Stability and Hopf bifurcation analysis on a delayed Leslie-Gower predator–prey system incorporating a prey refuge

In this paper, a modified Leslie-Gower predator–prey system with time delays is investigated, where the time delays are regarded as bifurcation parameters. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is considered. Moreover, we show that Hopf bifurcations occur when time delay crosses some critical values. By deriving the equation describing the flow on the center manifold, we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. In addition, we also try on the global existence of periodic solutions by using the global Hopf bifurcation result of Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838.] for functional differential equations. Numerical simulations are carried out to illustrate the theoretical results and they show that the time delays in the system under consideration can destroy the stability of the system.