Stability and Bifurcation in a Leslie–Gower Predator–Prey Model with Allee Effect

Abstract

In this paper, we consider a Leslie–Gower predator–prey model with Allee effect on the prey and a linear functional response. Here the Allee effect impacts the birth rate of the prey, which is different from the common multiplicative and additive Allee effects. The model is well-posed, that is, all solutions are bounded. Applying the blow-up method indirectly, we prove that the origin which is not an equilibrium of the system is an attractor. Then we study the existence and stability of equilibria, which indicate that the system undergoes bifurcations. With the help of Sotomayor’s theorem, we show the occurrence of saddle-node bifurcation. Moreover, there is degenerate Hopf bifurcation of codimension at least three. By choosing two (three) parameters of the system as bifurcation parameters and calculating a versal unfolding near the cusp, we demonstrate that the system undergoes Bogdanov–Takens bifurcation of codimension two (three). These theoretical results are supported with numerical simulations.