## Herbivore dynamics

General models of the type given in eqn [1] assume that the herbivore population is uncoupled from the dynamics of the vegetation and held at a constant density. This assumption may be reasonable for commercial ranches which buy and sell animals to keep a stocking rate; however, it does not hold, for example, in traditional pastoral grazing systems, where animal numbers are allowed to fluctuate. The simplest way to add herbivore dynamics to the model is to assume that herbivore density changes as dH ¡dt = ec{V) - d [4]

where d is the herbivore mortality rate and e a parameter describing the growth efficiency of the herbivore population. The resulting models are of the predator-prey type intensively studied in theoretical ecology, for example, by M. L. Rosenzweig and R. H. MacArthur, which may show stable equilibria, limit cycles, and dual stability. However, allowing for herbivore population dynamics tends to stabilize the grazing system in comparison with fixed stocking rates. This is due to the fact that deterioration of the vegetation is likely to lead to a decrease of the herbivore population size due to overexploitation of the resource. When grazing pressure is relaxed, vegetation recovers before herbivores are able to recover.

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