Explosive tritrophic food chain models with interference: A comparative study

Abstract

Depending upon the choice of food, availability of resource and growth structure, food uptake process of higher trophic level species are significantly complicated and gives interesting dynamical impacts on community food chain. The effect of top predator interference on the dynamics of a tritrophic food chain model is the principal goal to study general framework for three species food chain models in which intermediate and top predators are specialist and generalist types respectively where the top predator grows by sexual reproduction. This concept is used to design five different models in this paper by five different combinations of functional responses of specialist intermediate predator (food uptake process follows either prey dependent, Holling type III/IV or prey-predator dependent, Beddington-DeAngelis (BD)-functional response) and sexually reproductive generalist top predator (food uptake follows strictly prey-predator dependent, BD/Crowley–Martin (CM) functional response) respectively assuming the fact that the growth of intermediate predator is mediated by gestation delay. We establish the stability of each equilibrium point of the non-delayed counterpart of each model and investigate the existence of Hopf-bifurcation for the coexistence equilibrium point of all delayed systems. We also show that both the delayed and non delayed models can blow-up in finite time under sufficient conditions on the initial data. Furthermore, some dynamic behaviors of the systems are clarified by some numerical tests. The numerical simulations show that the gestation delay can act as a damping mechanism and prevent blow-up in certain critical range of gestation period. Ecological implications of this phenomenon are discussed.