Optimal tax policy of a one‐predator–two‐prey system with a marine protected area

Abstract

This paper is devoted to handle a dynamic one‐predator–two‐prey model. In order to protect fish population from over exploitation, we assume that marine protected area (MPA) is established and the fisherman only harvest the prey in the unreserved area, the predator consumes the prey in both the MPA and the unreserved area. And a tax is imposed in the process of harvesting. To begin with, boundeness of the system is discussed. Following this, we studied the existence of the possible equilibrium along with their local stability for both of the unexploited system (8) and exploited system (5). After that, we analyzed the global stability of the positive equilibrium of the exploited system and how the tax τ affects the positive equilibrium. Then, the optimal tax policy is obtained by using the Pontryagin's maximum principle. Finally, some numerical simulations are given to support the analytical findings.