Dynamic Analysis of Multi-factor Influence on a Holling Type II Predator-Prey Model

This paper concerns a Holling type II predator-prey system. We pay particular attention to the multi-factor influence including the Allee effect, fear effect, prey refuge and delay. The existence and stability of the equilibria of the system are investigated. Taking the prey refuge as a bifurcating parameter, a threshold condition is given for the local stability of the system without delay and show that the system may occur a supercritical Hopf bifurcation. Taking the delay as a bifurcating parameter, the delayed system undergoes a Hopf bifurcation at the positive equilibrium. The direction of Hopf bifurcation and the stability of bifurcating periodic solution are investigated by the center manifold theorem and normal form theory. We show that the delay, Allee effect, fear effect and prey refuge can enrich the dynamic behaviors. Our mathematical analysis shows that the influence of fear effect and Allee effect is negative, while the impact of the prey refuge is positive. Moreover, it shows that the delay can switch the stability of the system. Examples with their numerical simulations are given to illustrate our theoretical results.