An Exploration of the Effects of Periodic Top Predator Interference and Hunting on a Predator-Prey System

We introduce two models for the investigation of damages that periodic passages of migrating animals may have on established ecosystems of a smaller scale. In the first one, the damage is only of “physical” nature, in which the population at the highest trophic layer ravages the territory where the other interacting species live, thereby reducing their survival capabilities but not obtaining any benefit out of this. In the second one, it is assumed that the periodic damage comes from a moving top predator, periodically crossing the habitat of the lower trophic level populations, chasing both of them and therefore getting a reward from this hunt. The underlying model is assumed to be of minimalist predator-prey type, in order to better elucidate the effect of the action of the top predator. While the pure interference model does not seem to produce relevant effects other than the presence in a two-dimensional subspace of neutrally stable trajectories, a behavior that is forbidden in the underlying basic predator-prey model, the superpredation of a migrating population in suitable conditions induces a stable equilibrium, which may undergo a Hopf bifurcation with specific parameter choices. This elementary investigation elucidates therefore a possible invasion mechanism where the migrating population spots an attracting ecosystem in which thriving is possible and settles in it, substantially altering it and affecting the native populations.