Abstract
The present paper describes a prey-predator type fishery model with two predators in competition. The aim of the paper is to maximize the net economic revenue earn from the fishery through implementing the sustainable properties of the fishery to keep the ecological balance. The existence of the steady states and the stability of the interior equilibrium point is studied using Routh Hurwitz criterion. The problem of determining the fishing effort that maximizes the net economic revenue of each fisherman results in a Generalized Nash Equilibrium Problem. More precisely, we are interested in equilibrium of mathematical game given by the situation where all fishermen try to optimize their strategies according to the strategies of all other fishermen. The importance of marine reserve is analyzed through the obtained results of the numerical simulations of proposed model system. The results depict that reserves will be most effective when the coefficient of catchability decreases.
Highlights
In the present study, we analyse a prey predator model, with the existence of one prey- and two predators populations
We propose to study a bioeconomic model of three fish populations
What is the influence of the catchability coefficients on the net economic revenue? We take as a case of study two fishermen who catch the three fish populations having the following characteristics k1 = 20, k2 = 10, k3 = 5, r1 = 5, r2 = 4, r3 = 3, α = 0.9, β = 0.7, α = 0.5, β = 0.4, δ1 = 0.2, δ2 = 0.3, p1 = 10, p2 = 12, p3 = 20, c1 = 0.01, c2 = 8.10−3, q1 = 0.99, q2 = 0.5, q3 = 0.8
Summary
We analyse a prey predator model, with the existence of one prey- and two predators populations. The differential system is based on the Lotka-Volterra schema, considering a logistic growth for each population. This work is an attempt to study the effect of intrinsic growth rates changes in the net economic rents of several fishermen exploiting the three marine populations. To achieve this objective, we have to solve the problem of maximization of fishermen’s net economic rents by using the generalized Nash equilibrium problem and linear complementarity problem. Our study indicates that balance between harvesting and biodiversity should be maintained for better ecosystem management
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