Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System

Abstract

The present paper examines a predator-prey fishery system by taking into account the toxin released by the fish which can lead to polluted system. Both predator and prey fish species obey the logistic population growth with their respective environmental carrying capacities. In the proposed model, both fish species produce toxins that contribute to mutual infection which is detrimental to each other. Different harvesting efforts are applied on the predator and prey fish populations, respectively. The equilibria existed in the model are studied together with the local stability properties. We consider the threshold conditions which trigger the bifurcation that occurred in the steady states. The global stability properties of coexistence equilibrium are studied by constructing an appropriate Lyapunov function. Bendixson-Dulac criterion is applied to rule out the existence of limit cycle in the system. From the bifurcation analysis, the dynamical behaviors of the system are observed as well as the persistence and extinction properties. It is shown that harvesting parameters are most likely to drive a fish population towards extinction compared to toxicant parameters which are less influential. Regions of optimal harvesting strategies were found to guarantee the persistence of both fisheries. Finally, the existence of a bionomic equilibrium solution has been examined with three possible cases.