Optimal control of a phytoplankton-zooplankton spatiotemporal discrete bioeconomic model

Recently, some planktonic organisms have been put to use in biotechnological applications, and their usefulness has been discovered in the development of alternative and healthy foods, natural medicines, and cosmetics. Therefore, the management of plankton production is a major challenge for the development of aquaculture. In order to achieve this goal, chlorophyll a, a pigment present in all photosynthetic organisms, is generally and historically used as an estimator of the biomass of planktonic organisms. In this work, we propose a bioeconomic spatiotemporal discrete model in a multi-fishing zone to describe the predation interaction between phytoplankton and zooplankton (Crustacean) organisms by taking into consideration the harvesting activity. To guarantee the survival of two organisms, we consider two harvesting control strategies. The existence of optimal controls and their characterization are proved by using the discrete version of Pontryagin's maximum principle. Based on the concentration of chlorophyll a in the maritime zones of Morocco, we control and compare the biomass of the planktonic organisms in two situations (without and with control). As a major result, we found that after controlling the exploitation of planktonic organisms, their biomasses achieve a level that can ensure their sustainability. The achieved outcomes in the numerical simulations are given by using the forward-backward sweep method (FBSM).