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

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 equilibriumpoint thatmaximizes 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.


Introduction
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 long-term 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 [1,2].Clark and Munro [1] demonstrated that, with the aid of optimal control theory, fisheries economics can without difficulty be cast in a capital-theoretic framework yielding results that are both general and readily comprehensible.Chaudhuri [3] discussed the problem of combined harvesting of two competing fish species, each of which obeys the law of logistic growth; it is shown that the openaccess fishery may possess a bioeconomic equilibrium which drives one species to extinction.In this context, Chaudhuri [4] considered the problem of dynamic optimization of the exploitation policy connected with the combined harvesting of two competing fish species, each of which obeys the logistic growth law.Models on the combined harvesting of a two-species prey-predator fishery have been discussed by Chaudhuri and Ray [5].Kar and Chaudhuri [6] studied the problem of harvesting two competing species in the presence of a predator species which feeds on both the competing species; a combined harvesting effort is devoted to the exploitation of the first two (prey) species while the third (predator) species is not harvested.Mchich et al. [7] proposed a specific stock-effort dynamic model; the stock corresponds to two fish populations growing and moving between two fishing zones, on which they are harvested by two different fleets; the effort represents the number of fishing vessels of the two fleets which operate on the two fishing zones; the bioeconomic model is a set of four ordinary differential equations governing the stocks and the fishing efforts in the two fishing areas; fish migration, as well as vessels displacements, between the two zones is assumed to take place at a faster time scale than the variation of the stocks and the changes of fleets sizes, respectively; the vessels movements between the two fishing areas are assumed to be stock dependent, that is, the larger the stock density is in a zone, the more the vessels tend to remain in it.
International Journal of Differential Equations Many mathematical models have been developed to describe the dynamics of fisheries; we can refer, for example, to El Foutayeni et al. [8] who in their work have built a bioeconomic equilibrium model for several fishermen who catch two fish species; in this work, the authors have showed that the problem of determining the equilibrium point that maximizes the profit of each fisherman is solved by using linear complementarity problem.El Foutayeni et al. [9] have also defined a bioeconomic equilibrium model for "" fishermen who catch three species; these species compete with each other for space or food; the natural growth of each species is modeled using a logistic law; the objective of their work is to calculate the fishing effort that maximizes the profit of each fisherman at biological equilibrium by using the generalized Nash equilibrium problem.
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 [10,11] have presented a bioeconomic model of fish populations taking into consideration the fact that the prices of fish populations vary according to the quantity harvested.But in these articles they assumed the existence of a single fisherman.
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 2, we give a description of the biological model of fish populations; we will define the mathematical model and study the stability of the equilibrium of our system.In Section 3, we give the bioeconomic model of the fish populations taking into consideration the fact that the prices of fish populations vary according to the quantity harvested; in this section we prove that the resolution of bioeconomic equilibrium model of the three fish populations is equivalent to solving a Nash equilibrium problem and then we show that the latter problem is equivalent to a linear complementarity problem, then we prove that the obtained problem (LCP) admits a unique solution that represents the Nash equilibrium of our problem.Some numerical simulations are given in Section 4 to illustrate the results.Finally, in Section 5 we give a conclusion.

The Biological Model of Fish Populations
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 Routh-Hurwitz stability criterion.

The Mathematical Model and
Hypotheses.The evolution of the biomass of the first species is given by the following mathematical equation: where  1 () is the biomass of population 1;  1 is the intrinsic growth rate of species 1;  1 is the carrying capacity for species 1;  12 is the coefficient of competition between species 2 and species 1; and  13 is the coefficient of competition between species 3 and species 1.
The evolution of the biomass of the second population is given by the following mathematical equation: where  2 () is the biomass of population 2;  2 is the intrinsic growth rate of species 2;  2 is the carrying capacity for species 2;  21 is the coefficient of competition between species 1 and species 2; and  23 is the coefficient of competition between species 3 and species 2.
The evolution of the biomass of the third species is given by the following mathematical equation: where  3 () is the biomass of population 3;  3 is the intrinsic growth rate 3;  3 is the carrying capacity for the species of species 3;  31 is the coefficient of competition between species 1 and species 3; and  32 is the coefficient of competition between species 2 and species 3.
Theorem 1.All the solutions of system (5) which start in R 3 + are uniformly bounded.
Proof.We define the function Therefore, the time derivative along a solution of ( 5) is For each  > 0, we have We can easily show that Then Therefore, we can deduce that So the right-hand side is positive; therefore it is bounded for all ( 1 ,  2 ,  3 ) ∈ R 3 + .Therefore we find a  > 0 with / +  < .Using the theory of differential inequality [13], we obtain which, upon letting  → ∞, yields 0 ≤  ≤ /.Then, we have where  is the region in which all the solutions of system of (5) that start in R 3 + are confined.

The Stability of the Steady States.
The variational matrix of system ( 5) is where Proposition 2. The point  1 (0, 0, 0) is unstable.
Proof.The variational matrix of system (5) at the steady state  4 (0, 0,  3 ) is The eigenvalues of  4 are then, the point  4 (0, 0,  3 ) is unstable; if not, it is stable.
) is locally asymptotically stable.
Proof.We proof this theorem by using Routh-Hurwitz stability criterion.
The variational matrix of system (5) in the steady state where Using the fact that by (16) we have then The characteristic polynomial of the variational matrix is  where (51) we have   > 0, ∀ = 0, 1, 2, 3.In fact, (i)  0 = 1 > 0, (iii) using the fact that by (4) we have so (iv) From ( 4) we deduce that Then, using the Routh-Hurwitz stability criterion we conclude that the steady state point  8 ( * 1 ,  * 2 ,  * 3 ) is locally asymptotically stable.

Bioeconomic Model of Fishery
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 where   is the catches of fish population  by the fisherman ,   is the fishing effort to exploit a fish population  by the fisherman , and   is the catchability coefficient of fish population , the model for the evolution of fish populations is given by the following mathematical system of equations: On one hand, we denote by   =  1 +  2 the total catches of species  by all fishermen; on the other hand, we denote by   =  1 +  2 the total fishing effort dedicated to species  by all fishermen, and we denote by  () = ( 1 ,  2 ,  3 )  the vector fishing effort which must be provided by the fisherman  to catch the three species.
In what follows of this paper, the product of two vectors  ∈ R 3 and  ∈ R 3 is the vector noted by  or  and is defined by The scalar product is noted by   .The division of the vector  ∈ R 3 and the not null vector  ∈ R 3 (i.e.,   ̸ = 0, ∀ = 1, 2, 3) is the vector noted by / and is defined by International Journal of Differential Equations 13 The product of the vector  ∈ R 3 and the matrix  ∈ R 3×3 is noted by  and is defined by Now we give the expression of biomass as a function of fishing effort.

Expression of the Total Revenue.
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  exploited by the fisherman  is given by   =   /  +  0 , where   and  0 are given positive parameters for all  = 1, 2, 3.Under these more realistic assumptions the total revenue of the fisherman  is where  0 = diag( 0 ).

Expression of the Total Effort Cost.
In accordance with many standard fisheries models, we consider that expression of the total effort cost is where  () is the constant cost per unit of harvesting and  ()  is the total effort of the fisherman .

Expression of the Profit.
The profit of each fisherman   is equal to total revenue (TR)  minus total cost (TC)  ; it is represented by the following function: (67)

Proposition 10. The matrix
is -matrix.
Proof.We have   < 0 for all  = 1, 2, 3 and Δ > 0 so, if we note by (  ) =1,...,9 the submatrix of , we obtain Then, the matrix  is -matrix and therefore the linear complementarity problem LCP(, ) admits one and only one solution.
The unique solution of LCP(, ) represents the Nash equilibrium point of our problem and it is given by where Then, the fishing effort that maximizes the profit of the first fisherman for catching the first species is International Journal of Differential Equations the fishing effort that maximizes the profit of the first fisherman for catching the second species is the fishing effort that maximizes the profit of the first fisherman for catching the third species is The fishing effort that maximizes the profit of the second fisherman for catching the first species is the fishing effort that maximizes the profit of the second fisherman for catching the second species is the fishing effort that maximizes the profit of the second fisherman for catching the third species is

Numerical Simulations and Discussion of the Results
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 (5) as shown in Table 1.Let us consider the economic parameters such as that shown in Table 2.
Using the parameters cited in Tables 1 and 2, thereafter we will see how changes in the minimum prices can affect effort fishing, catches, and profit.3, 4, and 5 we will discover how changes in the price can affect the fishing effort, catches, and profit.According to Tables 3, 4, and 5, one can remark that an increase in the price leads to an increase in fishing effort, catches, and profit.But it is clear that when the price level increases significantly, that is, when it varies in a large amplitude interval, the fishing effort and the catches increase by varying in an interval of small amplitude.More precisely, when the price is between 1 and 1090, the fishing effort varies between 16,51 and 17,68, and the catches vary between 234,4 and 246,7.This is justified by the need for conservation of marine species even if the price increases significantly.
From Table 5 one can see that the level of profit increases, which allows fishermen to have highest returns through more reasonable catches, taking into account the conservation of biodiversity.
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.

Conclusions
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 Routh-Hurwitz criterion.

Figure 1 :
Figure 1: Dynamical behaviors and phase space trajectories of the three marine species.

Figure 2 :
Figure 2: Dynamical behaviors and phase space trajectories of the three marine species.

Figure 3 :
Figure 3: Dynamical behaviors and phase space trajectories of the three marine species.

Figure 4 :
Figure 4: Dynamical behaviors and phase space trajectories of the three marine species.

Figure 5 :
Figure 5: Dynamical behaviors and phase space trajectories of the three marine species.

Figure 6 :
Figure 6: Dynamical behaviors and phase space trajectories of the three marine species.

Figure 8 :
Figure 8: Dynamical behaviors and phase space trajectories of the three marine species.
one can see that the steady state point  8 is locally asymptotically stable.More precisely, beginning with different initial values we can confirm that the three marine species tend to point  8 , and according to the phase space trajectories given by International Journal of Differential Equations

Table 1 :
Characteristics of the three marine species.

Table 2 :
Economic parameters of the model.

Table 3 :
The influence of the price on the fishing effort.

Table 4 :
The influence of the price on the catches.
4.1.The Influence of the Price on the Fishing Effort, Catches, and Profit.By Tables

Table 5 :
The influence of the price on the profits.