Dynamic Behaviours and Bifurcation Analysis of a Filippov Ecosystem with Fear Effect

Evidences of the fear effect of the prey are well documented, which can greatly affect the dynamics of the predator-prey system. In this study, considering that the fear effect of the prey is triggered on as the density of the predator reaches and exceeds a threshold value, we develop a Filippov system of predator-prey model with the fear effect. In addition, we also include a modify factor of the growth rate of the prey when they adopt the antipredator behaviours due to the fear effect. We initially analyze the dynamics of the two subsystems, including the existence and stability of the equilibria. Utilizing the theory of the Filippov system, we discuss the sliding dynamics, i.e., the existence of sliding region and sliding equilibria. By choosing the threshold as the bifurcation parameter, we investigate the bifurcations near the regular equilibria. The solution curve has three cases: crossing the threshold curve, sliding on the threshold curve, and approaching the pseudoequilibrium. Finally, we numerically verified the existence of the global sliding bifurcation near the regular equilibrium and also the touching bifurcation.