Stability analysis of a delayed predator-prey model with cosner-type functional response

Abstract

We consider a predator-prey model with a Cosner-type functional response. We incorporate time delay in the predation process. Delay played an important role in the population dynamics. The analysis of positive solution in the delayed model is challenging. We will prove the positivity and boundedness of solutions of both the models. Apart from trivial and boundary equilibrium under some conditions, the system consists of either no interior equilibrium, a unique interior equilibrium, or two distinct equilibria between boundary equilibria. We discuss different dynamic behaviors due to variation of time delay. Based on the parameter conditions, the stable co-existing equilibrium of the non-delayed model remains stable for increasing time delay. For some other parameter restrictions the stable equilibrium may experience instability through a Hopf bifurcation at a critical delay threshold.There does not exists any delay induced stability switching phenomena in this system.