Complex dynamics of sexually reproductive generalist predator and gestation delay in a food chain model: double Hopf-bifurcation to Chaos

Abstract

Food uptake ability of higher trophic level species are more complicated and interesting due to their choice and availability of food and consequently their growth and also the effect of top predator interference on the dynamics of a tritrophic food chain model. In this paper, we consider the general framework for calculating the stability of equilibria, Hopf and double Hopf-bifurcation of a prey–predator system with Holling type IV and Beddington–DeAngelis type functional responses of intermediate and top predator respectively. The top predator is of generalist type and its growth is considered due to sexual reproduction. Firstly, we have shown feasibility and boundedness of the solutions of the considered model system, behavior of equilibria and the existence of Hopf-bifurcation. Conditions are determined under which the coexistence equilibrium point remains globally asymptotically stable. We identify the critical values of delay parameter for which stability switches and nature of the Hopf-bifurcation by using normal form theory and center manifold theorem. We investigate the occurence of double Hopf bifurcation at positive equilibrium point when we choose appropriate measure of the predator’s immunity or tolerance of the prey. Furthermore, some dynamic behaviors, such as stability switches, chaos, bifurcation and double Hopf-bifurcation scenarios are observed using numerical simulations. The chaotic behavior of the system is clarified by standard numerical tests.