Trophic and halieutic dynamics of grazer–predator fishes: harvesting optimal control policies for the environmental sustainability and bioeconomic cases

Abstract

Interactions between trophic and halieutic populations, have often been ignored in most mathematical models that are interested in prey–predator dynamics. However, the presence of vegetation in hydraulic environments, is important, and it has a direct impact on the life cycle of many fishes. In fact, aquatic populations are essential for providing oxygen, food and also shelter for grazers. For this, we suggest a mathematical and optimization approach, in an attempt to discuss the possibility of finding effective harvesting control strategies that aim to optimize the fishing efforts without affecting the trophic-halieutic populations and compromising with the interests of fishermen. The model is in the form of three ordinary differential equations, and which describes dynamics of multi-species of a fishery that includes grazer–predator fishes and aquatic plants, during fishing periods. We study the stability of the proposed differential system, and we suggest harvesting optimal control approaches for the environmental sustainability and bioeconomic cases after the introduction of two control functions in the model and which represent the efforts of fishing in grazer and predator populations respectively. The two optimal controls are characterized in the first case, based on Pontryagin’s maximum principle, and we seek their analytical formulations in the second case, when they are singular functions. Hence, these two optimal control approaches lead to two two-point boundary value problems which are resolve numerically based on the forward-backward sweep method with the incorporation of iterative Runge-Kutta fourth order progressive-regressive schemes.