Regional optimal harvesting controls of a spatiotemporal prey-predator three species fishery model

Abstract

In this paper, we propose another extension of the basic prey-predators model. We incorporate the spatial behavior of the fish population and a term of regional control to provide a realistic description of the prey effects of two predators. We present a study of regional optimal control strategies of a spatiotemporal prey-predator model. Based on an existing model, we add the Laplacian to describe the spatial mobility of its individual. Our main objective is to characterize two harvesting strategies of control, which allow users to increase prey density and decrease the density of predators and super predators to maintain a differential chain system and ensure sustainability. Firstly, by applying the semigroup theory, we investigate the existence of the solution and estimations of the unique strong global solution for the controlled system. Secondly, we prove the existence of a pair of controls and characterize the controls in terms of state and assistant functions according to Pontryagin’s maximum principle. Finally, some numerical simulations for several cases to verify the theoretical analysis are obtained. Also, the results show the effectiveness of the controls if the harvesting strategies of the regional controls are applied simultaneously.