STRONG ALLEE EFFECTS IN PREY FOR SOME MODELS OF PREDATOR-PREY WITH A CLASS OF FUNCTIONAL RESPONSES

The main goal of this paper is to study a predator-prey model with general functional response. The model contains the impact of strong Allee effects upon the predator-prey interaction when the prey population is subject to strong Allee effects. We illustrate that the existence of a separatrix curve in the phase plane, determined by the stable manifold of the equilibrium point associated to the strong Allee effect, implying that the solutions are highly sensitive to the initial conditions. Orbits starting at one side of this separatrix curve have the equilibrium point (0, 0) - both the species become extinct - as their ω − limit set, whereas orbits starting at the other side will approach to one of the following attractors: a stable periodic solution, a unique stable interior equilibrium point or the equilibrium point (1, 0) in which the predators disappear and the preys attain their carrying capacity. We prove also that the existence of a stable periodic solution associated to occurrence of Hopf bifurcation when the unique interior equilibrium point loses its stability. In order to complete the investigation of this model, by applying the particular conditions, we show that both populations become extinct. Finally, the analysis of some predator-prey models with strong Allee effects and different functional responses are showed via numerical simulations.