In the present paper, we investigate the impact of fear in a tri-trophic food chain model. We propose a three-species food chain model, where the growth rate of middle predator is reduced due to the cost of fear of top predator, and the growth rate of prey is suppressed due to the cost of fear of middle predator. Mathematical properties such as equilibrium analysis, stability analysis, bifurcation analysis and persistence have been investigated. We also describe the global stability analysis of the equilibrium points. Our numerical simulations reveal that cost of fear in basal prey may exhibit bistability by producing unstable limit cycles, however, fear in middle predator can replace unstable limit cycles by a stable limit cycle or a stable interior equilibrium. We observe that fear can stabilize the system from chaos to stable focus through the period-halving phenomenon. We conclude that chaotic dynamics can be controlled by the fear factors. We apply basic tools of nonlinear dynamics such as Poincaré section and maximum Lyapunov exponent to identify the chaotic behavior of the system.
Existence of limit cycles in a tritrophic food chain model with Holling functional responses of type II and III