Dynamical behaviours of prey-predator fishery model with two reserved area for prey in the presence of toxicity and response function Holling type IV

Abstract

This paper proposes a model with two type of preys and one predator of fishery with Holling type IV function response. The effect of harvesting was incorporated to both populations and thoroughly analysed. We study the dynamics as the prey-predator system of fishing in two fishing zones: a free zone and a reserved zone. The equilibrium points are calculated and the local and global stability conditions of the system are obtained. The local stability conditions were obtained by the Routh-Hurwitz criterion. In addition, the global stability of the coexistence equilibrium point is proved by defining an appropriate Lyapunov function. The optimal harvesting policy is discussed by using the Pontryagin’s Maximal Principle. Finally, numerical simulations are carried to verify the analytical results.