Neimark–Sacker Bifurcation of a Discrete-Time Predator–Prey Model with Prey Refuge Effect

Abstract

In this paper, we deduce a predator–prey model with discrete time in the interior of R+2 using a new discrete method to study its local dynamics and Neimark–Sacker bifurcation. Compared with continuous models, discrete ones have many unique properties that help to understand the changing patterns of biological populations from a completely new perspective. The existence and stability of the three equilibria are analyzed, and the formation conditions of Neimark–Sacker bifurcation around the unique positive equilibrium point are established using the center manifold theorem and bifurcation theory. An attracting closed invariant curve appears, which corresponds to the periodic oscillations between predators and prey over a long period of time. Finally, some numerical simulations and their biological meanings are given to reveal the complex dynamical behavior.