Stability Switching in a Cooperative Prey-Predator Model with Transcritical and Hopf-bifurcations

Abstract

In nature organisms attempt to adopt new techniques to diminish the possibilities of being falling prey. Interspecies cooperation is one of these approaches which two different types of prey can use against a common predator. Inspired by this, we purpose a prey-predator model having two prey who cooperate with each other while interacting with a predator. For making the model more general and realistic, the interactions between prey and predator are handled through general Holling type-IV and Crowley-Martin functional responses. For well-posedness of the proposed model, firstly, its boundedness is investigated which is followed by the vigorous proofs for the existence of equilibrium points, their stability analysis, evaluation of conditions for occurrence of transcritical and Hopf-bifurcations. Numerically, we observe that as the inverse measure of predator’s immunity from first prey and coefficient of cooperation from first prey to second prey crosses some respective critical values, there is occurrence of Hopf-bifurcation.Transcritical bifurcation is also depicted numerically for the intrinsic growth rate of first prey and the death rate of predator species. Several phase portraits, bifurcation diagrams are drawn to support our analytical findings. We also endorse the attribute of bistability, and basins of attraction for both stable equilibrium points are also drawn.