## Modeling Grazing Processes

There have been many different developments of mathematical models to simulate or analyze grazing and its effects. A compelling modeling framework should include what happens to the plants, what happens to the grazers and, where appropriate, the consequences of predators influencing the numbers or activities of grazers. One example from Caughley is briefly considered here.

The rate of growth of biomass of a population of vegetation, V, can be modeled as a logistic equation rv = rmv (1 - V/K)

where K is the carrying capacity, that is, the amount of vegetation in some area when it reaches the maximum

This modeling shows, among other things, that the equilibrium population of plants in a grazed habitat is not affected by the intrinsic rate of increase of the plants (rm is not in the equation for V' ). Instead, increases in rm lead to greater populations of herbivores (c2 and/or a1) leads to changes in equilibrial values of populations of plants and herbivores (c2 and a1 are in the equation for V' ; V affects H ).

Finally, changes of populations of predators feeding on herbivores are modeled by rP = -a2 + c3(l -e-d,H) /H

where rP is the rate of change of the population of predators. H is the size of the population of herbivores (the prey); a2, c3, d4 are as explained above for herbivores, that is, al, c2, d2.

Predators are also reducing the rate of growth of herbivores which is now rn = -«1 + -e-d2V)/V-fP(1 -e-d'H)/H [8]

where f is the rate of consumption per capita of herbivores by predators, when there are no shortages of food for predators, P is the size of the population of predators, and d3 is a constant describing how f is altered as the population of herbivores changes.

This type of modeling can produce a steady-state (an equilibrium) or oscillating populations of plants, herbivores, and predators, depending on the values of various constants. It is therefore quite a general tool for analyzing various grazer/plant, predator/grazer interactions, and to examine 'bottom-up' (V matters most) versus 'top-down' (P matters most) controls of the populations of herbivores.

This type of modeling makes many assumptions (e.g., that growth of populations fits a logistic trajectory). It also assumes that the ecology of the grazing system is simple and that there are no complex indirect interactions, multi-species influences of competition among species of grazers, etc.

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