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International Journal of Differential Equations
Volume 2017 (2017), Article ID 3157294, 17 pages
https://doi.org/10.1155/2017/3157294
Research Article

Optimization of the Two Fishermen’s Profits Exploiting Three Competing Species Where Prices Depend on Harvest

Analysis, Modeling and Simulation Laboratory, Hassan II University, Casablanca, Morocco

Correspondence should be addressed to Youssef El Foutayeni

Received 21 August 2017; Accepted 13 November 2017; Published 26 December 2017

Academic Editor: Abid A. Lashari

Copyright © 2017 Imane Agmour et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Bioeconomic modeling of the exploitation of biological resources such as fisheries has gained importance in recent years. In this work we propose to define and study a bioeconomic equilibrium model for two fishermen who catch three species taking into consideration the fact that the prices of fish populations vary according to the quantity harvested; these species compete with each other for space or food; the natural growth of each species is modeled using a logistic law. The main purpose of this work is to define the fishing effort that maximizes the profit of each fisherman, but all of them have to respect two constraints: the first one is the sustainable management of the resources and the second one is the preservation of the biodiversity. The existence of the steady states and their stability are studied using eigenvalue analysis. The problem of determining the equilibrium point that maximizes the profit of each fisherman leads to Nash equilibrium problem; to solve this problem we transform it into a linear complementarity problem (LCP); then we prove that the obtained problem (LCP) admits a unique solution that represents the Nash equilibrium point of our problem. We close our paper with some numerical simulations.