A density dependent delayed predator–prey model with Beddington–DeAngelis type function response incorporating a prey refuge

Abstract

This paper describes a predator–prey model incorporating a prey refuge. The feeding rate of consumers (predators) per consumer (i.e. functional response) is considered to be of Beddington–DeAngelis type. The Beddington–DeAngelis functional response is similar to the Holling-type II functional response but contains an extra term describing mutual interference by predators. We investigate the role of prey refuge and degree of mutual interference among predators in the dynamics of system. The dynamics of the system is discussed mainly from the point of view of permanence and stability. We obtain conditions that affect the persistence of the system. Local and global asymptotic stability of various equilibrium solutions is explored to understand the dynamics of the model system. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional. The dynamical behaviour of the delayed system is further analyzed through incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. The analytical results found in the paper are illustrated with the help of numerical examples.