Modeling Allee Effect in the Leslie-Gower Predator–Prey System Incorporating a Prey Refuge

This paper considers a Leslie-Gower predator–prey system with Allee effect and prey refuge. By considering the prey refuge constant as a parameter, we analyze the stability of the equilibria in the system, and find that there are abundant dynamic behaviors. It is shown that the model can undergo a sequence of bifurcations including saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of codimension two or three as the parameters vary. Moreover, the model underdoes a degenerate Hopf bifurcation of codimension two and has two limit cycles, where the inner one is stable and the outer one is unstable. Through some numerical simulations, the occurrence of Bogdanov–Takens bifurcation and Hopf bifurcation of codimension two are confirmed.