Dynamics and Optimal Harvesting for Fishery Models with Reserved Areas

Abstract

This paper analyzes the dynamic behavior of a fishery model described by differential algebraic equations. Two patches, namely free fishing area and protected area, are included in the model. The migration of fish is symmetrical, i.e., the fish can migrate between the two patches. It is observed that a singularity-induced bifurcation occurs when the economic benefit of harvesting changes. When the economic benefit is positive, a state feedback controller is added to stabilize the system. Some examples and numerical simulations are presented to verify the theoretical results. In addition, harvesting of prey populations is used as a control measure to obtain the maximum economic benefits and ecological sustainability. The optimal solution is derived by using Pontryagin’s maximum principle. Through extensive numerical simulations, it is shown that the optimal solution is capable of achieving ecosystem sustainability.