The effect of the fear factor on the dynamics of an eco-epidemiological system with standard incidence rate

In order to protect endangered prey, ecologists suggest introducing parasites into predators which have achieved the expected goal in practice. Then how to explain the inherent mechanism and validate the effectiveness of this approach theoretically? In response to this question, we propose an eco-epidemiological system with the standard incidence rate and the anti-predator behavior in this paper, where the predator population is infected by parasites. We show the existence and local stability of equilibria for the system, and verify the occurrence of Hopf bifurcation. Theoretical and numerical results suggest that the fear effect reduces the density of the predator population but has no effect on the density of prey population. In addition, the cost of fear may not only break the stability of the equilibrium of the system, but also induce the equilibrium to change from unstable to stable. Based on the theoretical analysis, we confirm that introducing parasites into the predator population is an effective method to protect endangered prey.