Fear and delay effects on a food chain system with two kinds of different functional responses

Abstract

For food chain system with three populations, direct predation is the basic interaction between species. Different species often have different predation functional responses, so a food chain system with Holling-II response for middle predator and Beddinton–DeAngelis response for top predator is proposed. Apart from direct predation, predator population can significantly impact the survival of prey population by inducing the prey’s fear, but the impact often possesses a time delay. This paper is concentrated to explore how the fear and time delay affect the system stability and the species persistence. By use of Lyapunov functional method and bifurcation theory, the positiveness and boundedness of solutions, local and global behavior of species, the system stability around the equilibrium states and various kinds of bifurcation are investigated. Numerically, some simulations are carried out to validate the main findings and the critical values of the bifurcation parameters of fear and conversion rate are obtained. It is observed that fear and delay can not only stabilize, but also destabilize the system, which depends on the magnitude of the fear and delay. The system varies from unstable to stable due to the continuous increase of the prey’s fear by middle predator. Small fear induced by top predator or small delay of the prey’s fear can stabilize the system, while they are sufficiently large, the system stability is to be destroyed. Simultaneously, the conversion rate can also change the stability and even make the species to be extinct. Some rich dynamics like multiple stabilities and various types of bistability behaviors are also exhibited, which results in the convergence of the species from one stable equilibrium to another.