energy-based entropy and its generalisation

ENERGY-BASED ENTROPY AND ITS GENERALISATION Part 2. Extensions of concept and application in regional vegetation survey László Orlóci* and Kürşad Özkan**

*Department of Biology, Western University, London, Ontario, Canada **Department of Soil and Forest Ecology, Faculty of Forestry, Süleyman Demirel University, Isparta 32260, Turkey Access: [email protected], [email protected]

CONTENTS CONTENTS...............................................................................................................1 KEY CONCEPTS AND IDENTITIES .............................................................................3 STRUCTURES WITHIN THE EUCLIDEAN FRAME ......................................................6 Co-ordinate data.................................................................................................6 Eigenanalysis.......................................................................................................7 Cluster analysis .................................................................................................10 STRUCTURES WITHIN THE KANIADAKI FRAME .....................................................13 GQE analysis .........................................................................................................14 Three-way partitions ........................................................................................14 The  effects .....................................................................................................22 The dominance effect .......................................................................................23 Instability scalar w ............................................................................................24 GQES instability and the dominance scalars ....................................................25

2 TRANSECT I........................................................................................................... 26 Instability by altitude ....................................................................................... 26 Instability and the environment ...................................................................... 29 Instability and the dominance scalars ............................................................. 31 More on order scaler  .................................................................................... 34 EUMS AND GQES JAXTAPOSED ............................................................................ 36 SUMMARY............................................................................................................ 39 REFERENCES ......................................................................................................... 40 APPENDIX ............................................................................................................. 41 Manuals and data source reference ................................................................ 41 Beyşehir research site and sampling ............................................................... 41 General conditions ....................................................................................... 41 Site selection ................................................................................................ 42 Description of the transects ............................................................................ 42 Dedegul Mountains ..................................................................................... 42 Sultan Mountains:........................................................................................ 43 Evaluation of species performance ................................................................. 48 Phytosociological tables .................................................................................. 49 Gediklli Transect I......................................................................................... 49 Mada Transect II .......................................................................................... 50 Yenisar Transect III ....................................................................................... 51 Modus Transect IV ....................................................................................... 52 Erenkilit Transect V ...................................................................................... 53 Erenkilit Transect VI ..................................................................................... 54 Environmental data ......................................................................................... 55 Gedikli Transect I ......................................................................................... 55 Mada Transect II .......................................................................................... 55 Yenişar Transect III ....................................................................................... 56 Modus Transect IV ....................................................................................... 56 Erenkilit Transect V ...................................................................................... 56 Kafadag Transect VI ..................................................................................... 57 RQES instability by sample plot ....................................................................... 57 Instability matrix .......................................................................................... 57 Transect II .................................................................................................... 58 Transect III ................................................................................................... 59 Transect IV ................................................................................................... 60 Transect V .................................................................................................... 61 Transect VI ................................................................................................... 62

3 Abstract. Building on the contents presented in Part I, the essay takes the next step in new directions. Two major propositions give context to the discourse. The first suggests that the plant community structure, detectable by Euclidean metrics, and the structure detectable by the generalised quantum entropy scalar, are independent sources of structural information. If this is true, then the message to the practitioner is clear: the structures made visible by the two analytical frames are not redundant. The second proposition is concerned with the dominance effect issuing from the qualitative and quantitative components of the data, namely population richness and performance. The second proposition suggests that structures detectable by the generalised quantum entropy scalar are sensitive indicators of the dominance effect issuing from the data components. If this is true, then the practitioner will know that the way to the ecological interpretation of a generalised quantum entropy structure is through the identification of historic sources and current environmental effects responsible for richness and performance. Considering that the generalised quantum entropy is a proxy for potential energy, the propositions clarify further the essence of the generalised quantum entropy approach in the study of plant community energetics, and sharpen the contrast with the classical approach in which the focus is on calorific flow, which is indeed proxy for energy flow. Conceptual and practical implications are raised and argued. Hands-on examples are used for illustration, all based on survey type data from the Beyşehir Lake watershed in Turkey. Tables and graphs in main text and Appendix take up a major portion of the contents.

KEY CONCEPTS AND IDENTITIES A brief review should save the readers time by not having to visit Part 1 or other publications before moving on to the main text. We focus on structural scalars in the Euclidean (EUM) frame (Orlóci 2012 v14) and in the Kaniadakis generalised quantum entropy (GQE) frame (Kaniadakis 2001, Santos et al. 2011). Our objects are metacommunities1 of plant populations2. The number of metacommunities are designated by s, the number of populations by n, and the populations’ total

1

The metacommunity’s size is open. The basic level is the vegetation stand (VS). A population is a collection whose members are joined by common inheritance (species) or by common convergent form or function (growth form, functional type, etc.). 2

4 performance in a metacommunity by T. EUM has two associated scalars. One is the Euclidean distance (EUD) and the other the product moment (PRM). The structures themselves are captured in the manner of an s x s matrix for EUD and an n x n matrix for PRM. EUD is defined for the metacommunity duplet j,k by djk 

n

(X h1

hj

 Xhk )2 . Symbol Xhj stands for the

performance of population h in metacommunity j. PRM is defined for the plant s

population duplet h,i by Shi  (Xhj  X h )(Xij  X i ) . Bar-on-X is symbol for mean j1

value. When we use the resonator complex model analogy (Planck 1901) for the metacommunity3, EUDS and PRMS are mathematically linked as resonator level dual structures. We use EUMS as a common label for EUDS and PRMS. GQE is defined in Kaniadakis’ frame (Plank 1901). GQE is a generalisation of energy-based entropy EBE in Max Plank’s resonator complex frame. EBE’s equation is EQ0  -ln P in which P =

1 (n  T  1)! (n  T)n T 4 and C   n T . Read symbol EQ0 (n  1)!T! nT C

as GQE of order 0, namely ESE. The values of the order scalar fall within the range 0 >  < 1. The handling of the Kaniadakis calculus in the general case of EQE is explained by example in Part 1 (Orlóci and Özkan 2018). When the resonator complex model is applied to the metacommunity, n is the number of plant populations and T is the populations’ total performance. We

3

The resonators are populations, and the metacommunity is qualified as the ‘metacommunity complex’. 4 We use the last term, known as ‘Stirling’s approximation’, in all calculation involving EBE. In this we fallow general practice in the field.

5 mention that EBE is the paradigm-defining quantity in Statistical Quantum Ecology (Orlóci 2014b v16, 2017b). The GQE structure of the metacommunity (GQES) is represented by a 3-valued vector of the terms taken from the right-hand side Q

Q

Q

Q

of the generalised quantum entropy equation, E = E Phy + E Env + E Rnd (Orlóci 2014 v16, 2017, Orlóci and Özkan 2018). Subscript  identifies GQE’s order, superscript Q refers to GQE. Other symbols identify potential entropy sources: Phy for phylogeny or evolutionary convergence, Env for environmental mediated sorting of plant populations into distinct communities, and Rnd a grab-bag of chance related unidentified effects. Any of the EQ values divided by the corresponding n, is the one-resonator, one population, or average GQE, symbolically HQ . Note the convertibility principle (Planck 1901) according to which any of the EQ values is a proxy for potential energy in the plant community (Orlóci 2014 v16, 2015 v16). GQES is a high-level structure. It is completely defined by two parameters, the number of populations5 n and the total performance of populations T at any given . The mathematical frame of GQES is the -algebra, a mathematical manoeuvre that makes EBE equal to GQE of order 0. In the essay order 0 implies  =0.00001 or less. Scalar  has practical utility in managing the dilemma in the estimation of GQES level in the metacommunity.

5

Scalar n has different designation depending on context. It can be a count of plant taxa of any type on any hierarchical level. It can be a count of states assumed or possessed by a population, community, or environmental trait. It may be a count of occupied cells or all cells in the phytosociologist’s contingency table - depending on the problem. When n is a count of all cells, it implies, from a sampling point of view, that cells are left unoccupied by chance, and not by some other mechanism of exclusion.

6 The data set (Appendix) for illustration is Özkan’s (2003). It comes from a regional survey of the Beyşehir Lake watershed in Turkey. The dataset contains phytosociological and environmental relevés (record sets) from 54 vegetation stands, completed within 20m x 20m sample plots. Sample plots are located at equal intervals by increasing altitude in equal number (9) in six transects. The altitudinal range extends from 1200m to 2000m above mean sea level. The actual survey area is within 380 02' 60'' and 370 25' 60'' northern latitudes and 310 15' 01'' and 310 46' 10'' eastern longitudes. The transects are portioned out equally between two districts, the Dedegul Mountains (DEDE) west and south to Beyşehir Lake, and the Sultan Mountains (SULTAN) to the east. The altitude of Beyşehir Lake is 1121 m. The highest mountain of the watershed is Dedegul (2993 m). General description of transects, sampling design, species checklist and systematic status, selected sets of species cover-abundance estimates, and types of environmental data are presented in tables, organized by sample plot (elevation) in transect, are in the Appendix.

STRUCTURES WITHIN THE EUCLIDEAN FRAME Co-ordinate data We already explained that EUMS has two versions. One version is scaled by the product moment PRM, and the other by the Euclidean distance EUD. PRM and EUD take cover/abundance data (Table 1) as metric co-ordinates in the same reference space. Table 1. Species cover-abundance totals are given, sorted by transects (Roman numerals) and sample plots at the altitudes listed. Scalar n is the number of species recorded per transect. T is the total cover-abundance value of the n species. Altitude 2000 1900

I 19 35

II 20 29

III 24 45

IV 19 17

V 24 35

VI 29 20

7 1800 1700 1600 1500 1400 1300 1200 T n

57 42 22 38 28 31 17

45 36 35 31 44 31 19

30 38 37 30 21 19 53

20 20 15 17 26 27 13

31 17 18 14 25 18 19

23 17 36 16 12 14 23

289 30

290 40

297 43

174 18

201 28

190 24

The PRM and EUD define dual structures. This is so when the co-ordinates are numerically identical. This condition can be destroyed by standardization. The PRM matrix (Table 2) is set up for ordination by eigenanalysis and the EUD matrix (Table 5) for cluster analysis by sum of squares agglomeration.

Eigenanalysis Eigenanalysis is the matrix algebraic core of Hoteling’s principal component analysis, unburdened by data manipulations statisticians use to force the data into conformity with preconceptions of what the data set’s distribution properties should be. The data set in Table 2 is typical for presentation of Euclidean co-ordinates. Note that the original data are left as measured. In this way, the natural PRM structure is retained. The output includes eigenvalues, eigenvectors, and eigenscores (Tables 2, 3). In the present case the eigenvalues are generalised sum of squares. The elements in the eigenvectors are direction cosines. The eigenscores are co-ordinates on linearly independent axes, therefore parsimonious descriptors of the metacommunity’s EUM structure. Table 2. Result of ordination by eigenanalysis. Table 1 contains the raw co-ordinate data. The perimeters of the analysis are discussed in the main text. Program PCAR is uploaded to ResearchGate, DOI: 10.13140/RG.2.2.12732.33924 PROGRAM: PCAR ===================== COORDINATE FILE NUMBER OF ROWS (VARIABLES): 9 NUMBER OF COLUMNS (INDIVIDUALS): 6 OPTION: SUM OF SQUARES AND PRODUCTS

8 SUM OF SQUARES AND PRODUCTS MATRIX ----------------------------------------------------------------------------77.500 -4.500 -141.000 -114.000 99.500 -100.000 -152.000 -134.000 105.000 - 4.500 544.833 320.667 370.667 137.833 273.667 52.000 -15.333 533.000 -141.000 320.667 991.333 635.333 28.667 590.333 421.000 345.333 -152.000 -114.000 370.667 635.333 665.333 189.667 565.333 287.000 257.333 306.000 99.500 137.833 28.667 189.667 494.833 147.667 -33.000 -100.333 453.000 -100.000 273.667 590.333 565.333 147.667 493.333 246.000 237.333 176.000 -152.000 52.000 421.000 287.000 -33.000 246.000 550.000 311.000 -230.000 -134.000 -15.333 345.333 257.333 -100.333 237.333 311.000 265.333 -222.000 105.000 533.000 -152.000 306.000 453.000 176.000 -230.000 -222.000 1062.000 SUBROUTINE EIGENEIGENVECTORS ADJUSTED TO UNIT LENGTH. EIGENVALUE ----------------------------------------------------------------------------EIGENVALUE 1 = 2513.9227 OR 48.87% (CUMULATIVE 48.87%) EIGENVECTOR: -0.089 .304 .554 .499 .126 .428 .271 .206 .169 EIGENVALUE 2 = 1757.3672 OR 34.16% (CUMULATIVE 83.03%) EIGENVECTOR: .131 .303 -0.261 .063 .340 -0.003 -0.297 -0.260 .739 EIGENVALUE 3 = 378.7117 OR 7.36% (CUMULATIVE 90.39%) EIGENVECTOR: .069 -0.532 -0.156 .060 .757 .080 .309 .090 -0.046 EIGENVALUE 4 = 301.8989 OR 5.87% (CUMULATIVE 96.26%) EIGENVECTOR: -.236 .142 -.491 .026 -.257 -.130 .677 .247 .285 EIGENVALUE 5 = 192.5995 OR 3.74% (CUMULATIVE 100.00%) EIGENVECTOR: -.211 -.442 -.310 .394 -.277 .359 -.430 .312 .141 EIGENVALUE 6 = .0000 OR .00% (CUMULATIVE 100.00%) EIGENVECTOR: .058 .213 .069 -.284 .210 -.222-.252 .841 .047 EIGENVALUE 7 = .0000 OR .00% (CUMULATIVE 100.00%) EIGENVECTOR: .083 .483 -.471 .415 .211 .026 -.112 .002 -.555 COMPONENT SCORES: ----------------------------------------------------------------------------SET 1 : 27.297 19.053 13.126 -21.565 -13.962 -23.948 SET 2 : -13.613 -1.377 33.826 -4.440 -3.288 8.892 SET 3 : -6.697 12.199 -1.913 -0 .467 -10.950 7.827 SET 4 : - 8.447 5.169 4.664 7.624 1.950 -10.961 SET 5 : 3.844 -5.077 2.481 8.051 -8.999 -0.301 COMPONENT SCORES STORED IN TEXT FILE (.txt or .TRU)

Reader attempting reproduction of the results of L.O.’s PCA program, note the following settings: no normalization or other transformation of the variables, sum-of-squares and products matrix for data rows (altitudes), normalization of eigenvectors to make the elements qualify as direction cosines, computation of eigenscores for Transects I to VI, and setting the length of each eigenscores vector on each axis equal to the square root of the corresponding eigenvalue. The  is a sum of squares,  /(n-1) is a variance.

9 Table 3. Eigenvalues and eigenscores for three eigenaxes, extracted from Table 2. Roman numerals identify transects. The raw co-ordinate data is in Table 1. Axis



%

Cum %

I

II

III

IV

V

VI

X:

2513.923

48.87%

48.87

27.297

19.053

13.126

-21.565

-13.962

-23.948

Y:

1757.367

34.16%

83.03

-13.613

-11.377

33.826

-14.440

-3.288

8.892

Z:

378.7117

7.36%

90.39

-6.697

12.199

-1.913

-0.467

-10.950

7.827

The level of efficiency of an eigenaxis is proportional to the associated eigenvalue relative to the sum of all eigenvalues. The two-dimensional plot’s efficiency exceeds 83% (Figure 1).

Figure 1. Eigen plot in the XY plane. Roman numerals are transect labels. Efficiency of the axes in accounting for sums-of-squares: X 49% and Y 34%. The numerics are in body of Table 2.

The statistical significance of point distances can be tested for statistical significance. We perform the test in the x, y, z space (co-ordinates in Table 3). The criterion scar is a probability associated with standard distances (Table 4). The latter are Euclidean distances calculated from standardized eigenscores. This is how the calculus goes for transect duplet I and II: 1. dX I,II 

|XI  XII | |27.297  19.053|   0.36767 X / 5 2513.923 / 5

2. dy I,II 

|YI  YII | | 13.613  (11.377)|   0.11929 Y / 5 1757.367 / 5

3. dZ I,II 

|ZI  ZII | | 6.697  12.199|   2.17115 Z / 5 378.7117 / 5

10 4. dI,II = (d2X I,II  d2Y I,II  d2Z I,II  2.20529 Table 4. Standard distance values and probability points are shown. Roman numerals identify transects. See the explanation in the main text. Transects

II

III

IV

V

VI

I

2.20529

2.665447

2.294124

1.981885

3.073942

2.917673

2.329369

3.070668

2.258057

3.008231

2.695214

2.545479

1.385353

1.629902

II III IV V Probability Probability points

2.296875 0.1

0.05

0.01

0.001

1.28

1.65

2.33

3.1

What is the use for having the last two rows in Table 4? Utility comes from each d being a standard normal variate, assuming specified regularity conditions which must be fulfilled (Orlóci 1912 v.1914). Take the value dI,II = 2.20529. Its magnitude exceeds the 0.05 probability point (1.65) of the Normal distribution. On this basis, we declare dI,II statistically significant at that probability. This is the same as saying, Transect I and II are statistically distinguishable. Each d value in Table 4 exceeds the chosen decision point of 0.05, except dIV,V and dIV,VI. But these miss significance only by small margin within rounding errors. These are important facts, clearing the way for the analysis of the six transects as statistically distinct objects.

Cluster analysis In the next step we probe the 9 x 6 co-ordinate data (Table 1) for group structure in the 6-transect collection. A dendrogram is created (Figure 2). The tips of the dendrogram branches are the transects. The clustering algorithm is SSA (Orlóci 2012 v.1914). It takes the upper half of the distance matrix created by algorithm METRIC. Metrics takes input from Table 1. The SSA results are in Tables 5 and 6. Table 5. Transect distance calculation by METRICS. SSA is uploaded to ResearchGate DOI 10.13140/RG.2.1.1196.3601

11 PROGRAM: METRICS ======================== DATA FILE:data9x6.tru NUMBER OF VARIABLES = 9 NUMBER OF OBSERVATIONS PER VARIABLE = 6 DISTANCE OPTION (3): EUCLIDEAN DISTANCES FILE: 0

35.594943 0

52.392748 49.979996 0

31.352831 37.973675 44.899889 0

18.411953 34.322005 54.662601 27.239677 0

34.132096 36.414283 52.42137 40.816663 31.874755 0

DISTANCES FILE: dist6x6.txt Time: 23:04:12 Today is 18-03-30

Table 6. Clustering results by SSA. The data input includes the upper half of the distance matrix in Table 5. SSA is uploaded to ResearchGate DOI 10.13140/RG.2.1.1196.3601 PROGRAM: SSA ==================== DISTANCES FILE: dist6x6.txt NUMBER OF OBJECTS: 6 CLUSTERING PASS 1 -------------------------------------------------------NUMBER OF INDIVIDUALS IN FUSION GROUP: 2 FUSION SUM OF SQUARES: 169.50001 INDIVIDUALS: 1 5 CLUSTERING PASS 2 -------------------------------------------------------NUMBER OF INDIVIDUALS IN FUSION GROUP: 3 FUSION SUM OF SQUARES: 688.00001 INDIVIDUALS: 1 5 4 CLUSTERING PASS 3 -------------------------------------------------------NUMBER OF INDIVIDUALS IN FUSION GROUP: 2 FUSION SUM OF SQUARES: 663. INDIVIDUALS: 2 6 CLUSTERING PASS 4 -------------------------------------------------------NUMBER OF INDIVIDUALS IN FUSION GROUP: 5 FUSION SUM OF SQUARES: 2224.8 INDIVIDUALS: 1 5 4 2 6 CLUSTERING PASS 5 -------------------------------------------------------NUMBER OF INDIVIDUALS IN FUSION GROUP: 6 FUSION SUM OF SQUARES: 4019.8333 INDIVIDUALS: 1 5 4 2 6 3 DENDROGRAM DATA STORED IN FILE dendro6x6b.txt RUN PROGRAM TREE TO PLOT THE DENDROGRAM!

12

Figure 2. Dendrogram showing hierarchical structure of the transect collection. The data base is Table 1. Roman numerals identify transects. Arabic numerals are within group sum of squares.

The graph is depicting clusters in the full-dimensional sample. Differences of numerical values indicate emergent sums of squares. For example, the difference 2224-688-663=873 is the emergent sum of squares generated by pooling two clusters, I,V,IV and II,VI into one. A simple analysis reveals the two groups’ statistical standing: Source

Sum of squares

Group size

Variance

F statistic

Between groups (Emergent)

873

2

873

1.939

Within groups

1351

4

450.33

Total

2224

6

To be noted is the enormous size of the emergent sum of squares, 39% of the total. This speaks of sharp isolation transect groups I, V, IV and II, VI. But are they really that sharply separated statistically? This should be tested. We observe that F=1.939 at 1 and 3 degrees of freedom. The question to be asked is this: where does an F value as large as 1.939 put the measured distance of the two transect groups on a probability scale. We construct the scale based on the assumption that the distribution of F is ruled by chance. Under the circumstances the area integral of F gives us guidance. The solution is P(FRND  1.939)  0.258 . This is far too

13 large a probability to consider the separation of the two groups significant. We would like to have a probability much smaller, say 0.05 or less, which is the same as wishing for an F value at least 10.117, before we would consider the two groups’ separation statistically significant. In any case, it is best to regard the test inconclusive, considering the very small group sizes. The reader may have noted that in distance terms (Tables 4 and 5) Transect I is closest to Transect II in Figure 1. Yet in Figure 2, Transect I is closest to Transect V. What is the reason for the discrepancy? The reason should be found in that 10% shortfall of information caused by using a two-dimensional plot. We emphasise, the basic data were not adulterated by a priori transformations. Therefore, the results reflect what exists and not what should exist.

STRUCTURES WITHIN THE KANIADAKI FRAME The natural vegetation stand’s GQES is best regarded as a high-level construct, shaped in community assembly by effects issuing from three independent sources: phylogeny, current environment mediation, and the always present, ubiquitous chance events. Historic phylogeny is the ultimate source of species richness. Current environmental mediation is responsible for sorting the available flora elements among the sites. Chance events, by definition, defy prediction and the sources cannot be identified. Yet, the effects can be measured indirectly analytically by the residuals. In a technical sense, EBE has been shown by Max Plank to be a proxy scalar for the unmeasurable potential energy of a resonator complex. The theory has been worked out at the particle level (fermions, bosons), but the principle of convertibility, energy to entropy or entropy to energy, is universal. Whenever a probability is given, say P, EBE is defined by -ln P. Building on its scale independence, intuitions tell us that since we have P we can calculate EBE, and since we know from

14 Max Plank the linkage of EBE to energy via the Normal distribution, and since GQE is a generalization of EBE, the fundamental properties of EBES should be true for GQES too at any order . On this basis, we have rewritten the energy-based entropy equation for any vegetation stand in GQE terms, EQ = EQ Phy + EQ Env + EQ Rnd . Recall, the superscript Q refers to GQE, and subscript  is the Kaniadakis order variable with limits 0 EQ ,Env T, resonator richness (n) takes dominance and the graphs are convergent. This is the case of the Phy graphs of transects I and II. For Transects I and II, we have I for A and II for B: n,AB 

nsup,AB ninf,AB



T 290 40  1.13333 and T ,AB  sup,AB   1.0035 Tinf,AB 289 30

The comparison of n,AB and T,AB is telling us that the richness data component is dominant and the Phy curves of transects I and II are convergent. The Phy curves of transect I and V (Figure 3) present a case for divergence. We have I for A and V for V: n,AB 

nsup,AB ninf,AB



T 30 289  1.4378  1.0714 and T ,AB  sup,AB  Tinf,AB 201 28

In this case, the quantitative data component is dominant. The complete table of dominance coefficients is given below: n I II III IV V T I II III

II 1.3333

III 1.4333 1.0750

IV 1.6667 2.2222 2.3889

V 1.0714 1.4286 1.5357 1.5556

II 1.0035

III 1.0277 1.0241

IV 1.6609 1.6667 1.7069

V 1.4378 1.4428 1.4776

VI 1.2500 1.6667 1.7917 1.3333 1.1667 VI 1.5211 1.5263 1.5632

24 IV V

1.1552

1.0920 1.0579

We intend to perform comparisons of n, T and EkQ0.00001,Tot (Table 8). For this purpose, we take the EkQ0.00001,Tot vector from Table 8: Transect EkQ0.00001,Tot

I

II

III

IV

V

VI

387.1463

446.7691

468.1744

232.6831

311.1187

280.5847

We need the EkQ0.00001,Tot differences matrix: dif EkQ0.00001,Tot I II III IV V

II 59.6228

III 81.0281 21.4053

IV 154.4632 214.0860 235.4913

V 76.0276 135.6504 157.0557 232.6831

VI 106.5616 166.1844 187.5897 232.6831 30.5340

The product moment correlation values are: r(dif EkQ0.00001,Tot ,n)= 0.7327 and r(dif

EkQ0.00001,Tot ,T)= 0.5212 . These are pointing to an interesting fact. The dominance scalar n and EBE, i.e. RGE of order one, are correlated more highly than T and EBE. This further elevates the importance of species richness on the regional level.

Instability scalar w The general equation is w = 1 - P2 - (1 - P)2 . We link w to the one-resonator RGE by P = e

-HQ

. This makes w a standard, one-resonator instability scalar. This makes

w a sample-nonspecific having direct comparability between cases. The instability idea is simple. If a vegetation stand, described by the n,T couplet has P probability to have assembled in a purely chance-ruled process, exactly the way it actually have, there must be an alternative state into which it could reassemble by chance with probability 1-P. The alternative state is called ‘ghost state’. It is intuitive that in a chance ruled process the ghost state could in fact

25 materialise. Therefore, we are justified to speak of instability in the vegetation stand’s GQRS and measure it by scaler w. The value of w is equal to 0.5 at its possible maximum and zero at its possible minimum. Indeed, w is a squared probability. We write w=

2 and ω = 2w . This 2

is an instability measure on the 0    1 probability scale. We can use ω as a statistical test criterion. In general, we shall consider any GQES stabile when ω is small, say 0.05 or less.

GQES instability and the dominance scalars We consider the metacommunity o the transect level. Instability values (w) are calculated, and from that, a ‘dif w’ matrix is constructed for transect pairs. Transect totals (n, T) and the intermediate steps of calculation are in the following table for =0.00001: Transect T n EQ0 HQ0 P 1-P w

I

II

II

289 30 99.46273 3.315424 0.036319 0.963681 0.069999

290 40 121.8799301 3.046998254 0.047501297 0.952498703 0.090489848

297 43 129.0715 3.001662 0.049704 0.950296 0.094468

IV 174 18 59.7368 3.318711 0.036199 0.963801 0.069778

V 201 28 85.05633 3.037726 0.047944 0.952056 0.09129

VI 190 24 75.111 3.129625 0.043734 0.956266 0.083643

The dif w matrix is given next, followed by the matrices of the two dominance scalars: The =0.00001 dif w matrix Transect I II III IV V

II 0.0095002

III 0.005522 0.810432

IV 0.030212 0.835122 0.02469

V 0.0087 0.81361 0.003178 0.023512

VI 0.016346976 0.821256976 0.010824976 0.015865024 0.007646976

III 1.433333 1.075

IV 1.666667 2.222222 2.388889

V 1.071429 1.428571 10.60714

VI 1.25 1.666666667 12.375

Dominance matrix n Transect I II III

II 1.3333333

26 IV V

1.555556

1.333333333 1.166666667

V 10.32143 1.442786 1.477612 1.155172

VI 12.04166667 1.526315789 1.563157895 1.091954023 1.057894737

Dominance matrix T Transect I II III IV V

II 1.0034602

III 6.72093 1.024138

IV 16.05556 1.666667 1.706897

0.229095 2.838254 3.990376

r(dif w,n) r(dif w,T)

The mean values correlations: Mean w Mean n Mean T

-0.22371 -0.32115

We see from n |t| or P>F “the probability of an at least as large t or F value”.

Perusal of Table 10 points to high statistical precision (r2=0.9596). It should be noted that there are the 90% confidence limits about the regression coefficients a, b. The limits do not enclose zero. This is an indication of significant departure in the coefficient’s magnitude from zero expectation at 0.10 P. In other cases (coefficients c to g), the confidence limits include zero. Why should we not discard coefficients c to g? The reason is that the 6th order polynomial defines an ecologically meaningful trend which coefficient b alone could not. In view of the small

29 sample size, to retain the curve in its exact form, we need to retain all coefficients from b to g. L.O. expects that future expansion of sample size by laying additional new quadrats on the transect will smoothen the trend and narrow the confidence limits. The 1st derivative is highly informative in picking out characteristic points on the integral curves namely the minima, maxima and inflexion points. The 1st derivative’s sign is added on by the application to indicate whether the point is in the descending or ascending portion of the integral curve. It is advisable to track the rate of the rate-of-change by the 2nd differential equation (Figure 5). Characteristic points located by rate change or rate of the rate-of-change help to locate points where the environmental conditions are the most destabilising.

Instability and the environment The reader may be wondering, after inspection of the instability curves in Figure 5, where could evidence be found to verify that the instability curve is a manifestation of environmental effects. The evidence is offered by examination of the environment within the sample plots located on the transect where the extreme points of the integral curve reside. It should be clear that the evidence cannot be more than statistical and requires environmental data. Table 11 holds topographic and soil data for Transect I. A separate environmental table is included in the Appendix for Transects II to VI. Table 11. State of environmental descriptors with sample plots on Transect I. Seven physical variables are scored. Sample plot Gedikli Aşağı Karamuklu stream Aşağı Karamuklu stream Aşağı Karamuklu stream Kahır area

Altitude (m) 1200 1300

Slope (%) 25 70

Aspect E S

Slope position Lower Middle

Parent material Limestone Limestone

Soil stoniness* Stony Very stony

Soil depth** Medium Medium

1400

70

WE

Lower

Limestone

Stony

Medium

1500

80

WE

Lower

Limestone

Medium

1600

40

S

Lower

Limestone

Slightly stony Stony

Medium

30 Kahır area Kahır area

1700 1800

40 100

SW NW

Middle Upper

Limestone Limestone

Kahır area Kahır area

1900 2000

4 10

SE W

Ridge Ridge

Limestone Limestone

Very stony Slightly stony Very stony Very stony

Shallow Shallow Medium Medium

* Slightly stony < 10%, Stony 10%-%30, Very stony 30%-75, Skeletal =t) 0.1032 0.0310