Impacts of strong Allee effect and hunting cooperation for a Leslie-Gower predator-prey system

Abstract

In this paper, we propose a Leslie-Gower predator-prey system with strong Allee effect in prey and hunting cooperation among predators. To discuss the impacts of Allee effect and hunting cooperation, we choose the severity of Allee effect and the cooperative hunting coefficient as the main control parameters. First, using the topology equivalent method, type of singular point (0,0) is obtained. Moreover, the basin of attraction of (0,0) is studied. Then, the stability of all nonnegative equilibrium points, including type of degenerate equilibrium point, is discussed. Based on Sotomayor’s theorem, the existence of saddle-node bifurcation is derived. To determine the stability of limit cycles arising from the Hopf bifurcation, the first Lyapunov number is calculated. Meanwhile, by the rigorous mathematical proofs, we obtain that the bifurcating limit cycle is stable if the hunting cooperation coefficient and the severity of Allee effect are sufficiently small. Finally, numerical simulations are presented to validate the theoretical results and further explore the influences of Allee effect and hunting cooperation. These results show that strong Allee effect and hunting cooperation have significant impacts on dynamical behaviors of the system.