  We develop a mathematical model of extinction and coexistence in a generic
predator-prey ecosystem composed of two herbivores in asymmetrical competition
and a hunter exerting a predatory pressure on both species. With the aim of
representing the satiety of hunters when preys are overabundant, we introduce
for the predation behavior a dependence on preys density. Specifically,
predation is modeled as growing proportionally to the presence of herbivores at
low density, and saturating when the total population 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 parameters space the solutions oscillate in
time, both as a transient phenomena 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.
