Dynamic behavior of a class of predator–prey model with two time delays

Abstract

In this paper, three modified Leslie–Gower predator–prey models with two time delays are considered based on the original Leslie–Gower predator–prey model. Taking the time delay as a bifurcation parameter, when Hopf bifurcation occurs, the critical value corresponding to time delay is obtained. By using normal form theory and central manifold argument, the direction of Hopf bifurcation and the stability of bifurcation periodic solution can be determined. The time delay affects the stability of the positive equilibria. When the time delay exceeds the critical value, the positive equilibria change from stable to unstable and bifurcate out a set of periodic solutions. Finally, numerical simulation is performed to support theoretical analysis.