Analysis of a tritrophic food chain model with fear effect incorporating prey refuge

Abstract

In this work, we have introduced a tritrophic food-chain model where consumer hunt for prey with Holling type-III functional response. The birth rate of the prey population has been reduced due to the fear of predation, i.e., a fear effect is considered in the prey population. Moreover, a fraction of the prey is available to the consumer for consumption and this has been done by incorporation of prey refuge term. The predation between consumer and predator follows Beddington-DeAngelis response. Boundedness and positivity of the system prove that the proposed model is well-posed. Also, there are some parametric restrictions under which the system is permanent. Routh-Hurwitz criterion shows the local stability conditions of the equilibrium points and on the other hand Lyapunov LaSalle theorem guarantees that the locally stable equilibrium points are globally stable. Also, Matlab validates the analytical results with the help of diagrams. The occurrence of transcritical bifurcations have been shown and conditions for the existence of a limit cycle in the system through Hopf bifurcation also have been stated. Both the analytical and numerical results suggest that a certain amount of fear can make the system steady. It is also noted that the prey refuge has both stabilizing and destabilizing effect on the system.