The Impact of Allee Effect on a Leslie–Gower Predator–Prey Model with Hunting Cooperation

Abstract

A Leslie–Gower predator–prey model with Allee effect on prey and hunting cooperation on predator is considered. We show the solution of model is positive and ultimately upper bounded, and prove the origin is an attractor by applying the blow-up method. The model has at most two positive equilibria, one is always a hyperbolic saddle and the other is a weak focus of multiplicity at least two. Moreover, we confirm that the degenerate equilibrium can be a cusp of codimension at most 3. A series of bifurcations can occur, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Selecting Allee effect and hunting cooperation as bifurcation parameters, we investigate the influence of Allee effect and hunting cooperation on the dynamics of the model. Finally, through numerical simulations, we illustrate the Allee effects (or hunting cooperation) is detrimental to the coexistence of two species when the strength of the Allee parameter (or hunting cooperation) increases.