Qualitative and Bifurcation Analysis in a Leslie-Gower Model with Allee Effect

In this paper, we consider a Leslie-Gower model with weak Allee effect in the prey. By analysing the dynamics near the origin, we show that both predator and prey will tend to extinction if the intensity of Allee effect is strong enough. Meanwhile, we provide some sufficient conditions on the global asymptotic stability of the unique positive equilibrium. In addition, Allee effect can change the stability of positive equilibrium, which leads to the occurrence of a supercritical Hopf bifurcation and one stable limit cycle. It is interesting to note that there exists at least one limit cycle around the unstable positive equilibrium. In particular, sufficient conditions for the existence of a unique stable limit cycle have been presented. Numerical simulations are conducted to validate the main results.