The stability of grazing systems revisited: spatial models and the role of heterogeneity

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 Noy-Meir’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? 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. . 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. 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 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. We derive a model to describe the dynamics of bite-sized 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 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: 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: 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: 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: 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. 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 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): 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: 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 bites taken are assumed to be of identical size, the number of bites taken per animal per day is: 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: 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: 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 where current biomass is the of a defoliation or the of a defoliation that place some time et al. Parsons et al. and models (Johnson & Parsons 1985; Parsons et al. & Johnson of grass growth that these two can produce different growth defoliation, net growth rate per unit biomass tends to be because much of the standing biomass of which to a high rate of defoliation, and are all removed and so there is a of young in the Thus, when vegetation has to the same biomass as after the defoliation, the rate of is and net growth rate per unit biomass is growth models the of defoliation but here we a We introduce a function in which the rate of growth after defoliation depends not only on current patch state b, but also on the for the patch state after the The variable the rate of growth with as the maximal biomass. growth rate is maximal and the function from the logistic values of more defoliation and growth rates the logistic is constant for one growth interval, can be taken to be a parameter and equation 10 can be the logistic We can the function for the growth g of equation In the stochastic case, defoliation intervals and conditions for regrowth between patches, so we the growth function to each of the 2000 patches following each defoliations. We the deterministic and stochastic to distinguish the of discrete patch per from the of in patch selection. We that our between and stochasticity to the patch at the field scale the basis of that a field a large number of both deterministic dynamics. 2 is the equivalent of Noy-Meir’s (1975) graphical analysis now based on the deterministic patch In the deterministic case we can from the patch scale to the field scale, by the consumption function c (equation by the interval tint that would animals to return to a patch (equation and the growth function g (equation by the interval that would to to the biomass before We can do this in the deterministic case, any harvested patch is of all patches that are harvested at any the of Noy-Meir’s (1975) stability model are in the that stability arises at intermediate stock densities and when the portion of the consumption the rate of digestion the growth . Stability analysis on the field scale for the discrete model with deterministic patch selection. plant growth rates (solid lines) and rates of consumption (dotted lines) as functions of patch biomass prior to Intersections mark equilibrium points. The defoliation fraction is to f and stock densities to and an other parameters are to the patch dynamics to the of stability in The equilibrium is by long defoliation intervals (in this example and patches that recover to maximum The equilibrium is by intervals and biomass densities for the two equilibria are kg and kg stability implies on conditions, all patches in the field fall into either one or the other of the two . of patch dynamics in the stability for the deterministic patch model with parameter values to for the stochastic patch model with parameter values to We can now parameters of grazing not considered by Noy-Meir (1975). In Fig. we the of the defoliation fraction f on growth and In our this parameter not only consumption but also because our growth function the of grazing In the consumption function, defoliation fraction f the of the where consumption limited by the rate of digestion by to the This is at the bite mass, where bites are large to animals in the maximal time to Thus, when bite are biomass to be at the In the growth function, small defoliation move the biomass density that is for regrowth after defoliation to the f the of the consumption function and of growth in conditions for stability (i.e. where the portion of the consumption function with the growth become that stability can exist for f at any stock can that stability at f where the of the growth with the for animal Thus, Noy-Meir (1975) the of this relationship between the growth and the consumption on we can now a with the stability be grazing take of the standing biomass. . The of defoliation fraction f on plant growth rates (solid lines) and rates of consumption (dotted lines) as functions of patch biomass prior to for all parameters are to values is the equivalent of and now for the stochastic case, random patch selection. The growth in this by 2000 patches to a state. on the a with a different number of defoliations per The consumption are calculated based on the of patch . Stability analysis for the model with random patch selection. plant growth rates (solid lines) and rates of consumption (dotted lines) are based on solutions for fixed bite taken at random from 2000 patches and presented as functions of patch biomass. The defoliation fraction is to f and stock densities to and an other parameters are to In the stochastic case, stability is in a much of parameter at f stability exists only for a small of stock densities between and an and at f it does not exist for any stock The patch scale dynamics associated with stability in the stochastic case are in As in the deterministic case, the equilibrium (the biomass is kg is by defoliation intervals and the equilibrium kg by intervals patch dynamics at these alternative stable states are quite different from one total animal intake is (see Fig. In the stochastic case, defoliation intervals at equilibrium have a distribution they in a and so a to of 1. A distribution for defoliation intervals has been in that it is not to assume random patch under some we considered only two of patch one where all patches are harvested in exactly the same and one where patches are at we consider a case where animals for patches of a biomass. by large for either or grass patches has been in & et al. and so it is to consider both To the of we assume that patch encounter is still random but that patches are with a of when they are either or a biomass Thus, animals patches either they are kg or the are kg In patch is associated with an time one of & Hobbs 1992; Parsons et al. et al. 1994). In a model that is not we cannot time, for with the by we a fixed time with patch that is and the of we assume that patch is associated either with time or with an equivalent to and In Fig. we assumed a defoliation fraction of f to in this the amount of biomass that animals can these conditions, patch has large on vegetation growth and animal animals high biomass patches animal intake can be to a when animals do not in the the reduction of animal intake is not so much the of the time associated with but to the state of the which supports have this on vegetation by patches a they (1) a portion of the field to approach yield and to and (2) increase the grazing pressure in the portion of the which also the rate of animals low biomass patches, the on the vegetation is The maximal vegetation growth rate is now so that more animals be at a intake to animals that do not animals also at vegetation provided the of patch is . Stability analysis for three of in patch selection. plant growth rates (solid lines) and rates of consumption (dotted are based on solutions for fixed bite taken from patches and presented as functions of patch biomass. The defoliation fraction is to f and stock density is an other parameters are to of of all patches kg are of all patches kg are to different of and the increase in the direction of the arrows (see for analysis of a spatial model that (i.e. not be as as Noy-Meir’s analysis The present model is different from Noy-Meir’s model in (1) it describes grazing as a discrete to process at the patch scale, (2) it a different growth function, which growth rates to vegetation after defoliation and (3) it in patch selection. In and we the of each on the stability . stability for continuous and deterministic The parameter (in defoliation fraction f and stock that other parameters are to . stability for deterministic and random patch with discrete The parameter (in defoliation fraction f and stock that support other parameters are to . stability for different of in patch selection. Patch encounter is random and there is associated with patch The parameter (in defoliation fraction f and stock that other parameters are to patches kg are with or A of does not give patches kg are with 50% or In Fig. we the of modelling grazing as a discrete continuous process per on the stability To a continuous of our the amount of biomass per day uniformly from across all patches, instead of it only from patches where bites taken is in some grazing models that assume spatial homogeneity, 1998). The the of and stock densities that grazing as a discrete process per to have on the of the stability but for both growth the stability stock In Fig. we assume defoliation is a discrete process and we now consider the of deterministic stochastic patch selection. both growth random patch the stability to deterministic patch and the growth function has an even stability the logistic under random patch selection. Fig. that here as the to patches or a biomass can also The of high biomass patches tends to the stability of low biomass patches tends to do the near f stability in all both (Noy-Meir and more grazing models for temperate (e.g. Johnson & Parsons 1985; Parsons, Harvey & have the that stability can in grazed systems, for this is and is limited to where the of the productivity & & in that the grazing system is in a state, following an equilibrium that with This it to the stability of a system stability a as it is to the system is a or an state. in in stock density, the an state. As Noy-Meir to recover from an state would then a reduction in stock density across the stability and into the where there exists only one equilibrium at or near the maximum be only of stock density in response to in (Noy-Meir 1975). We have the stability of a model that to previous models in assumptions and parameter but that grazing as a discrete process at the bite we made assumption the ability to when we assumed that animals from any patch they and encounter random with to patch we that stability much a for previous models This we a logistic or the growth per bite to be in of 50% of the grazeable biomass for stability to and even it a much of stock In the of stability because the productivity of the two stable equilibria Thus, the two equilibria would be in a field we assumed that animals for patches of states, the different stability when animals patches of large but more when they patches of low biomass a for large biomass provided the time of such as for random patch stability when the defoliation fraction 50% or defoliation for large have been to be in that same of 50% (Ungar et al. 1991; Laca et al. to our stability only be a Thus, we that in part by the stochastic and spatial of the of We have seen that grazing is to the and of spatial heterogeneity in pasture. is a that spatial heterogeneity is to productivity and this is to a that spatial at all times (e.g. instead of spatial per is not the case where grazing at the patch scale is deterministic with constant defoliation We can such a grazing process to be by a so that patches, instead of grazed are simultaneously at that same defoliation state would be heterogeneous, under it would be but the productivity of the two systems would be exactly because the patch states and defoliation intervals the In Fig. we the equilibrium animal that can be under a of stock We also the biomass for patches at stock density, where yield is In deterministic and random patch we that random patch heterogeneity in patch state. yield is from kg under deterministic grazing to kg under random patch selection. of this of yield can be when animals a for high biomass patches low biomass by the in patch states a more The maximal growth rate for the vegetation growth function is kg when kg and tint This illustrates that intake which are the only yield in the deterministic case, have as much to do with yield from as has patch heterogeneity by random patch selection. the other it also be that for stock random patch allows either equal or yield deterministic Thus, in the and stochastic of grazing not a to . The rates of growth and consumption in the equilibrium as a function of stock density for deterministic (solid random of high biomass patches (dotted and for low biomass patches The of is other parameters are to more to the maximal yield random is a patch process that patches with low biomass the of high biomass there is a for such a system to have stability alternative stable states, at different the system instead has a to and so the of two patch by This has been in in systems (e.g. Laca et al. 1992; et al. and in for low biomass patches is of that it may increase the of the it is also as per ha, because the animals on only a fraction of the total grassland The we as for this of large biomass has not been but our model that may in this the into both removed from In et stock density the of the grazed area and the of grazed area also the case in our of and patches to form and This that some of at scales that of bites. Noy-Meir’s analysis of a grazing model an important for grazing His analysis suggested to yield (i.e. animal intake per in grazing systems, stock density has to be to the that still supports the equilibrium state animals are or this also seen to be associated with a large of the equilibrium state and the equilibrium state by after in vegetation through the of the two of Thus, the to a grazing system to be a between yield and the of analysis to a different any stock density, when the defoliation fraction is at or there is only one stable state (i.e. the spatial of continuously and any change in stock density to a change in Thus, there not be any to conditions or to recover from an state without stock and a to by the for vegetation growth would be a to be a is the of growth We did not from our that any defoliation fraction we did a close between foraging and vegetation In all of foraging that we considered we a on state and are many and spatial of foraging & & which take the of different foraging into but these of the of foraging on resource now a explicit of grazing that the in this with in foraging We and two for their on an of the