Optimal Harvesting of a Holling Type 2 Gordon–Schaefer Bioeconomic Model

Techniques of bioeconomic modeling of the fisheries, developed from the end of the 50s, call primarily upon the neoclassic economic theory. Their use allowed significant theoretical projections in the field of adjustment of fishing, but their operational virtues remain weak, in particular, because the assumptions which underlie them are far away from reality. We propose in this article an improved model of Gordon–Smith for which the function of rejuvenation of the biomass of the stock of the halieutic resources is a logistic function and the intensity of the captures compared to the efforts of fishing is related to the type Holling 2 fascinating of account the saturation of the machine of fishing and guaranteeing nonextinction of the halieutic resources. To determine the bioeconomic conditions of the system allowing a viable management of the activity of fishing, we must control the maximum value m1 of the intensity of the captures necessary to generate a positive total J profit for the fisherman and to avoid overfishing. The significant results of this study are on the one hand, the boundedness of the solutions and the stability of the equilibrium points, and on the other hand, the optimal control of the intensity of the captures m1 that maximizes net profit J of the fisherman.