Abstract
We develop a mathematical model of extinction and coexistence in a generic predator–prey ecosystem composed of two herbivores in asymmetrical competition and a predator of both. With the aim of representing the satiety of predators when preys are overabundant, we introduce for the predation behavior a dependence on prey abundance. Specifically, predation is modeled as growing proportionally to the presence of herbivores at low densities, and saturating when the total metapopulation of prey is sufficiently large. The model predicts the existence of different regimes depending on the parameters considered: survival of a single species, coexistence of two species and extinction of the third one, and coexistence of the three species. But more interestingly, in some regions of parameters space the solutions oscillate in time, both as a transient phenomenon and as persistent oscillations of constant amplitude. The phenomenon is not present for the more idealized linear predation model, suggesting that it can be the source of real ecosystems oscillations.
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