grass to defoliation (Johnson & Parsons 1985;. Parsons, Johnson & Harvey 1988). Briefly, in the logistic equation, the rate of growth is determined solely on the ...
ORIGINAL ARTICLE OA Functional Ecology 1999 13, 737–747
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EN
ESSAY REVIEW
The stability of grazing systems revisited: spatial models and the role of heterogeneity S. SCHWINNING* and A. J. PARSONS† *Department of Biology, University of Utah, Salt Lake City, UT 84112, USA and †AgResearch Grasslands, Private Bag 11008 Palmerston North, New Zealand
Introduction
© 1999 British Ecological Society
In 1975, Noy-Meir presented a comprehensive, graphical stability analysis for plant–herbivore interactions. Inspired by the analysis of predator–prey interactions by Rosenzweig & MacArthur (1963), he cast the model in the form of ordinary differential equations, representing grazing as a homogeneous process in space and continuous in time (Noy-Meir 1975). His analysis demonstrated that a continuously grazed ecosystem may have multiple stable states and may be ‘discontinuously stable’ (see also reviews by May 1977; Tainton, Morris & Hardy 1996). The term implies that the grazing system may move discontinuously from a stable state of high productivity to a stable state of low productivity with only a small increase in grazing pressure or with a minor disturbance. This and other harvesting models (Clark 1981) have provided an important paradigm for bioeconomics, namely, that without change in management, a biological resource can become overexploited and fall into a low-productivity state from which it can recover only through dramatic reduction in harvest rate. Since Noy-Meir’s (1975) analysis, much progress has been made in refining understanding of the grazing process in temperate grassland. Particularly, advances have been made in understanding grazing at the bite scale (Illius & Gordon 1987; Spalinger & Hobbs 1992; Parsons et al. 1994). Domestic ruminants remove a relatively fixed proportion (some 40–70%) of standing vegetation with each bite (Ungar, Genizi & Demment 1991; Laca et al. 1992). If ruminants harvested 50% of the standing vegetation uniformly across an entire grazeable area each day, this would exceed the ability of the vegetation to recover from defoliation (the estimated sustainable homogeneous removal rate is 2·5% of standing vegetation per day; Parsons & Chapman 1998). Thus, for a grazing system to be sustainable, grazing must take place from only a small portion of the total grazeable area each day (some 5%; Wade 1991; Parsons & Chapman 1998). This simple calculation illustrates that grazed grassland is necessarily heterogeneous, because at any time, different patches will be in different states of recovery from previous defoliations. As has long been recognized, ‘continuous’ grazing is not a continuous process, but involves a succession of discrete defoliations at the bite scale, each followed by a
variable period of regrowth (Hodgson & Ollerenshaw 1969; Morris 1969; Clark et al. 1984). Furthermore, it is not a deterministic process: because animals take bites from only a small part of the total area on any day, there is some uncertainty as to exactly where animals choose to take bites. Thus, when considering grazing at the bite scale, we must deviate from NoyMeir’s assumptions in three ways: we must assume (1) spatial heterogeneity instead of homogeneity, (2) discrete instead of continuous defoliation and (3) stochasticity instead of determinism. Below we introduce an implicitly spatial model with these characteristics, where we treat a field as a collection of bite-sized patches but ignore their explicit spatial relationships. We use the model to address the following questions: (1) does a consideration of grazing at the bite scale alter our current understanding of the stability and sustainability of grazing systems and (2) how does spatial heterogeneity and variance in defoliation intervals affect the yield of grazing systems?
Stability of grazing according to Noy-Meir Noy-Meir’s (1975) analysis demonstrated that a highly simplified representation of a biological process can still produce complex dynamics. In this case, the analysis suggested that continuously grazed ecosystems may be discontinuously stable. The graphical analysis that led to this insight is represented in Fig. 1. High and low stock densities support only one stable equilibrium vegetation state, but at intermediate stock densities the model allows three equilibria (i.e. states of zero net change; the points of intersection of growth and consumption curves). The two outer points of intersection are stable equilibria. Thus, as the arrows indicate, vegetation states near these equilibria approach them more closely. The intermediate equilibrium is unstable and nearby vegetation states are repelled and then attracted by one of the two stable equilibria. The existence of two alternative stable equilibria at one stock density has also been called ‘dual stability’ which, in Noy-Meir’s analysis, arises under the following conditions: (1) the consumption function must display non-linearity between zero and the vegetation state that supports the maximal instantaneous rate of 737
738 S. Schwinning & A. J. Parsons
vegetation growth and (2) there exists an ungrazeable plant reserve. These conditions are quite realistic in many grazing systems, notably in temperate zone pastures, and so we concentrate our analysis on this case.
bite scale, grazing is a discrete process with dynamics governed by instantaneous consumption c and subsequent plant growth g in the time tint between defoliations: b(j + 1) = b(j) – c(b(j)) + g(b'(j),tint),
A bite scale model for grazing We derive a model to describe the dynamics of bitesized patches. Variables are summarized in Table 1. Patch state is represented by a single state variable, b, the vegetation biomass per unit ground area. At the
eqn 1
where b(j) is the biomass immediately prior to the j’th defoliation and b'(j) is the biomass immediately after the j’th defoliation, so: b'(j) = b(j) – c(b(j)).
eqn 2
Formulating the grazing process by looking at patch state only once during the interval between defoliations allows us to identify equilibrium conditions, even though it is understood that biomass density in a patch is continually changing. Patch equilibria, b*, exist where growth in the interval between defoliations is exactly equal to the amount of biomass removed in each defoliation: g(b* – c(b*),tint) = c(b*).
eqn 3
DEFOLIATION
The consumption function, c, describes the biomass that is removed in a single bite. It captures the instantaneous and local functional response of animals to patch state. Herbivores remove variable amounts of foliage in a single bite, but removal tends to be proportional to the amount of standing vegetation (Ungar et al. 1991; Laca et al. 1992; Edwards et al. 1996), although a portion of the biomass may be considered ungrazable, thus: c = 0 if b ≤ bc,min c = f(b – bc,min) if b > bc,min,
Fig. 1. Stability analysis based on Noy-Meir (1975). Plant growth rates (solid lines) and rates of consumption (dotted lines) as functions of vegetation state. Intersections of the growth and consumption functions mark equilibrium points. The arrows indicate the direction of state change.
Meaning
Unit
PATCH-SCALE VARIABLES
b b′ c g mbite tbite tint
biomass (dry mass) density just before grazing in patch biomass density just after grazing in a patch biomass density removed in one bite biomass density increase between defoliations biomass per bite handling time per bite time in the interval between successive defoliations
kg m–2 kg m–2 kg m–2 kg m–2 kg s days
ANIMAL- OR FIELD-SCALE VARIABLES
Ma Na Ta
biomass intake per animal per day number of bites taken per animal per day grazing time per animal per day
where bc,min is the ungrazeable portion of the standing biomass (which precludes grazing to extinction). The parameter f is the fraction of biomass in the grazeable horizon (b – bc,min) that is removed by the grazer in one bite. Default values for all parameters are listed in Table 2 using the example of grazing by young cattle. GRAZING INTERVAL (DURATION OF REGROWTH)
Table 1. List of variables Symbol
eqn 4
kg – s
In contrast to the consumption function, which captures the instantaneous response of animals to local patch states, the function determining the grazing interval depends on how animals interact with the rest of the pasture. We can envisage the grazing interval as the time taken by animals to repeat the present defoliation of a single patch in all other patches in the pasture, thus, the time taken to return to the first defoliated patch. This return time depends on stock density, intake demand per animal and physical constraints such as handling time per bite. But the relationship is not simple, because animals may encounter patches in variable states and exploit them in variable ways. They may also not graze from
739 The stability of grazing systems
all the other patches before they return and so return times may be variable. To address this uncertainty, we consider two contrasting alternatives. First we cast grazing at the patch scale as a deterministic process in which all patches are defoliated and when they are in exactly the same state. This case can be seen as equivalent to a systematic, sequential defoliation of a large number of patches. In this way, we can derive equilibrium solutions based on the dynamics of a single patch (equation 3), because by assumption all patches in this case have identical (though time-displaced) dynamics. Second, we assume that animals defoliate patches at random and independent of patch state. To do this we must expand the model to consider many patches simultaneously (we chose 2000). Steady-state solutions for this stochastic model are identified by simulation. In both the deterministic and stochastic cases, the relationship between handling time per bite, tbite, and bite mass, mbite, is linear (e.g. Spalinger & Hobbs 1992; Parsons et al. 1994): tbite = tpreh + kmast mbite,
eqn 5
where tpreh is the time required to open and close the jaws to prehend a bite and kmast is the mastication (chewing) time per unit bite mass. We ignore for now other time-consuming activities associated with foraging. Bite mass is calculated by multiplying consumption c from a patch (in units of kg biomass per m2) with bite (patch) area abite: mbite = cabite.
eqn 6
Bite area is considered here to be constant and independent of patch state (Demment, Laca & Greenwood 1987; Ungar & Noy-Meir 1988). As did Noy-Meir (1975), we assume that animal intake per day is limited either by a maximum grazing time per animal Ta,max or by the rate of digestion to give a maximum daily intake of mass per animal Ma,max. Hence, in the deterministic case, where all
Table 2. List of parameters Symbol
Meaning
abite bmax bc,min
bite area maximal biomass density residual biomass density, animals cannot deplete biomass in patch to less than this defoliation fraction for grazeable biomass mastication time per unit biomass prehension time per bite maximal relative growth rate maximal daily intake per animal stock density maximal daily grazing time per animal
f kmast tpreh µ Ma,max S Ta,max
Default value 0·004 m2 0·321 kg m–2 0·0153 kg m–2 0·5 2000 s kg–1 0·82 s 0·1 day–1 8 kg 6 an ha–1 48000 s
Source, Parsons et al. (1994); intake parameters set here for young cattle (Penning et al. 1997)
bites taken are assumed to be of identical size, the number of bites taken per animal per day is: Ma,max Na = –––––– if Na tbite < Ta mbite or Ta,max Na = ––––– if Na mbite < Ma tbite
eqn 7
and the defoliation interval, tint, is derived as the time taken for all animals to bite once from all bite-size patches per ha, which is:
(
S tint = Na ––––– abite 10 000
)
–1
,
eqn 8
where S is stock density in animals per ha and 10 000 scales from animals per ha to animals per m2. In the stochastic case, mbite, tbite and tint are not identical across patches, but depend on random patch selection. We determine their distribution by simulating defoliation in 2000 patches. A daily foraging bout ends when one of two conditions is satisfied: Na
S 10 000
∑ mbite,j ≥ Ma,max ––––– 2000 abite
j=1
or Na
S 10 000
∑ tbite,j ≥ Ta,max ––––– 2000 abite .
j=1
eqn 9
PLANT GROWTH
We depart from the convention of using a two-parameter logistic equation and use a modified growth function that describes features of growth that have been shown to be important in modelling the response of grass to defoliation (Johnson & Parsons 1985; Parsons, Johnson & Harvey 1988). Briefly, in the logistic equation, the rate of growth is determined solely on the basis of current vegetation biomass. Thus, the function cannot distinguish between a situation where current biomass is the result of a recent lenient defoliation or the result of a severe defoliation that took place some time ago. Yet, observation (Grant et al. 1983; Parsons et al. 1983, 1988) and leaf-agestructured models (Johnson & Parsons 1985; Parsons et al. 1988; Thornley & Johnson 1990) of grass growth suggest that these two situations can produce very different specific growth rates. After lenient defoliation, net growth rate per unit biomass tends to be slow, because much of the standing biomass consists of older leaves, which contribute to a high rate of senescence. After severe defoliation, old and new leaves are almost all removed and so there is a predominance of young leaves in the regrowing vegetation. Thus, when vegetation has recovered to the same biomass as after the lenient defoliation, the rate of senescence is lower and therefore net growth rate per unit biomass is larger. Leaf-age-structured growth models incorporate the effect of defoliation severity explicitly but here we seek
740 S. Schwinning & A. J. Parsons
a simpler formulation. We introduce a function in which the rate of growth after defoliation depends not only on current patch state b, but also on the initial condition for regrowth, the patch state b' just after the last defoliation. The variable b' modifies the intrinsic rate of growth µ:
(
db b' –– = µb 1 – –––– bmax dt
)(
b 1 – –––– bmax
)
eqn 10
with bmax as the maximal biomass. When b'
