Dynamics of a modified Leslie-Gower model with double Allee effects

Bifurcation analysis of a class of modified Leslie-Gower model with Allee effects in both predator and prey species is given in detail. We show the existence of a heteroclinic separatrix that divides the dynamics of the predator and prey populations and considers the Hopf bifurcation around the interior positive equilibrium. We show that when there are two interior equilibria, the smaller equilibrium is always a saddle, and the larger equilibrium can be either an attractor or a repeller surrounded by a limit cycle. Combining mathematical analysis and numerical simulation, we show that the double Allee effects greatly alter the outcome of the survival of both species.