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.
Overfishing leads to resource destruction, that is why there is an increasing need for the bioeconomic modeling tool that evaluates the biological and economic effects of different harvesting strategies directed at extracting the longterm maximum sustainable production while avoiding the risk of recruitment overfishing. The techniques and issues associated with the bioeconomic modeling for the exploitation of marine resources have been discussed in detail by Clark and Munro [
Many mathematical models have been developed to describe the dynamics of fisheries; we can refer, for example, to El Foutayeni et al. [
Most bioeconomic models do not take into account the variational of the price of fish population. Usually, the existing models consider that the prices of the fish populations are constants. In this context, El Foutayeni and Khaladi [
This paper is situated in this general context; in this work we present a bioeconomic model for three species which compete with each other for space or food and each of which obeys the law logistic growth. These species are caught by two fishermen. We will assume that the price of the fish population increases with decreasing harvest and the price of the fish population decreases with the increase of the harvest, but the minimum price is equal to a fixed positive constant. The aim of this paper consists in determining the fishing effort strategy adopted by each fisherman to maximize its income under two assumptions; the first one is the sustainable management of the resources, and the second one is the preservation of the biodiversity.
The paper is structured as follows. In Section
The aim of this section is to define a biological model of three marine species that compete with each other for space or food and whose natural growth of each is obtained by means of a logistic law. We study the existence of the steady states and their stability using eigenvalue analysis and RouthHurwitz stability criterion.
The evolution of the biomass of the first species is given by the following mathematical equation:
The evolution of the biomass of the second population is given by the following mathematical equation:
The evolution of the biomass of the third species is given by the following mathematical equation:
It is interesting to note that to assure the existence of the three species and their stability we should assume that
The evolution of the biomass of fish populations is modeled by the following equations:
Let
All the solutions of system (
We define the function
Therefore, the time derivative along a solution of (
Then, we have
The steady states of the system of (
This system of equations has eight solutions
and
The system of (
The variational matrix of system (
The point
The variational matrix of system (
Let
Dynamical behaviors and phase space trajectories of the three marine species.
The point
The variational matrix of system (
Let
Dynamical behaviors and phase space trajectories of the three marine species.
The point
The variational matrix of system (
Let
Dynamical behaviors and phase space trajectories of the three marine species.
The point
The variational matrix of system (
Let
Dynamical behaviors and phase space trajectories of the three marine species.
The point
The variational matrix of system (
Let
Dynamical behaviors and phase space trajectories of the three marine species.
The point
The variational matrix of system (
Let
Dynamical behaviors and phase space trajectories of the three marine species.
The point
The variational matrix of system (
Let
Dynamical behaviors and phase space trajectories of the three marine species.
The point
We proof this theorem by using RouthHurwitz stability criterion.
The variational matrix of system (
using the fact that by (
so
Let
Dynamical behaviors and phase space trajectories of the three marine species.
More precisely, beginning with different initial values we can confirm that the three marine species tend to point
The main purpose of this section is to define and study a bioeconomic equilibrium model for two fishermen who catch three fish populations.
More specifically, this bioeconomic model includes three parts: a biological part connecting the catch to the biomass stock, an exploitation part connecting the catch to the fishing effort, and an economic part connecting the fishing effort to the profit.
So, introducing the fishing by reducing the rate of fish population growth by the amount
On one hand, we denote by
In what follows of this paper, the product of two vectors
The biomasses at biological equilibrium are the solutions of the system
where
It is interesting to note that there exist many different variables that affect the fish price; in this paper, we will consider that the price of the fish population depends on the quantity harvested; specifically we assumed that the price of the marine species increases with the decreasing harvest and the price of the marine species decreases with the increase of the harvest, but the minimum price is equal to a fixed positive constant. More precisely, the price of marine species
In accordance with many standard fisheries models, we consider that expression of the total effort cost is
The profit of each fisherman
The biological model has a meaning if and only if the biomass of all the marine species are strictly positive, then we have
The problem of determining the fishing effort that maximizes the profit of each fisherman leads to a Nash equilibrium problem. By definition a Nash equilibrium exists when there is no unilateral profitable deviation from any of the fishermen involved. In other words, no fisherman would take a different action as long as every other fisherman remains the same. This problem can be translated into the following two mathematical problems.
The first fisherman must solve the problem
The point
The essential conditions of KarushKuhnTucker applied to the problem
In the same way, the essential conditions of KarushKuhnTucker applied to the problem
To maintain the biodiversity of species, it is natural to assume that all biomasses remain strictly positive; that is,
As the scalar product of
We denote
The following proposition confirms that
The matrix
We have
Then, the matrix
The unique solution of
In this section, we take as case of study two fishermen who catch three fish species competing with each other for space or food. In order to assure the existence and stability of the locally asymptotically stable state of the three fish populations, we consider the parameters of the model system (
Characteristics of the three marine species.
Species 1  Species 2  Species 3 













Let us consider the economic parameters such as that shown in Table
Economic parameters of the model.
Species 1  Species 2  Species 3 
















Using the parameters cited in Tables
By Tables
The influence of the price on the fishing effort.






1  2  3  17,0451  16,5151 
11  17  23  17,5943  17,5314 
16  27  48  17,6383  17,6073 
31  47  78  17,6552  17,6363 
51  70  108  17,6627  17,6492 
84  101  273  17,6734  17,6677 
106  133  327  17,6749  17,6702 
340  378  427  17,6769  17,6736 
574  577  606  17,6783  17,6760 
808  811  914  17,6794  17,6778 
917  956  981  17,6795  17,6781 
1000  1079  1090  17,6797  17,6784 
The influence of the price on the catches.






1  2  3  245,0957  234,4651 
11  17  23  246,4411  245,9382 
16  27  48  246,5725  246,2429 
31  47  78  246,6298  246,3781 
51  70  108  246,6552  246,5718 
84  101  273  246,6865  246,5974 
106  133  327  246,6923  246,6334 
340  378  427  246,7020  246,6334 
574  577  606  246,7063  246,6582 
808  811  914  246,7095  246,6775 
917  956  981  246,7101  246,6804 
1000  1079  1090  246,7107  246,6839 
The influence of the price on the profits.






1  2  3  282  269 
11  17  23  2959  2942 
16  27  48  4513  4500 
31  47  78  8479  8465 
51  70  108  13584  13567 
84  101  273  23130  23118 
106  133  327  29111  29099 
340  378  427  85598  85572 
574  577  606  142017  141987 
808  811  914  200558  200531 
917  956  981  227721  227692 
1000  1079  1090  249295  249266 
According to Tables
From Table
These results allow us to deduce that our model is pertinent since it allows us to determine the fishing effort that maximizes the profit of each fisherman without being obliged to make more catches that lead to the overexploitation of these marine species.
Let us add that when the price tends to infinity, the fishing efforts of the two fishermen are equal and they do not exceed 18, as well as the catches which do not exceed 250; contrariwise the profit is always increasing thanks to the increase of the price. Then we can deduce the effect of the price change on the fishing effort, catches, and profit.
It is very interesting to note that if the price tends to infinity and the fishing effort is superior to 18, then the catches and the profit decrease.
In this paper, we have developed a bioeconomic model for three species catches by two fishermen. In one hand, we have assumed that the evolution of these species is described by a density dependent model taking into account the competition between the species which compete with each other for space or food. The natural growth of each species is modeled using a logistic law. On the other hand, we have assumed that the prices of these species vary according to the quantity harvested. In this work we have calculated fishing effort that maximizes the income of each fisherman at biological equilibrium by using the Nash equilibrium problem. The existence of the steady states and their stability are studied using eigenvalue analysis and RouthHurwitz criterion.
The authors declare that there are no conflicts of interest.