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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; moc.liamg@ineyatuof

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.

1. 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 open-access 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.

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.

2. 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.

2.1. The Mathematical Model and Hypotheses

The evolution of the biomass of the first species is given by the following mathematical equation:where is the biomass of population ; is the intrinsic growth rate of species ; is the carrying capacity for species ; is the coefficient of competition between species and species ; and is the coefficient of competition between species and species .

The evolution of the biomass of the second population is given by the following mathematical equation:where is the biomass of population ; is the intrinsic growth rate of species ; is the carrying capacity for species ; is the coefficient of competition between species and species ; and is the coefficient of competition between species and species .

The evolution of the biomass of the third species is given by the following mathematical equation:where is the biomass of population ; is the intrinsic growth rate ; is the carrying capacity for the species of species ; is the coefficient of competition between species and species ; and is the coefficient of competition between species and species .

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 be the solution of system (5). Then all the solutions of the system (5) are nonnegative. To demonstrate that, we must recall that by [12] the system of equation is a positive system if and only if In our case, for ,  , we have By the same, for ,  , we have Also for ,  , we have Therefore, all the solutions of system (5) are nonnegative.

Theorem 1. All the solutions of system (5) which start in are uniformly bounded.

Proof. We define the function Therefore, the time derivative along a solution of (5) is For each , 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 Therefore we find a with Using the theory of differential inequality [13], we obtain which, upon letting , yields
Then, we have where is the region in which all the solutions of system of (5) that start in are confined.

2.2. The Steady States of the System

The steady states of the system of (5) are obtained by solving the system of equations

This system of equations has eight solutions,  , ,, where , where , where and , where

The system of (16) has several solutions, but only one of them can give the coexistence of the biomass of the three species; this solution is the point

2.3. The Stability of the Steady States

The variational matrix of system (5) is where

Proposition 2. The point is unstable.

Proof. The variational matrix of system (5) at the steady state is The eigenvalues of are then, the point is unstable.

Let ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,   in appropriate units. Figure 1 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values ,  ,  . By Figure 1 we find that the steady state point is unstable, and more precisely this point tends to the point

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

Proposition 3. The point is unstable if the conditions of existence given by (4) hold; if not, it is stable.

Proof. The variational matrix of system (5) at the steady state is The eigenvalues of are if then, the point is unstable; if not, it is stable.

Let ,  , ,  ,  ,  ,  ,  , ,  ,  ,   in appropriate units. Figure 2 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values ,  ,  . By Figure 2 we can see that the steady state point is unstable, and more precisely this point tends to the point too.

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

Proposition 4. The point is unstable if the conditions of existence given by (4) hold; if not, it is stable.

Proof. The variational matrix of system (5) at the steady state is The eigenvalues of are if therefore, the point is unstable; if not, it is stable.

Let ,  , ,  ,  ,  ,  ,  , ,  ,  ,   in appropriate units. Figure 3 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values ,  ,  . By Figure 3 we can see that the steady state point is also unstable and tends to the point

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

Proposition 5. The point is unstable if the conditions of existence given by (4) hold; if not, it is stable.

Proof. The variational matrix of system (5) at the steady state is The eigenvalues of are if then, the point is unstable; if not, it is stable.

Let ,  , ,  ,  ,  ,  ,  ,  ,  ,  ,   in appropriate units. Figure 4 indicates the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values ,  ,  . Following Figure 4 we can see that the steady state point is unstable and also tends to the point

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

Proposition 6. The point is unstable.

Proof. The variational matrix of system (5) at the steady state is The eigenvalues of are where If then, ; if not, then Therefore, the point is unstable in all cases.

Let ,  , ,  ,  ,  ,  ,  ,  ,  ,  ,   in appropriate units. Figure 5 represents the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values ,  ,  . Following Figure 5 we can deduce that the steady state point is unstable and also tends to the point

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

Proposition 7. The point is unstable.

Proof. The variational matrix of system (5) at the steady state is The eigenvalues of are where If then, ; if not, then ; therefore, point is unstable.

Let ,  , ,  ,  ,  ,  ,  ,  ,  ,  ,   in appropriate units. Figure 6 indicates the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values ,  . Following Figure 6 we can deduce that the steady state point is unstable and also tends to the point

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

Proposition 8. The point is unstable.

Proof. The variational matrix of system (5) at the steady state is The eigenvalues of are where If then, ; if not, then . Therefore, point is unstable.

Let ,  , ,  ,  ,  ,  ,  ,  ,  ,  ,   in appropriate units. Figure 7 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values ,  . By Figure 7 we can conclude that the steady state point is unstable and also tends to point

Figure 7: Dynamical behaviors and phase space trajectories of the three marine species.

Theorem 9. The point 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 is where Using the fact that by (16) we have then The characteristic polynomial of the variational matrix is where we have ,  . In fact,(i),(ii),(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 is locally asymptotically stable.

Let ,  , ,  ,  ,  ,  ,  ,  ,  ,  ,   in appropriate units. Figure 8 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values ,  . By Figure 8 one can see that the steady state point is locally asymptotically stable.

Figure 8: 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 , and according to the phase space trajectories given by Figures 17 we can confirm that the steady state point is a global attractor.

3. 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 the total catches of species by all fishermen; on the other hand, we denote by the total fishing effort dedicated to species by all fishermen, and we denote by 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 and is the vector noted by or and is defined by The scalar product is noted by The division of the vector and the not null vector (i.e., , ) is the vector noted by and is defined by The product of the vector and the matrix is noted by and is defined by Now we give the expression of biomass as a function of fishing effort.

The biomasses at biological equilibrium are the solutions of the system The solutions of this system are given by

where or in matrix form , where , , and

3.1. 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 , where and are given positive parameters for all . Under these more realistic assumptions the total revenue of the fisherman is so, where .

3.2. 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 .

3.3. Expression of the Profit

The profit of each fisherman is equal to total revenue minus total cost ; it is represented by the following function:

3.4. Constraints of the Model

The biological model has a meaning if and only if the biomass of all the marine species are strictly positive, then we have In other words, for the fisherman ,

3.5. Nash Equilibrium Problem

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 : and the second fisherman must solve the problem :

The point is called Nash equilibrium point if and only if is a solution of problem for given, and is solution of problem for given.

The essential conditions of Karush-Kuhn-Tucker applied to the problem confirm that if is a solution of the problem then there exist constants ,  , and such that