Stability of fixed points for a discrete grazer-predator model and optimal controls for the environmental sustainability and bioeconomy

In this paper, we devise a discrete-time trophic-halieutic model in the form of three discrete equations which describe grazer-predator dynamics in the presence of a variable vegetation biomass. Further, we study the stability of the fixed points of the proposed system, and seek two optimal control approaches that are associated to the environmental sustainability and bioeconomic cases and aim to minimize two fishing efforts presented as discrete control functions applied to grazers and predators respectively. The first optimization approach aims to find an optimal control strategy which focuses more on the harvest of only one type of fishes rather than overfishing of both grazers and predators populations, while the second optimization approach aims to maximize the profits of fishermen. The two sought optimal fishing efforts are characterized based on a discrete version of Pontryagin’s maximum principle, while the two obtained two-point boundary value problems, are resolved based on discrete progressive-regressive iterative schemes which converge following a test related to the forward-backward sweep computational method.