Global Hopf Bifurcation of a Diffusive Modified Leslie–Gower Predator–Prey Model with Delay and Michaelis–Menten Type Prey Harvesting

In this paper, we investigate the global Hopf bifurcation of a diffusive modified Leslie–Gower predator–prey model with delay and Michaelis–Menten type prey harvesting. First, we obtain the stability of positive steady state and the existence of local Hopf bifurcation under certain conditions. Second, we get the permanence of the system by using the comparison theorem. Moreover, by constructing a suitable Lyapunov function, we derive sufficient conditions for the global attractivity of the unique positive steady state for the system without delay. Then, the global existence of positive periodic solutions is established by using the global Hopf bifurcation result of Wu. Finally, the results are verified by numerical simulation.