Dynamics of a predator-prey model with fear effects and gestation delays

Recent studies have shown that, in addition to direct predation, fear of predators alters the physiological behavior of prey. Based on this fact, this paper investigates a three-species food chain based on ratio-dependent and Beddington-DeAngelis type functional responses, which incorporates fear effects and two gestation delays. The positivity, boundedness and existence of equilibrium points of the system are investigated, and the local stability behavior of the equilibrium points and the occurrence of Hopf-bifurcation when the time lag parameters exceed the critical values are studied by analyzing the corresponding characteristic equations. The main results show that Hopf-bifurcation occurs when the time delay parameters attain the thresholds. Finally, numerical simulations are performed to verify our main results.