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In 1975, Noy-Meir presented a comprehensive, graphical stability analysis for plant–herbivore interactions. Inspired by the analysis of predator–prey interactions by Rosenzweig Tainton, Morris Spalinger Parsons et al. 1994). Domestic ruminants remove a relatively fixed proportion (some 40–70%) of standing vegetation with each bite (Ungar, Genizi 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 Wade 1991; Parsons 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 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 Ungar Parsons, Johnson Parsons et al. 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, Harvey 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. 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Schwinning et al. (Wed,) studied this question.