Global analysis of a delayed density dependent predator-prey model with Crowley-Martin functional response

In this paper, we present and study a two-dimensional continuous time dynamical system modelling a predator-prey with discrete delay incorporating Crowley-Martin functional response. Crowley-Martin functional response is similar to the Beddington-DeAngelis functional response but contains an extra term describing mutual interference by predator at high prey density. We consider the permanence, non-permanence, local asymptotic stability behaviour of various equilibrium points and global asymptotic stability of positive equilibrium to understand the dynamics of both delayed and non-delayed model systems. Global asymptotic stability is discussed constructing a suitable Lyapunov functional. We also show that increasing delays may cause bifurcations into periodic solutions. It is found that fluctuations in population levels arising due to gestation delay of predator can be prevented under certain parametric conditions. The direction and stability of Hopf bifurcation is also discussed by using normal form method and center manifold theory. In the end, some numerical simulations have been performed to substantiate our analytical findings.