Hopf Bifurcation Analysis of a Predator–Prey Model with Prey Refuge and Fear Effect Under Non-diffusion and Diffusion

In this paper, we propose a predator–prey model with prey refuge and fear effect under non-diffusion and diffusion. For the non-diffusion ODE model, we first analyze the existence and stability of equilibria. Then, the existence of transcritical bifurcation, Hopf bifurcation and limit cycle is discussed, respectively. We find that when the cost of minimum fear η is taken as the bifurcation parameter, it not only influence the occurrence of Hopf bifurcation but also alters its direction. For diffusion predator–prey model under homogeneous Neumann boundary conditions, we observe that the Turing instability does not occur, but the Hopf bifurcation will manifest near the interior equilibrium. By considering η as the bifurcation parameter, the direction and stability of spatially homogeneous periodic orbits are established. At last, the validity of the theoretical analysis are verified by a series of numerical simulations. The results indicate that prey refuge and fear effect play an key role in the stability of populations.