Dynamics of a Delayed Predator-Prey Model with Prey Refuge, Allee Effect and Fear Effect

In this paper, we consider a Holling type II predator-prey system with prey refuge, Allee effect, fear effect and time delay. The existence and stability of the equilibria of the system are investigated. Under the variation of the delay as a parameter, the system experiences a Hopf bifurcation at the positive equilibrium when the delay crosses some critical values. We also analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution by the center manifold theorem and normal form theory. We show that the influence of fear effect and Allee effect is negative, while the impact of the prey refuge is positive. In particular, the birth rate plays an important role in the stability of the equilibria. Examples with associated numerical simulations are provided to prove our main results.