Complex Dynamic Behaviors of a Modified Discrete Leslie–Gower Predator–Prey System with Fear Effect on Prey Species

Abstract

A discrete modified Leslie–Gower prey-predator model considering the effect of fear on prey species is proposed and studied in this paper. First, we discuss the existence of equilibria and the local stability of the model. Second, we use the iterative method and comparison principle to obtain the set of conditions which ensures the global attractivity of positive equilibrium point. The results show that prey and predator can coexist stably when the intrinsic growth rates of both prey and predator are maintained within a certain range. Then, we study the global attractivity of the boundary equilibrium point. Our results suggest that when the intrinsic rate of prey is small enough or the fear factor is large enough, the prey will tend to go extinct, while the predator can survive stably due to the availability of other food sources. Subsequently, we discuss flip bifurcation, transcritical bifurcation at the equilibria of the system, by using the center manifold theorem and bifurcation theory. We find that system changes from chaotic state to four-period orbit, two-period orbit, stable state, and finally prey species will be driven to extinction, while predator species survive in a stable state for enough large birth rate of prey species with the increasing of fear effect. Finally, we verify the feasibility of the main results by numerical simulations, and discuss the influence of the fear effect. The results show that the fear effect within a certain range can enhance the stability of the system.