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.

Automated summary

This paper investigates a Holling type II predator-prey system with prey refuge, Allee effect, fear effect, and time delay. The existence and stability of the system's equilibria are studied. The system undergoes a Hopf bifurcation at the positive equilibrium when the delay exceeds certain critical values. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solution are analyzed using the center manifold theorem and normal form theory. The results show that the fear effect and Allee effect have negative impacts, while the prey refuge has a positive impact. The birth rate plays a significant role in the equilibrium's stability. Numerical simulations are provided to validate the main results.