Global Hopf bifurcation of a delayed diffusive predator-prey model with Michaelis-Menten-type prey harvesting

This paper is concerned with a delayed diffusive predator–prey model with Michaelis–Menten-type prey harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, transcritical saddle-node, Hopf bifurcations, which can cause the behavior mutations of the system, such as the change of the stability, the change of the number of equilibria, and the emergence of the periodic orbits. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations near the positive steady state are given by the normal form theory and center manifold theorem. Meanwhile, the approximate expressions of periodic orbits are given. Furthermore, we derive the sufficient condition for global existence of the spatial homogeneous periodic orbits. Finally, some numerical simulations are carried out to support our analytical results.