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

Authors: Imane Agmour; Meriem Bentounsi; Naceur Achtaich; Youssef El Foutayeni
Journal: International Journal of Differential Equations
 (2017)

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## 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.


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## Body

## 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 “n” 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 Section2, 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:(1)x˙1t=r1x1t1-x1tK1-c12x1tx2t-c13x1tx3t,where x1(t) is the biomass of population 1; r1 is the intrinsic growth rate of species 1; K1 is the carrying capacity for species 1; c12 is the coefficient of competition between species 2 and species 1; and c13 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:(2)x˙2t=r2x2t1-x2tK2-c21x1tx2t-c23x2tx3t,where x2(t) is the biomass of population 2; r2 is the intrinsic growth rate of species 2; K2 is the carrying capacity for species 2; c21 is the coefficient of competition between species 1 and species 2; and c23 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:(3)x˙3t=r3x3t1-x3tK3-c31x1tx3t-c32x2tx3t,where x3(t) is the biomass of population 3; r3 is the intrinsic growth rate 3; K3 is the carrying capacity for the species of species 3; c31 is the coefficient of competition between species 1 and species 3; and c32 is the coefficient of competition between species 2 and species 3.It is interesting to note that to assure the existence of the three species and their stability we should assume that(4)ri>cijKj,∀i,j=1,2,3,withi≠j.The evolution of the biomass of fish populations is modeled by the following equations:(5)x˙1t=r1x1t1-x1tK1-c12x1tx2t-c13x1tx3t,x˙2t=r2x2t1-x2tK2-c21x1tx2t-c23x2tx3t,x˙3t=r3x3t1-x3tK3-c31x1tx3t-c32x2tx3t.Letx(t)=(x1(t),x2(t),x3(t)) 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 (6)x˙=fx1,x2,…,xnwithxt=0=x0is a positive system if and only if (7)x˙i=fix1≥0,…,xi=0,…,xn≥0≥0;∀i∈1⋯n. In our case, for x1=0,  x2,x3≥0, we have dx1/dt=0≥0. By the same, for x2=0,  x1,x3≥0, we have dx2/dt=0≥0. Also for x3=0,  x1,x2≥0, we have dx3/dt=0≥0. Therefore, all the solutions of system (5) are nonnegative.Theorem 1.
All the solutions of system (5) which start in R+3 are uniformly bounded.Proof.
We define the function(8)W=x1+x2+x3.
Therefore, the time derivative along a solution of (5) is (9)dWdt=r1x11-x1K1+r2x21-x2K2+r3x31-x3K3-c12x1x2-c13x1x3-c21x1x2-c23x2x3-c31x1x3-c32x2x3. For each ϑ>0, we have (10)dWdt+ϑW=r1x11-x1K1+r2x21-x2K2+r3x31-x3K3-c12x1x2-c13x1x3-c21x1x2-c23x2x3-c31x1x3-c32x2x3+ϑx1+ϑx2+ϑx3≤r1x11-x1K1+r2x21-x2K2+r3x31-x3K3+ϑx1+ϑx2+ϑx3=x1r11-x1K1+ϑ+x2r21-x2K2+ϑ+x3r31-x3K3+ϑ=x1r1+ϑ-r1K1x12+x2r2+ϑ-r2K2x22+x3r3+ϑ-r3K3x32. We can easily show that (11)-r1K1x12+x1r1+ϑ-K14r1r1+ϑ2≤0,-r2K2x22+x2r2+ϑ-K24r2r2+ϑ2≤0,-r3K3x32+x3r3+ϑ-K34r3r3+ϑ2≤0. Then (12)-r1K1x12+x1r1+ϑ≤K14r1r1+ϑ2,-r2K2x22+x2r2+ϑ≤K24r2r2+ϑ2,-r3K3x32+x3r3+ϑ≤K34r3r3+ϑ2. Therefore, we can deduce that (13)dWdt+ϑW≤K14r1r1+ϑ2+K24r2r2+ϑ2+K34r3r3+ϑ2. So the right-hand side is positive; therefore it is bounded for all (x1,x2,x3)∈R+3. Therefore we find a θ>0 with dW/dt+ϑW<θ. Using the theory of differential inequality [13], we obtain (14)0≤Wx1,x2,x3≤θϑ+Wx10,x20,x30-θϑe-ϑt which, upon letting t→∞, yields 0≤W≤θ/ϑ.
Then, we have(15)B=x1,x2,x3∈R+3:W<θϑ+ε,for anyε>0, where B is the region in which all the solutions of system of (5) that start in R+3 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(16)r1x11-x1K1-c12x1x2-c13x1x3=0,r2x21-x2K2-c21x1x2-c23x2x3=0,r3x31-x3K3-c31x1x3-c32x2x3=0.This system of equations has eight solutionsP1(0,0,0),P2(K1,0,0),  P3(0,K2,0), P4(0,0,K3),P5(x1(5),x2(5),0), where (17)x15=K1r2r1-c12K2r1r2-c12c21K2K1,x25=K2r1r2-c21K1r1r2-c12c21K2K1,P6(x1(6),0,x3(6)), where (18)x16=K1r3r1-c13K3r1r3-c13c31K3K1,x36=K3r1r3-c31K1r1r3-c13c31K3K1,P7(0,x2(7),x3(7)), where (19)x27=K2r3r2-c23K3r3r2-c32c23K2K3,x37=K3r2r3-c32K2r3r2-c32c23K2K3,andP8(x1∗,x2∗,x3∗), where (20)x1∗=K1r1r2r3-r1c23c32K2K3+r3c12c23K2K3-r2r3c12K2-r2r3c13K3+r2c13c32K2K3Δ,x2∗=K2r1r2r3-r2c13c31K1K3+r1c23c31K1K3-r1r3c21K1-r1r3c23K3+r3c13c21K1K3Δ,x3∗=K3r1r2r3-r3c12c21K1K2-r1r2c31K1+r1c21c32K1K2+r2c12c31K1K2-r1r2c32K2Δ,Δ=r1r2r3-r1c23c32K2K3-r2c13c31K1K3-r3c12c21K1K2+c12c23c31K1K2K3+c13c21c32K1K2K3.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 P8(x1∗,x2∗,x3∗).


### 2.3. The Stability of the Steady States

The variational matrix of system (5) is (21)J=J11-c12x1-c13x1-c21x2J22-c23x2-c31x3-c32x3J33, where (22)J11=r11-2K1x1-c12x2-c13x3,J22=r21-2K2x2-c21x1-c23x3,J33=r31-2K3x3-c31x1-c32x2.Proposition 2.
The pointP1(0,0,0) is unstable.Proof.
The variational matrix of system (5) at the steady state P1(0,0,0) is (23)J1=r1000r2000r3.The eigenvalues of J1 are (24)λ1=r1>0,λ2=r2>0,λ3=r3>0,then, the point P1(0,0,0) is unstable.Letr1=2,  r2=1,  r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=0.01,  y(0)=0.01,  z(0)=0.01. By Figure 1 we find that the steady state point P1 is unstable, and more precisely this point tends to the point P8.Figure 1
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 3.
The pointP2(K1,0,0) 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 P2(K1,0,0) is (25)J2=-r1-c12K1-c13K10r2-c21K1000r3-c31K1. The eigenvalues of J2 are (26)λ1=-r1<0,λ2=r2-c21K1,λ3=r3-c31K1if (27)r2>c21K1r3>c31K1, then, the point P2(K1,0,0) is unstable; if not, it is stable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002, c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=70,  y(0)=0.01,  z(0)=0.01. By Figure 2 we can see that the steady state point P2 is unstable, and more precisely this point tends to the point P8 too.Figure 2
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 4.
The pointP3(0,K2,0) 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 P3(0,K2,0) is (28)J3=r1-c12K200-c21K2-r2-c23K200r3-c32K2.The eigenvalues of J3 are (29)λ1=r1-c12K2,λ2=-r2<0,λ3=r3-c32K2 if (30)r1>c12K2,r3>c32K2; therefore, the point P3(0,K2,0) is unstable; if not, it is stable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002, c32=0.001,  K1=70,  K2=50, K3=40  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 x(0)=0.01,  y(0)=50,  z(0)=0.01. By Figure 3 we can see that the steady state point P3 is also unstable and tends to the point P8.Figure 3
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 5.
The pointP4(0,0,K3) 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 P4(0,0,K3) is (31)J4=r1-c13K3000r2-c23K30-c13K3-c32K3-r3. The eigenvalues of J4 are (32)λ1=r1-c13K3,λ2=r2-c23K3,λ3=-r3<0 if (33)r1>c13K3,r2>c23K3, then, the point P4(0,0,K3) is unstable; if not, it is stable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=0.01,  y(0)=0.01,  z(0)=40. Following Figure 4 we can see that the steady state point P3 is unstable and also tends to the point P8.Figure 4
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 6.
The pointP5(x1(5),x2(5),0) is unstable.Proof.
The variational matrix of system (5) at the steady state P5(x1(5),x2(5),0) is (34)J5=-r1K1x15-c12x15-c13x15-c21x25-r2K2x25-c23x2500r3-c31x15-c32x25. The eigenvalues of J5 are (35)λ1=-12K1K2M-N,λ2=-12K1K2M+N,λ3=r3-c31x15-c32x25, where (36)M=r1x15K2+K1r2x25,N=r1x15K2-K1r2x252+4K12K22c21x25c12x15. If (37)r1>c12K2,r2>c21K1 then, λ3>0; if not, then λ1>0. Therefore, the point P5(x1(5),x2(5),0) is unstable in all cases.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=60,  y(0)=28,  z(0)=0.01. Following Figure 5 we can deduce that the steady state point P5 is unstable and also tends to the point P8.Figure 5
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 7.
The pointP6(x1(6),0,x3(6)) is unstable.Proof.
The variational matrix of system (5) at the steady state P6(x1(6),0,x3(6)) is (38)J6=-r1k1x16-c12x16-c13x160r2-c21x16-c23x360-c31x36-c32x36-r3K3x36. The eigenvalues of J6 are (39)λ1=-12K1K3G-L,λ2=r2-c21x16-c23x36,λ3=-12K1K3G+L, where (40)G=r1x16K3+K1r3x36,L=r1x16K3-K1r3x362+4K12K32c31x36c13x16. If (41)r1>c13K3,r3>c31K1 then, λ2>0; if not, then λ1>0; therefore, point P6(x1(6),0,x3(6)) is unstable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=59,y(0)=0.01,  z(0)=38. Following Figure 6 we can deduce that the steady state point P6 is unstable and also tends to the point P8.Figure 6
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 8.
The pointP7(0,x2(7),x3(7)) is unstable.Proof.
The variational matrix of system (5) at the steady state P7(0,x2(7),x3(7)) is (42)J7=r1-c12x27-c13x3700-c21x27-r2K2x27-c23x27-c31x37-c32x37-r3K3x37. The eigenvalues of J7 are (43)λ1=r1-c12x27-c13x37,λ2=-12K2K3R-S,λ3=-12K2K3R+S, where (44)R=r2x27K3+K2r3x37,S=r2x27K3+K2r3x372+4K22K32c23x27c32x37. If (45)r2>c23K3,r3>c32K2 then, λ1>0; if not, then λ2>0. Therefore, point P7(0,x2(7),x3(7)) is unstable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=0.01,y(0)=48,  z(0)=39. By Figure 7 we can conclude that the steady state point P7 is unstable and also tends to point P8.Figure 7
Dynamical behaviors and phase space trajectories of the three marine species.Theorem 9.
The pointP8(x1∗,x2∗,x3∗) 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 P8(x1∗,x2∗,x3∗) is (46)J8=J11-c12x1∗-c13x1∗-c21x2∗J22-c23x2∗-c31x3∗-c32x3∗J33, where (47)J11=r11-2K1x1∗-c12x2∗-c13x3∗,J22=r21-2K2x2∗-c21x1∗-c23x3∗,J33=r31-2K3x3∗-c31x1∗-c32x2∗. Using the fact that by (16) we have (48)r11-2K1x1∗-c12x2∗-c13x3∗=-r1K1x1∗,r21-2K2x2∗-c21x1∗-c23x3∗=-r2K2x2∗,r31-2K3x3∗-c31x1∗-c32x2∗=-r3K3x3∗ then (49)J8=-r1K1x1∗-c12x1∗-c13x1∗-c21x2∗-r2K2x2∗-c23x2∗-c31x3∗-c32x3∗-r3K3x3∗. The characteristic polynomial of the variational matrix is (50)Pλ=a0λ3+a1λ2+a2λ+a3, where (51)a0=1,a1=r1K1x1∗+r2K2x2∗+r3K3x3∗,a2=r1K1x1∗r2K2x2∗+r1K1x1∗r3K3x3∗+r2K2x2∗r3K3x3∗-c23x2∗c32x3∗-c12x1∗c21x2∗-c13x1∗c31x3∗,a3=r1K1x1∗r2K2x2∗r3K3x3∗+c12x1∗c23x2∗c31x3∗+c13x3∗c32x2∗c21x1∗-c12x1∗c21x2∗r3K3x3∗-c23x2∗c32x3∗r1K1x1∗-c13x3∗c31x1∗r2K2x2∗;we have ai>0,  ∀i=0,1,2,3. In fact,(i)
a0=1>0,(ii)
a1=r1/K1x1∗+r2/K2x2∗+r3/K3x3∗>0,(iii)
using the fact that by (4) we have (52)r1r2>c12K2c21K1,r2r3>c23K3c32K2,r1r3>c13K3c31K1so(53)a2=r1K1x1∗r2K2x2∗+r1K1x1∗r3K3x3∗+r2K2x2∗r3K3x3∗-c23x2∗c32x3∗-c12x1∗c21x2∗-c13x1∗c31x3∗>0,a3=r1K1x1∗r2K2x2∗r3K3x3∗+c12x1∗c23x2∗c31x3∗+c13x3∗c32x2∗c21x1∗-c12x1∗c21x2∗r3x3∗K3-c23x2∗c32x3∗r1K1x1∗-c13x3∗c31x1∗r2K2x2∗>0,(iv)
(54)a1a2-a0a3=r1K1x1∗+r2K2x2∗+r3K3x3∗r1K1x1∗r2K2x2∗+r1K1x1∗r3K3x3∗+r2K2x2∗r3K3x3∗-c23x2∗c32x3∗-c12x1∗c21x2∗-c13x1∗c31x3∗-r1K1x1∗r2K2x2∗r3K3x3∗+c12x1∗c23x2∗c31x3∗+c13x3∗c32x2∗c21x1∗-c12x1∗c21x2∗r3x3∗K3-c23x2∗c32x3∗r1K1x1∗-c13x3∗c31x1∗r2K2x2∗=r1K1x1∗r2K2x2∗r3K3x3∗-c12x1∗c23x2∗c31x3∗+r1K1x1∗r2K2x2∗r3K3x3∗-c13x3∗c32x2∗c21x1∗+r12K12x1∗2r2K2x2∗-r1K1x1∗2c12c21x2∗+r22K22x2∗2r3K3x3∗-r2K2x2∗2c32c23x3∗+r12K12x2∗2r3K3x3∗-r1K1x1∗2c13c31x3∗+r32K32x3∗2r1K1x1∗-r3K3x3∗2c13c31x1∗+r22K22x2∗2r1K1x1∗-r2K2x2∗2c12c21x1∗+r32K32x3∗2r2K2x2∗-r3K3x3∗2c23c32x2∗. From (4) we deduce that (55)a1a2-a0a3>0. Then, using the Routh-Hurwitz stability criterion we conclude that the steady state point P8(x1∗,x2∗,x3∗) is locally asymptotically stable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=49,y(0)=30,  z(0)=38. By Figure 8 one can see that the steady state point P8 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 pointP8, and according to the phase space trajectories given by Figures 1–7 we can confirm that the steady state point P8 is a global attractor.


## 2.1. The Mathematical Model and Hypotheses

The evolution of the biomass of the first species is given by the following mathematical equation:(1)x˙1t=r1x1t1-x1tK1-c12x1tx2t-c13x1tx3t,where x1(t) is the biomass of population 1; r1 is the intrinsic growth rate of species 1; K1 is the carrying capacity for species 1; c12 is the coefficient of competition between species 2 and species 1; and c13 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:(2)x˙2t=r2x2t1-x2tK2-c21x1tx2t-c23x2tx3t,where x2(t) is the biomass of population 2; r2 is the intrinsic growth rate of species 2; K2 is the carrying capacity for species 2; c21 is the coefficient of competition between species 1 and species 2; and c23 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:(3)x˙3t=r3x3t1-x3tK3-c31x1tx3t-c32x2tx3t,where x3(t) is the biomass of population 3; r3 is the intrinsic growth rate 3; K3 is the carrying capacity for the species of species 3; c31 is the coefficient of competition between species 1 and species 3; and c32 is the coefficient of competition between species 2 and species 3.It is interesting to note that to assure the existence of the three species and their stability we should assume that(4)ri>cijKj,∀i,j=1,2,3,withi≠j.The evolution of the biomass of fish populations is modeled by the following equations:(5)x˙1t=r1x1t1-x1tK1-c12x1tx2t-c13x1tx3t,x˙2t=r2x2t1-x2tK2-c21x1tx2t-c23x2tx3t,x˙3t=r3x3t1-x3tK3-c31x1tx3t-c32x2tx3t.Letx(t)=(x1(t),x2(t),x3(t)) 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 (6)x˙=fx1,x2,…,xnwithxt=0=x0is a positive system if and only if (7)x˙i=fix1≥0,…,xi=0,…,xn≥0≥0;∀i∈1⋯n. In our case, for x1=0,  x2,x3≥0, we have dx1/dt=0≥0. By the same, for x2=0,  x1,x3≥0, we have dx2/dt=0≥0. Also for x3=0,  x1,x2≥0, we have dx3/dt=0≥0. Therefore, all the solutions of system (5) are nonnegative.Theorem 1.
All the solutions of system (5) which start in R+3 are uniformly bounded.Proof.
We define the function(8)W=x1+x2+x3.
Therefore, the time derivative along a solution of (5) is (9)dWdt=r1x11-x1K1+r2x21-x2K2+r3x31-x3K3-c12x1x2-c13x1x3-c21x1x2-c23x2x3-c31x1x3-c32x2x3. For each ϑ>0, we have (10)dWdt+ϑW=r1x11-x1K1+r2x21-x2K2+r3x31-x3K3-c12x1x2-c13x1x3-c21x1x2-c23x2x3-c31x1x3-c32x2x3+ϑx1+ϑx2+ϑx3≤r1x11-x1K1+r2x21-x2K2+r3x31-x3K3+ϑx1+ϑx2+ϑx3=x1r11-x1K1+ϑ+x2r21-x2K2+ϑ+x3r31-x3K3+ϑ=x1r1+ϑ-r1K1x12+x2r2+ϑ-r2K2x22+x3r3+ϑ-r3K3x32. We can easily show that (11)-r1K1x12+x1r1+ϑ-K14r1r1+ϑ2≤0,-r2K2x22+x2r2+ϑ-K24r2r2+ϑ2≤0,-r3K3x32+x3r3+ϑ-K34r3r3+ϑ2≤0. Then (12)-r1K1x12+x1r1+ϑ≤K14r1r1+ϑ2,-r2K2x22+x2r2+ϑ≤K24r2r2+ϑ2,-r3K3x32+x3r3+ϑ≤K34r3r3+ϑ2. Therefore, we can deduce that (13)dWdt+ϑW≤K14r1r1+ϑ2+K24r2r2+ϑ2+K34r3r3+ϑ2. So the right-hand side is positive; therefore it is bounded for all (x1,x2,x3)∈R+3. Therefore we find a θ>0 with dW/dt+ϑW<θ. Using the theory of differential inequality [13], we obtain (14)0≤Wx1,x2,x3≤θϑ+Wx10,x20,x30-θϑe-ϑt which, upon letting t→∞, yields 0≤W≤θ/ϑ.
Then, we have(15)B=x1,x2,x3∈R+3:W<θϑ+ε,for anyε>0, where B is the region in which all the solutions of system of (5) that start in R+3 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(16)r1x11-x1K1-c12x1x2-c13x1x3=0,r2x21-x2K2-c21x1x2-c23x2x3=0,r3x31-x3K3-c31x1x3-c32x2x3=0.This system of equations has eight solutionsP1(0,0,0),P2(K1,0,0),  P3(0,K2,0), P4(0,0,K3),P5(x1(5),x2(5),0), where (17)x15=K1r2r1-c12K2r1r2-c12c21K2K1,x25=K2r1r2-c21K1r1r2-c12c21K2K1,P6(x1(6),0,x3(6)), where (18)x16=K1r3r1-c13K3r1r3-c13c31K3K1,x36=K3r1r3-c31K1r1r3-c13c31K3K1,P7(0,x2(7),x3(7)), where (19)x27=K2r3r2-c23K3r3r2-c32c23K2K3,x37=K3r2r3-c32K2r3r2-c32c23K2K3,andP8(x1∗,x2∗,x3∗), where (20)x1∗=K1r1r2r3-r1c23c32K2K3+r3c12c23K2K3-r2r3c12K2-r2r3c13K3+r2c13c32K2K3Δ,x2∗=K2r1r2r3-r2c13c31K1K3+r1c23c31K1K3-r1r3c21K1-r1r3c23K3+r3c13c21K1K3Δ,x3∗=K3r1r2r3-r3c12c21K1K2-r1r2c31K1+r1c21c32K1K2+r2c12c31K1K2-r1r2c32K2Δ,Δ=r1r2r3-r1c23c32K2K3-r2c13c31K1K3-r3c12c21K1K2+c12c23c31K1K2K3+c13c21c32K1K2K3.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 P8(x1∗,x2∗,x3∗).


## 2.3. The Stability of the Steady States

The variational matrix of system (5) is (21)J=J11-c12x1-c13x1-c21x2J22-c23x2-c31x3-c32x3J33, where (22)J11=r11-2K1x1-c12x2-c13x3,J22=r21-2K2x2-c21x1-c23x3,J33=r31-2K3x3-c31x1-c32x2.Proposition 2.
The pointP1(0,0,0) is unstable.Proof.
The variational matrix of system (5) at the steady state P1(0,0,0) is (23)J1=r1000r2000r3.The eigenvalues of J1 are (24)λ1=r1>0,λ2=r2>0,λ3=r3>0,then, the point P1(0,0,0) is unstable.Letr1=2,  r2=1,  r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=0.01,  y(0)=0.01,  z(0)=0.01. By Figure 1 we find that the steady state point P1 is unstable, and more precisely this point tends to the point P8.Figure 1
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 3.
The pointP2(K1,0,0) 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 P2(K1,0,0) is (25)J2=-r1-c12K1-c13K10r2-c21K1000r3-c31K1. The eigenvalues of J2 are (26)λ1=-r1<0,λ2=r2-c21K1,λ3=r3-c31K1if (27)r2>c21K1r3>c31K1, then, the point P2(K1,0,0) is unstable; if not, it is stable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002, c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=70,  y(0)=0.01,  z(0)=0.01. By Figure 2 we can see that the steady state point P2 is unstable, and more precisely this point tends to the point P8 too.Figure 2
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 4.
The pointP3(0,K2,0) 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 P3(0,K2,0) is (28)J3=r1-c12K200-c21K2-r2-c23K200r3-c32K2.The eigenvalues of J3 are (29)λ1=r1-c12K2,λ2=-r2<0,λ3=r3-c32K2 if (30)r1>c12K2,r3>c32K2; therefore, the point P3(0,K2,0) is unstable; if not, it is stable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002, c32=0.001,  K1=70,  K2=50, K3=40  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 x(0)=0.01,  y(0)=50,  z(0)=0.01. By Figure 3 we can see that the steady state point P3 is also unstable and tends to the point P8.Figure 3
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 5.
The pointP4(0,0,K3) 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 P4(0,0,K3) is (31)J4=r1-c13K3000r2-c23K30-c13K3-c32K3-r3. The eigenvalues of J4 are (32)λ1=r1-c13K3,λ2=r2-c23K3,λ3=-r3<0 if (33)r1>c13K3,r2>c23K3, then, the point P4(0,0,K3) is unstable; if not, it is stable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=0.01,  y(0)=0.01,  z(0)=40. Following Figure 4 we can see that the steady state point P3 is unstable and also tends to the point P8.Figure 4
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 6.
The pointP5(x1(5),x2(5),0) is unstable.Proof.
The variational matrix of system (5) at the steady state P5(x1(5),x2(5),0) is (34)J5=-r1K1x15-c12x15-c13x15-c21x25-r2K2x25-c23x2500r3-c31x15-c32x25. The eigenvalues of J5 are (35)λ1=-12K1K2M-N,λ2=-12K1K2M+N,λ3=r3-c31x15-c32x25, where (36)M=r1x15K2+K1r2x25,N=r1x15K2-K1r2x252+4K12K22c21x25c12x15. If (37)r1>c12K2,r2>c21K1 then, λ3>0; if not, then λ1>0. Therefore, the point P5(x1(5),x2(5),0) is unstable in all cases.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=60,  y(0)=28,  z(0)=0.01. Following Figure 5 we can deduce that the steady state point P5 is unstable and also tends to the point P8.Figure 5
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 7.
The pointP6(x1(6),0,x3(6)) is unstable.Proof.
The variational matrix of system (5) at the steady state P6(x1(6),0,x3(6)) is (38)J6=-r1k1x16-c12x16-c13x160r2-c21x16-c23x360-c31x36-c32x36-r3K3x36. The eigenvalues of J6 are (39)λ1=-12K1K3G-L,λ2=r2-c21x16-c23x36,λ3=-12K1K3G+L, where (40)G=r1x16K3+K1r3x36,L=r1x16K3-K1r3x362+4K12K32c31x36c13x16. If (41)r1>c13K3,r3>c31K1 then, λ2>0; if not, then λ1>0; therefore, point P6(x1(6),0,x3(6)) is unstable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=59,y(0)=0.01,  z(0)=38. Following Figure 6 we can deduce that the steady state point P6 is unstable and also tends to the point P8.Figure 6
Dynamical behaviors and phase space trajectories of the three marine species.Proposition 8.
The pointP7(0,x2(7),x3(7)) is unstable.Proof.
The variational matrix of system (5) at the steady state P7(0,x2(7),x3(7)) is (42)J7=r1-c12x27-c13x3700-c21x27-r2K2x27-c23x27-c31x37-c32x37-r3K3x37. The eigenvalues of J7 are (43)λ1=r1-c12x27-c13x37,λ2=-12K2K3R-S,λ3=-12K2K3R+S, where (44)R=r2x27K3+K2r3x37,S=r2x27K3+K2r3x372+4K22K32c23x27c32x37. If (45)r2>c23K3,r3>c32K2 then, λ1>0; if not, then λ2>0. Therefore, point P7(0,x2(7),x3(7)) is unstable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=0.01,y(0)=48,  z(0)=39. By Figure 7 we can conclude that the steady state point P7 is unstable and also tends to point P8.Figure 7
Dynamical behaviors and phase space trajectories of the three marine species.Theorem 9.
The pointP8(x1∗,x2∗,x3∗) 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 P8(x1∗,x2∗,x3∗) is (46)J8=J11-c12x1∗-c13x1∗-c21x2∗J22-c23x2∗-c31x3∗-c32x3∗J33, where (47)J11=r11-2K1x1∗-c12x2∗-c13x3∗,J22=r21-2K2x2∗-c21x1∗-c23x3∗,J33=r31-2K3x3∗-c31x1∗-c32x2∗. Using the fact that by (16) we have (48)r11-2K1x1∗-c12x2∗-c13x3∗=-r1K1x1∗,r21-2K2x2∗-c21x1∗-c23x3∗=-r2K2x2∗,r31-2K3x3∗-c31x1∗-c32x2∗=-r3K3x3∗ then (49)J8=-r1K1x1∗-c12x1∗-c13x1∗-c21x2∗-r2K2x2∗-c23x2∗-c31x3∗-c32x3∗-r3K3x3∗. The characteristic polynomial of the variational matrix is (50)Pλ=a0λ3+a1λ2+a2λ+a3, where (51)a0=1,a1=r1K1x1∗+r2K2x2∗+r3K3x3∗,a2=r1K1x1∗r2K2x2∗+r1K1x1∗r3K3x3∗+r2K2x2∗r3K3x3∗-c23x2∗c32x3∗-c12x1∗c21x2∗-c13x1∗c31x3∗,a3=r1K1x1∗r2K2x2∗r3K3x3∗+c12x1∗c23x2∗c31x3∗+c13x3∗c32x2∗c21x1∗-c12x1∗c21x2∗r3K3x3∗-c23x2∗c32x3∗r1K1x1∗-c13x3∗c31x1∗r2K2x2∗;we have ai>0,  ∀i=0,1,2,3. In fact,(i)
a0=1>0,(ii)
a1=r1/K1x1∗+r2/K2x2∗+r3/K3x3∗>0,(iii)
using the fact that by (4) we have (52)r1r2>c12K2c21K1,r2r3>c23K3c32K2,r1r3>c13K3c31K1so(53)a2=r1K1x1∗r2K2x2∗+r1K1x1∗r3K3x3∗+r2K2x2∗r3K3x3∗-c23x2∗c32x3∗-c12x1∗c21x2∗-c13x1∗c31x3∗>0,a3=r1K1x1∗r2K2x2∗r3K3x3∗+c12x1∗c23x2∗c31x3∗+c13x3∗c32x2∗c21x1∗-c12x1∗c21x2∗r3x3∗K3-c23x2∗c32x3∗r1K1x1∗-c13x3∗c31x1∗r2K2x2∗>0,(iv)
(54)a1a2-a0a3=r1K1x1∗+r2K2x2∗+r3K3x3∗r1K1x1∗r2K2x2∗+r1K1x1∗r3K3x3∗+r2K2x2∗r3K3x3∗-c23x2∗c32x3∗-c12x1∗c21x2∗-c13x1∗c31x3∗-r1K1x1∗r2K2x2∗r3K3x3∗+c12x1∗c23x2∗c31x3∗+c13x3∗c32x2∗c21x1∗-c12x1∗c21x2∗r3x3∗K3-c23x2∗c32x3∗r1K1x1∗-c13x3∗c31x1∗r2K2x2∗=r1K1x1∗r2K2x2∗r3K3x3∗-c12x1∗c23x2∗c31x3∗+r1K1x1∗r2K2x2∗r3K3x3∗-c13x3∗c32x2∗c21x1∗+r12K12x1∗2r2K2x2∗-r1K1x1∗2c12c21x2∗+r22K22x2∗2r3K3x3∗-r2K2x2∗2c32c23x3∗+r12K12x2∗2r3K3x3∗-r1K1x1∗2c13c31x3∗+r32K32x3∗2r1K1x1∗-r3K3x3∗2c13c31x1∗+r22K22x2∗2r1K1x1∗-r2K2x2∗2c12c21x1∗+r32K32x3∗2r2K2x2∗-r3K3x3∗2c23c32x2∗. From (4) we deduce that (55)a1a2-a0a3>0. Then, using the Routh-Hurwitz stability criterion we conclude that the steady state point P8(x1∗,x2∗,x3∗) is locally asymptotically stable.Letr1=2,  r2=1, r3=3,  c12=0.009,  c21=0.007,  c13=0.008,  c23=0.001,  c31=0.002,  c32=0.001,  K1=70,  K2=50,  K3=40 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 x(0)=49,y(0)=30,  z(0)=38. By Figure 8 one can see that the steady state point P8 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 pointP8, and according to the phase space trajectories given by Figures 1–7 we can confirm that the steady state point P8 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(56)Hij=qjEijxj, where Hij is the catches of fish population j by the fisherman i, Eij is the fishing effort to exploit a fish population j by the fisherman i, and qj is the catchability coefficient of fish population j, the model for the evolution of fish populations is given by the following mathematical system of equations: (57)x˙1=r1x11-x1K1-c12x1x2-c13x1x3-q1E1x1,x˙2=r2x21-x2K2-c21x1x2-c23x2x3-q2E2x2,x˙3=r3x31-x3K3-c31x1x3-c32x2x3-q3E3x3.On one hand, we denote byHj=H1j+H2j the total catches of species j by all fishermen; on the other hand, we denote by Ej=E1j+E2j the total fishing effort dedicated to species j by all fishermen, and we denote by Ei=(Ei1,Ei2,Ei3)T the vector fishing effort which must be provided by the fisherman i to catch the three species.In what follows of this paper, the product of two vectorsα∈R3 and β∈R3 is the vector noted by αβ or βα and is defined by (58)αβ≔α1y1,α2y2,α3y3T∈R3. The scalar product is noted by αTβ. The division of the vector α∈R3 and the not null vector β∈R3 (i.e., βi≠0, ∀i=1,2,3) is the vector noted by α/β and is defined by (59)αβ≔α1β1,α2β2,α3β3T∈R3. The product of the vector α∈R3 and the matrix A∈R3×3 is noted by αA and is defined by (60)αA≔diagα·A∈R3×3. Now we give the expression of biomass as a function of fishing effort.The biomasses at biological equilibrium are the solutions of the system(61)r11-x1K1=c12x2+c13x3+q1E1,r21-x2K2=c21x1+c23x3+q2E2,r31-x3K3=c31x1+c32x2+q3E3. The solutions of this system are given by (62)x1=a11E1+a12E2+a13E3+x1∗,x2=a21E1+a22E2+a23E3+x2∗,x3=a31E1+a32E2+a33E3+x3∗,where(63)a11=K1c32K2K3c23q1-r3r2q1Δ,a12=K1-c32K2q2c13K3+K2q2c12r3Δ,a13=K1-K2K3c23c12q3+q3r2c13K3Δ,a21=K2-K3c23q1K1c31+K1c21r3q1Δ,a22=K2q2c13K1K3c31-q2r1r3Δ,a23=K2K3c23r1q3-K1c21q3c13K3Δ,a31=K3-q1K1c32K2c21+q1K1r2c31Δ,a32=K3r1c32K2q2-c12K1K2q2c31Δ,a33=K3c12K1K2c21q3-r1r2q3Δ or in matrix form X=-AE+X∗, where A=(-aij)1≤i,j≤3, E=(E1,E2,E3)T, and X∗=(x1∗,x2∗,x3∗)T.


### 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 speciesj exploited by the fisherman i is given by pij=aj/Hij+p0j, where aj and p0j are given positive parameters for all j=1,2,3. Under these more realistic assumptions the total revenue of the fisherman i is (64)TRi=pi1Hi1+pi2Hi2+pi3Hi3=a1Hi1+p01Hi1+a2Hi2+p02Hi2+a3Hi3+p03Hi3=p01Hi1+p02Hi2+p03Hi3+∑k=13ak=p01q1Ei1x1+p02q2Ei2x2+p03q3Ei3x3+∑k=13ak=P0,qEiX+∑k=13ak=P0,qEi-AE+X∗+∑k=13ak=P0,qEi-A∑i=12Ei+X∗+∑k=13ak=Ei,qP0-A∑i=12Ei+X∗+∑k=13ak,so, (65)TRi=Ei,-P0qAEi+Ei,P0qxX∗-P0qAEj+∑k=13ak, where P0=diag(p0).


### 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(66)TCi=ci,Ei, where ci is the constant cost per unit of harvesting and Ei is the total effort of the fisherman i.


### 3.3. Expression of the Profit

The profit of each fishermanπi is equal to total revenue (TR)i minus total cost (TC)i; it is represented by the following function: (67)πiEi=TRi-TCi=Ei,-P0qAEi+Ei,P0qX∗-P0qAEj+∑k=13ak-ci,Ei=Ei,-P0qAEi+Ei,P0qX∗-ci-P0qAEj+∑k=13ak.


### 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(68)X=-AE+X∗≥X0>0. In other words, for the fisherman i, (69)AEi≤-AEj+X∗.


### 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 problemP1: P1maxπ1E1=E1,-P0qAE1+P0qX∗-c1-P0qAE2+∑k=13aksubject toAE1≤-AE2+X∗E1≥0E2given and the second fisherman must solve the problem P2: P2maxπ2E2=E2,-P0qAE2+P0qX∗-c2-P0qAE1+∑k=13aksubject toAE2≤-AE1+X∗E2≥0E1given.The pointE1,E2 is called Nash equilibrium point if and only if E1 is a solution of problem P1 for E2 given, and E2 is solution of problem P2 for E1 given.The essential conditions of Karush-Kuhn-Tucker applied to the problemP1 confirm that if E1 is a solution of the problem P1 then there exist constants u1∈R+3,  v1∈R+3, and λ1∈R+3 such that KKT12P0qAE1+c1-P0qX∗+P0qAE2-u1+ATλ1=0,AE1+v1=-AE2+X∗,u1,E1=λ1,v1=0.In the same way, the essential conditions of Karush-Kuhn-Tucker applied to the problemP2 confirm that if E2 is a solution of the problem P2 then there exist constants u2∈R+3,v2∈R+3, and λ2∈R+3 such that KKT22P0qAE2+c2-P0qX∗+P0qAE1-u2+ATλ2=0,AE2+v2=-AE1+X∗,u2,E2=λ2,v2=0; we remark that KKT1 and KKT2 lead to the following system: (70)u1=2P0qAE1+c1-P0qX∗+P0qAE2+ATλ1,u2=2P0qAE2+c2-P0qX∗+P0qAE1+ATλ2,v1=-AE1-AE2+X∗,v2=-AE1-AE2+X∗,ui,Ei=λi,vi=0∀i=1,2,Ei,ui,λi,vi≥0∀i=1,2.To maintain the biodiversity of species, it is natural to assume that all biomasses remain strictly positive; that is,xj>0 for all j=1,2,3; therefore v1=v2>0.As the scalar product of(λi)i=1,2 and (vi)i=1,2 is zero, λi=0 for all i=1,2. So, we denote v=v1=v2. Then we have the following expressions: (71)u1=2P0qAE1+P0qAE2+c1-P0qX∗,u2=2P0qAE2+P0qAE1+c2-P0qX∗,v=-AE1-AE2+X∗,ui,Ei=0∀i=1,2,Ei,ui,v≥0∀i=1,2. Thus (72)u1u2v=2P0qAP0qAATP0qA2P0qA0-A-A0E1E20+c1-P0qX∗c2-P0qX∗X∗.


### 3.6. Linear Complementarity Problem

We denote(73)z=E1E20,w=u1u2v,M=2P0qAP0qAATP0qA2P0qA0-A-A0,b=c1-P0qX∗c2-P0qX∗X∗. The Nash equilibrium problem is equivalent to the Linear Complementarity Problem LCP(M,b). Find vectors z,w∈R6 such that (74)w=Mz+b≥0,z,w≥0,zTw=0.The following proposition confirms thatLCP(M,b) has a unique solution.Proposition 10.
The matrix(75)M=2P0qAP0qAATP0qA2P0qA0-A-A0 is P-matrix.Proof.
We haveaii<0 for all i=1,2,3 and Δ>0 so, if we note by (Mi)i=1,…,9 the submatrix of M, we obtain (76)detM1=-2p01q1a11>0,detM2=4p01q1p02q2q2q1K1r3q2K2Δ>0,detM3=8p01q1p02q2p3q3q3K3q1K1q2K2Δ2>0,detM4=-12a11p012q12p02q2p03q3q3K3q1K1q2K2Δ2>0,detM5=18p012q12p022q22p03q3q1K1r3q2K2q3K3q1K1q2K2Δ3>0,detM6=27p012q12p022q22p032q32q3K3q1K1q2K2Δ22>0,detM7=-9p01q1p022q22p032q32a11q3K3q1K1q2K2Δ22>0,detM8=3p01q1p02q2p032q32q1K1r3q2K2Δq3K3q1K1q2K2Δ22>0,detM=p01q1p02q2p03q3q3K3q1K1q2K2Δ22>0.
Then, the matrixM is P-matrix and therefore the linear complementarity problem LCP(M,b) admits one and only one solution.The unique solution ofLCP(M,b) represents the Nash equilibrium point of our problem and it is given by (77)E1=13A-1X∗-c1P0q,E2=13A-1X∗-c2P0q,where (78)A-1=r1K1q1c12q1c13q1c21q2r2K2q2c23q2c31q3c32q3r3K3q3. Then, the fishing effort that maximizes the profit of the first fisherman for catching the first species is (79)E11=13r1K1q1x1∗-c1P01q1+c12q1x2∗-c1P02q2+c13q1x3∗-c1P03q3; the fishing effort that maximizes the profit of the first fisherman for catching the second species is (80)E12=13r2K2q2x2∗-c1P02q2+c21q2x1∗-c1P01q1+c23q2x3∗-c1P03q3; the fishing effort that maximizes the profit of the first fisherman for catching the third species is (81)E13=13r3K3q3x3∗-c1P03q3+c31q3x1∗-c1P01q1+c32q3x2∗-c1P02q2.The fishing effort that maximizes the profit of the second fisherman for catching the first species is (82)E21=13r1K1q1x1∗-c2P01q1+c12q1x2∗-c2P02q2+c13q1x3∗-c2P03q3; the fishing effort that maximizes the profit of the second fisherman for catching the second species is (83)E22=13r2K2q2x2∗-c2P02q2+c21q2x1∗-c2P01q1+c23q2x3∗-c2P03q3; the fishing effort that maximizes the profit of the second fisherman for catching the third species is (84)E23=13r3K3q3x3∗-c2P03q3+c31q3x1∗-c2P01q1+c32q3x2∗-c2P02q2.


## 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 speciesj exploited by the fisherman i is given by pij=aj/Hij+p0j, where aj and p0j are given positive parameters for all j=1,2,3. Under these more realistic assumptions the total revenue of the fisherman i is (64)TRi=pi1Hi1+pi2Hi2+pi3Hi3=a1Hi1+p01Hi1+a2Hi2+p02Hi2+a3Hi3+p03Hi3=p01Hi1+p02Hi2+p03Hi3+∑k=13ak=p01q1Ei1x1+p02q2Ei2x2+p03q3Ei3x3+∑k=13ak=P0,qEiX+∑k=13ak=P0,qEi-AE+X∗+∑k=13ak=P0,qEi-A∑i=12Ei+X∗+∑k=13ak=Ei,qP0-A∑i=12Ei+X∗+∑k=13ak,so, (65)TRi=Ei,-P0qAEi+Ei,P0qxX∗-P0qAEj+∑k=13ak, where P0=diag(p0).


## 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(66)TCi=ci,Ei, where ci is the constant cost per unit of harvesting and Ei is the total effort of the fisherman i.


## 3.3. Expression of the Profit

The profit of each fishermanπi is equal to total revenue (TR)i minus total cost (TC)i; it is represented by the following function: (67)πiEi=TRi-TCi=Ei,-P0qAEi+Ei,P0qX∗-P0qAEj+∑k=13ak-ci,Ei=Ei,-P0qAEi+Ei,P0qX∗-ci-P0qAEj+∑k=13ak.


## 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(68)X=-AE+X∗≥X0>0. In other words, for the fisherman i, (69)AEi≤-AEj+X∗.


## 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 problemP1: P1maxπ1E1=E1,-P0qAE1+P0qX∗-c1-P0qAE2+∑k=13aksubject toAE1≤-AE2+X∗E1≥0E2given and the second fisherman must solve the problem P2: P2maxπ2E2=E2,-P0qAE2+P0qX∗-c2-P0qAE1+∑k=13aksubject toAE2≤-AE1+X∗E2≥0E1given.The pointE1,E2 is called Nash equilibrium point if and only if E1 is a solution of problem P1 for E2 given, and E2 is solution of problem P2 for E1 given.The essential conditions of Karush-Kuhn-Tucker applied to the problemP1 confirm that if E1 is a solution of the problem P1 then there exist constants u1∈R+3,  v1∈R+3, and λ1∈R+3 such that KKT12P0qAE1+c1-P0qX∗+P0qAE2-u1+ATλ1=0,AE1+v1=-AE2+X∗,u1,E1=λ1,v1=0.In the same way, the essential conditions of Karush-Kuhn-Tucker applied to the problemP2 confirm that if E2 is a solution of the problem P2 then there exist constants u2∈R+3,v2∈R+3, and λ2∈R+3 such that KKT22P0qAE2+c2-P0qX∗+P0qAE1-u2+ATλ2=0,AE2+v2=-AE1+X∗,u2,E2=λ2,v2=0; we remark that KKT1 and KKT2 lead to the following system: (70)u1=2P0qAE1+c1-P0qX∗+P0qAE2+ATλ1,u2=2P0qAE2+c2-P0qX∗+P0qAE1+ATλ2,v1=-AE1-AE2+X∗,v2=-AE1-AE2+X∗,ui,Ei=λi,vi=0∀i=1,2,Ei,ui,λi,vi≥0∀i=1,2.To maintain the biodiversity of species, it is natural to assume that all biomasses remain strictly positive; that is,xj>0 for all j=1,2,3; therefore v1=v2>0.As the scalar product of(λi)i=1,2 and (vi)i=1,2 is zero, λi=0 for all i=1,2. So, we denote v=v1=v2. Then we have the following expressions: (71)u1=2P0qAE1+P0qAE2+c1-P0qX∗,u2=2P0qAE2+P0qAE1+c2-P0qX∗,v=-AE1-AE2+X∗,ui,Ei=0∀i=1,2,Ei,ui,v≥0∀i=1,2. Thus (72)u1u2v=2P0qAP0qAATP0qA2P0qA0-A-A0E1E20+c1-P0qX∗c2-P0qX∗X∗.


## 3.6. Linear Complementarity Problem

We denote(73)z=E1E20,w=u1u2v,M=2P0qAP0qAATP0qA2P0qA0-A-A0,b=c1-P0qX∗c2-P0qX∗X∗. The Nash equilibrium problem is equivalent to the Linear Complementarity Problem LCP(M,b). Find vectors z,w∈R6 such that (74)w=Mz+b≥0,z,w≥0,zTw=0.The following proposition confirms thatLCP(M,b) has a unique solution.Proposition 10.
The matrix(75)M=2P0qAP0qAATP0qA2P0qA0-A-A0 is P-matrix.Proof.
We haveaii<0 for all i=1,2,3 and Δ>0 so, if we note by (Mi)i=1,…,9 the submatrix of M, we obtain (76)detM1=-2p01q1a11>0,detM2=4p01q1p02q2q2q1K1r3q2K2Δ>0,detM3=8p01q1p02q2p3q3q3K3q1K1q2K2Δ2>0,detM4=-12a11p012q12p02q2p03q3q3K3q1K1q2K2Δ2>0,detM5=18p012q12p022q22p03q3q1K1r3q2K2q3K3q1K1q2K2Δ3>0,detM6=27p012q12p022q22p032q32q3K3q1K1q2K2Δ22>0,detM7=-9p01q1p022q22p032q32a11q3K3q1K1q2K2Δ22>0,detM8=3p01q1p02q2p032q32q1K1r3q2K2Δq3K3q1K1q2K2Δ22>0,detM=p01q1p02q2p03q3q3K3q1K1q2K2Δ22>0.
Then, the matrixM is P-matrix and therefore the linear complementarity problem LCP(M,b) admits one and only one solution.The unique solution ofLCP(M,b) represents the Nash equilibrium point of our problem and it is given by (77)E1=13A-1X∗-c1P0q,E2=13A-1X∗-c2P0q,where (78)A-1=r1K1q1c12q1c13q1c21q2r2K2q2c23q2c31q3c32q3r3K3q3. Then, the fishing effort that maximizes the profit of the first fisherman for catching the first species is (79)E11=13r1K1q1x1∗-c1P01q1+c12q1x2∗-c1P02q2+c13q1x3∗-c1P03q3; the fishing effort that maximizes the profit of the first fisherman for catching the second species is (80)E12=13r2K2q2x2∗-c1P02q2+c21q2x1∗-c1P01q1+c23q2x3∗-c1P03q3; the fishing effort that maximizes the profit of the first fisherman for catching the third species is (81)E13=13r3K3q3x3∗-c1P03q3+c31q3x1∗-c1P01q1+c32q3x2∗-c1P02q2.The fishing effort that maximizes the profit of the second fisherman for catching the first species is (82)E21=13r1K1q1x1∗-c2P01q1+c12q1x2∗-c2P02q2+c13q1x3∗-c2P03q3; the fishing effort that maximizes the profit of the second fisherman for catching the second species is (83)E22=13r2K2q2x2∗-c2P02q2+c21q2x1∗-c2P01q1+c23q2x3∗-c2P03q3; the fishing effort that maximizes the profit of the second fisherman for catching the third species is (84)E23=13r3K3q3x3∗-c2P03q3+c31q3x1∗-c2P01q1+c32q3x2∗-c2P02q2.


## 4. 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.Table 1
Characteristics of the three marine species.
Species 1
Species 2
Species 3
r
1 = 0,5
r
2 = 0,3
r
3 = 0,2
K
1
=
1000
K
2
=
700
K
3
=
600
c
12
=
2
⋅
1
0
-
4
c
21
=
1
0
-
5
c
31
=
1
0
-
4
c
13
=
3
⋅
1
0
-
4
c
23
=
2
⋅
1
0
-
5
c
32
=
1
0
-
4Let us consider the economic parameters such as that shown in Table2.Table 2
Economic parameters of the model.
Species 1
Species 2
Species 3
a
1 = 0,1
a
2 = 0,2
a
3 = 0,3
p
01 = 1
p
02
=
2
p
03
=
3
q
1 = 0,1
q
2 = 0,02
q
3 = 0,004
c
1 = 0,1
c
1 = 0,1
c
1 = 0,1
c
2 = 0,2
c
2 = 0,2
c
2 = 0,2Using the parameters cited in Tables1 and 2, thereafter we will see how changes in the minimum prices can affect effort fishing, catches, and profit.


### 4.1. The Influence of the Price on the Fishing Effort, Catches, and Profit

By Tables3, 4, and 5 we will discover how changes in the price can affect the fishing effort, catches, and profit.Table 3
The influence of the price on the fishing effort.
p
01
p
02
p
03
E
1
E
2
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
17,6784Table 4
The influence of the price on the catches.
p
01
p
02
p
03
H
1
H
2
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
246,6839Table 5
The influence of the price on the profits.
p
01
p
02
p
03
π
1
π
2
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
249266According to Tables3, 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 Table5 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.


## 4.1. The Influence of the Price on the Fishing Effort, Catches, and Profit

By Tables3, 4, and 5 we will discover how changes in the price can affect the fishing effort, catches, and profit.Table 3
The influence of the price on the fishing effort.
p
01
p
02
p
03
E
1
E
2
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
17,6784Table 4
The influence of the price on the catches.
p
01
p
02
p
03
H
1
H
2
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
246,6839Table 5
The influence of the price on the profits.
p
01
p
02
p
03
π
1
π
2
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
249266According to Tables3, 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 Table5 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.


## 5. 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.



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*Source: 3157294-2017-12-26.xml*