − γ1 )Q2 JP

1.2 exp(0.0633(T − 25))Xb (Nd /(0.5 + S + Nd )1.2 exp(0.0633(T − 25))B (S/(0.5 + S + Nd )1.2 exp(0.0633(T − 25))B S = min(Nr , 0.6Nd ) 1.2Q2 exp(0.0633(T − 25))P

zooplankton

uptake

βG3

2.4fZ exp(0.1(T − 25))Z

DON

breakdown

µ4 D

0.25 exp(0.0633(T − 25))D

nitrate

respiration

r3 F3 + r4 F4 + r5 F5 + r6 F6

(f3 X3 + f4 X4 )(2.06 exp(0.1(T − 25)))+ (f5 X5 + f6 X6 )(0.15 exp(0.1(T − 25)))

Fasham

(not included)

Fasham Laws Fasham Laws Fasham Laws Fasham Laws Fasham Laws Laws Fasham Fasham Laws Fasham Laws Laws Fasham

343

flow from nitrate nitrate nitrate phytoplankton large phyto small phyto phytoplankton phytoplankton phytoplankton phytoplankton zooplankton zooplankton flagellates ciliate zooplankton filter feeders carnivores (not included) bacteria bacteria bacteria (not included) DOM DON ammonium

G.1 Comparing the Fasham Model and the Laws Model

Table G.3: Comparing the nutrient flow terms in the Fasham Model and the Laws Model. Model Fasham Laws

344


G.2

Documentation of the Fasham Model Fasham Model Section Page

Overview of the ss8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

345

Detritus – Finding the steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

346

Detritus – Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

346

Using differential equations to find steady states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

347

Zooplankton parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

349

Bacteria parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

349

The Q parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

350

Solve for phytoplankton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

351

The introduced f1 parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

351

Solve for zooplankton and bacteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

352

Solve for nutrient compartments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

353

Nutrient compartment parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

354

Restrictions upon parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

355

Solve each compartment for the steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

355

Find the resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

356

Create the Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

357

Differentiate common terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

359

Resilience equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

360

Return goal function value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

360

Overview of maxgoalfinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

363

Overview of oceanp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

365

G.2 Documentation of the Fasham Model

1.

Overview of the ss8.

345

The following sections tersely describe the reasoning and process used

implement the Fasham et al. (1990) model. Table G.4 lists the functions that are invoked by the code. The objective of the code is to find the value of a user-specified goal function for the system. For reasons that will be apparent later, the goal function is evaluated for a user-specified input value of Q2 and ef . The process has the following structure: 1. Solve each compartment for the steady state If the steady state is feasible 2. Use the steady state to find the resilience If the system is stable 3. Return the value of the goal function. h ss8.m

1i

≡

function res = ss8 (goalId , oceanParams , Q2 , ef , unstabFlag , unfeasFlag ) % A function that, when given ocean-specific parameters, Q2, ef and % error flags, returns the goal function value of the system as % found with Fasham et al.’s (1990) model. % % Designed for Octave % In conjunction with documentation % % INPUT % —– % function res=ss7(oceanParams,Q2,ef,unstabFlag,unfeasFlag) % oceanParams: Parameter values specific to the ocean under % consideration [T M L] % ef: Ratio of loading rate to new production % Q2: Nr /(K + Nr ) % unxxFlag: Error flags for unfeasible and unstable systems. % Set to zero if unfeasible/unstable systems need to be % evaluated. h Solve each compartment for ss h Find the resilience

15 i

14 i

346

2.


Detritus – Finding the steady state.

We begin by finding the steady state of the detritus

compartment. It is assumed that there is no loss of nutrients from the system except through the sinking of detritus. This means that terms of the form (m + h(t))P , M that appear in each of the differential equations in Fasham et al. (1990) may be ignored. This implies that the steady state of the detritus compartment may be found by taking a mass balance around the entire system D⋆ = LM/V. h Solve for detritus

(G.1) 2i

≡

D = L∗(M /V ); This code is used in section 14.

3.

Detritus – Parameters.

In order to find the steady state of the detritus compartment as

described above, the parameter values must be specified first. They are detailed in Table G.5. h Detritus parameters

3i

≡

% Detritus parameters – see Table G.5 of documentation. V = 1; L = oceanParams (3);

% Nutrient loading rate

% Convert to mMol N m−2 day for consistency with units in Fasham et al. (1990) L = L/(14); M = oceanParams (1); T = oceanParams (2); This code is used in section 14.

% Mixed layer depth % Temperature


4.

347

Using differential equations to find steady states. After the detritus compartment is solved,

we are left with 6 differential equations with which to find the steady state of the other compartments. Determining the best course of action required some trial and error, so part of the reasoning will be discussed here. First consider the zooplankton differential equation dZ = β1 G1 + β2 G2 + β3 G3 − (µ2 + µ5 )Z, dt where the Gi describe predation of the zooplankton compartment upon the phytoplankton, bacteria and detritus compartments respectively of the form Gi = gZ

pi Xi K3 + (Σpj Xj )

After Laws (n.d.), this may be reformulated as Gi = gZfi,4 where fi,4 = [0, 1], so dZ = β1 gf1,4 Z + β2 gf2,4 Z + β3 gf3,4 Z − (µ2 + µ5 )Z dt It can be seen that, when dZ/dt = 0 at the steady state, this differential equation does not reduce to a form useful for finding the steady state of the zooplankton compartment, Z ⋆ . With this in mind, consider the differential equation describing the phytoplankton compartment (Fasham et al. 1990) dP = (1 − γ1 )σP − G1 − µ1 P, dt where σ = J(Q1 + Q2 ). At the steady state, this may be reformulated as 0 = (1 − γ1 )σP ⋆ − gP ⋆ f1 − µ1 P ⋆ .

(G.3)

In this case, f1 does not have a simple 0 to 1 range, however f1 may be found for the steady state by rearranging Equation G.3 to f1 =

K3 + (p1

P⋆

p1 Z ⋆ (1 − γ1 )σ − µ1 = . + p2 B ⋆ + p3 D ⋆ ) g

(G.4)

The phytoplankton differential equation is now in a form that is not convenient for for solving P ⋆ , however the zooplankton differential equation may be reformulated in terms of f1 dZ = β1 gf1 P + β2 gf1 B + β3 gf1 D − (µ2 + µ5 )Z, dt

(G.5)

348


which has the potential to solve for P ⋆ , Z ⋆ , and B ⋆ . We will consider finding B ⋆ first, as it happens that this will solve P ⋆ as a matter of course. Consider the bacteria differential equation dB = U1 + U2 − G2 − µ3 B dt

(G.6a)

where Vb BNd K 4 + S + Nd Vb BS U2 = K 4 + S + Nd

(G.6b)

U1 =

(G.6c)

We can see that, if alternate expressions are found for U1 and U2 that do not involve the nutrient compartments, Equation G.6a can be used to find solutions for the steady states of the biotic compartments. An appropriate expression may be found by performing a mass balance around the DON and Ammonium compartments. At steady state, the differential equation describing DON may be rearranged to U1 = µ4 D⋆ + γ1 J(Q1 + Q2 )P ⋆ + (1 − ε)µ2 Z ⋆ ,

(G.7)

and the differential equation for Ammonium rearranged to U2 = −JQ2 P ⋆ + µ3 B ⋆ + εµ2 Z ⋆ + (1 − ω)µ5 Z ⋆ .

(G.8)

From the above, we can begin to work our way toward solving the biotic compartments for the steady state. h Solve other equations

4i

≡

h Biotic compartment parameters h Solve for biotic compartments This code is used in section 14.

5i

8i


5.

349

Zooplankton parameters. The parameter values used to find the steady state of the zooplankton

compartment are described in Table G.6. h Biotic compartment parameters

5i

≡

% Zooplankton parameters – see Table G.6 of documentation. beta = .35; g = (2.4/beta )∗exp(.1∗(T − 25)); mu2 = beta ∗exp(.1∗(T − 25)); mu5 = .05∗exp(.1∗(T − 25)); gamma1 = .3; J = (1.2/(1 − gamma1 ))∗exp(.0633∗(T − 25)); mu1 = .045∗exp(0.0633∗(T − 25)); See also sections 6, 7, and 9. This code is used in section 4.

6.

Bacteria parameters.

The parameter values used to find the steady state of the bacteria

compartment are described in Table G.7. h Biotic compartment parameters

5i

+≡

% Bacteria parameters – see Table G.7 of documentation. Vb = 1.2∗exp(0.0633∗(T − 25)); mu3 = .05∗exp(.0633∗(T − 25)); mu4 = .25∗exp(.0633∗(T − 25)); K4 = 0.5; eps = .75; omega = .33;

350

7.


The Q parameters. Q1 and Q2 are Nutrient limitation factors of phytoplankton uptake of nitrate

and ammonium respectively. Fasham et al. (1990) gives descriptions for both in which ammonium is taken up preferentially Nn e−ΨNr K 1 + Nn Nr Q2 = K 2 + Nr

(G.9a)

Q1 =

(G.9b)

However, our objective is to quantify Qi in order to solve the system of equations given in Equation G.15, so it is preferable to avoid involving more state variables at this point. To achieve this, we consider the inputs into the phytoplankton compartment. It can be shown that, at steady state, Q1 is related to the loading rate by L = JQ1 P ⋆ ,

(G.10)

and, if the ratio ef is specified, Q = Q1 + Q2 is related to ef by ef =

L JQP ⋆

(G.11)

To solve for the steady state, ef and Q2 are specified as input variables. Q2 has been chosen over Q because it is known (by examining Equations G.9) that Q1 and Q2 have the range [0, 1], whereas the dependence of Q1 upon Nr (and therefore Q2 ) makes the range of Q difficult to predict. It can be shown that ef =

Q1 , Q1 + Q2

(G.12)

so Q1 is solved by Q1 =

ef Q2 1 − ef

h Biotic compartment parameters Q1 = ef ∗Q2 /(1 − ef ); Q = Q1 + Q2 ;

(G.13) 5i

+≡


8.

Solve for phytoplankton.

351

By specifying ef and Q2 , the phytoplankton steady state can be

found by rearranging Equation G.11 to P⋆ =

L , ef JQ

(G.14)

which serves as an input to system of Equations G.15. h Solve for biotic compartments

8i

≡

P = L/(ef ∗J∗(Q)); See also section 10. This code is used in section 4.

9.

The introduced f1 parameter. Having specified Q, we are now ready to evaluate f1 .

This model arbitrarily uses pi = 1 ∀i, which means that zooplankton does not have any food preference. h Biotic compartment parameters

5i

+≡

f1 = ((1 − gamma1 )∗J∗Q − mu1 )/g; p1 = 1; p2 = 1; p3 = 1;

% No preference for zooplankton feeding

352


10.

Solve for zooplankton and bacteria.

So far, we have two differential equations which may

be used to solve for the steady state values of bacteria and zooplankton as follows. 0 = β1 gf1 P ⋆ + β2 gf1 B ⋆ + β3 gf1 D⋆ − (µ2 + µ5 )Z ⋆ ,

(G.15a)

is found by setting Equation G.5 to zero. By setting Equation G.6a to zero, and substituting in Equation G.7 and G.8 0 = µ4 D⋆ + γ1 J(Q1 + Q2 )P ⋆ + µ2 Z ⋆ − JQ2 P ⋆ + (1 − ω)µ5 Z ⋆ − gB ⋆ f1 .

(G.15b)

Now, both of the above equations may be solved via matrix methods, provided that values are given for Qi and P ⋆ h Solve for biotic compartments

8i

+≡

% Zooplankton row 1 B(1, 1) = −(mu2 + mu5 ); B(1, 2) = beta ∗g∗f1 ; LHS (1, 1) = beta ∗g∗f1 ∗(P + D); % Bacteria row 2 (after mass balance) B(2, 1) = mu2 + (1 − omega )∗mu5 ; B(2, 2) = −g∗f1 ; LHS (2, 1) = mu4 ∗D + gamma1 ∗J∗Q1 ∗P − (1 − gamma1 )∗J∗Q2 ∗P ; if cond (B) > 1 · 105 res = unfeasFlag ; return end % Solve linear system for bacteria and zooplankton X = B\(−LHS ); B = X(2, 1); Z = X(1, 1);

353


11.

Solve for nutrient compartments. Having solved for D⋆ , P ⋆ , B ⋆ and Z ⋆ , these steady state

values can now be used to solve for Nr⋆ , Nd⋆ and Nn⋆ . By Equation 14 of Fasham et al. (1990), S = min(Nr , ηNd ).

(G.16)

By Equation G.6b, it can be seen that if S = Nd , U1 = U2 , where U1 and U2 are calculated by Equations G.7 and G.8. It has been observed that the probability of this occurring is infinitesimally small, so Nr⋆ = S ⋆ .

(G.17)

Now, three sets of equations may be found by rearranging Equations G.6, and Equations G.9, which will solve for the three unknowns. First, rearranging each of the expressions for bacterial uptake gives α1 =

Nd⋆ U1 = , VB B ⋆ K4 + Nr⋆ + Nd⋆

(G.18a)

α2 =

Nr⋆ U2 = . VB B ⋆ K4 + Nr⋆ + Nd⋆

(G.18b)

and

The differential equation describing the change in nutrient in the bacteria compartment gives α3 =

G2 + µ3 B ⋆ Nd⋆ + Nr⋆ . = ⋆ ⋆ K 4 + Nr + Nd VB B ⋆

(G.18c)

Now the above equations may be combined to isolate Nr⋆ and Nd⋆ Nd⋆ =

K4 α1 , 1 − α3

(G.19a)

Nr⋆ =

K4 α2 . 1 − α3

(G.19b)

Substituting in the value of Nr⋆ into Equation G.9 and rearranging gives Nn⋆ =

K 1 Q1 ⋆ −ΨN r − Q e

.

(G.20)

1

h Solve for N compartments

11 i

≡

U1 = gamma1 ∗J∗Q∗P + mu4 ∗D + (1 − eps )∗mu2 ∗Z; U2 = −J∗Q2 ∗P + mu3 ∗B + eps ∗mu2 ∗Z + (1 − omega )∗mu5 ∗Z; G1 = g∗f1 ∗P ; G2 = g∗f1 ∗B; G3 = g∗f1 ∗D;

354


alpha1 = U1 /(Vb ∗B); alpha2 = U2 /(Vb ∗B); alpha3 = (G2 + mu3 ∗B)/(Vb ∗B); % Solve other state variables for steady state Nd = K4 ∗alpha1 /(1 − alpha3 ); Nr = K4 ∗alpha2 /(1 − alpha3 ); Nn = K1 ∗Q1 /(exp(−psi ∗Nr ) − Q1 ); This code is used in section 14.

12.

Nutrient compartment parameters.

The parameter values used to find the steady state of

the nutrient compartments are described in Table G.8. h N parameters

12 i

≡

% Nutrient compartments parameters – see Table G.8 of documentation. psi = 1.5; K1 = 0.5; K2 = 1; eta = 0.6; This code is used in section 14.

355


13.

Restrictions upon parameter values. There are some restrictions upon the values that the

nutrient expressions may take. Recalling the derivation of Equation G.17, the first restriction would be that Nr < η. Nd

(G.21)

In addition to this it can be seen that Nd eta ) ∨ (Nd /(K4 + Nr + Nd ) > 1) ∨ ((Nr + Nd )/(K4 + Nr + Nd ) > 1) % This is not allowed res = unfeasFlag ; return end This code is used in section 15.

14.

Solve each compartment for the steady state.

been solved for the steady state. An overview is provided. h Solve each compartment for ss h Detritus parameters h Solve for detritus

3i

2i

h Solve other equations h N parameters

14 i

4i

12 i

h Solve for N compartments This code is used in section 1.

11 i

≡

Now that all of the compartments have

356


15.

Find the resilience.

Once each of the steady states have been found, the resilience of the

system may be determined. First, because it is computationally expensive to find eigenvalues, we only proceed if the steady state is feasible, unless the user indicates otherwise with the unfeasFlag input. Second, previously discussed (Section 13) restrictions upon parameter values are tested for. Provided that we do have positive values for nutrient concentration in each of the compartments, and the restrictions are satisfied, we can proceed to create the Jacobian Matrix and apply the definition of resilience to the matrix. Once resilience is calculated, the user defines whether or not a negative value for resilience will be accepted. Provided that the resilience is positive, or the user indicates that they will accept an unstable system, the value of the goal functions may be returned. h Find the resilience

15 i

≡

if min([P B D Z Nr Nd Nn ]) > 0 ∨ unfeasFlag ≡ 0 % Only want feasible systems unless told otherwise h If satisfy par range h Create Jacobian

13 i

16 i

h Resilience equation

18 i

if R > 0 ∨ unstabFlag ≡ 0

% Only want stable systems unless told otherwise

h Return goal function value

19 i

return else res = unstabFlag ;

% User defined error flag

return end else

% System was not feasible

res = unfeasFlag ; return end This code is used in section 1.

% User defined error flag

357


16.

Create the Jacobian Matrix. Before creating the complete Jacobian Matrix, it is convenient to

differentiate each of the terms that commonly occur in the differential equations. Once this is done, each compartment’s differential equation is differentiated with respect to each compartment and evaluated at the steady state. h Create Jacobian

16 i

≡

h Diff common terms

17 i

% Now find each element of the Jacobian % Phytoplankton Jac (1, 1) = (1 − gamma1 )∗sigma − dG1dP − mu1 ; Jac (1, 2) = −dG1dZ ;

% dP’/dZ

Jac (1, 3) = −dG1dB ;

% dP’/dB

Jac (1, 4) = (1 − gamma1 )∗J∗P ∗(dQ1dNn );

% dP’/dP

% dP’/dNn

Jac (1, 5) = (1 − gamma1 )∗J∗P ∗(dQ1dNr + dQ2dNr ); Jac (1, 6) = 0;

% dP’/dNr

% dP’/dNd

Jac (1, 7) = −dG1dD ;

% dP’/dD

% Zooplankton Jac (2, 1) = beta ∗(dG1dP + dG2dP + dG3dP );

% dZ’/dP

Jac (2, 2) = beta ∗(dG1dZ + dG2dZ + dG3dZ ) − (mu2 + mu5 ); Jac (2, 3) = beta ∗(dG1dB + dG2dB + dG3dB );

% dZ’/dB

Jac (2, 4 : 6) = zeros (1, 3); Jac (2, 7) = beta ∗(dG1dD + dG2dD + dG3dD );

% dZ’/dD

% Bacteria Jac (3, 1) = −dG2dP ;

% dB’/dP

Jac (3, 2) = −dG2dZ ;

% dB’/dZ

Jac (3, 3) = dU1dB + dU2dB − dG2dB − mu3 ; Jac (3, 4) = 0;

% dB’/dNn

Jac (3, 5) = dU1dNr + dU2dNr ;

% dB’/dNr

Jac (3, 6) = dU1dNd + dU2dNd ;

% dB’/dNd

Jac (3, 7) = −dG2dD ;

% dB’/dD

% Nitrate nitrogen Jac (4, 1) = −J∗Q1 ;

% dNn’/dP

Jac (4, 2 : 3) = zeros (1, 2); Jac (4, 4) = −J∗P ∗dQ1dNn ;

% dNn’/dNn

Jac (4, 5) = −J∗P ∗dQ1dNr ;

% dNn’/dNr

Jac (4, 6 : 7) = zeros (1, 2);

% dB’/dB

% dZ’/dZ

358


% Ammonium nitrogen Jac (5, 1) = −J∗Q2 ;

% dNr’/dP

Jac (5, 2) = eps ∗mu2 + (1 − omega )∗mu5 ; Jac (5, 3) = −dU2dB + mu3 ; Jac (5, 4) = 0;

% dNr’/dZ

% dNr’/dB

% dNr’/dNn

Jac (5, 5) = −J∗P ∗dQ2dNr − dU2dNr ; Jac (5, 6) = −dU2dNd ; Jac (5, 7) = 0;

% dNr’/dNr

% dNr’/dNd

% dNr’/dD

% DON Jac (6, 1) = gamma1 ∗J∗Q;

% dNd’/dP

Jac (6, 2) = (1 − eps )∗mu2 ;

% dNd’/dZ

Jac (6, 3) = −dU1dB + mu3 ;

% dNd’/dB

Jac (6, 4) = gamma1 ∗J∗P ∗dQ1dNn ;

% dNd’/dNn

Jac (6, 5) = gamma1 ∗J∗P ∗(dQ1dNr + dQ2dNr ) − dU1dNr ; Jac (6, 6) = −dU1dNd ; Jac (6, 7) = mu4 ;

% dNd’/dNr

% dNd’/dNd % dNd’/dD

% Detritus Jac (7, 1) = (1 − beta )∗dG1dP + (1 − beta )∗dG2dP − beta ∗dG3dP + mu1 ;

% dD’/dP

Jac (7, 2) = (1 − beta )∗dG1dZ + (1 − beta )∗dG2dZ − beta ∗dG3dZ ;

% dD’/dZ

Jac (7, 3) = (1 − beta )∗dG1dB + (1 − beta )∗dG2dB − beta ∗dG3dB ;

% dD’/dB

Jac (7, 4 : 6) = zeros (1, 3); Jac (7, 7) = (1 − beta )∗dG1dD + (1 − beta )∗dG2dD − beta ∗dG3dD − mu4 − (V /M ); This code is used in section 15.


17.

Differentiate common terms.

Common terms in the partial differential equations are

differentiated before constructing the Jacobian Matrix. h Diff common terms

17 i

≡

sigma = J∗(Q1 + Q2 ); K3plsF = p1 ∗Z/f1 ;

% K3+F

K3 = K3plsF − p1 ∗P − p2 ∗B − p3 ∗D; F = K3plsF − K3 ; dG1dZ = (g∗p1 ∗P )/(K3 + F ); dG2dZ = (g∗p2 ∗B)/(K3 + F ); dG3dZ = (g∗p3 ∗D)/(K3 + F ); dG1dP = g∗Z∗p1 ∗(K3 + F − p1 ∗P )/(K3 + F )ˆ2; dG1dB = (K3 + F − g∗Z∗p1 ∗p2 ∗P )/(K3 + F )ˆ2; dG1dD = (K3 + F − g∗Z∗p1 ∗p3 ∗P )/(K3 + F )ˆ2; dG2dB = g∗Z∗p2 ∗(K3 + F − p2 ∗B)/(K3 + F )ˆ2; dG2dP = (K3 + F − g∗Z∗p2 ∗p1 ∗B)/(K3 + F )ˆ2; dG2dD = (K3 + F − g∗Z∗p2 ∗p3 ∗B)/(K3 + F )ˆ2; dG3dD = g∗Z∗p3 ∗(K3 + F − p3 ∗D)/(K3 + F )ˆ2; dG3dP = (K3 + F − g∗Z∗p3 ∗p1 ∗D)/(K3 + F )ˆ2; dG3dB = (K3 + F − g∗Z∗p3 ∗p2 ∗D)/(K3 + F )ˆ2; dQ1dNn = K1 ∗exp(−psi ∗Nr )/(K1 + Nn )ˆ2; dQ1dNr = −psi ∗Q1 ; dQ2dNr = K2 /(K2 + Nr )ˆ2; dU1dB = U1 /B; dU1dNd = Vb ∗B∗(K4 + Nr )/(K4 + Nr + Nd )ˆ2; dU1dNr = (−Vb ∗B∗Nd )/(K4 + Nr + Nd )ˆ2; dU2dB = U2 /B; dU2dNr = Vb ∗B∗(K4 + Nd )/(K4 + Nr + Nd )ˆ2; dU2dNd = (−Vb ∗B∗Nr )/(K4 + Nr + Nd )ˆ2; This code is used in section 16.

359

360


18.

Resilience equation. The resilience of a stable system is defined as the negative of the maximum

(most positive) real part of the eigenvalues of the Jacobian matrix. This can now be evaluated. h Resilience equation

18 i

≡

R = −max(real(eig(Jac ))); This code is used in section 15.

19.

Return goal function value. Each of the goal functions are defined in the body of the paper.

Which goal function is returned depends upon the user-specified goalId . h Return goal function value

19 i

≡

switch goalId case 1

% Biomass

res = sum([P B Z]); case 2

% Production

res = J∗P ∗Q; case 3

% Production/Biomass

res = J∗P ∗Q/sum([P B Z]); case 4

% Resilience

res = R; otherwise error (’Goal function not defined’) end This code is used in section 15.


eig exp max min real sum

361

Table G.4: List of functions. In-built Octave functions Eigenvalues of a matrix ex Finds the maximum Finds the minimum Real parts of a vector Sum the elements of a vector

User-created functions ss8 Section 1 maxgoalfinder Section 20 oceanp Section 21

Table G.5: Parameters used for finding the steady state of detritus. Parameter Value Units Comments V 1 m d−1 Sinking rate of detritus. Standard value (Fasham et al. 1990). L – mMol N m−2 d−1 Nutrient loading. Dependent upon location (Laws et al. 2000). M – m Mixed layer depth. Dependent upon location (Laws et al. 2000).

Table G.6: Parameters used for finding the steady state with the zooplankton differential equation Parameter Value Units Comments β 0.75 – Zooplankton assimilation efficiency (Fasham et al. 1990). 2.4 (0.1(T −25)) d−1 Maximum specific grazing rate of zooplankton. g β e Analogous to A3 and A4 of Laws et al. (2000). µ2 βe(0.1(T −25)) d−1 Zooplankton excretion (Laws et al. 2000). µ5 0.05e(0.1(T −25)) d−1 Zooplankton death (Laws et al. 2000). γ1 0.3 – Fraction of total net primary production exuded by phytoplankton to the DON compartment. Analogous to (1 − q2s ) = (1 − q2L ) = 0.3 of Laws et al. (2000). 1.2 (0.0633(T −25)) J – Maximal growth rate assuming that 1−γ e the environment is not nutrient limited. analogous to A2s = A2L of Laws et al. (2000). µ1 0.045e(0.0633(T −25)) d−1 Phytoplankton death (Laws et al. 2000).

362


Table G.7: Parameters used for finding the steady state with the bacteria differential equation Parameter Value Units Comments Vb 1.2e(0.0633(T −25)) d−1 Bacteria maximum growth rate. (Laws et al. 2000). K4 0.5 mMol N m−3 Bacteria half-saturation rate. (Fasham et al. 1990). (0.0633(T −25)) −1 µ3 0.05e d Bacteria excretion rate. µ4 0.25e(0.0633(T −25)) d−1 Detrital breakdown rate. ε 0.75 – Ammonium fraction of zooplankton excretions (Fasham et al. 1990). ω 0.33 – Detrital fraction of zooplankton mortality (Fasham et al. 1990).

Table G.8: Parameters used for finding the steady state for the nutrient compartments. Parameter Value Units Comments Ψ 1.5 mMol N−1 Strength of ammonium inhibition of nitrate uptake (Fasham et al. 1990). K1 0.5 mMol m−3 Nitrate half saturation constant (Fasham et al. 1990). K2 1 mMol m−3 Ammonium half saturation constant (Fasham et al. 1990). η 0.6 – NH4 /DON uptake ratio (Fasham et al. 1990).


363

20. Overview of maxgoalfinder. Previously, the function ss8 was described, which finds the value of a goal function given an input of ef and Q2 . The purpose of maxgoalfinder is to find the ef and Q2 on a grid of spacing del that maximises the user-specified goal function. maxgoalfinder is used as a front-end to ss8 . h maxgoalfinder.m

20 i

≡

function res = maxgoalfinder(goalId , oceanId , efmin , efmax , Q2min , Q2max , gsize ); % Function that determines the ef ratio for a given ocean that % maximises the specified goal function % Simultaneously generates a list of ef and Q2 in the feasible stable % region % % USAGE % —– % maxgoalfinder(goalId,oceanId,del); % % INPUTS % —— % goalId: Goal function’s ID no % oceanId: Ocean’s ID no % del: Step size for ef and Q2 grid if goalId ≡ 4 ; fName = [’allpoints’, num2str (oceanId ), ’.m’]; fId = fopen (fName , ’w’); fprintf (fId , ’%% The following is the result of a grid search \n’); fprintf (fId , ’%% created ’); fprintf (fId , [date , ’\n’]); fprintf (fId , ’%% with the file maxgoalfinder.m \n’); fprintf (fId , ’%% And with gsize’); fprintf (fId , [num2str (gsize ), ’\n’]); fprintf (fId , ’%% on ocean ’); fprintf (fId , [num2str (oceanId ), ’\n’]); fprintf (fId , ’%% with [efmin,efmax,Q2min,Q2max] \n’); fprintf (fId , [’%% [’, num2str (efmin ), ’,’, num2str (efmax ), ’,’, num2str (Q2min ), ’,’, num2str (Q2max ), ’] \n’]); fprintf (fId , ’%% Results in the form [ef Q2 goalval] \n’);

364


fprintf (fId , ’%% \n’); fprintf (fId , ’results=[ \n’); end % Ocean-specific parameter values oceanParams = oceanp(oceanId , 1); store = 0; goalMax = −Inf ;

% Initialise

efrange = linspace (efmin , efmax , gsize ); Q2range = linspace (Q2min , Q2max , gsize ); for countef = 1 : gsize for countQ2 = 1 : gsize goalNow = ss8 (goalId , oceanParams , Q2range (countQ2 ), efrange (countef ), −Inf , −Inf ); if goalNow 6= −Inf if goalId ≡ 4

% Only generating one set

spare = [efrange (countef ) Q2range (countQ2 ) goalNow ]; fprintf (fId , ’%14.4e %14.4e %14.4e \n’, spare ′ ); end if goalNow > goalMax goalMax = goalNow ; store = [efrange (countef ) Q2range (countQ2 )]; end end end end if goalId ≡ 4

% Only generating one set

fprintf (fId , ’]; \n’); fclose (fId ); end res = [store , goalMax ];


365

21. Overview of oceanp. For convenience, the euphotic zone depth (z), temperature (T ), nutrient loading rate (L) and name of each ocean is specified in a separate function. h oceanp.m

21 i

≡

function res = oceanp(oceanId , flag ); % A function which takes the ocean’s ID and returns the % the depth of the euphotic % zone (z), the temperature (T) and the nutrient loading rate (L) % OR the name of the ocean % % USAGE % —– % oceanmaker(oceanId,flag) % % INPUT % —– % oceanId: Ocean’s id no % flag: =1 return values (else return name) % % OUTPUT % —— % z: Euphotic zone depth % T: Temperature % L: Nutrient loading % oname: Ocean’s name switch oceanId case 1;

% BATS

z = 140; T = 21; L = 7.8; oname = ’BATS’; case 2;

% HOT

z = 150; T = 25; L = 12.2; oname = ’HOT’;

366


case 3;

% NABE

z = 35; T = 12.5; L = 98; oname = ’NABE’; case 4;

% EqPac-normal

z = 120; T = 24; L = 32.1; oname = ’EqPac-normal’; case 5;

% EqPac-El Nino

z = 120; T = 27; L = 12.3; oname = ’EqPac-El Nino’; case 6;

% Arabian Sea

z = 65; T = 25; L = 29.2; oname = ’Arabian Sea’; case 7;

% Ross Sea

z = 40; T = 0; L = 165; oname = ’Ross Sea’; case 8;

% Subarctic Pacific Station P - correct

%z=120; T=6; L=229; oname=’Subarctic Pacific Station P’; z = 120; T = 6; L = 40.3; oname = ’Subarctic Pacific Station P’; case 9;

% Peru-normal

z = 25.5; T = 16.8; L = 339; oname = ’Peru-normal’;

367


case 10;

% Peru-El Nino

z = 17.8; T = 17.4; L = 256; oname = ’Peru-El Nino’; case 11;

% Greenland polynya

z = 50; T = 0; L = 35.6; oname = ’Greenland polynya’; otherwise error (’Invalid ocean ID’); end if flag ≡ 1 ;

% Return values

res = [z T L]; else

% flag == 0 Return name

res = [oname ]; end return

Index of fashammodel alpha1 : 11

dG2dB : 16, 17

alpha2 : 11

dG2dD : 16, 17

alpha3 : 11

dG2dP : 16, 17

beta : 5, 10, 16

dG2dZ : 16, 17

cond : 10

dG3dB : 16, 17

countef : 20

dG3dD : 16, 17

countQ2 : 20

dG3dP : 16, 17

date : 20

dG3dZ : 16, 17

del : 20

dQ1dNn : 16, 17

dG1dB : 16, 17

dQ1dNr : 16, 17

dG1dD : 16, 17

dQ2dNr : 16, 17

dG1dP : 16, 17

dU1dB : 16, 17

dG1dZ : 16, 17

dU1dNd :

16, 17

368


dU1dNr : 16, 17

linspace :

dU2dB : 16, 17

max : 1

dU2dNd : 16, 17

maxgoalfinder : 1, 20

dU2dNr : 16, 17

min : 1

ef : 1, 7, 8

mu1 : 5, 9, 16

efmax : 20

mu2 : 5, 10, 11, 16

efmin : 20

mu3 : 6, 11, 16

efrange : 20

mu4 : 6, 10, 11, 16

eig : 1

mu5 : 5, 10, 11, 16

eps : 6, 11, 16

Nd : 11, 13, 15, 17

error : 19, 21

Nn : 11, 15, 17

eta : 12, 13

Nr : 11, 13, 15, 17

exp : 1

num2str : 20

fclose : 20

oceanId : 20, 21

fId : 20

oceanp : 1, 20

flag : 21

oceanParams : 1, 3, 20

fName : 20

omega : 6, 10, 11, 16

fopen : 20

oname :

fprintf : 20

psi : 11, 12, 17

f1 : 9, 10, 11, 17

p1 : 9, 17

gamma1 : 5, 9, 10, 11, 16

p2 : 9, 17

goalId : 1, 19, 20

p3 : 9, 17

goalMax : 20

Q1 : 7, 10, 11, 16, 17

goalNow : 20

Q2 : 1, 7, 10, 11, 16, 17

gsize : 20

Q2max : 20

G1 : 11

Q2min : 20

G2 : 11

Q2range :

G3 : 11

real : 1

Inf : 20

res : 1, 10, 13, 15, 19, 20, 21

Jac : 16, 18

sigma : 16, 17

K1 : 11, 12, 17

spare : 20

K2 : 12, 17

ss8 : 1, 20

K3 : 17

store : 20

K3plsF : 17

sum : 1

K4 : 6, 11, 13, 17

TeX : 1, 20

LHS :

unfeasFlag :

10

20

21

20

1, 10, 13, 15

G.3 Fasham Model comparisons between goal functions

unstabFlag : 1, 15

Vb : 6, 11, 17

U1 : 11, 17

zeros : 16

369

U2 : 11, 17

List of Refinements in fashammodel

h maxgoalfinder.m h oceanp.m h ss8.m

20 i

21 i

1i

h Biotic compartment parameters h Create Jacobian

16 i

5, 6, 7, 9 i

Used in section 15.

h Detritus parameters

3i

Used in section 14.

h Diff common terms

17 i

Used in section 16.

h Find the resilience

15 i

Used in section 1.

h If satisfy par range

13 i

h N parameters

Used in section 14.

12 i

h Resilience equation

18 i

Used in section 15.

Used in section 15.

h Return goal function value

19 i

h Solve each compartment for ss h Solve for N compartments

11 i

h Solve for biotic compartments h Solve for detritus

2i

h Solve other equations

G.3 G.3.1

Used in section 4.

Used in section 15. 14 i

Used in section 1.

Used in section 14. 8, 10 i

Used in section 4.

Used in section 14. 4i

Used in section 14.

Fasham Model comparisons between goal functions Comparison between goal functions

Figures G.1 to G.4 show the regions for which the system is both feasible and stable and the location of each of the goal functions in parameter space for the Fasham Model.

370


The Fasham Model can be used to provide some external validation of the Laws Model. It would be expected that, if both models are successful, they would not only give reasonable predictions of the ef ratio, but show inter-model consistency regarding other aspects of the system. One such consistency is apparent by comparing the relative sizes of the Laws Model’s feasible-stable regions (Figures 7.3 to 7.6) with those of the Fasham Model (Figures G.1 to G.4). An approximate ordering of the oceans with respect to the relative sizes of their feasible-stable regions shows inter-model agreement. For example, both models agree that BATS, HOT, Eq Pac and the Arabian Sea have much smaller feasible-stable regions than Ross Sea, Peru and Greenland. Similar to the Laws Model results, the Fasham Model maximises the resilience in the interior of the Q2 parameter space. However, resilience is maximised near the extremities of the feasible-stable ef range (for that Q2 ) for all oceans except Peru-Normal. Interestingly, this causes the model to under-predict the ef ratio. This is inconsistent with the Laws Model, where the parameter set that maximised resilience moved around the feasible-stable region in a way that corresponded to the field data. The maximisation of resilience at the extremities of the ef range is due to the way in which resilience is maximised. It was observed that, unlike the Laws Model, resilience in the Fasham Model was not maximised at the cusp of change between the dominance of the eigenvalues for all oceans except PeruNormal. For all oceans except Subarctic Pacific-Station P and Peru-Normal, resilience was maximised at the transition from real to complex-conjugate eigenvalues (cf. Kristensen et al. 2003), which can occur quite close to the feasible-stable boundary. For Subarctic Pacific Station P, resilience was maximised when the leading real eigenvalue was maximised.

371


Feasible and stable region in parameter space for BATS

1 oB P o P/B oR

0.9 0.8 0.7 0.6 Q2 0.5 0.4 0.3 0.2 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef

Feasible and stable region in parameter space for HOT

1

oB oR P o P/B

0.9 0.8 0.7 0.6 Q2 0.5 0.4 0.3 0.2 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef

Feasible and stable region in parameter space for NABE

1 0.9

o P/B P

0.8 0.7 0.6 0.5

Q2

0.4 0.3 oR

0.2 0.1

oB

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef

Figure G.1: The location of the parameter set that maximises each of the goal functions in the Fasham Model. The boundary of the feasible and stable region is marked as a dotted line.

372


Feasible and stable region in parameter space for EqPac-normal

1

oB oR o P/B P

0.9 0.8 0.7 0.6 Q2 0.5 0.4 0.3 0.2 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef

Feasible and stable region in parameter space for EqPac-El Nino

1

oB

0.9

oo PR o P/B

0.8 0.7 0.6 Q2 0.5 0.4 0.3 0.2 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef

Feasible and stable region in parameter space for Arabian Sea

1 0.9

oP ooP/B R

0.8 0.7 0.6 0.5

Q2

0.4 0.3 0.2

oB

0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef


373


Feasible and stable region in parameter space for Ross Sea

1 0.9 0.8

o P/B P

0.7 0.6 0.5

Q2

0.4 0.3 0.2

oR

0.1

oB

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef

Feasible and stable region in parameter space for Subarctic Pacific Station P

1 o P/B B P

0.9

oR

0.8 0.7 0.6 0.5

Q2

0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef

Feasible and stable region in parameter space for Peru-normal

1 0.9 o P/B P

0.8 0.7 0.6 0.5

Q2

0.4 0.3 oR

0.2

oB

0.1 0

0

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1

ef


374


Feasible and stable region in parameter space for Peru-El Nino

1 0.9 o oP P/B

0.8 0.7 0.6 0.5

Q2

0.4 0.3

oR

0.2 0.1

oB

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef

Feasible and stable region in parameter space for Greenland polynya

1 0.9 o oP P/B

0.8 0.7 0.6 0.5

Q2

0.4 0.3 0.2

oR

0.1

oB

0 0.1

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0.3

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0.5

0.6

0.7

0.8

0.9

1

ef


375

G.4 Variations on the Fasham Model

G.4 G.4.1

Variations on the Fasham Model Introduction

The purpose of this appendix is to summarise the variations on the Fasham Model that were investigated. Their are two purposes to these these investigations. First, we are interested in creating a Fasham Model that predicts the field data in the same way that the Laws Model does. Second, we are interested in determining which aspects of the models are significant to their ability to predict the ef-ratio field data.

G.4.2

Model

Details of variations on the Fasham Model In order to make the Fasham Model more like the Laws Model, the ammonium and DON compartments were combined into a single nutrient compartment, and the effect of removing the flow of nutrients from the detrital compartment to the DOM compartment. Figure G.5 shows the model diagrammatically. Variations on the model involved varying the strength of the flows shown by dashed lines in Figure G.5.

Xp

Xd

Phytoplankton

Detritus

Xz Xn

Zooplankton

Inorganic nutrient

Xc

Xb

DOM

Bacteria

Figure G.5: Fasham Model variations. Dotted lines indicate nutrient flows that were varied in the variations investigated. See Table G.9 for a summary.

The dynamical equations governing the system are as follows. For phytoplankton X˙ p = qp Fp − Fz

Xp − µp Xp . Xp + Xb + Xd

(G.24a)

376


For zooplankton X˙ z = qz Fz − µz Xz .

(G.24b)

For bacteria X˙ b = qb Fb − Fz

Xb . Xp + Xb + Xd

(G.24c)

For detritus X˙ d = gp sp Fp + gz sz Fz − Fz

V Xd Xd − µd Xd − + µz Xz + µp Xp , Xp + Xb + Xd M

(G.24d)

where gi is the proportion of i that in the large phytoplankton stream, that is, the proportion that excretes to the detritus compartment. For nitrate X˙ n = L − Fp + rz Fz .

(G.24e)

For the combined nutrient compartment X˙ c = (1 − gp )sp Fp + (1 − gz )sz Fz + µd Xd − Fb .

(G.24f)

The solution to the steady state is given in Appendix G.5.2. Similar to Appendix E.4, we used the ratios gp to describe the fraction of large phytoplankton, Xpl , versus small phytoplankton, Xps , gp =

⋆ Xpl ⋆ + X⋆ . Xpl ps

(G.25)

We have also used a ratio gz to describe the fraction of large zooplankton, Xzl , versus small phytoplankton, Xzs gz =

⋆ Xzl . ⋆ + Xzs

⋆ Xzl

(G.26)

Their differences are described in the body of the text (Section 10.3.2). It was observed that the local (rather than global) resilience optima in the Laws Model successfully predicted the field data (Chapter 7. This optima is associated with a predominance of large phytoplankton. The response of the Fasham Model to the following three changes were investigated: 1. Presence and absence of a flow from the detrital compartment to the DOM compartment. 2. Changes in the proportion of large phytoplankton to total phytoplankton, gp . 3. Changes in the value of threshold concentrations for nutrient uptake. Combinations of the latter two allow us to isolate the effects of waste direction from threshold concentration value, thus determining which aspect of large phytoplankton is important to the model’s predictive ability.

377

G.4 Variations on the Fasham Model

Table G.9: A summary of variations on the Fasham Model investigated. Model Name gp µd P2l Fasham 3 [0, 1] 0.05 75 nM Fasham 4 1 0.05 75 nM Fasham 5 1 0 75 nM Fasham 6 [0, 1] 0 75 nM Fasham 7 1 0 7.5 nM Solving for the steady state The steady-state detritus concentration is again found by a mass balance about the entire system Xd⋆ =

LM . V

(G.27)

Equation G.24a and Equation G.24c can be rearranged and the Fz terms equated such that µp = qp fp Ap − qb fb Ab ,

(G.28)

with the logical constraint µp ≥ 0. Equation G.24b can be rearranged at the steady state to give µz = qz Az fz ,

(G.29)

which is also constrained by µz ≥ 0. The phytoplankton uptake threshold concentration is taken as a weighted sum of the small and large phytoplankton threshold concentrations Pp = Pps + gp (Ppl − Pps ),

(G.30)

where the subscripts pl and ps denote large and small phytoplankton respectively. Three linear equations formed from Equations G.24a, G.24e and G.24f can be used to solve for Xp⋆ , Xz⋆ and Xb⋆ . From Equation G.24a 0 = qp fp Ap (Xp⋆ + Xb⋆ + Xd⋆ ) − qz Az fz Xz⋆ − µp .

(G.31a)

From Equation G.24e 0 = L − fp Ap Xp⋆ + rz fz Az Xz⋆ .

(G.31b)

From Equation G.24f 0 = (1 − gp )sp fp Ap Xp⋆ + (1 − gz )sz fz Az Xz⋆ − fb Ab Xb⋆ + µd Xd⋆ .

(G.31c)

378


From the definitions of fb and fp , Xc⋆ =

Pb , 1 − fb

(G.31d)

Xn⋆ =

Pp . 1 − fp

(G.31e)

and

The formulation above leaves five parameters unspecified; fp , fz , fb , gp , and gz , all of which are between 0 and 1.

G.4.3


Table G.10 summarises the predictive ability of each of the model variations by presenting the coefficients of a linear regression fitted to the predictions. Setting µd = 0 (Fasham 5, 6 and 7) increases the precision with which the model predicts the data. It increases Setting µd = 0 (Fasham 5, 6 and 7) not only increases the ef-ratios closer to the field data, as indicated by slope closer to 1 : 1. Reducing the P value of the large phytoplankton to that of the small phytoplankton (Fasham 5 to Fasham 7) decreased the slope and r2 , but maintained the high predictive ability of the model (with respect to slope and r2 ). Setting gp = 1 always moved the y-intercept closer to 0. It had a different effect on the slope and r2 , depending upon the µd . Comparing Fasham 3 and Fasham 4 (µd = 0.5), the constraint gp = 1 increased slope, and decreased r2 . Comparing Fasham 6 and 5 (µd = 0), the situation was the opposite. The constraint gp = 1 decreased slope, and increased r2 . The gp had no effect on the RSS for Fasham 3 versus Fasham 4, however the RSS was improved when gp = 1 for Fasham 5 and 6. Comparing Fasham 5 to Fasham 7, reducing the P value of the large phytoplankton to that of the small phytoplankton, removed some predictive ability (with respect to RSS, slope and r2 ), however Fasham 7 still performed better than all other model variations, and the decrease in predictive ability was not as great as Fasham 5 to Fasham 6, when gp was allowed to vary.

G.4.4

Concluding remarks

We have replicated the predictive ability of the Laws Model in a variation of the Fasham Model, Fasham 5. This model has exclusively large phytoplankton, and no transfer of nutrients from the detrital compartment to the DON compartment. It was found that, compared to a Fasham Model variation with

G.5 Maximal resilience versus ef-ratio profile – Fasham Model 6

379

Table G.10: Residual sum of square error, comparing model predictions to field data. A summary of linear regression on Fasham models. (x, y) =(Observed ef, Model ef). Maximise resilience. βˆ1 r2 Model Residual sum βˆ0 Name of square y-intercept slope coefficient of error determination Ideal 0 0 1 1 Fasham 3 0.364 -0.004107 0.55246 0.77323 Fasham 4 0.364 -0.000483 0.55854 0.70052 Fasham 5 0.063 0.015972 0.80905 0.92198 Fasham 6 0.159 -0.049521 0.85929 0.86423 Fasham 7 0.127 0.010393 0.73110 0.87948

none of these attributes, the most significant improvement was gained by preventing nutrient transfer from detritus to DON, and a smaller improvement by making all phytoplankton large phytoplankton.

G.5

Maximal resilience versus ef-ratio profile – Fasham Model 6

G.5.1

Introduction

This appendix presents the maximal resilience versus ef-ratio profiles of a variation on the Fasham Model: Fasham 6. A description of the model may be found in Appendix G.4. As discussed in the body of the text, Fasham Model 6 was chosen because it has structural and parameter attributes intermediate between the Laws Model and the Fasham Model.

G.5.2

Approach

We investigated the maximal resilience profile of Fasham Model 6 (see Appendix G.4). For each ocean region, a grid search was performed to find the maximum resilience value associated with each ef-ratio value. In order to do this, the ef-ratio was made a free-parameter in the place of gp . The method used to solve for the steady state is described in Section G.5.2. Thus, the model had the following free-parameters: fp , fb and fz , which are the fraction of maximal uptake by phytoplankton, bacteria, and zooplankton respectively, gz , which is the ratio of large zooplankton to total zooplankton, and ef , which is the ef-ratio. The ef-ratio range between 0.01 and 0.99 was divided into 30 increments. At each increment, the other free-parameter ranges were divided into a grid of 20 increments, also between 0.01 and 0.99. At each ef-ratio grid-point, the value of resilience at every grid-point was evaluated, and the maximal resilience value obtained. The maximal resilience versus ef-ratio profile was then presented in graphs.

380


Solving for the steady state To efficiently find the profile of maximal resilience against the ef-ratio, we replaced gp with the ef-ratio as a free parameter. This meant that the steady-state solution had to be re-derived. The steady-state is found by the following equations. As done in Section G.4.2, the steady-state detritus nutrient concentration is found by a mass balance about the entire system Xd⋆ =

LM . V

(G.32a)

The combined nutrient compartment concentration is found by utilising the definition of fb , Xc⋆ =

Pb . 1 − fb

(G.32b)

The steady-state nutrient concentration of the phytoplankton compartment is found using the definition of the ef-ratio. As ef = L/Fp , Xp⋆ =

L . ef Ap fp

(G.32c)

Equation G.24e can be rearranged to give the zooplankton steady-state nutrient concentration Xz⋆ =

Fp − L . rz Az fz

(G.32d)

Equation G.24c can be rearranged to give the bacteria steady-state nutrient concentration Xb⋆ =

Fz − qb Ab fb (Xp⋆ + Xd⋆ ) . qb Ab fb

(G.32e)

Specifying ef determines the value of gp , which must be found in order to evaluate the nutrient concentration of the nitrate compartment at steady state. Equation G.24f can be rearranged at steady-state to give gp =

1 sp Fp

(sp Fp + (1 − gz )sz Fz − Fb ) ,

(G.32f)

which is used to find the uptake threshold for phytoplankton Pp = Pps + gp (Ppl − Pps ),

(G.32g)

which is then, in turn, used to find the steady-state nutrient concentration of the nitrate compartment Xn =

G.5.3

Pp . 1 − fp

Results

Figures G.6 and G.7 show the maximal resilience versus the ef-ratio for Fasham Model 6.

(G.32h)

381

G.5 Maximal resilience versus ef-ratio profile – Fasham Model 6

Fasham Model 6. Bermuda Atlantic Time-series Study.

Fasham Model 6. Hawaiian Ocean Time-series.

0.12

0.16

0.14 0.1 0.12

Maximal resilience

Maximal resilience

0.08

0.06

0.1

0.08

0.06

0.04 0.04 0.02 0.02

0

0 0

0.1

0.2

0.3

0.4

0.5 ef-ratio

0.6

0.7

0.8

0.9

1

0

0.1

0.2

Fasham Model 6. North Atlantic Bloom Experiment.

0.3

0.4

0.5 ef-ratio

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Fasham Model 6. Pacific equatorial upwelling - normal.

0.1

0.16

0.09

0.14

0.08 0.12

Maximal resilience

Maximal resilience

0.07 0.06 0.05 0.04

0.1

0.08

0.06

0.03 0.04 0.02 0.02

0.01 0

0 0

0.1

0.2

0.3

0.4

0.5 ef-ratio

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

Fasham Model 6. Pacific equatorial upwelling - El Nino.

0.4

0.5 ef-ratio

0.6

Fasham Model 6. Arabian Sea.

0.2

0.18

0.18

0.16

0.16

0.14


Maximal resilience

0.12 0.12 0.1 0.08

0.1 0.08 0.06

0.06 0.04

0.04

0.02

0.02 0

0 0

0.1

0.2

0.3

0.4

0.5 ef-ratio

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 ef-ratio

Figure G.6: Maximal resilience for the Fasham Model 6.

0.6

382


Fasham Model 6. Ross Sea.

Fasham Model 6. Subarctic Station P.

0.045

0.045

0.04

0.04

0.035

0.035 0.03 Maximal resilience

Maximal resilience

0.03 0.025 0.02 0.015

0.025 0.02 0.015

0.01

0.01

0.005

0.005

0

0 0

0.1

0.2

0.3

0.4

0.5 ef-ratio

0.6

0.7

0.8

0.9

1

0

0.1

0.2

Fasham Model 6. Peru upwelling - normal.

0.3

0.4

0.5 ef-ratio

0.6

0.7

0.8

0.9

1

0.8

0.9

1

Fasham Model 6. Peru upwelling - El Nino.

0.14

0.16

0.14

0.12

0.12

Maximal resilience

0.08

0.06

0.1

0.08

0.06

0.04 0.04 0.02

0.02

0

0 0

0.1

0.2

0.3

0.4

0.5 ef-ratio

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 ef-ratio

Fasham Model 6. Greenland polynya. 0.04

0.035

0.03

Maximal resilience

Maximal resilience

0.1

0.025

0.02

0.015

0.01

0.005

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ef-ratio

Figure G.7: Maximal resilience for the Fasham Model 6.

0.6

0.7

Appendix H

Appendices relating to Chapter 11 H.1

Documentation of the permanence food web building code

383

384


permbuild Section Page Permanence food web building algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

386

permbuild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

387

Preamble of output file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

391

Print A and D to file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

392

Print time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

393

Find a species that can invade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

394

Restrict predator diets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

395

Find all subsystems of system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

398

Retain only those subs with those elm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

399

Remove specified branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

400

addspp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

401

Determine noMoreAuto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

403

Create autotroph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

404

Create heterotroph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

405

Make sppInd pred of new . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

405

Make sppInd prey of new . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

406

netsperm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

407

Create bchVect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

410

Functions relating to finding subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

411

findsubs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

412

Remove isolated autotrophs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

414

Determine if single autotroph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

415

Determine if member of chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

415

Add to subs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

415

Add to list of subs to go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

416

Create subIgnoreVec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

417

Find subsystems of the subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

418

Append subsubsystems to subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

418

Adjust indices of subsD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

419

Pad smaller of subs and newSubs with zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

420

findsubspost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

421

findnonchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

424

findnonfloat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

427

H.1 Documentation of the permanence food web building code

385

Create multABak, augment A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

429

Create the multABak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

430

Augment A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

431

Accessory functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

433

userparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

434

makea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

436

jumble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

437

remone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

438

remonevec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

439

nadrepm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

440

Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

441

postprocess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

442

Initialise empty storage vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

445

Initialise storage vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

446

Classify basal, intermediate, top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

447

Count link types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

448

Count omnivores and max chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

450

largestweb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

452

Identify basal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

453

Identify largest web from basal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

454

Find compliment of largest web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

455

Remove compliment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

455

pathfinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

456

ischaino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

458

rigidcircuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

460

Create niche overlap graph, M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

462

Remove single isolated nodes from M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

463

Rose et al label and number scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

464

ID adjacency set of vert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

465

Append numbEntry to labl of adjacent vertices . . . . . . . . . . . . . . . . . . . . . . . . . . 63

465

ID next vert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

466

Create dedLabl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

467

Find lexico highest dedLabl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

468

Print output from postprocess to file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

469

Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

471

controlbuild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

471

386

1.


Permanence food web building algorithm.

The following sections document the permanence food web building code. It is described in Section 11.2.2. It is designed for GNU Octave.


2.

387

permbuild.

As described in the body, the purpose of permbuild is to build a food web using incremental invasions, with permanence as a constraint. It accepts three variables: 1. Asz : Integer. The size of the initial community. 2. its : Integer. The number of successful invasions that will be performed upon the community. 3. fName : String. If specified, the name of the output file to which results will be printed. It returns five variables: 1. A: n + 1 × n + 1 matrix. The final community interaction matrix. 2. D: n + 1 × 1 vector. The final mortality vector. 3. X: n × 1 vector. The final steady state nutrient concentration of the biotic compartments. It outputs one output file: 1. eachstep .m: An m-file containing matrices of the form [A, D] for each iteration. User specified parameters are read from the file userparameters. An output file is created, named fName , into which information about each iteration will be stored. A preamble is printed to this file. Given an initial community size, Asz , the initial food chain is created using the function makea. The variable maxAutoDi was specified in userparameters. The food web building code is run for the number of invasions specified by the user, its . The flagStartAgain ≡ 1 condition refers to a previous case, in which the number of subsystems of the systems exceeds the maximum specified by the user, maxSzSubs . For each iteration, species are randomly generated until one is found that can invade the system. Once a successful invader is found, all subsystems of the new system are found. These subsystems are stored in a matrix called subs . Each nonzero element in subs is the row index of a removed compartment. Each row of subs represents a particular subsystem of the new system, or branch of the system removed. For example, if the fifth row of subs has the form [0 0 1 2], this means that the fifth subsystem collected is the new system with compartments 1 and 2 removed. If subs is not empty, and larger than 1, the order of the branches in subs is randomised using the function jumble. The reason for doing this is that, later in this function, impermanent branches will be

388


removed in the order in which they are encountered. maxJumbleInd refers to a user-specified parameter in userparameters, which determines how much of subs is randomised. Once subs is determined, the permanence of the system is tested against each branch in subs . The function netsperm tests the permanence of the system specified, with respect to the subs specified. If the system is permanent, it returns 0. If it is not permanent, it returns the row index, or branch number, corresponding to the first impermanent branch in subs . While impermanent branches exist in the new system, these branches are removed, and the permanence of the system retested. This is continued until a permanent system results. After a permanent system is found, all information about the system is updated for the next iteration. The resulting system is stored in the output file. Once all iterations are completed, the time is printed is the user requires it, and the output file is closed. h permbuild.m

2i

≡

function [A, D, X] =permbuild(Asz , its , fName ) userparameters;

% User specified parameters

if nargin ≡ 2 fName = ’eachstep.m’; end fid = fopen(fName , ’w’); h Preamble of output file

3i

flagStartAgain = 0; startAsz = Asz ; if specADinit ≡ 1 ; Asz = size(AD ); Asz = Asz (1, 1); A = AD (1 : Asz , 1 : Asz ); D = AD (: , Asz + 1); X = A\ − D; else [A, D, X] = makea(Asz , maxAutoDi ); end itn = 0; while itn < its itn = itn + 1;

% Create first system


if (flagStartAgain ≡ 1) fprintf(fid , ’%% Started again \n’); [A, D, X] = makea(startAsz , maxAutoDi ); flagStartAgain = 0; Asz = startAsz ; end h Find a species that can invade

6i

h Find all subsystems of system

8i

% Randomise order of subs

szSubs = size(subs ); szSubs = szSubs (1, 1); if szSubs > maxSzSubs ;

% userparameters

flagStartAgain = 1; else flagStartAgain = 0;

% May revert after a few repeats

end if szSubs 6= 0 ; if (szSubs 6= 1) ;

% Move first to last

inv = subs (1, :); subs = subs (2 : szSubs , : ); subs = jumble(subs , maxJumbleInd ); subs = [subs ; inv ]; end bchRemRowInd = netsperm(Anew , Dnew , subs );

% Determine if permanent

if bchRemRowInd 6= 0 while (bchRemRowInd 6= 0) h Retain only those subs with those elm

9i

bchRemRowInd = netsperm(Anew , Dnew , subs ); end h Remove specified branch

10 i

end end szAnew = size(Anew ); szAnew = szAnew (1, 1); A = Anew ; D = Dnew ;

% Update for next iteration

% Determine if permanent

389

390


Asz = size(A); Asz = Asz (1, 1); X = A\ − D; h Print A and D to file end

4i

% while itn < its

if printTime ≡ 1 ; timeNow = clock; fprintf(fid , ’%% Time ’); fprintf(fid , ’%1.4g’, timeNow (1, 4)); fprintf(fid , ’:’); fprintf(fid , ’%1.4g’, timeNow (1, 5)); fprintf(fid , ’ ’); fprintf(fid , ’%1.2g’, round (timeNow (1, 5))); fprintf(fid , ’ s \n’); fprintf(fid , ’ \n’); end fclose(fid );

% Close file


3.

391

Preamble of output file.

The creation of the preamble of the output file provides information about how the file was generated, what the initial system used was, and how the data is stored. This ensures that output files can be easily identified. h Preamble of output file

3i

≡

timeNow = clock; fprintf(fid , ’%% Generated by permbuild.m \n’); fprintf(fid , ’%% Date: ’); fprintf(fid , ’%1.3g’, timeNow (1, 4)); fprintf(fid , ’ day of the ’); fprintf(fid , ’%1.3g’, timeNow (1, 2)); fprintf(fid , ’ month, ’); fprintf(fid , ’%1.4g\n’, timeNow (1, 1)); fprintf(fid , ’%% Time: ’); fprintf(fid , ’%1.4g’, timeNow (1, 4)); fprintf(fid , ’:’); fprintf(fid , ’%1.4g’, timeNow (1, 5)); fprintf(fid , ’ ’); fprintf(fid , ’%1.2g’, round (timeNow (1, 5))); fprintf(fid , ’ s \n’); fprintf(fid , ’%% Information about run: \n’); fprintf(fid , ’%% Started with size: ’); fprintf(fid , ’%1.0g\n’, Asz ); fprintf(fid , ’%% and a chain structure. \n’); fprintf(fid , ’%% The total number of iterations performed was: \n’); fprintf(fid , ’its = ’); fprintf(fid , ’%7.7g’, its ); fprintf(fid , ’; \n ’); fprintf(fid , ’%% The data below is the A and D matrices for \n’); fprintf(fid , ’%% each iteration. \n’); fprintf(fid , ’%% Output of the form: [A,D] \n’); fprintf(fid , ’%% Information about userparameters. \n’); fprintf(fid , ’%% Refer to userparameters.m for details. \n’); fprintf(fid , ’%% maxAuto = ’); fprintf(fid , ’%7.7g\n’, maxAuto);

392


fprintf(fid , ’%% maxSzSubs = ’); fprintf(fid , ’%7.7g\n’, maxSzSubs ); fprintf(fid , ’%% constFlag = ’); fprintf(fid , ’%7.7g\n’, constFlag ); fprintf(fid , ’%% maxBasPred = ’); fprintf(fid , ’%7.7g\n’, maxBasPred ); fprintf(fid , ’%% maxAutoDi = ’); fprintf(fid , ’%7.7g\n’, maxAutoDi ); This code is used in sections 2 and 69.

4.

Print A and D to file.

Each matrix entry in the output file is of the form [A, D]. The iteration to which it belongs is indicated by the name of the matrix. For example, the fourth iteration will be given in the matrix ADitn4 . It is necessary to create a string, matstr , such that it contains the same number of print instructions as there are columns in the matrix. This is achieved with a for loop. h Print A and D to file

4i

≡

fprintf(fid , ’ \n’); h Print time

5i

fprintf(fid , ’ADitn’); fprintf(fid , ’%1.4g’, itn ); fprintf(fid , ’ = [ \n’); % Adjust length of command to suit size of [A,D] matstr = [’%7.4e ’]; for sz = 2 : Asz matstr = [matstr , ’%7.4e ’]; end matstr = [matstr , ’%7.4e \n’]; fprintf(fid , matstr , [A, D]′ ); fprintf(fid , ’ ]; \n’); This code is used in sections 2 and 69.


5.

393

Print time.

The user may specify, in userparameters, that the time should be printed to the output file at the completion of each invasion. This is useful for debugging, and for determining what size maxSzSubs should be given a particular computer. h Print time

5i

≡

if printTime ≡ 1 ; timeNow = clock; fprintf(fid , ’%% Time ’); fprintf(fid , ’%1.4g’, timeNow (1, 4)); fprintf(fid , ’:’); fprintf(fid , ’%1.4g’, timeNow (1, 5)); fprintf(fid , ’ ’); fprintf(fid , ’%1.2g’, round (timeNow (1, 5))); fprintf(fid , ’ s \n’); fprintf(fid , ’ \n’); end This code is used in section 4.

394

6.


Find a species that can invade.

While a successful invader has not been found (flagInvad =0), a new compartment is generated using the function addspp. addspp returns the matrices A and D, in which the first row corresponds to the new compartment. maxBasPred is a user-specified parameter from userparameters, which gives the maximum trophic level which is permitted to predate upon basal compartments. If this is specified, the interactions of predators in the new A and D are altered to conform to this constraint. Success of invasion is tested using the transversal eigenvalue test, as described in the body of the text. h Find a species that can invade flagInvad = 0;

6i

≡

% Initialise can’t invade

Asznew = Asz + 1; while flagInvad ≡ 0 % Create system with new species (given index 1) [Anew , Dnew ] = addspp(A, D, maxAuto , constFlag , maxAutoDi ); if maxBasPred 6= 0 ; h Restrict predator diets

7i

end Xnew = [0; X(1 : Asz , 1)];

% Check sign of transversal eigenvalue

tran = Anew (1, : )∗Xnew + Dnew (1, 1); if tran > 0 ; flagInvad = 1; if rank(Anew ) ≡ Asz + 1 ; Xnew = Anew \ − Dnew ; if Xnew (1, 1) < 0 flagInvad = 0; end end end end This code is used in section 2.


7.

395

Restrict predator diets.

Specifying maxBasPred in userparameters allows the user to specify whether or not predator diets should be restricted according to their trophic level. The purpose of this is to emulate the architectural constraints imposed upon consumers which restrict their ability to feed upon both plant and animal matter1 . Each compartment is given a trophic level troph , which is defined as the longest distance from the basal level to the compartment of interest. If a compartment is basal, it is given a troph of 1. The restriction is dependant upon the parameter maxBasPred , which is an integer greater than zero. maxBasPred is defined as the maximal trophic level that is permitted to predate upon basal compartments, as well as other compartment levels. The constraint is that the introduction of the new compartment may not cause a violation of this restriction, but only for compartments with which the new compartment interacts. This is achieved by augmenting the interactions of the new species with the incumbents. This approach does not rule out the possibility that that the introduction of the new compartment causes a violation of the restriction in a predator with which it does not directly interact. We have decided to allow this, as the alternative is a retrospective alteration of predator relationships for incumbent compartments. Two possibilities exist for the new compartment: that it is a basal compartment, or a consumer. If the new compartment is basal, we identify those incumbents which prey upon it, predsNew . The trophic level of each compartment is identified by pulsing nutrient through the system, using the Augmented A and the matrix multA. Details of this approach may be found in Subsection 36. Once the trophic levels have been identified, two approaches corresponding to the two different types of new compartment are employed. If the new compartment is a consumer, and if its trophic level exceeds maxBasPred , all interactions between the new consumer and incumbent basal compartments are removed. If the new compartment is a basal compartment, for each of its predators in predsNew , if the trophic level of that predator exceeds maxBasPred , the interaction between the new basal compartment and that incumbent predator is removed. h Restrict predator diets

7i

≡

Abak = A; AnewBak = Anew ; A = Anew ; 1 Note

to reviewers: This option was not used in this study, but is included for completeness.

396


Aszbak = Asz ; Asz = Asz + 1; troph = zeros(Asz , 1); % Find basal basal = find(diag (A)); szBas = size(basal ); szBas = szBas (1, 1); troph (basal , : ) = 1; if troph (1, 1) ≡ 1

% New spp a basal spp

% Identify those that predate upon new basal predsNew = find(Anew (2 : Asz , 1)) + 1; szPreds = size(predsNew ); szPreds = szPreds (1, 1); flagBasalNew = 1; else flagBasalNew = 0; predsNew = 1; szPreds = 1; end h Augment A

36 i

multA = zeros(Asz , Asz ); for row = 1 : Asz if troph (row , 1) ≡ 1 ; multA(row , :) = ones(1, Asz ); end end % First pass (up web) count = 1; szNut = 42;

% Initialise

while (szNut > 0) ∧ (count ≤ Asz ) count = count + 1; multA = A∗multA; nut = find(multA(: , 1)); if isempty (nut ) ≡ 0 troph (nut , : ) = count ;


end end % If new consumer if (troph (1, 1) > maxBasPred ) ∧ (flagBasalNew ≡ 0) Anew (1, basal ) = 0; Anew (basal , 1) = 0; end % If new basal if flagBasalNew ≡ 1 for predInd = 1 : szPreds predOne = predsNew (predInd , 1); if troph (predOne , 1) > maxBasPred ; Anew (1, predOne ) = 0; Anew (predOne , 1) = 0; end end end A = Abak ; Asz = Aszbak ; This code is used in section 6.

397

398

8.


Find all subsystems of system.

Finding the subsystems of a system takes two steps. The first step involves the function findsubs, which uses a depth first search to identify non-singular, non-floating, feasible subsystems. It is described in more detail in Section 20. The second step involves the function findsubspost to remove double entries from the results of findsubs. It is described in more detail in Section 31. If the system is larger than a single autotroph, we find all subsystems that do not begin with the removal of the new compartment. The reason for this is that the subsystem resulting from the removal of the new compartment is dealt with in the body of the code. It is appended onto the start of the matrix subs . The output of findsubs often contains multiple instances of the same subsystem. They are removed using the function findpostsubs , to reduce the number of branches the permanence must be tested against. h Find all subsystems of system

8i

≡

szAnew = size(Anew ); szAnew = szAnew (1, 1); if szAnew ≡ 1 ;

% Single autotroph

subs = 1; else ignoreVec = 1;

% Tells it to ignore all subsystems of 1

subs = findsubs(Anew , Dnew , ignoreVec ); szSubsCol = size(subs ); szSubs = szSubsCol (1, 1); szSubsCol = szSubsCol (1, 2); if szSubs ≡ 0 subs = 1; else if szSubsCol ≡ 1 subs = [1; subs ]; else subs = [[1, zeros(1, szSubsCol − 1)]; subs ]; end end szSubs = szSubs + 1; subs = findsubspost(subs ); end This code is used in section 2.

% Remove double entries, note also sorts entries


9.

399

Retain only those subs with those elm.

The vector bchRemRowInd specifies one set of elements that form an impermanent branch. This implies that either the removal of this branch will ensure permanence, or that the removal of this branch plus some other compartments will ensure permanence. The set of elements specified by bchRemRowInd is a subset of a set of elements which, when removed, will ensure the permanence of the system. Therefore, we need only test permanence of the full system against supersets of the bchRemRowInd set. Elements of subs are removed until only supersets of the bchRemRowInd set remain. bchRemVect is the vector of the impermanent branch found – the bchRemRowInd set of subs . For each element, elm , in bchRemVect such that elm is not zero, if a particular row of subs does not contain elm , it is removed. Removals are performed in the following way. For each row of subs , a row vector bchComp is formed of that row. match is created, which is a vector of indices where an element of bchComp ≡ elm . If match is empty, that row is removed from subs . h Retain only those subs with those elm

9i

≡

bchRemVect = subs (bchRemRowInd , :); szSubsCol = size(subs ); szSubsCol = szSubsCol (1, 2); subs = remonevec(bchRemRowInd , subs ); for bchRemColInd = 1 : szSubsCol elm = bchRemVect (1, bchRemColInd ); if elm 6= 0 szSubs = size(subs ); szSubs = szSubs (1, 1); subRow = 0; while subRow < szSubs ; subRow = subRow + 1; bchComp = subs (subRow , :); match = find(bchComp ≡ elm ); if isempty (match ) subs = remonevec(subRow , subs ); subRow = subRow − 1; szSubs = szSubs − 1; end

400


end end end This code is used in section 2.

10.

Remove specified branch.

bchRemVec is an impermanent branch. To remove this branch from the system, each compartment in the branch is removed from Anew and Dnew . h Remove specified branch

10 i

≡

% Create subsystem szSubsCol = size(subs ); szSubsCol = szSubsCol (1, 2); update = 0; for subsInd = 1 : szSubsCol ; elm = bchRemVect (1, subsInd ); elm = elm − update ; if elm 6= 0 [Anew , Dnew ] = remone(elm , Anew , Dnew ); update = update + 1; end end This code is used in section 2.


11.

401

addspp.

The purpose of addspp is to add a randomly generated compartment to a system described by an interaction matrix, A, and a mortality vector, D. The new compartment is random with respect to three attributes: its trophic type (autotroph or heterotroph), which other compartments it interacts with, and the coefficient values in A and D. It accepts three variables: 1. A: n × n matrix. The community interaction matrix. 2. D: n × 1 vector. The mortality vector. 3. maxAuto : Integer. The maximum number of independent autotrophs allowed, as specified by the user. 4. constFlag : 0 or 1. A flag to indicate that the strength of negative interaction upon the prey must be less than or equal to the positive interaction upon the predator. This represents conservation of matter. 5. maxAutoDi : Real positive number. The maximum value of the di range for autotrophs. Increasing this increases the productivity of the autotrophic compartments. It returns two variables: 1. newA: n + 1 × n + 1 matrix. The community interaction matrix. 2. newD : n + 1 × 1 vector. The mortality vector. The user-specified parameter maxAuto restricts the number of unconnected autotrophs permitted in the system. If specified (i.e. if not set to zero), the system will be checked for the number of independent autotrophs, and if this is equal to or exceeds maxAuto , this will be indicated by noMoreAuto = 1. Else, noMoreAuto = 0. Details can be found in Section 12. We define probAuto , which is the probability that the new compartment added will be an autotroph. Two matrices, intnVectRow and intnVectCol , are initialised. These matrices will specify the interactions between the new compartment and old compartments. With probability probAuto , the new compartment is an autotroph. If it is an autotroph, this is indicated by setting firstPreyInd = 1. If it is not an autotroph, it must be assigned a prey, firstPreyInd 6= 1. Note that an else is not used to separate the autotroph and heterotroph statements. This allows an autotroph to be retrospectively changed to a heterotroph in the rare instance described in Section 13. A factor, factr , is specified to scale the probability of an interaction between the new species and other compartments.

402


For each incumbent compartment, if the compartment considered is not the new compartment’s first prey, and a random number is less than factr , the new compartment may interact with the compartment considered. Prey and predator relationships are given equal probability, except in the case when the old compartment considered is an autotroph compartment. In this case, the new compartment is not permitted to be a prey of the nutrient compartment. When all interaction terms have been assigned to intnVectRow and intnVectCol , the new compartment is added to the old system. It is given the index 1. h addspp.m

11 i

≡

function [newA, newD ] =addspp(A, D, maxAuto , constFlag , maxAutoDi ) Asz = size(A); Asz = Asz (1, 1);

% Size of old system

h Determine noMoreAuto

12 i

probAuto = .2; intnVectRow = zeros(Asz + 1, Asz + 1); intnVectCol = zeros(Asz + 1, Asz + 1);

% Initialise new interaction row % and new interaction col

autoFlag = rand; if autoFlag < probAuto

% If autotroph

firstPreyInd = 1; h Create autotroph

% Flags autotroph 13 i

end if autoFlag ≥ probAuto

% If heterotroph

firstPreyInd = floor(rand∗(Asz )) + 2; h Create heterotroph

14 i

end factr = .2;

% Scales the probability of prey/pred

for sppInd = 2 : Asz + 1 if (sppInd 6= firstPreyInd ) ∧ (rand < factr ) randno = rand; if (randno > .5) ∧ (firstPreyInd 6= 1) h Make sppInd prey of new

16 i

elseif (randno ≤ .5) ∧ (A(sppInd − 1, sppInd − 1) ≡ 0) h Make sppInd pred of new end end

15 i


403

end newA = [zeros(1, Asz ); A]; newA = [zeros(Asz + 1, 1), newA]; newA = intnVectRow + intnVectCol + newA;

12.

Determine noMoreAuto. .

The purpose of this routine is to determine the value of noMoreAuto . noMoreAuto is a flag that, when set to 1, indicates that no more independent autotrophs are permitted in the system. The number of independent autotrophs is determine as follows. For each entry in A, if that entry corresponds to an autotrophic compartment, and if all other entries in that row are zero, that autotroph is independent of the rest of the web. If the number of independent autotrophs is equal to, or exceeds, the user-specified value maxAuto , this is indicated by setting the flag noMoreAuto to 1. Else, the flag noMoreAuto remains equal to 0. h Determine noMoreAuto

12 i

≡

noMoreAuto = 0; if maxAuto > 0 countAuto = 0; for Aind = 1 : Asz if A(Aind , Aind ) < 0 spare = A(Aind , : ); spare (1, Aind ) = 0; if isempty (find(spare )) ; countAuto = countAuto + 1; end end end if countAuto ≥ maxAuto ; noMoreAuto = 1; end end This code is used in section 11.

404


13.

Create autotroph.

The purpose of this routine is to create an autotroph. By setting itnVectRow (1, 1) < 0, the autotroph is self limiting, that is a1,1 < 0. This ensures that the system is dissipative. The first entry in newD is set to a positive value, corresponding to the requirement that, for autotrophs, di > 0. All values are selected randomly. If noMoreAuto = 1, this implies that a new independent autotroph is not permitted in the system. Therefore, the new compartment is assigned a predator. Only heterotrophs are permitted to predate, therefore, all heterotrophs in the system are identified as potential predators, in hetVect . If there are heterotrophs in the system, one is randomly selected, with index sppInd , to predate upon the new compartment. If there are no heterotrophs in the system, it is not possible to permit a new autotroph in the system. Therefore, the autotroph is retrospectively changed to a heterotroph by setting autoFlag = 1. This allows the algorithm to enter the next if statement in the body of the algorithm. h Create autotroph

13 i

≡

intnVectRow (1, 1) = −rand;

% Self limiting

newD = [maxAutoDi ∗rand; D]; if noMoreAuto ≡ 1 ;

% Positive self term

% Must have pred

diagA = diag (A); hetVect = find(diagA ≡ 0); szHetVect = size(hetVect ); szHetVect = szHetVect (1, 1); if szHetVect 6= 0 sppInd = floor(rand∗(szHetVect )) + 1; sppInd = hetVect (sppInd , 1) + 1; h Make sppInd pred of new

15 i

else intnVectRow (1, 1) = 0; autoFlag = 1; end end This code is used in section 11.

% Change back


14.

405

Create heterotroph.

The purpose of this routine is to create a heterotroph. The first entry in newD is set to a negative value, corresponding to the requirement that, for heterotrophs, di < 0. The value is selected randomly. The compartment corresponding to the index firstPreyInd is made the prey of the new compartment, to ensure that the system does not float. h Create heterotroph

14 i

≡

newD = [−rand; D];

% Negative self term

sppInd = firstPreyInd ; h Make sppInd prey of new

16 i

This code is used in section 11.

15.

Make sppInd pred of new.

The purpose of this subroutine is to make the compartment with the index sppInd the prey of the new compartment. If constFlag is equal to zero, this implies that we are interpreting the state variables as populations numbers, and therefore there are no constraints upon the relative values of positive and negative interaction terms. Else, the state variables are nutrient concentrations, in which case the positive flow to the predator compartment must be less than the negative flow from the prey compartment. h Make sppInd pred of new

15 i

≡

intnVectRow (1, sppInd ) = −rand; if constFlag ≡ 0 intnVectCol (sppInd , 1) = +rand; else intnVectCol (sppInd , 1) = −rand∗intnVectRow (1, sppInd ); ; end This code is used in sections 11 and 13.

406


16.

Make sppInd prey of new.

The purpose of this subroutine is to make the compartment with the index sppInd the prey of the new compartment. If constFlag is equal to zero, this implies that we are interpreting the state variables as populations numbers, and therefore there are no constraints upon the relative values of positive and negative interaction terms. Else, the positive flow to the predator compartment must be less than the negative flow from the prey compartment. h Make sppInd prey of new

16 i

≡

intnVectCol (sppInd , 1) = −rand; if constFlag ≡ 0 intnVectRow (1, sppInd ) = rand; else intnVectRow (1, sppInd ) = −rand∗intnVectCol (sppInd , 1); end This code is used in sections 11 and 14.


17.

407

netsperm.

The purpose of this function is to determine if the system is permanent with respect to the removal of a specified branch. It accepts three variables: 1. A: n × n matrix. The community interaction matrix. 2. D: n × 1 vector. The mortality vector. 3. subs : k1 × m matrix. Specifies the subsystems found so far. Each row is a different subsystem. Each element is the index of the compartment that is removed from A to create the subsystem. It returns one variable: 1. res : Integer. The index of the first branch in subs encountered that is not permanent. If all branches are permanent, the function returns 0. Consider a particular branch, bchNo . First, we create a vector, bchVect , that only contains the nonzero elements of subs in row bchNow . This vector has length szBch . For each element of bchVect , denoted elm , we remove this compartment from the system, to create a new system with interaction matrix subA and mortality vector subD . This is achieved using the function remone. Once every compartment in bchVect has been removed from the system, the size of the system, two checks are made. First, we check to ensure that the subsystem is non-singular. Next, we check to ensure that the subsystem is feasible. If it fails either of these checks, we know that the subs matrix provided was in error, and so a message is returned to indicate this. subX is the steady state solution for the subsystem. We create subXFull , which is subX with zero entries where the absent compartments are. subXFull is then used to calculate the the transversal eigenvalues. If all of these eigenvalues are non-positive, the system cannot recover from this subsystem, and so the system is not permanent with respect to the removal of this branch. Therefore, netsperm returns this branch number. If at least one of the eigenvalues are positive, we continue to the next iteration of the while loop. h netsperm.m

17 i

≡

function res = netsperm(A, D, subs ); Asz = size(A); Asz = Asz (1, 1);

408


% Check permanence of each branch szSubs = size(subs );

% [Number of branches, max elms per branch]

flagP = 0;

% Indicates system is permanent so far

flagB = 0;

% Indicates system failed due to infeasable subsystem

bchNo = 0;

% Initialise

res = 0;

% Initialise all perm

while (flagP ≡ 0) ∧ (flagB ≡ 0) ∧ (bchNo < szSubs (1, 1)) ; bchNo = bchNo + 1; h Create bchVect

18 i

subA = A; subD = D;

% This will be the subsystem

subsz = Asz ; subBchVect = bchVect ; for elmInd = 1 : szBch ;

% For each element of bchVect

elm = subBchVect (elmInd ); [subA, subD ] = remone(elm , subA, subD ); % Update all because removal from previous subsz = subsz − 1; subBchVect = subBchVect − 1; end

% Check feas/sing

rankSubA = rank(subA); if rankSubA < subsz ;

% Error check

disp (bchVect (bchNo , : )); error(’subs includes singular subsystem’); flagB = 1; res = bchNo ; else subX = subA\ − subD ; if (min(subX ) < 0)

% Error check

flagB = 1; disp (bchVect (bchNo , : )); error(’subs includes infeasible subsystem’); res = bchNo ; end end subXFull = subX ;

% Each branch


szsubX = Asz − szBch ; for elmInd = 1 : szBch ;

% Put zeros back in

elm = bchVect (elmInd ); if elm ≡ 1 ; subXFull = [0; subXFull ]; elseif elm ≡ szsubX + 1; subXFull = [subXFull ; 0]; else subXFull = [subXFull (1 : elm − 1, 1); 0; subXFull (elm : szsubX , 1)]; end szsubX = szsubX + 1;

% Because just added spp

end % Check each transversal eigenvalue tranVect = [ ];

%

for elmInd = 1 : szBch ; elm = bchVect (elmInd ); tranVect (elmInd , 1) = A(elm , : )∗subXFull + D(elm , 1); end if max(tranVect ) < 0 flagP = 1;

% Not permanent

res = bchNo ; end end

% Check of each branch

409

410


18.

Create bchVect.

The purpose of this routine is to take all non-zero elements of a row of subs , and create a vector bchVect from these elements. h Create bchVect

18 i

bchInd = 0; elm = 1;

≡

% Initialise index % Initialise value

bchVect = [ ];

% Initialise vector containing branch of interest

bchVectInd = 0; while (bchInd < szSubs (1, 2)) ; bchInd = bchInd + 1; elm = subs (bchNo , bchInd ); if elm 6= 0 bchVectInd = bchVectInd + 1; bchVect (bchVectInd , 1) = elm ; end end szBch = size(bchVect ); szBch = szBch (1, 1); This code is used in section 17.

% For each elm


19.

411

Functions relating to finding subsystems.

Following from the work of Kirlinger (Kirlinger 1986, Kirlinger 1988), all possible subsystems must be found before the transversal eigenvalues may be tested. While finding subsystems of a system is relatively simple with pen and paper, the algorithms required for computational searches are surprisingly involved. The main issue is that not all subsystems need to be generated. There are two types of subsystems that may be excluded from consideration. The first type is those subsystems that are singular, and the second type we will call floating subsystems, which is described below. For a singular system, both the A matrix, and the Jacobian matrix, are singular. As such, if we were to allow the algorithm to blindly compute steady solutions to this system, Octave will persist until terminating when the round-off error becomes too small, thus wasting CPU time, and giving an inaccurate solution. A floating subsystem is one for which there does not exist a path for nutrients to flow from an autotroph to one or more of the heterotrophs. We will call these unconnected heterotrophs floating heterotrophs. Because they have no nutrient supply, floating heterotrophs always have a population size (or biomass) of zero. As a result, testing the permanence of the full system with respect to the removal of floating heterotrophs is redundant. The algorithm described in this section is a combination of a depth-first search to identify all subsystems, functions testing for the constraints described above, and post-processing routines to eliminate double entries of particular subsystems. I have found it convenient to define subsystems by the compliment of the subsystem (i.e. the set of compartments absent from the full system) rather than by the subsystem itself. This convention is used for the key variables subs and substg , which are lists of subsystems against which permanence is tested.

412

20.


findsubs. (Recursive)

The purpose of this function find all of the subsystems of a specified system. This function uses two constraints upon which subsystems are included. First, no part of the system may ‘float’. That is, there must be at least one path by which nutrients can flow from a autotroph to every heterotroph compartment. Second, the community interaction matrix, A, cannot be singular. It accepts three variables: 1. A: n × n matrix. The community interaction matrix. 2. D: n × 1 vector. The mortality vector. 3. ignoreVec : m × 1 vector. A vector of compartment indices which we want disregarded. It returns three variables: 1. subs : k1 × m matrix. Specifies the subsystems found so far that satisfy the constraints. Each row is a different subsystem. Each element is the index of the compartment removed from A to create the subsystem. The function findnonfloat is used to determine which compartments can be removed without resulting in a floating system. These compartments are stored in nonFloatVect . They provide a starting point for a recursive depth-first search. Compartments in nonFloatVect which correspond to an isolated autotroph are removed. This is done to save CPU time. There are two lists of subsystems (defined by the compliment of the subsystem relative to the full system defined by A) that are used in the algorithm: subs , and substg . subs contains all subsystems that satisfy various constraints. Ultimately, subs will contain all subsystems for which the permanence of the full system is tested against. substg contains subsystems for which further subsubsystems should be found. For each compartment, remelm , in nonFloatVect , we create a subsystem, subA, by removing that compartment. The qualities of subA determine whether that compartment is added to subs , substg , or both. The rules are summarised below. For each substg , subsystems of that subsystem (subsubsystems) are found recursively. At each step of the unfolding of the recursion, the indices of subsubsystems are appended onto the end of the subsystems. Finally, findsubs returns subs , a matrix of all subsystems. h findsubs.m

20 i

≡

function [subs , substg ] =findsubs(A, D, ignoreVec ) Asz = size(A); Asz = Asz (1, 1);

% Size of system


nonFloatVect = findnonfloat(A, ignoreVec );

% Which can be rem

szNonFloatVect = size(nonFloatVect ); szNonFloatVect = szNonFloatVect (1, 1); % Initialisations subs = [ ]; subsInd = 0; substg = [ ]; substgInd = 0; h Remove isolated autotrophs

21 i

for nonFloatInd = 1 : szNonFloatVect ; remelm = nonFloatVect (nonFloatInd , 1); [subA, subD ] = remone(remelm , A, D);

% Fnc removes 1 compart

szsubA = size(subA); szsubA = szsubA(1, 1); if rank(subA) ≡ szsubA ;

% If not singular

subX = subA\ − subD ; if min(subX ) > 0 ; h Add to subs

% If feasible

24 i

h Determine if single autotroph

22 i

h Determine if member of chain

23 i

if indicEnd ≡ 0

% If not at end

h Add to list of subs to go

25 i

end else h Add to list of subs to go

25 i

end else h Add to list of subs to go

25 i

end end szsubstg = size(substg ); szsubstg = szsubstg (1, 1); if szsubstg > 0

% Subsystems of subsystems to be found

for substgInd = 1 : szsubstg

% For each subsystem to go

substgElm = substg (substgInd , 1); h Create subIgnoreVec

26 i

413

414


h Find subsystems of the subsystem

27 i

h Append subsubsystems to subsystem

28 i

end end

21.

Remove isolated autotrophs.

The purpose of this routine is to remove isolated autotrophs from nonFloatVect . The reason for doing this is that, although valid subsystems (in the sense of nonsingular, non-floating subsystems) may formed by removal of these autotrophs, it is already known that the full system is permanent with respect to the removal of these autotrophs (see body of text). For each compartment i (Aind ) in nonFloatVect , if ai,i < 0 (A(Aind , Aind ) < 0), which implies that the compartment is an autotroph, and if ai,j = 0 ∀j 6= i, which implies that the autotrophic compartment does not interact with any other compartment, then the function remonevec is used to remove that compartment from nonFloatVect . h Remove isolated autotrophs

21 i

≡

nonFloatInd = 0; Aind = 0; while nonFloatInd < szNonFloatVect ; nonFloatInd = nonFloatInd + 1; Aind = nonFloatVect (nonFloatInd ); if A(Aind , Aind ) < 0 spare = A(Aind , : ); spare (1, Aind ) = 0; if isempty (find(spare )) ; nonFloatVect = remonevec(nonFloatInd , nonFloatVect ); szNonFloatVect = szNonFloatVect − 1; nonFloatInd = nonFloatInd − 1; end end end This code is used in section 20.


22.

415

Determine if single autotroph.

The depth-first-search is terminated in either of the following instances: if the resulting subsystem is a chain, or if the resulting subsystem is a single (autotrophic) compartment. The purpose of this routine is to test for the latter. If it is found that the subsystem is a single autotroph, this is indicated by setting indicEnd = 1. szsubA is the size of the subsystem. If this is equal to 1, then we have reached the end of the search (along this path). An additional test was used to ensure that this remaining compartment is indeed an autotroph, for debugging purposes. h Determine if single autotroph indicEnd = 0;

22 i

≡

% Initialise as not single autotroph

if szsubA ≡ 1 if subA(1, 1) < 0 indicEnd = 1; else error(’Ended with heterotroph.’); end end This code is used in section 20.

23.

Determine if member of chain.

The function findnonchain is used to determine if the compartment to be removed, remelm , is a member of a chain. Details may be found in Section 32. h Determine if member of chain

23 i

≡

indicEnd = findnonchain(remelm , A); This code is used in section 20.

24.

Add to subs.

This routine increments the counter subsInd , and appends remelm to subs . h Add to subs

24 i

≡

subsInd = subsInd + 1; subs (subsInd , :) = remelm ; This code is used in section 20.

416

25.


Add to list of subs to go.

This routine increments the counter substgInd , and appends remelm to substg . h Add to list of subs to go

25 i

≡

substgInd = substgInd + 1; substg (substgInd , :) = remelm ; This code is used in section 20.


26.

417

Create subIgnoreVec.

The purpose of this subroutine is to reduce the number of multiple instance of subsystems found. As discussed in Section 19, there is some subset of subsystems that are non-floating. Therefore, for each full system, there is some subset of all compartments for which the removal of each compartment will not result in a floating system. For convenience, we will refer to the compartments that possess this property as non-unique path bearing compartments (NPB), in recognition of the fact that nutrients can find other paths to all compartments when they are absent. That is, a compartment is a NPB if and only if the removal of that compartment from the full system results in a non-floating subsystem. Consider a system such that it has n NPB compartments, {i1 , i2 , . . . in }. Defining a subsystem by the set of compartments absent from the system, it is apparent that subsystem {ij , ik } and {ik , ij } are the same subsystem, and furthermore, that all of their subsubsystems, as found by the algorithm, will be the same. The following (arbitrary) constraint will avoid this redundancy: if ij and ik are both NPB compartments of a system, and if the indices are such that ij < ik , only subsystems of {ij , ik } will be found. This is achieved by defining ignoreVec such that, when it comes time to find subsystems of {ik } it contains only il satisfying ik < il . The designation of ignoreVec is slightly complicated by fact that the indices of the compartments will shift relative to the first NPB compartment removed, substgElm . The mapping is described as follows:   i if i < substgElm , (H.1) i→  i − 1 if i > substgElm .

Conveniently, however, defining ignoreVec such that it only includes those values satisfying the first equation avoids the need to shift the indices. h Create subIgnoreVec

26 i

≡

subIgnoreVec = [ignoreVec ; nonFloatVect ]; subIgnoreVec = sort(subIgnoreVec); lessThan = find(subIgnoreVec < substgElm ); lessThan (lessThan , 1) = subIgnoreVec (lessThan , 1); subIgnoreVec = lessThan ; This code is used in section 20.

418


27.

Find subsystems of the subsystem.

Each element of substg is a compartment that is removed to create a particular subsystem. Those subsystems included in substg are non-chains, which implies that we are interested in finding subsystems of these subsystems (subsubsystems). In order to find subsubsystems for each entry, substgElm in substg , subA is created such that compartment substgElm is removed. This is then fed into findsubs recursively. findsubs returns subsD , which is a matrix of subsystems of subA. h Find subsystems of the subsystem

27 i

≡

[subA, subD ] = remone(substgElm , A, D);

% Create sub-subsystem

[subsD , substgD ] = findsubs(subA, subD , subIgnoreVec);

% Recursion

This code is used in section 20.

28.

Append subsubsystems to subsystem.

In order to append subsD onto the end of the terminating subs , two steps are required. First, newSubs must be created, such that subsD is appended onto the end of substgElm . That is, newSubs = [substgElm ,subsD ]. Second, a matrix newSubs must be appended onto the end of the subs matrix. That is, subs = [subs ,newSubs ]T . Regarding the creation of newSubs , the indices returned in subsD will not match the indices used on this level of the recursion. For this reason, a subroutine is used to adjust the indices of subsD . Regarding appending newSubs to the end of subs , the number of columns of the two matrices may differ. For this reason, as subroutine is used to pad the smaller matrix with zeros. h Append subsubsystems to subsystem

28 i

≡

szsubsDrow = size(subsD ); szsubsDcol = szsubsDrow (1, 2); szsubsDrow = szsubsDrow (1, 1); if szsubsDrow > 0 ; h Adjust indices of subsD

29 i

substgElmRep = substgElm ∗ones(szsubsDrow , 1); newSubs = [substgElmRep , subsD ]; h Pad smaller of subs and newSubs with zeros subs = [subs ; newSubs ]; end This code is used in section 20.

30 i


29.

419

Adjust indices of subsD.

The indices returned in subsD will not match the indices used on this level of the recursion. For example, if the system is {a, b, c, d, e}, corresponding to indices {1, 2, 3, 4, 5}, and substgElm is {c}, subA in the level below will be {a, b, d, e}, with indices {1, 2, 3, 4}. Therefore, all indices returned in subsD with values greater than or equal substgElm are in fact that index plus one. To account for this, for each element of subsD , if the element has a value greater than or equal to substgElm , one is added to its value. h Adjust indices of subsD

29 i

≡

for subsDrow = 1 : szsubsDrow for subsDcol = 1 : szsubsDcol if subsD (subsDrow , subsDcol ) ≥ substgElm subsD (subsDrow , subsDcol ) = subsD (subsDrow , subsDcol ) + 1; end end end This code is used in section 28.

420


30.

Pad smaller of subs and newSubs with zeros.

The purpose of this subroutine is to pad either subs or newSubs with zeros, such that the number of columns in each are equal. First, the sizes of subs and newSubs are determined. If the number of columns in newSubs exceeds the number of columns in subs , the end of subs is padded with zeros to make up the difference. Alternately, the end of newSubs is padded with zeros to make up the difference. h Pad smaller of subs and newSubs with zeros

30 i

≡

szNewSubsrow = size(newSubs ); szNewSubscol = szNewSubsrow (1, 2); szNewSubsrow = szNewSubsrow (1, 1); szsubsrow = size(subs ); szsubscol = szsubsrow (1, 2); szsubsrow = szsubsrow (1, 1); if szsubscol < szNewSubscol subs = [subs , zeros(szsubsrow , szNewSubscol − szsubscol )]; elseif szsubscol > szNewSubscol newSubs = [newSubs , zeros(szNewSubsrow , szsubscol − szNewSubscol )]; end This code is used in section 28.


31.

421

findsubspost.

The purpose of this function is to remove double entries from subs . It accepts one variable: 1. subs : k1 × m matrix. Specifies the subsystems found. Each row is a different subsystem. Each element is the index of the compartment removed from A to create the subsystem. It returns one variable: 1. subsred : k2 × m matrix. Specifies the subsystems found. Each row is a different subsystem. Each element is the index of the compartment removed from A to create the subsystem. The number of rows and columns in subs is found. The sum of each row of subs is stored in sumSubs . In will be used in the while loop. subsred is set equal to subs . The algorithm will proceed by systematically removing entries from subsred . For each row index subInd of subsred , the row is stored as subComp . A first indication that other rows are equivalent to subComp is that the sum of their entries will be equal. The indices of rows satisfying this are stored in findEqSums . diffSubs is created using the indices stored in findEqSums . Ultimately, diffSubs becomes a column vector where each entry is the sum of the absolute value of the difference between each entry in subComp and each entry identified by findEqSums . An entry in diffSubs will be equal to zero if and only if a particular row is equivalent to subComp . The indices of such rows, relative to findEqSums , are stored in findEqElms . findEqSums is then used to convert these entries back into indices relative to subsred . h findsubspost.m

31 i

≡

function subsred = findsubspost(subs ) szsubsrow = size(subs ); szsubscol = szsubsrow (1, 2); szsubsrow = szsubsrow (1, 1); if szsubscol ≡ 1 sumSubs = subs ; else sumSubs = sum(subs ′ )′ ; end subs = sort(subs ′ )′ ; subsred = subs ;

422


subInd = 0; while subInd < szsubsrow

% For each row

% Choose index for comparison subInd = subInd + 1; subComp = subsred (subInd , :); sumSubComp = sumSubs (subInd , 1);

% Find rows with equal sum to comparison row

findEqSums = find(sumSubs ≡ sumSubComp ); szFindEqSums = size(findEqSums ); szFindEqSums = szFindEqSums (1, 1); subCompRep = nadrepm(subComp , szFindEqSums , 1); % Of the equal sum rows, identify rows with equiv elements diffSubs = subsred (findEqSums , : ); diffSubs = abs(diffSubs − subCompRep );

sumDiffSubs = sum(diffSubs ′ )′ ;

findEqElms = find(sumDiffSubs ≡ 0); % Convert indices in findEqElms to indices in subsred findEqElms = findEqSums (findEqElms , 1); szFindEqElms = size(findEqElms ); szFindEqElms = szFindEqElms (1, 1); if szFindEqElms > 1 eqElmInd = 2; while eqElmInd ≤ szFindEqElms eqElm = findEqElms (eqElmInd ); subsred = remonevec(eqElm , subsred ); findEqElms = findEqElms − 1; eqElmInd = eqElmInd + 1; end end szsubsrow = size(subsred ); szsubscol = szsubsrow (1, 2); szsubsrow = szsubsrow (1, 1); if szsubscol ≡ 1 sumSubs = subsred ; else sumSubs = sum(subsred ′ )′ ;


end end

423

424


32.

findnonchain.

The purpose of this function is to determine whether or not a specified compartment is a member of a chain. It accepts two variables: 1. elm : Integer. The index of the compartment to be tested. 2. A: n + 1 × n + 1 matrix. The community interaction matrix. It returns one variable: 1. chain : Integer 1 or 0. 1 indicates that the compartment is the member of a chain, and 0 indicates that it is not. The identification of a non-chain is performed by identifying branching in the web connected to elm . Because branches may either branch upward or downward (i.e. a prey with more than one predator, or a predator with more than one prey, respectively), it is necessary to make three passes up and down the trophic web, starting at the specified compartment elm . The first pass, up the web, begins at elm . Nutrient at elm is pulsed up the web using augmented A, and at each step, a test is performed to ensure that only one compartment holds nutrient at any one time. The first pass ensures that there are no branches upward above elm . If the web connected to elm is not found to be a non-chain after the first pass, a second pass is performed. The second pass, down the web, begins at the last compartment reached by the nutrient in the previous pass. Nutrient is pulsed down the web by use of the transpose of the Augmented A. Again, at each step, a test is performed to ensure that only one compartment holds nutrient at any one time. The second pass ensures that there are no branches downward. If the web connected to elm is not found to be a non-chain after the second pass, a third, and final, pass is performed. The third pass, up the web, begins at the last compartment reached by the nutrient in the previous pass. Nutrient is pulsed down the web by use of the Augmented A. Again, at each step, a test is performed to ensure that only one compartment holds nutrient at any one time. The third pass ensures that there are no branches upward below elm . If the web connected to elm is not found to be a non-chain after the third and final pass, it is a chain. h findnonchain.m

32 i

≡

function chain = findnonchain(elm , A);


Asz = size(A); Asz = Asz (1, 1); h Augment A

36 i

chain = 1; multA = zeros(Asz , Asz ); multA(elm , :) = ones(1, Asz ); % First pass (up web) count = 0; szNut = 42; recNut = elm ; while (szNut > 0) ∧ (chain ≡ 1) ∧ (count < Asz ) count = count + 1; multA = A∗multA; nut = find(multA(: , 1)); szNut = size(nut ); szNut = szNut (1, 1); if szNut ≡ 1 recNut = nut ; elseif szNut > 1 chain = 0; end end % Second pass (down web from recNut) elm = recNut ; A = A′ ; multA = zeros(Asz , Asz ); multA(elm , :) = ones(1, Asz ); count = 0; szNut = 42; recNut = elm ; while (szNut > 0) ∧ (chain ≡ 1) ∧ (count < Asz ) count = count + 1; multA = A∗multA; nut = find(multA(: , 1)); szNut = size(nut );

425

426


szNut = szNut (1, 1); if szNut ≡ 1 recNut = nut ; elseif szNut > 1 chain = 0; end end % Third pass (up web from recNut) elm = recNut ; A = A′ ; multA = zeros(Asz , Asz ); multA(elm , : ) = ones(1, Asz ); count = 0; szNut = 42; recNut = elm ; while (szNut > 0) ∧ (chain ≡ 1) ∧ (count < Asz ) count = count + 1; multA = A∗multA; nut = find(multA(: , 1)); szNut = size(nut ); szNut = szNut (1, 1); if szNut ≡ 1 recNut = nut ; elseif szNut > 1 chain = 0; end end


33.

427

findnonfloat.

The purpose of this function is to identify which compartments may be removed from the system without resulting in a ‘floating’ subsystem. By ‘floating’, we mean a system in which there does not exist a path for nutrients from an autotroph to one or more heterotrophs. It accepts two variables: 1. A: n × n matrix. The community interaction matrix. 2. ignoreVec : m × 1 vector. A vector of compartment indices which we want disregarded. It returns one variable: 1. nonFloatVect : m × 1 vector. Specifies the compartments which may be removed without resulting in a floating subsystem. First, an augmented A, and multABak , are created. The augmented A is the original A with all diagonal and negative entries removed, and all off-diagonal positive entries replaced with 1. multABak is a matrix of size Asz × Asz , such that each element of each row corresponding to an autotrophic compartment is equal to 1, and all other elements are zero. We test the removal of each compartment, rowrem , from 1 to Asz . For each subsystem, the two matrices described above are used to track the flow of nutrients from the autotroph compartment(s) to each of the other compartments in a subsystem. We define subA as the augmented A with compartment rowrem removed. For each time step, the operation multA=subA*multA simulates the flow of nutrients up the system. We are interested in ensuring that the nutrient flow visits every compartment. We define the following parameters • visitVect initialised as a vector of zeros. When nutrient flows visits a compartment, the corresponding element of visitVect is set to 1. • szVisitVect is an integer. Its value is equal to the number of nonzero elements in visitVect . • indV is a vector. It records which compartments contain nutrient at a given time step. While the number of time steps is less that Asz − 1 (which is the maximum number of iterations it should take for nutrient to flow through the entire system), and while szVisitVect < Asz − 1 (i.e. while there remain compartments to be visited), and while szindV > 0 (which indicates that nutrient remains in the system) the operation is performed. At the end of this while loop, if all compartments were visited by the flow of nutrients, the subsystem defined by the removal of rowrem contains no floating compartments, and therefore, is added to

428


nonFloatVect . h findnonfloat.m

33 i

≡

function nonFloatVect = findnonfloat(A, ignoreVec ); Asz = size(A); Asz = Asz (1, 1); h Create multABak, augment A

34 i

% Track nutrients through system nonFloatInd = 0; nonFloatVect = [ ]; for rowrem = 1 : Asz if any (find(ignoreVec ≡ rowrem )) ≡ 0 visitVect = zeros(Asz − 1, 1); szVisitVect = find(visitVect ); szVisitVect = size(szVisitVect ); szVisitVect = szVisitVect (1, 1); subA = remone(rowrem , A); multA = multABak ;

% Trial removal

% Initialise

multA = remone(rowrem , multA); indV = find(multA(: , 1)); szindV = size(indV ); szindV = szindV (1, 1); multCount = 0; while (multCount < Asz ) ∧ (szindV > 0) ∧ (szVisitVect < Asz − 1) ; multCount = multCount + 1; visitVect (indV , 1) = 1; szVisitVect = find(visitVect ); szVisitVect = size(szVisitVect ); szVisitVect = szVisitVect (1, 1); multA = subA∗multA; indV = find(multA(: , 1)); szindV = size(indV ); szindV = szindV (1, 1); end if szVisitVect ≡ Asz − 1 % All were visited

% Nutrients flow up food web


429

nonFloatInd = nonFloatInd + 1; nonFloatVect (nonFloatInd , 1) = rowrem ; end end end

34.

Create multABak, augment A.

The matrices multABak and the augmented A are used throughout the code. It is used to track nutrient flow through the system, which can be used to determine various qualities of the system. multABak is a matrix of ones and zeros, where a row of ones indicates that the nutrient is present in the corresponding compartment. The A matrix is also a matrix of ones and zeros, where a one indicates an (positive or negative) interaction term between the compartments corresponding to the row and column index of the entry. In general, nutrient flow is simulated by operations of the form multABak = A∗multABak (e.g. Section 33). Details may be found in the corresponding subroutines (Sections 35) and 36). h Create multABak, augment A h Create the multABak h Augment A

34 i

≡

35 i

36 i

This code is used in sections 33, 45, and 52.

430


35.

Create the multABak.

The purpose of this subroutine is to initialise the matrix multABak , such that each autotroph contains nutrient. For each element in A, if that element is on the diagonal and positive, this indicates that the compartment corresponding to the row in question is an autotroph. For each autotroph, a zero matrix multABak is augmented such that all elements in that row are ones. h Create the multABak

35 i

≡

multABak = zeros(Asz , Asz ); for row = 1 : Asz if A(row , row ) < 0 ;

% autotroph

multABak (row , :) = ones(1, Asz ); end end This code is used in section 34.


36.

431

Augment A.

The purpose of this subroutine is to augment the A matrix, such that its elements are all ones and zeros, where a one indicates a positive interaction term in A. When used in conjunction with multABak in the manner described in Section 34, it simulates the flow of nutrients up the food web. The reverse may be achieved by taking the transpose of the augmented A. For each element in A, if it is on the diagonal, it is set to 0. Else, if it is not on the diagonal, if it is positive, it is set to 1, else it is set to zero. Cohen (1989b) terms this matrix the predation matrix, P . It is a matrix in which each predator’s interaction is denoted by a 1.   0 if ai,j ≤ 0, pi,j =  1 if ai,j > 0.

This is exactly the same as the augmented A matrix. h Augment A

36 i

≡

for row = 1 : Asz for col = 1 : Asz if row ≡ col A(row , col ) = 0; else if A(row , col ) > 0 A(row , col ) = 1; else A(row , col ) = 0; end end end end This code is used in sections 7, 32, and 34.

(H.2)

432


singular? singular not singular not singular not singular not singular

Is the feasible? (not applic.) not feasible feasible feasible feasible

subsystem ... a single compartment? (not poss.) (not poss.) > 1 compartment 1 compartment > 1 compartment

a chain? (not poss.) (either) chain (chain) not chain

Then add remelm to ... substg substg subs subs subs & substg


37.

433

Accessory functions.

The following sections document functions that are not particularly interesting from a modelling point of view, but necessary nonetheless. They are included here for completeness.

434

38.


userparameters.

The purpose of this script is to allow the user an easy way to change certain parameters. As such, documentation of it is contained in comments. This script is reserved for parameters that are changed infrequently. h userparameters.m

38 i

≡

% Set to 0 for no, and 1 for yes % Food web building algorithm % ————————— % In the output file, would you like the time printed in comments for % each [A,D] generated? printTime = 1; % Would you like to use a specified starting matrix? specADinit = 0; % If yes, what is the matrix? AD = [ ]; % Would you like to restrict the number of independent autotrophs in % the system? If yes, specify the number maxAuto = 10; % Would you like the option of restricting the maximum index of subs % that is jumbled? maxJumbleInd = 0; % Would you like the option of restricting the maximum size of subs? maxSzSubs = 1000; % Would you like to restrict the interactions between predator and % prey such that the proportional growth of the predator compartment is greater % than or equal to the decline in the prey? constFlag = 1; % Generally, predators high on the trophic scale are unable to process % both animal and plant effectively. If we numbered each compartment % as t+1, when t is the highest trophic level of its prey, and basal = % 1, what is the highest predator you will permit to eat plants? maxBasPred = 0;


% Autotrophs have a positive di term, with a range % (0, maxAutoDi). What should the maxAutoDi value be (typical default % is maxAutoDi=1)? maxAutoDi = 1; % Post-processing algorithm % ———————— % Would you like to omit systems under a certain size? Specify number % if so minSz = 0;

435

436

39.


makea.

The purpose of this function is to generate a community of the size specified, with random interaction strengths and mortality. This function will generate food chains only. It accepts two variables: 1. Asz : Integer. The length of the food chain. Note that this must be greater than 1. 2. maxAutoDi : Real positive number. The maximum value of the di range for autotrophs. Increasing this increases the productivity of the autotrophic compartments. It returns three variables: 1. A: n + 1 × n + 1 matrix. The community interaction matrix. 2. D: n + 1 × 1 vector. The mortality vector. 3. X: n × 1 vector. The final steady nutrient concentration of the biotic compartments. While the system is not feasible (flagB =0), A is generated by assigning random variables to a food chain structure. Entries in D are also randomly assigned. h makea.m

39 i

≡

function [A, D, X] =makea(Asz , maxAutoDi ); factSelfNeg = 1;

% Scaling self limitation term autotroph

flagB = 0; while flagB ≡ 0 ;

% While the system is not feasible

pred = diag (rand(Asz − 1, 1), 1); prey = diag (−1∗rand(Asz − 1, 1), −1); self = diag ([zeros(Asz − 1, 1); −rand∗factSelfNeg ]); A = pred + prey + self ; D = [−rand(Asz − 1, 1); maxAutoDi ∗rand]; X = A\ − D; if min(X) > 0 ;

% Check minimum biomass is ¿ 0 % If system feas

flagB = 1; end end

% Create death

% If system feas % Found feasible system


40.

437

jumble.

The purpose of this function is to randomise the order of the rows of a matrix. It is applied to subs in the body of the code. It accepts two variables: 1. subs : k1 × m matrix. Specifies the subsystems. Each row is a different subsystem. Each element is the index of the compartment removed from A to create the subsystem. 2. maxJumbleInd : Integer. This restricts the guaranteed reordering to the first maxJumbleInd rows. If set to zero, all rows are reordered. It returns one variable: 1. res : subs above in a randomised order. The function operates by counting through each row of subs , denoted subsInd , and a comparison row, subsCompInd . With a probability of 0.5, the two rows are interchanged. h jumble.m

40 i

≡

function res = jumble(subs , maxJumbleInd ) szSubs = size(subs ); szSubs = szSubs (1, 1); if (maxJumbleInd ≡ 0) ∨ (szSubs < maxJumbleInd ) szSubsMax = szSubs ; else szSubsMax = maxJumbleInd ; end for subsInd = 1 : szSubsMax ; for subsCompInd = 1 : szSubs ; if rand > .5 ; spare = subs ;

% Swap two rows

subs (subsInd , :) = spare (subsCompInd , :); subs (subsCompInd , :) = spare (subsInd , :); end end end res = subs ;

438

41.


remone.

The purpose of this function is to accept a community matrix, and optionally an associated mortality vector, and remove the specified compartment from these systems. It accepts three variables: 1. elm : integer. The compartment to be removed. 2. A: n × n matrix. The community interaction matrix. 3. D: n × 1 vector. The mortality vector. It returns two variables: 1. newA: n − 1 × n − 1 matrix. The new community interaction matrix. 2. newD : n − 1 × 1 vector. The new mortality vector. h remone.m

41 i

≡

function [subA, subD ] =remone(elm , A, D) Asz = size(A); Asz = Asz (1, 1); % Error checking if nargin ≡ 2

% Only doing matrix

D = zeros(Asz , 1); else spare = size(D); spare = spare (1, 1); if Asz 6= spare error = (’Size of matrix and vector does not conform’); end end subA = A; subD = D; if elm ≡ 1 ; subA = subA(elm + 1 : Asz , :);

% rows

subA = subA(: , elm + 1 : Asz );

% cols

subD = subD (elm + 1 : Asz , :);

% rows

elseif elm ≡ Asz ; subA = subA(1 : elm − 1, :);

% rows

subA = subA(: , 1 : elm − 1);

% cols

subD = subD (1 : elm − 1, :);

% rows


else subA = [subA(1 : elm − 1, :); subA(elm + 1 : Asz , : )]; subA = [subA(: , 1 : elm − 1), subA(: , elm + 1 : Asz )]; subD = [subD (1 : elm − 1, : ); subD (elm + 1 : Asz , :)];

% rows % cols % rows

end

42.

remonevec.

The purpose of this function is to accept any vector, and and remove the specified rows from it. It accepts three variables: 1. elm : integer. The row to be removed. 2. M : n × m matrix. The matrix of interest. It returns one variable: 1. subM : n − 1 × m matrix. The matrix of interest minus the specified rows. h remonevec.m

42 i

≡

function subM = remonevec(elm , M ) Msz = size(M ); Msz = Msz (1, 1); subM = M ; if elm ≡ 1 ; subM = subM (elm + 1 : Msz , :);

% rows

elseif elm ≡ Msz ; subM = subM (1 : elm − 1, : );

% rows

else subM = [subM (1 : elm − 1, :); subM (elm + 1 : Msz , : )]; end

% rows

439

440

43.


nadrepm.

This function was written by Paul Kienzle. It is included in this documentation for completeness. h nadrepm.m

43 i

≡

% Author: Paul Kienzle (pkienzle at kienzle.powernet.co.uk) % Created: July 2000 % 2001-06-27 Paul Kienzle (pkienzle at users.sf.net) % * cleaner, slightly faster code function x = nadrepm(b, m, n) if ( nargin < 2 ∨ ∨ nargin > 3 ) usage (’repmat (a, m, n)’); endif if nargin ≡ 2 if is scalar (m) n = m; elseif ( is vector (m) ∧ ∧ length (m) ≡ 2 ) n = m(2); m = m(1); else error(’repmat: only builds 2D matrices’) endif endif [rb , cb ] = size(b); if (isempty (b)) x = zeros(m∗rb , n∗cb ); else x = b([1 : rb ]′ ∗ones(1, m), [1 : cb ]′ ∗ones(1, n)); endif endfunction


44.

441

Post-processing.

The functions described in the following sections are concerned with the calculation of various metrics and emergent properties of the systems generated by permbuild (Section 2). The process is described in Section 11.2.4.

442

45.


postprocess.

The purpose of postprocess is to calculate the values of various emergent properties of the output systems, as described in the body of the text. They are specified in a list below. It accepts three variables: 1. itnStart : Integer. Emergent properties are calculated for iterations itnStart to itn . This allows the algorithm some time to initialise if the user so chooses. 2. eachstepName : String. The name of the file from which output from permbuild is read. The default often used is eachstep .m. 3. fName : String. The name of the file to which output from postprocess can be stored. Optional. It returns one variable: 1. res : Matrix. The result of post-processing, each row of res corresponds to the largest food web present in the system with the same iteration number as the row index. Each column is the integer value (or NaN ) of some quality of the food web, as specified in the list below. The following properties are measured for the largest food web in the system, where the item number corresponds to the column number of res , the output: 1. isCh : 0 or 1. Indicates (by a 1) that the food web is a chain. 2. noTop : Integer. The number of top compartments. 3. noInt : Integer. The number of intermediate compartments. 4. noBas : Integer. The number of basal compartments. 5. noLtot : Integer. The total number of links. 6. noLbi : Integer. The number of links from the basal to the intermediate compartments. 7. noLbt : Integer. The number of links from the basal to the top compartments. 8. noLii : Integer. The number of links between two intermediate compartments. 9. noLit : Integer. The number of links from the intermediate to the top compartments. 10. szEch : Integer. The size of the system. 11. noOm1 : Integer. The number of omnivores of Type 1 (see corresponding subroutine). 12. noOm2 : Integer. The number of omnivores of Type 2 (see corresponding subroutine). 13. maxChnVec : Integer. The maximum chain length, measured in compartments.


443

14. cycleVec : 0 or 1. Indicates (by a 1) the presence of cycles in the food web. 15. rigCircVec : 0 or 1. Indicates (by a 1) that the niche-overlap graph is a rigid-circuit graph. First, we read in the output file, eachstepName , with the option of reading an alternate, user specified, output file. Empty storage vectors for each of the properties listed above are created. As described above, eachstep will make the following variables available: (1) its , the total number of iterations, and (2) ADitn1, ADitn2, ... ADitnn, matrices of the form [AD] for each iteration 1 . . . its . For each iteration, the function largestweb is used to identify the largest food web in the system. If the size of the food web is greater than the user specified minSz , the algorithm proceeds to calculated its properties. Storage vectors are given the same names as those specified in the above list, so the line starting with isCh = is used to determine if the web is a chain, etc. To deduce some of the properties, the augmented A matrix, and multABak , are used to track nutrient flow through through the system. Specifically, this method is used to determine noLtot , noOm1 , noOm2 , maxChnVec, cycleVec , and the link types. Provided that the results are not empty (that is, that at least one of the iterations met the minSz criteria), all of the vectors above are put into the matrix res and returned. If specified by the user, the matrix is also written in a file. h postprocess.m

45 i

≡

function res = postprocess(itnStart , eachstepName , fName ) userparameters; eval(sprintf (eachstepName )); posn = 0; h Initialise empty storage vectors

46 i

for itn = itnStart : its ; eval(sprintf ([’A=ADitn’, num2str (itn ), ’;’])); Asz = size(A); Asz = Asz (1, 1); D = A(: , Asz + 1);

% Number of rows

%!

A = A(: , 1 : Asz ); [A, D] = largestweb(A, D); Asz = size(A); Asz = Asz (1, 1); if (Asz ≥ minSz ) posn = posn + 1;

% Number of rows

444


h Initialise storage vectors

47 i

isCh (posn , 1) = ischaino(A); szEch (posn , 1) = Asz ; ABak = A; h Classify basal, intermediate, top h Create multABak, augment A

48 i

34 i

rigCirc (posn , 1) = rigidcircuit(A); noLtot (posn , 1) = sum(sum(A)); multA = multABak ; h Count omnivores and max chain

50 i

noOm1 (posn , 1) = omn1 ; noOm2 (posn , 1) = omn2 ; maxChnVec(posn , 1) = maxChn ; cycleVec (posn , 1) = cycle ; h Count link types

49 i

end end if isempty (isCh ) disp (’No web met the minimum size criteria’); disp (’Minimum size used was:’); disp (minSz ); res = 0; else res = [isCh , noTop , noInt , noBas , noLtot , noLbi , noLbt , noLii , . . . noLit , szEch , noOm1 , noOm2 , maxChnVec , cycleVec , rigCirc ]; if nargin ≡ 3 h Print output from postprocess to file end end

67 i


46.

Initialise empty storage vectors.

h Initialise empty storage vectors rigCirc = [ ]; cycleVec = [ ]; maxChnVec = [ ]; noTop = [ ]; noInt = [ ]; noBas = [ ]; noLtot = [ ]; noLbi = [ ]; noLbt = [ ]; noLii = [ ]; noLit = [ ]; szEch = [ ]; isCh = [ ]; noOm1 = [ ]; noOm2 = [ ]; This code is used in section 45.

46 i

≡

445

446

47.


Initialise storage vectors.


47 i

rigCirc (posn , 1) = 0; cycleVec (posn , 1) = 0; maxChnVec (posn , 1) = 0; noTop (posn , 1) = 0; noInt (posn , 1) = 0; noBas (posn , 1) = 0; noLtot (posn , 1) = 0; noLbi (posn , 1) = 0; noLbt (posn , 1) = 0; noLii (posn , 1) = 0; noLit (posn , 1) = 0; szEch (posn , 1) = 0; isCh (posn , 1) = 0; noOm1 (posn , 1) = 0; noOm2 (posn , 1) = 0; This code is used in section 45.

≡


48.

447

Classify basal, intermediate, top.

The purpose of this routine is to classify each compartment in the web as either basal, intermediate, or top. For each compartment in the system, the corresponding element of the vector levVect is given a value denoting the compartment’s trophic status. Basal species are given value 3, intermediate species value 2, and top species value 1. At the same time, the vectors noBas , noInt and noTop are altered accordingly. For each row (each compartment) in the web, if the diagonal element is less than zero, the compartment is an autotroph, and therefore a basal compartment. Else, further checks are required to identify it. For heterotrophs, for each column in the row, if the entry is less than zero, this implies that the compartment is prey for some other compartment. Therefore, the compartment is an intermediate compartment. For compartments for which no negative entry can be found, the entry in levVect remains at the initialised value of 1. h Classify basal, intermediate, top

48 i

≡

levVect = zeros(Asz , 1); for row = 1 : Asz if A(row , row ) < 0 levVect (row , 1) = 3; noBas (posn , 1) = noBas (posn , 1) + 1; else col = 0; levVect (row , 1) = 1;

% Initialise as top

noTop (posn , 1) = noTop (posn , 1) + 1; while (col < Asz ) ∧ (levVect (row , 1) 6= 2) ; col = col + 1; if A(row , col ) < 0 levVect (row , 1) = 2; noInt (posn , 1) = noInt (posn , 1) + 1; noTop (posn , 1) = noTop (posn , 1) − 1; end end end end This code is used in section 45.

% Initialise as top

448

49.


Count link types.

The purpose of this routine is to count the number of each of the link types present in the food web. Link types are identified by examining the vector levVect . levVect is created in Section 48, and it specifies the trophic level of each compartment. Recall that basal species are given value 3, intermediate species value 2, and top species value 1. For each row and each column of the augmented A (or each prey and predator pair), if a predation link exists between the pair, an ordered pair linkType is formed, which contains the trophic level of each member of the pair. Checking is then performed by a logical filtering process, starting with the largest index in linkType . h Count link types

49 i

≡

for row = 1 : Asz for col = 1 : Asz if A(row , col ) 6= 0 linkType = sort([levVect (row , 1), levVect (col , 1)]); switch linkType (1, 2) case 3

% Bx

switch linkType (1, 1) case 2 noLbi (posn , 1) = noLbi (posn , 1) + 1; case 1 noLbt (posn , 1) = noLbt (posn , 1) + 1; otherwise error(’Bx linkType’); end case 2

% Ix

switch linkType (1, 1) case 2 noLii (posn , 1) = noLii (posn , 1) + 1; case 1 noLit (posn , 1) = noLit (posn , 1) + 1; otherwise error(’Ix linkType’); end otherwise error(’Could not get second index of linkType’)


end end end end This code is used in section 45.

449

450


50.

Count omnivores and max chain.

The purpose of this routine is to count the number of omnivores in the food web. We have two definitions for omnivory. Omnivory 1 We define an omnivore (1) as a compartment that feeds upon both flesh and plant matter. For each compartment with more than one prey, if any two prey are not both basal or both intermediate, the predator is designated an omnivore (1). Omnivory 2 We define an omnivore (2) as a compartment that feeds upon two (or more) compartments that have differing shortest distances to a basal compartment. This definition has been chosen for its convenience. By tracking a single nutrient pulse from the basal compartment up the chain, one need only record how many time steps were required to reach each compartment trophdist . Then, for each compartment with more than one prey, if any two prey have differing trophdist values, the predator is designated an omnivore (2). h Count omnivores and max chain

50 i

≡

% Omnivory 1 omn1 = 0; for row = 1 : Asz if levVect (row , 1) 6= 3

prey = find(A(row , : ))′ ;

prey = levVect (prey , 1); szPrey = size(prey ); szPrey = szPrey (1, 1); if szPrey > 1 prey = prey − prey (1, 1); prey = find(prey ); if isempty (prey ) 6= 1 omn1 = omn1 + 1; end end end end % Omnivory 2

%’


trophdist = ones(Asz , 1); count = 1; szNut = 42; while (szNut > 0) ∧ (count ≤ Asz + 1) count = count + 1; multA = A∗multA; nut = find(multA(: , 1)); szNut = size(nut ); szNut = szNut (1, 1); for nutInd = 1 : szNut ; row = nut (nutInd , 1); if trophdist (row , 1) ≡ 1 trophdist (row , 1) = count ; end end end % Compartments visited = count-1 if (count − 1) > Asz ; maxChn = NaN ; cycle = 1; else maxChn = count − 1; cycle = 0; end omn2 = 0; for row = 1 : Asz if levVect (row , 1) 6= 3

prey = find(A(row , :))′ ; prey = trophdist (prey , 1);

szPrey = size(prey ); szPrey = szPrey (1, 1); if szPrey > 1 prey = prey − prey (1, 1); prey = find(prey ); if isempty (prey ) 6= 1

%’

451

452


omn2 = omn2 + 1; end end end end This code is used in section 45.

51.

largestweb.

The food web building algorithm creates systems of one or more food webs. The food web of interest is the larges in the system. The purpose of this function is to accept an interaction matrix, A, and determine which food web in the system has the largest number of members. It accepts two variables: 1. A: n × n matrix. The community interaction matrix. 2. D: n × 1 vector. The mortality vector. It returns one variables: 1. A: m × m matrix. The community interaction matrix, augmented such that only the largest food web remains. 2. D: m × 1 vector. The mortality vector, augmented such that only the largest food web remains. An augmented A and multABak is used to identify the basal compartments, and to track nutrients through the system to identify the largest web. When the largest web is found, the compliment of the largest web is found, and removed from the system. The function then returns the largest food web’s community interaction matrix, and mortality vector. h largestweb.m

51 i

≡

function [A, D] =largestweb(A, D); Asz = size(A); Asz = Asz (1, 1); h Identify basal

52 i

h Identify largest web from basal h Find compliment of largest web h Remove compliment

55 i

53 i 54 i


52.

453

Identify basal.

The purpose of this routine is to return a vector whose indices correspond to the indices of basal compartments. multABak , generated in a previous routine in largestweb, is used to achieve this. h Identify basal

52 i

≡

ABak = A; h Create multABak, augment A basVect = find(multABak (: , 1)); This code is used in sections 51 and 57.

34 i

454


53.

Identify largest web from basal.

The purpose of this routine is to identify the largest web in the system. It makes use of the recursive function pathfinder to achieve this. The augmented A is added to its transpose, as required by pathfinder. For each basal compartment, the ordered vector path is generated by pathfinder, where each entry is the index associated with a compartment which is eventually connected the the basal compartment specified. For each path generated, if the size of that path exceeds the size of a previously generated path , it is stored in pathLargst . If the new path is of the same size as a previously found path , the which path is stored is decided randomly. h Identify largest web from basal A = A + A′ ;

53 i

≡

%’ Search in both directions

szBasVect = size(basVect ); szBasVect = szBasVect (1, 1); pathLargst = basVect (1, 1); szPath = 1; for countBas = 1 : szBasVect basElm = basVect (countBas , 1); multA = zeros(Asz , Asz ); multA(basElm , :) = ones(1, Asz ); path = basElm ; path = pathfinder(path , A, multA); szPathNew = size(path ); szPathNew = szPathNew (1, 1); if (szPathNew > szPath ) ∨ ((szPathNew ≡ szPath ) ∧ (rand > 0.5)) ; szPath = szPathNew ; pathLargst = path ; end end This code is used in section 51.


54.

455

Find compliment of largest web.

The purpose of this routine is to find the compliment of the largest web. The largest web is stored in an ordered vector, pathLargst , where each entry is the index of the compartment. For each compartment elm in the system, if that compartment does not appear in pathLargst , as determined by taking the difference between pathLargst and elm , that elm is a member of the compliment compment . h Find compliment of largest web

54 i

≡

compment = [ ]; szCompment = 0; for elm = 1 : Asz diff = pathLargst − elm ; diff = find(diff ); szDiff = size(diff ); szDiff = szDiff (1, 1); if szDiff ≡ szPath ; compment (szCompment + 1, 1) = elm ; szCompment = szCompment + 1; end end This code is used in section 51.

55.

Remove compliment.

The purpose of this routine is to remove the compliment of the largest web from the system A. A, the interaction matrix, is recovered from ABak . For each entry, elm , in the compliment, compment , the function remone is used to remove that elm from both the interaction matrix A, and the morality vector D. h Remove compliment

55 i

≡

A = ABak ; for countCompment = 1 : szCompment elm = compment (countCompment , 1); [A, D] = remone(elm , A, D); compment = compment − 1; end This code is used in section 51.

456

56.


pathfinder. (Recursive)

The purpose of this function is to identify all compartments that are connected to compartment(s) of interest. It accepts three variables: 1. path : m × 1 vector. A sorted list of compartments that are eventually connected to the compartment(s) of interest. 2. A: n × n matrix. The augmented community interaction matrix plus its transpose. 3. multA: n × n matrix. Has ones in the row(s) corresponding to the compartment(s) of interest. It returns three variables: 1. path : m × 1 vector. A sorted list of compartments that are eventually connected to the compartment(s) of interest. 2. A: n × n matrix. The augmented community interaction matrix plus its transpose. 3. multA: n × n matrix. Has ones in the row(s) corresponding to the compartment(s) of interest. multA is a matrix of zeros and ones, such that there are ones in the row(s) corresponding to the compartment(s) of interest. Pathfinder finds compartments that are connected to the compartment(s) of interest by following nutrient through the (undirected) food web. First the sizes of the path determined up to this point, path , and the food web, are found. The operation multA = A∗multA is then used to find which compartments are connected (either predator or prey) of the compartment(s) of interest. nut records their indices. If at least one compartment is connected to the compartment(s) of interest, and it has not been encountered before in path , it must be appended to the path . First, we determine if the elements in nut have been encountered before in path . For each element, countPath , of path , and for each element, countNut of nut , if the two elements are equal, the compartment has been encountered before. Therefore it is removed from nut . Once all compartments already in path are removed from nut , we append nut to path , and sort path . If any elements remained in nut after the removal process, all compartments connected to these compartments are found by a recursive call of pathfinder. h pathfinder.m

56 i

≡

function [path , A, multA] =pathfinder(path , A, multA);


szPath = size(path ); szPath = szPath (1, 1); Asz = size(A); Asz = Asz (1, 1); multA = A∗multA;

% Follow nutrient one step

nut = find(multA(: , 1)); nut = sort(nut ); szNut = size(nut ); szNut = szNut (1, 1); if szNut > 0 ;

% Has it been encountered before in path ?

for countPath = 1 : szPath ; elmPath = path (countPath , 1); countNut = 1; elmNut = 1; while (elmNut ≤ elmPath ) ∧ (countNut ≤ szNut ) ; elmNut = nut (countNut , 1); if elmNut ≡ elmPath

% Has been encountered before

nut = remonevec(countNut , nut ); szNut = szNut − 1; countNut = countNut − 1; end countNut = countNut + 1; end end end path = [path ; nut ];

% Append nut to path

path = sort(path ); % Follow remaining nut for countNut = 1 : szNut multA = zeros(Asz , Asz ); multA(nut (countNut , 1), :) = ones(1, Asz ); path = pathfinder(path , A, multA); end

% Remove from nut

457

458

57.


ischaino.

The purpose of this function is to determine whether or not a food web (not a system) is a chain. It accepts one variable: 1. A: n × n matrix. The community interaction matrix. It returns one variable: 1. chain : Integer = 1 or 0. Indicates whether or not the food web is a chain (indicated by 1). First, the basal compartments are identified. Note that part of the routine “Identify basal” includes finding the augmented A and multABak . If there is more than one basal compartment, the food web is not a chain. Else, the algorithm continues with further checks. Nutrient is pulsed from the basal compartment up the web. If, at any iteration, the number of compartments containing nutrient exceeds 1, this implies that the food web contains a branch and is therefore not a chain. Finally, if the number of iterations exceeds Asz , this implies that nutrient is cycling somewhere in the system, and the food web is not a chain. h ischaino.m

57 i

≡

function chain = ischaino(A); Asz = size(A); Asz = Asz (1, 1); h Identify basal

52 i

szBasVect = size(basVect ); szBasVect = szBasVect (1, 1); if szBasVect > 1 chain = 0; else chain = 1; szNut = 42;

% Initialise as 1 % Initialise ¿ 0

basElm = basVect ; count = 0; while (szNut > 0) ∧ (chain ≡ 1) ∧ (count < (Asz + 5)) count = count + 1; multABak = A∗multABak ;


nut = find(multABak (: , 1)); szNut = size(nut ); szNut = szNut (1, 1); if szNut > 1 chain = 0; end end if count > Asz chain = 0; end end

459

460

58.


rigidcircuit.

The purpose of this function is to identify food webs (or communities) whose niche overlap graphs possess the rigid circuit property. It accepts one variable: 1. A: n × n matrix. The augmented community interaction matrix, such that a positive interaction term is indicated by a 1, and all other terms are 0. It returns one variables: 1. rcFlag : Integer = 1 or 0. Indicates whether or not the food web’s niche overlap graph possesses the rigid-circuit property. 1 indicates that it does, and 0 that it does not. After the niche overlap graph, M , is generated, and isolated nodes have been removed from the system, if the size of M is greater than zero, the web is tested for the rigid circuit property. The algorithm used was devised by Fulkerson & Gross (1965). It is implemented subject to the scheme devised by Rose, Tarjan & Leuker (1976), which provides an ordering, numb, for the Fulkerson & Gross (1965) algorithm, and is described in the routine “Rose et al label and number scheme”. numb is a vector of size Msz such that each row corresponding to a node in the nich-overlap graph M is given a unique value between 1 and Msz . numb describes an ordering of the nodes in M . The algorithm proceeds by checking, in the order given in numb , if the specified node is simplicial, which is to say, that the set of nodes adjacent to that node form a clique. If the specified node is simplicial, it is removed, and the algorithm proceeds to the next node. If it is not simplicial, it can be shown (Golumbic 1980, pp. 81–86) that the graph M is not a rigid circuit graph. posnInd is the integer value of the position of each vertex in M in the ordering of the algorithm. For each posnInd value, 1 to Msz , the vertex index vert corresponding to that position posnInd is identified in numb . The compliment of the set of vertices adjacent to vert , named compAdjSet , is identified by the absence of an entry in the vert column of M . A matrix describing the interactions between the adjacency set, AdjMat , is formed. AdjMat is initialised to M , then for each vertex in the compliment of the adjacency set, that vertex is removed from AdjMat , until only the adjacency set remains. A clique implies that an edge exists between every possible pair of vertices in AdjMat . That is, that every entry is a 1. If AdjMat does not form a clique, the niche-overlap graph is not a rigid circuit graph, and the while loop is terminated.


461

If AdjMat describes a clique, vert is removed from M , and the process repeated until no more vertices remain, or a non-clique adjacency set is found. h rigidcircuit.m

58 i

≡

function rcFlag = rigidcircuit(A) Asz = size(A); Asz = Asz (1, 1); rcFlag = 0;

% Initialise

h Create niche overlap graph, M

59 i

h Remove single isolated nodes from M

60 i

if Msz ≡ 0 rcFlag = 1; else h Rose et al label and number scheme

61 i

M = M + diag (ones(1, Msz )); posnInd = 0; rcFlag = 1;

% Now assume it is rigidcirc

while (posnInd < Msz ) ∧ (rcFlag ≡ 1) posnInd = posnInd + 1; vert = find(numb ≡ posnInd ); % Find adjacency set compAdjSet = find(M (: , vert ) ≡ 0); szCompAdjSet = size(compAdjSet ); szCompAdjSet = szCompAdjSet (1, 1); AdjMat = M ; for compInd = 1 : szCompAdjSet compl = compAdjSet (compInd , 1); AdjMat = remone(compl , AdjMat ); compAdjSet = compAdjSet − 1; end

% Determine if does not form a clique

notCliq = find(AdjMat ≡ 0); if isempty (notCliq ) ≡ 0 rcFlag = 0; end M = remone(vert , M ); numb = remonevec(vert , numb );

% find unconnected

462


Msz = Msz − 1; end end

59.

Create niche overlap graph, M.

The purpose of this routine is to create a niche overlap graph for a food web. A niche overlap graph is an undirected graph such that an edge between to compartments implies that those two compartments share a common prey compartment. For each prey species, if that prey has more than one predator (as deduced from sumA), for each pair of predators, their relationship is recorded by a 1 in M , the niche overlap graph. h Create niche overlap graph, M sumA = sum(A);

59 i

≡

% Indicate which prey has > 1 pred

M = zeros(Asz , Asz ); for prey = 1 : Asz if sumA(1, prey ) > 1 for pred1 = 1 : Asz if A(pred1 , prey ) ≡ 1 for pred2 = pred1 + 1 : Asz if A(pred2 , prey ) ≡ 1 M (pred1 , pred2 ) = 1; M (pred2 , pred1 ) = 1; end end end end end end This code is used in section 58.


60.

463

Remove single isolated nodes from M.

The purpose of this routine is to remove isolated nodes from the niche overlap graph, M . For each compartment (or node), if the sum of M by column, sumM , is equal to zero in column corresponding to that compartment, this implies that the compartment is not connected to any other compartment. If the compartment satisfies this test, it is removed from M . h Remove single isolated nodes from M node = 0; Msz = Asz ; while node < Msz node = node + 1; sumM = sum(M ); if sumM (1, node ) ≡ 0 Msz = Msz − 1; if Msz 6= 0 M = remone(node , M ); node = node − 1; end end end This code is used in section 58.

60 i

≡

464


61.

Rose et al label and number scheme.

The purpose of this routine is to determine the lexicographic (i.e. dictionary-style ordering) scheme devised by Rose et al. (1976), which gives the ordering for the algorithm devised by Fulkerson & Gross (1965) used in the body of the code. See Golumbic (1980, pp. 85) for a worked example. Two lists are maintained: labl , the lexicographic numbering scheme, and numb , a vector giving the numbering scheme corresponding to labl . The scheme begins arbitrarily at vertex 1. At each vertex, its corresponding numb is set to numbEntry . First, we identify the adjacency set of vert . Next, we append numbEntry to the labl of each adjacent vertex. Then we identify the next vert . numbEntry is decremented for each iteration. h Rose et al label and number scheme

61 i

≡

labl = zeros(Msz , Msz ); numb = zeros(Msz , 1); countNode = Msz ; numbEntry = Msz ; vert = 1; while countNode 6= 0 numb (vert , 1) = numbEntry ; h ID adjacency set of vert

62 i

h Append numbEntry to labl of adjacent vertices h ID next vert

64 i

% Update counters numbEntry = numbEntry − 1; countNode = countNode − 1; end This code is used in section 58.

63 i


62.

465

ID adjacency set of vert.

The purpose of this routine is to identify all vertices adjacent to the vertex vert . If a vertex vert in the niche-overlap graph M is connected to another vertex vert2 , this is indicated by a 1 in the vert2 row and vert column of M . Therefore, the adjacency set can be identified by using the find function on the vert column of M . h ID adjacency set of vert

62 i

≡

adjSet = M (: , vert ); adjSet = find(adjSet ); szAdjSet = size(adjSet ); szAdjSet = szAdjSet (1, 1); This code is used in section 61.

63.

Append numbEntry to labl of adjacent vertices.

Once the adjacency set of vert is found, the value numbEntry must be appended onto the labels of each vertex in the adjacency set, provided that they are unnumbered. For each adjacent vertex, adjNode , adjNode is unnumbered if the entry in numb corresponding to adjNode is zero. If it is zero, the value numbEntry is put in the first available space in labl . The first available space is identified, frstAvail , and its value is set to numbEntry . h Append numbEntry to labl of adjacent vertices

63 i

≡

for adjSetInd = 1 : szAdjSet adjNode = adjSet (adjSetInd , 1); if numb (adjNode , 1) ≡ 0

% if unumbered

frstAvail = labl (adjNode , :); frstAvail = find(frstAvail ≡ 0); frstAvail = frstAvail (1, 1);

% ID first avail space in labl

labl (adjNode , frstAvail ) = numbEntry ; end end This code is used in section 61.

% Place numbEntry in first available space

466


64.

ID next vert.

We create two vectors, with entries corresponding to vertices with values of 0 and 1 according to the following rules: If no unnumbered vertices have a label (isempty (whichLabUnumb ≡ 1 )), the next vertex is the next unnumbered vertex. If there are unnumbered vertices that have labels, we proceed to the else statement. We create dedLabl , which is the same as labl , except the entries in labl corresponding to a vertex that has already been numbered are removed. We find the lexicographically highest label in dedLabl , which we call vertDedLabl . It becomes the next vertex. h ID next vert

64 i

≡

labUnumb = zeros(Msz , 1); unlabNumb = zeros(Msz , 1); firstUnumb = 0; for count = 1 : Msz if (numb (count , 1) ≡ 0) ∧ (firstUnumb ≡ 0) firstUnumb = count ; end if (labl (count , 1) 6= 0) ∧ (numb (count , 1) ≡ 0) labUnumb (count , 1) = 1;

% has labl, no numb

else unlabNumb(count , 1) = 1;

% has either no labl, or numb

end end whichLabUnumb = find(labUnumb ); if isempty (whichLabUnumb ) ; vert = firstUnumb; else

% If there are unnumbered vertices that have labels

h Create dedLabl

65 i

h Find lexico highest dedLabl

66 i

vert = whichLabUnumb (vertDedLabl ); end This code is used in section 61.


65.

467

Create dedLabl.

The purpose of this routine is to create dedLabl . dedLabl is the same as labl , except all entries associated with vertices that have already been numbered are removed. This makes finding the unnumbered vertex with the highest label less complicated. First, we identify all numbered vertices, (both labelled and unlabelled), and put their indices in the vector wUnlabNumb . For each entry in wUnlabNumb , we use the function remonevec to remove that entry. h Create dedLabl

65 i

≡

dedLabl = labl ; szLabUnumb = size(whichLabUnumb ); szLabUnumb = szLabUnumb (1, 1); unlabNumbInd = 0; wUnlabNumb = find(unlabNumb ); szUnlabNumb = size(wUnlabNumb ); szUnlabNumb = szUnlabNumb (1, 1); while unlabNumbInd < szUnlabNumb ; unlabNumbInd = unlabNumbInd + 1; node = wUnlabNumb (unlabNumbInd , 1); dedLabl = remonevec(node , dedLabl ); wUnlabNumb = wUnlabNumb − 1; end This code is used in section 64.

% remove those already numbered

468


66.

Find lexico highest dedLabl.

The purpose of this routine is to find which row of dedLabl has the lexicographically highest entry, and make the index of the row vertDedLabl . szLabUnumb is the number of columns in vertDedLabl . For each column in vertDedLabl , we find the maximum value in that column maxLabl . We also find the index of dedLabl with entries equal to maxLabl . If there is only one entry equal to maxLabl , or if we are on the last column, vertDedLabl is equal to that index, and we have found the next vertex. If there is more than one entry equal to maxLabl , and we still have columns left to check, we reduce both dedLabl and whichLabUnumb to only those rows, and repeat the process on the next column. h Find lexico highest dedLabl

66 i

≡

% Find which dedLabl is largest vertDedLabl = 0; col = 0; while (col < szLabUnumb ) ∧ (vertDedLabl ≡ 0) col = col + 1; maxLabl = max(dedLabl (: , col )); findDedLabl = find(dedLabl (: , col ) ≡ maxLabl ); szFindDedLabl = size(findDedLabl ); szFindDedLabl = szFindDedLabl (1, 1); if ((szFindDedLabl > 1) ∧ (col ≡ szLabUnumb)) ∨ (szFindDedLabl ≡ 1) vertDedLabl = findDedLabl (1, 1); end if (szFindDedLabl > 1) ∧ (col < szLabUnumb ) % Do another dedLabl = dedLabl (findDedLabl , :); whichLabUnumb = whichLabUnumb (findDedLabl , :); end end This code is used in section 64.

469


67.

Print output from postprocess to file.

The purpose of this routine is to print the results of the post-processing to a file named fName . h Print output from postprocess to file

67 i

≡

fid = fopen(fName , ’w’); % Preamble fprintf(fid , ’%% Generated by postprocess.m \n’); fprintf(fid , ’%% Generated from results from permbuild.m \n’); fprintf(fid , ’%% found in a file named ’); fprintf(fid , eachstepName ); fprintf(fid , ’.m \n’); fprintf(fid , ’%% Entries are as follows: \n’); fprintf(fid , ’%% 1,

2,

3,

4,

5,

6,

7,

... \n’);

fprintf(fid , ’%% isCh, noTop, noInt, noBas, noLtot, noLbi, noLbt, ... \n’); fprintf(fid , ’%% 8,

9,

10,

11,

12,

13,

14,

15 \n’)

fprintf(fid , ’%% noLii, noLit, szEch,noOm1,noOm2,maxChnVec,cycleVec,rigCirc \n’) fprintf(fid , ’postout = [ \n’); % Extend size of command to fit size of res [szRowRes , szColRes ] = size(res ); matstr = [’%4.4g’]; for sz = 2 : szColRes − 1 matstr = [matstr , ’%4.4g’]; end matstr = [matstr , ’%4.4g\n’]; fprintf(fid , matstr , res ′ ); fprintf(fid , ’ ]; \n’); fclose(fid ); This code is used in section 45.

470


Is the vertex ... Labelled? Numbered? Yes Yes Yes

No

No

Yes

No

No

Then lablUnumb ≡ 0 unlabNumb ≡ 1 lablUnumb ≡ 1 unlabNumb ≡ 0 lablUnumb ≡ 0 unlabNumb ≡ 1 lablUnumb ≡ 0 unlabNumb ≡ 1


68.

Controls.

69.

controlbuild.

471

The purpose of this function is to provide a control algorithm. It is identical to permbuild, except that there are no permanence or feasibility constraints upon the ecosystems. h controlbuildv4.m

69 i

≡

function [A, D, X] =controlbuildv4 (Asz , its ) userparameters;

% User specified parameters

fid = fopen(’eachstepcontrol.m’, ’w’); h Preamble of output file

3i

flagStartAgain = 0; startAsz = Asz ; [A, D, X] = makea(Asz , maxAutoDi );

% Create first system

itn = 0; while itn < its itn = itn + 1; if (flagStartAgain ≡ 1) [A, D, X] = makea(startAsz , maxAutoDi ); flagStartAgain = 0; Asz = startAsz ; end

% Add species, irrespective of any constraints

[Anew , Dnew ] = addspp(A, D, maxAuto , constFlag , maxAutoDi ); A = Anew ; D = Dnew ;

% Update for next iteration

Asz = size(A); Asz = Asz (1, 1); h Print A and D to file

4i

if Asz > 14 flagStartAgain = 1; end end

% while itn ¡ its

fclose(fid );

% Close file

472


Index of permbuild Abak :

7

chain : 32, 57

ABak : 45, 52, 55

clock : 1

abs : 1

col : 36, 48, 49, 66

AD : 2, 38

compAdjSet : 58

addspp : 1

compInd : 58

ADitn4 : 4

compl : 58

AdjMat : 58

compment : 54, 55

adjNode : 63

cond : 1

adjSet : 62, 63

constFlag : 3, 6, 11, 15, 16, 38, 69

adjSetInd : 63

controlbuildv4 : 69

Aind : 12, 21

count : 7, 32, 50, 57, 64

Anew : 2, 6, 7, 8, 10, 69

countAuto : 12

AnewBak : 7

countBas :

any : 33

countCompment : 55

Asz : 2, 3, 4, 6, 7, 11, 12, 17, 20, 32, 33, 35,

countNode : 61

53

36, 39, 41, 45, 48, 49, 50, 51, 53, 54, 56,

countNut : 56

57, 58, 59, 60, 69

countPath : 56

Aszbak : 7

cycle : 45, 50

Asznew : 6

cycleVec : 45, 46, 47

autoFlag : 11, 13

dedLabl : 64, 65, 66

basal : 7

diag : 7, 13, 39, 58

basElm : 53, 57

diagA : 13

basVect : 52, 53, 57

diff : 54

bchComp : 9

diffSubs : 31

bchInd : 18

disp : 17, 45

bchNo : 17, 18

Dnew : 2, 6, 8, 10, 69

bchNow : 17

eachstep : 2, 45

bchRemColInd : 9

eachstepName : 45, 67

bchRemRowInd : 2, 9

eig : 1

bchRemVec : 10

elm : 9, 10, 17, 18, 32, 41, 42, 54, 55

bchRemVect : 9, 10

elmInd :

bchVect : 17, 18

elmNut : 56

bchVectInd : 18

elmPath :

cb : 43

endfunction : 43

17

56


endif : 43

intnVectCol : 11, 15, 16

eqElm : 31

intnVectRow :

eqElmInd : 31

inv : 2

error : 1

is scalar : 43

eval : 1

is vector : 43

factr : 11

isCh : 45, 46, 47

factSelfNeg : 39

ischaino : 1

fclose : 1

isempty : 7, 9, 12, 21, 43, 45, 50, 58, 64

fid : 2, 3, 4, 5, 67, 69

itn : 2, 4, 45, 69

find : 1

itnStart : 45

findDedLabl : 66

itnVectRow : 13

findEqElms : 31

its :

findEqSums : 31

jumble : 1

findnonchain : 1

labl : 61, 63, 64, 65

findnonfloat :

lablUnumb : 64

1

11, 13, 15, 16

2, 3, 45, 69

findpostsubs : 8

labUnumb : 64

findsubs : 1

largestweb : 1

findsubspost : 1

length : 43

firstPreyInd : 11, 14

lessThan : 26

firstUnumb : 64

levVect : 48, 49, 50

flagB : 17, 39

linkType : 49

flagBasalNew : 7

makea : 1

flagInvad : 6

match : 9

flagP : 17

matstr : 4, 67

flagStartAgain : 2, 69

max : 1

floor : 1

maxAuto : 3, 6, 11, 12, 38, 69

fName : 2, 45, 67

maxAutoDi : 2, 3, 6, 11, 13, 38, 39, 69

fopen : 1

maxBasPred : 3, 6, 7, 38

fprintf : 1

maxChn : 45, 50

frstAvail : 63

maxChnVec :

gset : 1

maxJumbleInd :

hetVect : 13

maxLabl : 66

hold : 1

maxSzSubs : 2, 3, 5, 38

ignoreVec : 8, 20, 26, 33

min : 1

indicEnd : 20, 22, 23

minSz : 38, 45

indV : 33

Msz : 42, 58, 60, 61, 64

45, 46, 47 2, 38, 40

473

474


multA : 7, 32, 33, 45, 50, 53, 56

permbuild : 1

multABak : 33, 34, 35, 36, 45, 51, 52, 57

plot : 1

multCount : 33

posn :

nadrepm :

posnInd : 58

1

45, 47, 48, 49

NaN : 45, 50

postprocess : 1

nargin : 2, 41, 43, 45

pred : 39

netsperm : 1

predInd : 7

newA : 11, 41

predOne : 7

newD : 11, 13, 14, 41

predsNew :

newSubs : 28, 30

pred1 : 59

noBas : 45, 46, 47, 48

pred2 : 59

node : 60, 65

prey :

noInt :

printTime :

45, 46, 47, 48

7

39, 50, 59 2, 5, 38

noLbi : 45, 46, 47, 49

probAuto : 11

noLbt : 45, 46, 47, 49

rand : 1

noLii : 45, 46, 47, 49

randno : 11

noLit : 45, 46, 47, 49

rank : 1

noLtot : 45, 46, 47

rankSubA : 17

noMoreAuto : 11, 12, 13

rb : 43

nonFloatInd : 20, 21, 33

rcFlag : 58

nonFloatVect : 20, 21, 26, 33

real : 1

noOm1 : 45, 46, 47

recNut : 32

noOm2 : 45, 46, 47

remelm : 20, 23, 24, 25

notCliq : 58

remone : 1

noTop : 45, 46, 47, 48

remonevec : 1

numb : 58, 61, 63, 64

res : 17, 40, 45, 67

numbEntry : 61, 63

rigCirc : 45, 46, 47

num2str : 45

rigCircVec : 45

nut : 7, 32, 50, 56, 57

rigidcircuit : 1

nutInd : 50

round : 2, 3, 5

omn1 : 45, 50

row : 7, 35, 36, 48, 49, 50

omn2 : 45, 50

rowrem : 33

ones : 1

self : 39

path : 53, 56

size : 1

pathfinder :

1

pathLargst : 53, 54

sort : 1 spare : 12, 21, 40, 41


specADinit : 2, 38

szAnew : 2, 8

sppInd : 11, 13, 14, 15, 16

szBas : 7

sprintf : 45

szBasVect : 53, 57

startAsz : 2, 69

szBch : 17, 18

subA : 17, 20, 22, 27, 29, 33, 41

szColRes : 67

subBchVect :

szCompAdjSet : 58

17

subComp : 31

szCompment : 54, 55

subCompRep : 31

szDiff : 54

subD : 17, 20, 27, 41

szEch : 45, 46, 47

subIgnoreVec : 26, 27

szFindDedLabl : 66

subInd : 31

szFindEqElms : 31

subM : 42

szFindEqSums : 31

subRow : 9

szHetVect : 13

subs : 2, 8, 9, 10, 17, 18, 19, 20, 24, 28, 30, 31, 40

szindV : 33

subsCompInd :

szLabUnumb : 65, 66

40

subsD : 27, 28, 29

szNewSubscol : 30

subsDcol : 29

szNewSubsrow : 30

subsDrow : 29

szNonFloatVect : 20, 21

subsInd : 10, 20, 24, 40

szNut : 7, 32, 50, 56, 57

subsred : 31

szPath : 53, 54, 56

substg : 19, 20, 25, 27

szPathNew : 53

substgD : 27

szPreds : 7

substgElm : 20, 26, 27, 28, 29

szPrey : 50

substgElmRep : 28

szRowRes : 67

substgInd : 20, 25

szsubA : 20, 22

subsz : 17

szSubs : 2, 8, 9, 17, 18, 40

subX : 17, 20

szSubsCol : 8, 9, 10

subXFull : 17

szsubscol : 30, 31

sum : 1

szsubsDcol : 28, 29

sumA : 59

szsubsDrow : 28, 29

sumDiffSubs : 31

szSubsMax : 40

sumM : 60

szsubsrow : 30, 31

sumSubComp : 31

szsubstg : 20

sumSubs : 31

szsubX : 17

sz : 4, 67

szUnlabNumb : 65

szAdjSet : 62, 63

szVisitVect : 33

475

476


TeX : 1

vert :

timeNow : 2, 3, 5

vertDedLabl : 64, 66

tran :

vert2 : 62

6

tranVect : 17

visitVect :

troph : 7 trophdist :

58, 61, 62, 63, 64

whichLabUnumb : 64, 65, 66

50

wUnlabNumb : 65

unlabNumb : 64, 65 unlabNumbInd :

xlabel : 1

65

update : 10

Xnew : 6

usage : 43

ylabel : 1

userparameters : 1

zeros : 1

List of Refinements in permbuild

h addspp.m

11 i

h controlbuildv4.m

69 i

h findnonchain.m

32 i

h findnonfloat.m

33 i

h findsubs.m

20 i

h findsubspost.m h ischaino.m h jumble.m

40 i 51 i

39 i

h nadrepm.m

43 i

h netsperm.m

17 i

h pathfinder.m h permbuild.m

56 i 2i

h postprocess.m h remone.m

31 i

57 i

h largestweb.m h makea.m

41 i

33

45 i


h remonevec.m

42 i

h rigidcircuit.m

58 i

h userparameters.m

38 i

h Add to list of subs to go h Add to subs

24 i

25 i

Used in section 20.

Used in section 20.

h Adjust indices of subsD

29 i

Used in section 28.

h Append numbEntry to labl of adjacent vertices h Append subsubsystems to subsystem h Augment A

36 i

28 i

63 i

Used in section 61.

Used in section 20.

Used in sections 7, 32, and 34.

h Classify basal, intermediate, top h Count link types

49 i

48 i

Used in section 45.

h Count omnivores and max chain h Create autotroph

Used in section 45.

13 i

50 i

Used in section 45.

Used in section 11.

h Create bchVect

18 i

Used in section 17.

h Create dedLabl

65 i

Used in section 64.

h Create heterotroph

14 i

Used in section 11.

h Create multABak, augment A

34 i

Used in sections 33, 45, and 52.

h Create niche overlap graph, M

59 i

Used in section 58.

h Create subIgnoreVec h Create the multABak

26 i

Used in section 20.

35 i

Used in section 34.

h Determine if member of chain

23 i

Used in section 20.

h Determine if single autotroph

22 i

Used in section 20.

h Determine noMoreAuto

Used in section 11.

12 i

h Find a species that can invade

6i

Used in section 2.

h Find all subsystems of system

8i

Used in section 2.

h Find compliment of largest web h Find lexico highest dedLabl

66 i

54 i

Used in section 51.

Used in section 64.

477

478


h Find subsystems of the subsystem h ID adjacency set of vert h ID next vert

64 i

h Identify basal

62 i

27 i

Used in section 20.

Used in section 61.

Used in section 61.

52 i

Used in sections 51 and 57.

h Identify largest web from basal

53 i

Used in section 51.

h Initialise empty storage vectors

46 i

Used in section 45.


47 i

Used in section 45.

h Make sppInd pred of new

15 i


h Make sppInd prey of new

16 i


h Pad smaller of subs and newSubs with zeros h Preamble of output file h Print A and D to file

3i

4i

5i

Used in sections 2 and 69. 67 i

Used in section 45.

Used in section 4.

h Remove compliment

55 i

Used in section 51.

h Remove isolated autotrophs

21 i

Used in section 20.

h Remove single isolated nodes from M h Remove specified branch h Restrict predator diets

10 i

7i

60 i

Used in section 58.

Used in section 2.

Used in section 6.

h Retain only those subs with those elm h Rose et al label and number scheme

H.2

Used in section 28.


h Print output from postprocess to file h Print time

30 i

9i

61 i

Used in section 2. Used in section 58.

All GLV chains are permanent

Below, we show that a feasible Lotka-Volterra food web (Equation 11.1) whose structure is a food chain is always permanent. The proof used is very similar to that presented in Section 6.4.5.

479

H.2 All GLV chains are permanent

Remark 1 A generalised Lotka-Volterra model that is a chain is permanent provided that the removal of a predator increases the steady state biomass of its prey. Using the definition of permanence (Hofbauer & Sigmund 1988, pp. 166–169), it can be shown that at least one transversal eigenvalue is positive for each subsystem of a n-chain on bd(Sn ), implying that the system is permanent, provided that this condition holds. We denote the subsystem p. We order the indices of the food chain such that i = j +1 when compartment i is the predator of compartment j. When testing the permanence of the full system against each subsystem, only subsystems of the form pTm = 0 0

. . . 0 x⋆m

x⋆m+1

. . . x⋆n ,

(where m denotes the highest trophic level of the subsystem) are considered, as these are the only subsystems on bd(Sn ) that exist. We denote each of the transversal eigenvalues by γi , such that n X x˙i γi = ai,j xj . = di + xi i=1

Consider pm , which is the first subsystem of pm−1 . The only non-zero transversal eigenvalue associated with the system is γm−1 = am−1,m x⋆m + dm−1 , which is greater than 0 (implying that the system is permanent with respect to that subsystem) if x⋆m >

−dm−1 . am−1,m

(H.3)

Note that if Equation H.3 was an equality, it would describe the steady state value of xm in the original system pm−1 . This implies that, in order for the transversal eigenvalue to be positive, the removal of compartment m − 1 to form the subsystem pm must result in an increase in the steady state value of xm . In other words, the removal of the predator must increase the steady-state size of the prey compartment. This argument holds for the test of permanence of each subsystem m through to n.

Remark 2 If the top predator of a feasible chain is removed, the steady state value of its prey will increase.

480


Consider the first two steps of the Gaussian elimination performed upon G = [A|d].     .. .. .. . . | .         gn−2,:  0 . . . an−2,n−3 0 a 0 | −d n−2,n−1 n−2   = ,     gn−1,:  0 . . . 0 an−1,n−2 0 an−1,n | −dn−1      gn,:

0 ...

0

0

an,n−1

an,n

|

−dn

where gi,j is the ith row and jth column of the Gaussian Elimination. To get G into (lower) triangular form, each of the ak,k+1 terms need to be set to zero. This is done by operations of the form gk−1,: = gk−1,: − gk,:

gk−1,k . gk,k

(H.4)

Consider the first operation, starting from the bottom. As all of the non-zero elements of gn,: are negative, the operation described in Equation H.4 will result in all elements of gn−1,: being negative, with the possible exception of the gn−1,n+1 term. However, if we assume that the system is feasible, gn−1,n+1 must be negative. By similar reasoning, each step of the Gaussian Elimination will cause the corresponding resultant gk−1,: to have all negative or zero elements. Consider the completed Gaussian    g1,1 0 ... g1,:       g2,:  = g2,1 g2,2 0    .. .

such that

elimination upon the chain system. The top two entries will be  0 | g1,n+1   ... 0 | g2,n+1  , (H.5)  .. .. . . |

g2,1 x⋆1 + g2,2 x⋆2 = g2,n+1 ,

(H.6)

where all gi,j have the same (negative) sign. Therefore setting the predator to zero, x⋆1 = 0, increases the size of the prey compartment, x⋆2 .

Corollary 1 All feasible Lotka-Volterra chains described are permanent. This follows from a combination of Remarks 1 and 2.

H.3 H.3.1

The effect of increasing productivity Approach

The high basal fraction predicted by the permanence algorithm implies that a large number of basal compartments are required to support the upper trophic levels. Would increasing the productivity of

481

H.3 The effect of increasing productivity

the basal compartments address this? We simulated this effect by increasing the range of di for the autotrophic compartments, and measuring changes in food web attributes.

H.3.2

Results

Trophic and link-type fractions Figures H.1 and H.2 show the trophic fractions for differing productivity. Figures H.3 to H.6 show the link-type fractions for the same. Figures H.2 and H.1 shows that the basal fraction decreases, and the intermediate fraction increases, with increasing productivity. Intermediate fraction for differing productivity. di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]


1

0.8

0.6

0.4

0.2

0 0

5


15

20

Figure H.1: Intermediate fraction versus total number of compartments. Comparison between food webs with differing productivity.

Figures H.4 and H.5 both show relatively strong trends of decrease and increase with increasing productivity for the basal-top and intermediate-intermediate link-type fractions respectively. For smaller food webs (less than approximately 8 compartments), a trend may be discerned in the basal-intermediate and intermediate-top link-type fractions as well. Both increase with increasing productivity. However, for larger food webs, both of these trends disappear.

482


Basal fraction for differing productivity. di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]


1

0.8

0.6

0.4

0.2

0 0

5


15

20

Figure H.2: Basal fraction versus total number of compartments. Comparison between food webs with differing productivity.

Basal-intermediate link type fraction for differing productivity. di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]


1

0.8

0.6

0.4

0.2

0 0

5


15

20

Figure H.3: Fraction of basal-intermediate links versus Total number of compartments. Comparison between food webs with differing productivity.

483


Basal-top link type fraction for differing productivity. di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]

1


0.8

0.6

0.4

0.2

0 0

5


15

20

Figure H.4: Fraction of basal-top links versus Total number of compartments. Comparison between food webs with differing productivity.

Intermediate-intermediate link type fraction for differing productivity. di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]


1

0.8

0.6

0.4

0.2

0 0

5


15

20

Figure H.5: Fraction of intermediate-intermediate links versus Total number of compartments. Comparison between food webs with differing productivity.

484


Intermediate-top link type fraction for differing productivity. di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]

1


0.8

0.6

0.4

0.2

0 0

5


15

20

Figure H.6: Fraction of intermediate-top links versus Total number of compartments. Comparison between food webs with differing productivity.

485


Food web size Figure H.7 shows a histogram of the relative frequency of food web sizes. As productivity is increased, food webs around size 5 become more dominant. Figure H.8 shows the maximum chain length in each food web for differing productivity. As productivity is increased, maximum chain length increases. The increase in the maximum chain length with increasing productivity indicates that increasing the productivity lengthens the food chain, in contrast with the short-but-broad webs characteristic of the low productivity models. This is not to say that food webs become larger, but that they become taller, for equivalent sized webs. Frequency histogram of web sizefor differing productivity 0.25 di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]

Relative frequency

0.2

0.15

0.1

0.05

0 0

2

4

6

8 10 12 14 Number of compartments

16

18

20

22

Figure H.7: Histogram of relative frequency of food web sizes. Comparison between food webs with differing productivity.

486


Maximum chain length for differing productivity 7 di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]


6

5

4

3

2 0

2

4

6

8 10 12 Number of compartments

14

16

18

20

Figure H.8: Maximum chain length versus number of compartments. Comparison between food webs with differing productivity.

487


Food web structure Figure H.9 shows the proportion of food webs that are chains for differing productivity. Increasing the productivity increases the web size range for which chains manifest, and increases the probability of those chains. Proportion of webs that are chains for differing productivity di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]

1

Proportion of webs that are chains

0.8

0.6

0.4

0.2

0 0

5


15

20

Figure H.9: Proportion of food webs that are chains. Comparison between food webs with differing productivity.

Figure H.10 shows that webs containing cycles are more probable with increasing productivity, however the trends are variable for large webs. As shown in Figure H.7, for food webs with high productivity, only a few data points exist for webs much larger than 10 compartments. Therefore, the results for these large webs should be treated with caution.

488


Proportion of webs with cycles for differing productivity di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]

1

Proportion of webs with cycles

0.8

0.6

0.4

0.2

0 0

5


15

20

Figure H.10: Proportion of food webs containing a cycle. Comparison between food webs with differing productivity.

Proportion of rigid circuit niche-overlap graphs for differingproductivity di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]

Proportion of webs with rigid circuit niche-overlap graphs

1

0.8

0.6

0.4

0.2

0 0

5


15

20

Figure H.11: Proportion of food webs that have rigid-circuit niche-overlap graphs. Comparison between food webs with differing productivity.

489


Links per compartment for differing productivity 1.5 di=[0,1] di=[0,5] di=[0,10] di=[0,20] di=[0,30]

1.4


1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0

2

4

6

8 10 12 Number of compartments

14

16

18

20

Figure H.12: Links per compartment versus number of compartments. Comparison between food webs with differing productivity.

490


Summary Table H.1 summarises the changes in food web attributes observed as productivity is increased. Increasing productivity decreases the basal fraction, increases the intermediate fraction, and decreases the basal-top fraction, which improves the agreement between the algorithm results and the empirical literature. The following figures show the relationships between the attributes and the productivity. Those attributes that showed no change are omitted for brevity. Table H.1: A summary of trends in various food web attributes with increasing productivity range. Attribute Trend with increasing productivity Web size Distribution becomes more peaked around size = 4. Maximum chain length Increases. Fraction that are chains Increases. Fraction with cycles Increases. Fraction rigid-circ. Increases? Top No trend. Intermediate Increases. Basal Decreases. Basal-intermediate Increase for small webs. No trend for large webs. Basal-top Decreases. Intermediate-intermediate Increases. Intermediate-top Increase for small webs. No trend for large webs. Omnivore 1 fraction No trend. Omnivore 2 fraction No trend.

Index Brusselator, 282–283

goal functions, 32–62

buffer capacity, 56

exergy, 46 maximise ascendency, 58–59

concordance, 90–93, 124, 127–129, 132–134

maximise biomass, 56–58

parameter range effect, 94–97

maximise exergy, 57, 56–57, 287–291

relative to all goal functions, 91

maximise flux, 36–38, 58–59, 285

relative to traditional goal functions, 92

Lotka (1922), 36

dissipative systems, 39, 38–43

maximise production to biomass ratio, 59–60

assumptions, 39

minimise entropy production, 40, 40–43 minimise production to biomass ratio, 59

ef-ratio, 27, 138

thermodynamic, 38–43

emergent property, 38

assumptions, 285–287

entropy, 39

Prigoginean, 39–40

ecological proxies, 40

traditional, 83, 104, 124, 130

equilibrium, 66 export production, 138 f-ratio, 138

invasion, 165, 231 Jacobian Matrix, 68, 104

food web attributes

CG Model, 302

cycles, 221, 223

generalised Lotka-Volterra, 105

generalisations, 227, 223–227, 237

chain, 107

criticisms, 223–227 link density, 223

least specific dissipation, 41, 284

link-type fraction, 221, 223

local equilibrium thermodynamics, 40, 40–43, 283– 284

maximum chain length, 221, 223 predator prey ratio, 221, 223

Lyapunov function, 70–72

scale-invariance, 219, 223, 226–227

Goh’s Lyapunov function, 70

scale-variance, 219, 227, 226–227

Rouche’s Lyapunov function, 70

trophic fraction, 221, 223 food web building algorithms, 79, 78–80, 228–235

models CG Model, 83, 82–87

general evolution criterion, 40, 41, 283

variations, 90, 298–302 491

492

Index

Fasham Model, 197, 195–217, 343–384 variations, 379–383 Fasham-Laws Model, 205, 205–216

resistance, 161, 161–162, 164 survival proxy, 64–65, 74, 159, 164–165 steady state, 66

generalised Lotka-Volterra, 100–103, 105–117

feasible, 66

Laws Model, 140, 137–157, 174–194

infeasible, 66

steady state, 331–333

non-trivial, 66, 105

variations, 334–340

trivial, 66 succession, 46–53

new production, 138 niche-overlap graph, 221 interval, 221, 223 rigid-circuit, 221, 223 non-equilibrium, 281 order through fluctuations, 281, 285

Clementsian, 47–49 climax, 47 E.P. Odum Tabular Model, 51, 50–53 facilitation, 48, 50 goal functions, 52–53 individualist, 49 inhibition, 49

primary production, 138

life strategies, 49 Markov process, 49

resilience, 68, 104–105 heuristic, 28, 89, 88–98, 123–135, 263

Odum brothers, 49–53 tolerance, 49

hypothesis, 28, 82, 82–88, 262 Routh-Hurwitz criteria, 107–109, 321–322

thermodynamic equilibrium, 281 transversal eigenvalue test, 115, 231–234

stability, 63–81, 158–173 balance of nature, 65 complexity, and, 75–80 eigenvalue analysis, 67–68 feasible-stable, 120, 151–153, 204–205 feedback, 44–46, 158–173 Goh-Lyapunov stability, 71, 111–114 local stability, 66–69 shortcomings, 69 Lyapunov stability, 66, 70 neighbourhood stability, 66 permanence, 72–74, 219, 232–234, 482–484 chains, 115–117, 482–484 transversal eigenvalue test, 115 resilience, 68–69, 80–81, 164–167, 171

universal evolution criterion, 283

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