# The Impact of Price on the Profits of Fishermen Exploiting Tritrophic Prey-Predator Fish Populations

Authors: Meriem Bentounsi; Imane Agmour; Naceur Achtaich; Youssef El Foutayeni
Journal: International Journal of Differential Equations
 (2018)

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## Abstract

We define and study a tritrophic bioeconomic model of Lotka-Volterra with a prey, middle predator, and top predator populations. These fish populations are exploited by two fishermen. We study the existence and the stability of the equilibrium points by using eigenvalues analysis and Routh-Hurwitz criterion. We determine the equilibrium point that maximizes the profit of each fisherman by solving the Nash equilibrium problem. Finally, following some numerical simulations, we observe that if the price varies, then the profit behavior of each fisherman will be changed; also, we conclude that the price change mechanism improves the fishing effort of the fishermen.


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## Body

## 1. Introduction

The problem of modelization is, perhaps, the most challenging in modern ecology, biology, chemistry, and many other sciences. In population dynamics, specially, in the dynamics of prey-predator marine species interactions modelization has gained a great importance. A predator is an organism that feeds another organism. The prey is the organism which the predator feeds. Predator always depends upon its prey and the predator dies if it does not get food. The first basic classic prey-predator model is renowned by Lotka-Volterra model and mathematical formulation of this model is directly related to the great work of Lotka (in 1925) and Volterra (in 1926). Thanks to this prey-predator model, other models have been proposed and studied [1–3]. In [1], the authors have considered predator-prey dynamics with predator “searching” and “handling” modes; they have derived a model that generalizes Holling’s functional responses and they have proved results concerning local and global properties, including for oscillations. In [2], the authors have formulated and studied a stage-structured predator-prey model of Beddington-DeAngelis type functional response to investigate the impact of predation over the immature prey by the juvenile predator. In [3], the authors have studied the global stability of diffusive predator-prey system of Holling-Tanner type in a bounded domain.Let us add that many researchers have studied extended tritrophic (prey, middle predator, and top predator) models to understand the interaction of different types of species [4–6]. In [4], the dynamics of a predator-prey model with disease in super predator are investigated. In [5], the authors have studied a prey-predator model with the concept of super predator under economic perspective. In [6], the authors have made a systematic analysis of the dynamics of a predator-prey system with type II functional response, in which the predator growth rate is affected by the presence of a super predator.In recent years, the biodiversity of marine populations is threatened by human impact, more precisely, by harvesting, which required many scientists to study bioeconomic models of fishery [7, 8]. In [7], the authors have made a mathematical study of a bioeconomic model of fishing for multisite, exploited by several fishermen, except one of them which is defined as not exploitable free fishing zone. In [8], the authors proposed and analyzed an extended model for the prey-predator-scavenger in presence of harvesting to study the effects of harvesting of predator as well as scavenger.In this paper, we have studied a tritrophic (prey, middle predator, and top predator) generalist model. The objective is to calculate the fishing effort that maximizes the profit of the fishermen, while respecting the conservation of the three fish populations, and also to study the effect of the variation of the price on each profit. The remaining part of this paper is organized as follows. In Section2, we introduce the biological tritrophic model. The existence and the stability of the steady states solutions are analyzed in Section 3. The bioeconomic model of the prey, middle predator, and top predator system is proposed in Section 4. In Section 5, we compute and solve the Nash equilibrium problem. In Section 6, we solve the linear complementarity problem. In Section 7, we present some numerical simulations to show the impact of price on the profits of fishermen. Finally, a brief conclusion is given in Section 8.


## 2. Biological Model

In this section, we consider a tritrophic prey-predator model which consists of three constituent populations, that is, prey, middle predator, and top predator. We impose that the population of preyx(t) grows in the logistic manner with birth rate constant and there exist interactions between the prey and middle and top predator due to defensive ability of prey; we impose that the population of middle predator y(t) grows also in the logistic manner with birth rate constant, prey x(t) is favorite food for middle predator y(t), and hence in the presence of favorite food the population density of middle predator y(t) will increase, and there are interactions between the middle predator and top predator due to defensive ability of middle predator; in the presence of favorite food (prey and middle predator) of top predator z(t) the population density of top predator z(t) will increase. Hence we can write this model in mathematical terms as(1)dxtdt=r1x1-x-αxy-βxz,dytdt=r2y1-y+α¯xy-δyz,dztdt=r3z1-z+β¯xz+δ¯yzwith positive initial conditions x(0)>0, y(0)>0, z(0)>0.Herer1, r2 and r3 are the per capita growth rate of prey, middle predator, and top predator, respectively; α, β, and δ are the maximum value which per capita reduction rate of x and y can attain, respectively; α¯ is the conversion rate of prey x into middle predator y, and β¯ and δ¯ are the conversion rate of prey x into top predator z and the conversion rate of middle predator y into top predator z, respectively.


## 3. The Steady States of the System

Since the focus is on the growth of marine species, there is need for the steady states of the system to satisfy conditions for nonnegativity. Furthermore, it is realized that the predators cannot survive in the complete absence of their prey. System (1) has eight biologically feasible nonnegative steady states. These steady states are obtained by solving the system of equations(2)r11-x-αy-βz=0,r21-y+α¯x-δz=0,r31-z+β¯x+δ¯y=0.(i)
P1=(0,0,0), P2=(1,0,0), P3=(0,1,0), P4=(0,0,1).(ii)
P5=(x5,y5,0), where x5=r2(r1+α)/r1r2+αα¯ and y5=r1(r2+α¯)/r1r2+αα¯.(iii)
P6=(x6,0,z6), where x6=r3(r1-β)/r1r3+ββ¯ and z6=r1(r3+β¯)/r1r3+ββ¯.(iv)
P7=(0,y7,z7), where y7=r3(r2-δ)/r2r3+δδ¯ and z7=r2(r3+δ)/r2r3+δδ¯.(v)
P8≔P∗=(x∗,y∗,z∗), where(3)x∗=r1r2r3+r1δδ¯+r3αδ-r2r3α-r2r3β-r2βδ¯Δ,y∗=r1r2r3+r2ββ¯-r3βα¯-r1r3δ+r1r3α¯-r1β¯δΔ,z∗=r1r2r3+r1r2β¯+r1r2δ¯+r1α¯δ¯-r2β¯α+r3αα¯Δ,whereΔ=r1r2r3+r1δδ¯+r2ββ¯+r3αα¯-αδβ¯+βα¯δ¯.One can see that the steady state equilibriumP∗ exists if r1>max⁡α,β, r2>δ, and r3>max⁡δ,β¯.


### 3.1. Analysis of Steady States

The Jacobian matrix for system (1) is given by (4)J=J11-αx-βxα¯yJ22-δyβ¯zδ¯zJ33,where (5)J11=r11-2x-αy-βz,J22=r21-2y+α¯x-δz,J33=r31-2z+β¯x+δ¯y.At any steady state solution, the Jacobian matrix is computed. LetJk=J denote the Jacobian evaluated at Pk for k=1,2,3,…,7, the corresponding entries, and J∗ denote the Jacobian evaluated at P∗.


#### 3.1.1. Local Stability of the Steady StateP1

For the equilibrium pointP1=(0,0,0) the Jacobian matrix is given by (6)J1=r1000r2000r3.The eigenvalues are found to be λ1=r1>0, λ2=r2>0, and λ3=r3>0, and then this point is unstable.According to Table1, Figure 1 shows the dynamical behaviors and phase space trajectory of the prey, middle predator, and top predator fish populations against time, beginning with the initial values x(0)=0.01, y(0)=0.01, and 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 P∗.Table 1
Characteristics of the three fish populations.
Prey
Middle predator
Top predator
r1=5
r
2
=
4
r
3
=
3
α=9.10-6
α
¯
=
8.1
0
-
6
β
¯
=
2.1
0
-
6
β=7.10-6
δ
=
6.1
0
-
6
δ
¯
=
1
0
-
6Figure 1
Dynamical behaviors and phase space trajectories of the three fish populations.


#### 3.1.2. Local Stability of the Steady StateP2

In the same way we can consider the stationary pointP2=(1,0,0) and find the Jacobian matrix (7)J2=-r1-α-β0r2+α¯000r3+β¯.The eigenvalues can easily be computed, λ1=-r1, λ2=r2+α¯>0, and λ3=r3+β¯>0. Therefore, the point P2=(1,0,0) is unstable.Following Table1, Figure 2 shows the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=1.01, and 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 P∗ too.Figure 2
Dynamical behaviors and phase space trajectories of the three fish populations.


#### 3.1.3. Local Stability of the Steady StateP3

For the pointP3=(0,1,0) we have the Jacobian matrix J3 which is written in the form (8)J3=r1-α00α¯-r2-δ00r3+δ¯.The eigenvalues are λ1=r1-α, λ2=-r2, and λ3=r3+δ¯>0. Then, the point P3=(0,1,0) is unstable.According to Table1, Figure 3 shows the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=0.01, y(0)=1.01, and z(0)=0.01. By Figure 3 we can see that the steady state point P3 is also unstable and tends to the point P∗.Figure 3
Dynamical behaviors and phase space trajectories of the three fish populations.


#### 3.1.4. Local Stability of the Steady StateP4

P
4
=
(
0,0
,
1
) is stable if r1-β<0 and r2-δ<0; if not, it is unstable.In fact, the Jacobian matrix of the system in this state is written as(9)J4=r1-β000r2-δ0β¯δ¯-r3.The eigenvalues are λ1=r1-β, λ2=r2-δ, and λ3=-r3. Therefore, if r1-β<0 and r2-δ<0, then the point P4=(0,0,1) is stable; if not, it is an unstable point.For the same values parameters in Table1, Figure 4 indicates the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=0.01, y(0)=0.01, and z(0)=1.01. Following Figure 4 we can see that the steady state point P3 is unstable and also tends to the point P∗.Figure 4
Dynamical behaviors and phase space trajectories of the three fish populations.


#### 3.1.5. Local Stability of the Steady StateP5

For the equilibrium pointP5=(x5,y5,0) the Jacobian matrix is given by (10)J5=-r1x5-αx5-βx5α¯y5-r2y5-δy500r3+β¯x5+δ¯y5.The eigenvalues are (11)λ1=-12r1x5+r2y5+r1x5-r2y52-4αα¯x5y5,λ2=-12r1x5+r2y5-r1x5-r2y52-4αα¯x5y5,λ3=r3+β¯x5+δ¯y5.We have λ3>0; then, the point P5=(x5,y5,0) is unstable.Following Table1, Figure 5 represents the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=1.01,y(0)=1.02, and z(0)=0.01. Following Figure 5 we can deduce that the steady state point P5 is unstable and also tends to the point P∗.Figure 5
Dynamical behaviors and phase space trajectories of the three fish populations.


#### 3.1.6. Local Stability of the Steady StateP6

For the equilibrium pointP6=(x6,0,z6) the Jacobian matrix is given by (12)J6=-r1x6-αx6-βx60r2+α¯x6-δz60β¯z6δ¯z6r3z6.the eigenvalues are (13)λ1=-12r1x6+r3z6-r1x6-r3z62-4ββ¯x6z6,λ2=-12r1x6+r3z6+r1x6-r3z62-4ββ¯x6z6,λ3=r2+α¯x6-δz6.We have λ1>0 and λ2>0. Therefore, the point P6=(x6,0,z6) is unstable.According to Table1, Figure 6 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values x(0)=1.05, y(0)=0.01, and z(0)=1.5. Following Figure 6 we can deduce that the steady state point P6 is unstable and also tends to the point P∗.Figure 6
Dynamical behaviors and phase space trajectories of the three marine species.


#### 3.1.7. Local Stability of the Steady StateP7

For the equilibrium pointP7=(0,y7,z7) the Jacobian matrix is given by (14)J7=r1-αy7-βz700α¯y7-r2y7-δy7β¯z7δ¯z7-r3z7.the eigenvalues are(15)λ1=-12r2y7+r3z7-r2y7+r3z72-4δδ¯y7z7,λ2=-12r2y7+r3z7+r2y7+r3z72-4δδ¯y7z7,λ3=r1-αy7-βz7.We have λ1>0 and λ2>0. Therefore, the point P7=(0,y7,z7) is unstable.Following Table1, 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)=1.4, and z(0)=1.65. By Figure 7 we can conclude that the steady state point P7 is unstable and also tends to the point P∗.Figure 7
Dynamical behaviors and phase space trajectories of the three fish populations.


#### 3.1.8. Local Stability of the Steady StateP∗

As usual, one can consider the corresponding linearized system (the Jacobian matrix) and determine the characteristic equation for the eigenvalues. The Jacobian matrix (in the equilibrium pointP∗) reads (16)J∗=-r1x∗-αx∗-βx∗α¯y∗-r2y∗-δy∗β¯z∗δ¯z∗-r3z∗resulting in the characteristic equation and trying to apply Routh-Hurwitz conditions. We find (17)Pλ=a0λ3+a1λ2+a2λ+a3,where (18)a0=1,a1=r1x∗+r2y∗+r3z∗,a2=r3z∗r1x∗+r2y∗+x∗y∗αα¯+r1r2+x∗z∗ββ¯+y∗z∗δδ¯,a3=z∗r1x∗+r2y∗x∗ββ¯+y∗δδ¯-δ¯y∗z∗δr2y-α¯βx∗-β¯x∗z∗βr1x∗+αδy∗+r3x∗y∗z∗αα¯+r1r2.We can easily verify that a0, a1, a2, a3, and a1a2-a0a3 are all positive. Thus, the Routh-Hurwitz conditions are satisfied. Therefore, the point P∗=x∗,y∗,z∗ is locally asymptotically stable.For the values parameters quoted in Table1, Figure 8 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values x(0)=46.7, y(0)=39.9, and z(0)=31.1. By Figure 8 one can see that the steady state point P∗ is locally asymptotically stable.Figure 8
Dynamical behaviors and phase space trajectories of the three fish populations.More precisely, beginning with different initial values we can note that the three fish populations tend to the pointP∗.


## 3.1. Analysis of Steady States

The Jacobian matrix for system (1) is given by (4)J=J11-αx-βxα¯yJ22-δyβ¯zδ¯zJ33,where (5)J11=r11-2x-αy-βz,J22=r21-2y+α¯x-δz,J33=r31-2z+β¯x+δ¯y.At any steady state solution, the Jacobian matrix is computed. LetJk=J denote the Jacobian evaluated at Pk for k=1,2,3,…,7, the corresponding entries, and J∗ denote the Jacobian evaluated at P∗.


### 3.1.1. Local Stability of the Steady StateP1

For the equilibrium pointP1=(0,0,0) the Jacobian matrix is given by (6)J1=r1000r2000r3.The eigenvalues are found to be λ1=r1>0, λ2=r2>0, and λ3=r3>0, and then this point is unstable.According to Table1, Figure 1 shows the dynamical behaviors and phase space trajectory of the prey, middle predator, and top predator fish populations against time, beginning with the initial values x(0)=0.01, y(0)=0.01, and 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 P∗.Table 1
Characteristics of the three fish populations.
Prey
Middle predator
Top predator
r1=5
r
2
=
4
r
3
=
3
α=9.10-6
α
¯
=
8.1
0
-
6
β
¯
=
2.1
0
-
6
β=7.10-6
δ
=
6.1
0
-
6
δ
¯
=
1
0
-
6Figure 1
Dynamical behaviors and phase space trajectories of the three fish populations.


### 3.1.2. Local Stability of the Steady StateP2

In the same way we can consider the stationary pointP2=(1,0,0) and find the Jacobian matrix (7)J2=-r1-α-β0r2+α¯000r3+β¯.The eigenvalues can easily be computed, λ1=-r1, λ2=r2+α¯>0, and λ3=r3+β¯>0. Therefore, the point P2=(1,0,0) is unstable.Following Table1, Figure 2 shows the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=1.01, and 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 P∗ too.Figure 2
Dynamical behaviors and phase space trajectories of the three fish populations.


### 3.1.3. Local Stability of the Steady StateP3

For the pointP3=(0,1,0) we have the Jacobian matrix J3 which is written in the form (8)J3=r1-α00α¯-r2-δ00r3+δ¯.The eigenvalues are λ1=r1-α, λ2=-r2, and λ3=r3+δ¯>0. Then, the point P3=(0,1,0) is unstable.According to Table1, Figure 3 shows the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=0.01, y(0)=1.01, and z(0)=0.01. By Figure 3 we can see that the steady state point P3 is also unstable and tends to the point P∗.Figure 3
Dynamical behaviors and phase space trajectories of the three fish populations.


### 3.1.4. Local Stability of the Steady StateP4

P
4
=
(
0,0
,
1
) is stable if r1-β<0 and r2-δ<0; if not, it is unstable.In fact, the Jacobian matrix of the system in this state is written as(9)J4=r1-β000r2-δ0β¯δ¯-r3.The eigenvalues are λ1=r1-β, λ2=r2-δ, and λ3=-r3. Therefore, if r1-β<0 and r2-δ<0, then the point P4=(0,0,1) is stable; if not, it is an unstable point.For the same values parameters in Table1, Figure 4 indicates the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=0.01, y(0)=0.01, and z(0)=1.01. Following Figure 4 we can see that the steady state point P3 is unstable and also tends to the point P∗.Figure 4
Dynamical behaviors and phase space trajectories of the three fish populations.


### 3.1.5. Local Stability of the Steady StateP5

For the equilibrium pointP5=(x5,y5,0) the Jacobian matrix is given by (10)J5=-r1x5-αx5-βx5α¯y5-r2y5-δy500r3+β¯x5+δ¯y5.The eigenvalues are (11)λ1=-12r1x5+r2y5+r1x5-r2y52-4αα¯x5y5,λ2=-12r1x5+r2y5-r1x5-r2y52-4αα¯x5y5,λ3=r3+β¯x5+δ¯y5.We have λ3>0; then, the point P5=(x5,y5,0) is unstable.Following Table1, Figure 5 represents the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=1.01,y(0)=1.02, and z(0)=0.01. Following Figure 5 we can deduce that the steady state point P5 is unstable and also tends to the point P∗.Figure 5
Dynamical behaviors and phase space trajectories of the three fish populations.


### 3.1.6. Local Stability of the Steady StateP6

For the equilibrium pointP6=(x6,0,z6) the Jacobian matrix is given by (12)J6=-r1x6-αx6-βx60r2+α¯x6-δz60β¯z6δ¯z6r3z6.the eigenvalues are (13)λ1=-12r1x6+r3z6-r1x6-r3z62-4ββ¯x6z6,λ2=-12r1x6+r3z6+r1x6-r3z62-4ββ¯x6z6,λ3=r2+α¯x6-δz6.We have λ1>0 and λ2>0. Therefore, the point P6=(x6,0,z6) is unstable.According to Table1, Figure 6 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values x(0)=1.05, y(0)=0.01, and z(0)=1.5. Following Figure 6 we can deduce that the steady state point P6 is unstable and also tends to the point P∗.Figure 6
Dynamical behaviors and phase space trajectories of the three marine species.


### 3.1.7. Local Stability of the Steady StateP7

For the equilibrium pointP7=(0,y7,z7) the Jacobian matrix is given by (14)J7=r1-αy7-βz700α¯y7-r2y7-δy7β¯z7δ¯z7-r3z7.the eigenvalues are(15)λ1=-12r2y7+r3z7-r2y7+r3z72-4δδ¯y7z7,λ2=-12r2y7+r3z7+r2y7+r3z72-4δδ¯y7z7,λ3=r1-αy7-βz7.We have λ1>0 and λ2>0. Therefore, the point P7=(0,y7,z7) is unstable.Following Table1, 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)=1.4, and z(0)=1.65. By Figure 7 we can conclude that the steady state point P7 is unstable and also tends to the point P∗.Figure 7
Dynamical behaviors and phase space trajectories of the three fish populations.


### 3.1.8. Local Stability of the Steady StateP∗

As usual, one can consider the corresponding linearized system (the Jacobian matrix) and determine the characteristic equation for the eigenvalues. The Jacobian matrix (in the equilibrium pointP∗) reads (16)J∗=-r1x∗-αx∗-βx∗α¯y∗-r2y∗-δy∗β¯z∗δ¯z∗-r3z∗resulting in the characteristic equation and trying to apply Routh-Hurwitz conditions. We find (17)Pλ=a0λ3+a1λ2+a2λ+a3,where (18)a0=1,a1=r1x∗+r2y∗+r3z∗,a2=r3z∗r1x∗+r2y∗+x∗y∗αα¯+r1r2+x∗z∗ββ¯+y∗z∗δδ¯,a3=z∗r1x∗+r2y∗x∗ββ¯+y∗δδ¯-δ¯y∗z∗δr2y-α¯βx∗-β¯x∗z∗βr1x∗+αδy∗+r3x∗y∗z∗αα¯+r1r2.We can easily verify that a0, a1, a2, a3, and a1a2-a0a3 are all positive. Thus, the Routh-Hurwitz conditions are satisfied. Therefore, the point P∗=x∗,y∗,z∗ is locally asymptotically stable.For the values parameters quoted in Table1, Figure 8 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values x(0)=46.7, y(0)=39.9, and z(0)=31.1. By Figure 8 one can see that the steady state point P∗ is locally asymptotically stable.Figure 8
Dynamical behaviors and phase space trajectories of the three fish populations.More precisely, beginning with different initial values we can note that the three fish populations tend to the pointP∗.


## 3.1.1. Local Stability of the Steady StateP1

For the equilibrium pointP1=(0,0,0) the Jacobian matrix is given by (6)J1=r1000r2000r3.The eigenvalues are found to be λ1=r1>0, λ2=r2>0, and λ3=r3>0, and then this point is unstable.According to Table1, Figure 1 shows the dynamical behaviors and phase space trajectory of the prey, middle predator, and top predator fish populations against time, beginning with the initial values x(0)=0.01, y(0)=0.01, and 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 P∗.Table 1
Characteristics of the three fish populations.
Prey
Middle predator
Top predator
r1=5
r
2
=
4
r
3
=
3
α=9.10-6
α
¯
=
8.1
0
-
6
β
¯
=
2.1
0
-
6
β=7.10-6
δ
=
6.1
0
-
6
δ
¯
=
1
0
-
6Figure 1
Dynamical behaviors and phase space trajectories of the three fish populations.


## 3.1.2. Local Stability of the Steady StateP2

In the same way we can consider the stationary pointP2=(1,0,0) and find the Jacobian matrix (7)J2=-r1-α-β0r2+α¯000r3+β¯.The eigenvalues can easily be computed, λ1=-r1, λ2=r2+α¯>0, and λ3=r3+β¯>0. Therefore, the point P2=(1,0,0) is unstable.Following Table1, Figure 2 shows the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=1.01, and 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 P∗ too.Figure 2
Dynamical behaviors and phase space trajectories of the three fish populations.


## 3.1.3. Local Stability of the Steady StateP3

For the pointP3=(0,1,0) we have the Jacobian matrix J3 which is written in the form (8)J3=r1-α00α¯-r2-δ00r3+δ¯.The eigenvalues are λ1=r1-α, λ2=-r2, and λ3=r3+δ¯>0. Then, the point P3=(0,1,0) is unstable.According to Table1, Figure 3 shows the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=0.01, y(0)=1.01, and z(0)=0.01. By Figure 3 we can see that the steady state point P3 is also unstable and tends to the point P∗.Figure 3
Dynamical behaviors and phase space trajectories of the three fish populations.


## 3.1.4. Local Stability of the Steady StateP4

P
4
=
(
0,0
,
1
) is stable if r1-β<0 and r2-δ<0; if not, it is unstable.In fact, the Jacobian matrix of the system in this state is written as(9)J4=r1-β000r2-δ0β¯δ¯-r3.The eigenvalues are λ1=r1-β, λ2=r2-δ, and λ3=-r3. Therefore, if r1-β<0 and r2-δ<0, then the point P4=(0,0,1) is stable; if not, it is an unstable point.For the same values parameters in Table1, Figure 4 indicates the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=0.01, y(0)=0.01, and z(0)=1.01. Following Figure 4 we can see that the steady state point P3 is unstable and also tends to the point P∗.Figure 4
Dynamical behaviors and phase space trajectories of the three fish populations.


## 3.1.5. Local Stability of the Steady StateP5

For the equilibrium pointP5=(x5,y5,0) the Jacobian matrix is given by (10)J5=-r1x5-αx5-βx5α¯y5-r2y5-δy500r3+β¯x5+δ¯y5.The eigenvalues are (11)λ1=-12r1x5+r2y5+r1x5-r2y52-4αα¯x5y5,λ2=-12r1x5+r2y5-r1x5-r2y52-4αα¯x5y5,λ3=r3+β¯x5+δ¯y5.We have λ3>0; then, the point P5=(x5,y5,0) is unstable.Following Table1, Figure 5 represents the dynamical behaviors and phase space trajectory of the three fish populations against time, beginning with the initial values x(0)=1.01,y(0)=1.02, and z(0)=0.01. Following Figure 5 we can deduce that the steady state point P5 is unstable and also tends to the point P∗.Figure 5
Dynamical behaviors and phase space trajectories of the three fish populations.


## 3.1.6. Local Stability of the Steady StateP6

For the equilibrium pointP6=(x6,0,z6) the Jacobian matrix is given by (12)J6=-r1x6-αx6-βx60r2+α¯x6-δz60β¯z6δ¯z6r3z6.the eigenvalues are (13)λ1=-12r1x6+r3z6-r1x6-r3z62-4ββ¯x6z6,λ2=-12r1x6+r3z6+r1x6-r3z62-4ββ¯x6z6,λ3=r2+α¯x6-δz6.We have λ1>0 and λ2>0. Therefore, the point P6=(x6,0,z6) is unstable.According to Table1, Figure 6 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values x(0)=1.05, y(0)=0.01, and z(0)=1.5. Following Figure 6 we can deduce that the steady state point P6 is unstable and also tends to the point P∗.Figure 6
Dynamical behaviors and phase space trajectories of the three marine species.


## 3.1.7. Local Stability of the Steady StateP7

For the equilibrium pointP7=(0,y7,z7) the Jacobian matrix is given by (14)J7=r1-αy7-βz700α¯y7-r2y7-δy7β¯z7δ¯z7-r3z7.the eigenvalues are(15)λ1=-12r2y7+r3z7-r2y7+r3z72-4δδ¯y7z7,λ2=-12r2y7+r3z7+r2y7+r3z72-4δδ¯y7z7,λ3=r1-αy7-βz7.We have λ1>0 and λ2>0. Therefore, the point P7=(0,y7,z7) is unstable.Following Table1, 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)=1.4, and z(0)=1.65. By Figure 7 we can conclude that the steady state point P7 is unstable and also tends to the point P∗.Figure 7
Dynamical behaviors and phase space trajectories of the three fish populations.


## 3.1.8. Local Stability of the Steady StateP∗

As usual, one can consider the corresponding linearized system (the Jacobian matrix) and determine the characteristic equation for the eigenvalues. The Jacobian matrix (in the equilibrium pointP∗) reads (16)J∗=-r1x∗-αx∗-βx∗α¯y∗-r2y∗-δy∗β¯z∗δ¯z∗-r3z∗resulting in the characteristic equation and trying to apply Routh-Hurwitz conditions. We find (17)Pλ=a0λ3+a1λ2+a2λ+a3,where (18)a0=1,a1=r1x∗+r2y∗+r3z∗,a2=r3z∗r1x∗+r2y∗+x∗y∗αα¯+r1r2+x∗z∗ββ¯+y∗z∗δδ¯,a3=z∗r1x∗+r2y∗x∗ββ¯+y∗δδ¯-δ¯y∗z∗δr2y-α¯βx∗-β¯x∗z∗βr1x∗+αδy∗+r3x∗y∗z∗αα¯+r1r2.We can easily verify that a0, a1, a2, a3, and a1a2-a0a3 are all positive. Thus, the Routh-Hurwitz conditions are satisfied. Therefore, the point P∗=x∗,y∗,z∗ is locally asymptotically stable.For the values parameters quoted in Table1, Figure 8 shows the dynamical behaviors and phase space trajectory of the three marine species against time, beginning with the initial values x(0)=46.7, y(0)=39.9, and z(0)=31.1. By Figure 8 one can see that the steady state point P∗ is locally asymptotically stable.Figure 8
Dynamical behaviors and phase space trajectories of the three fish populations.More precisely, beginning with different initial values we can note that the three fish populations tend to the pointP∗.


## 4. Bioeconomic Model

The main purpose of this section is to define and study a bioeconomic model for two fishermen who catch the three fish populations. The model for the evolution of these three fish populations becomes(19)dxtdt=r1x1-x-αxy-βxz-q1E1x,dytdt=r2y1-y+α¯xy-δyz-q2E2y,dztdt=r3z1-z+β¯xz+δ¯yz-q3E3z.The capturability coefficientq is a key parameter in the validation process of the fishing simulation model, which is assumed to be constant. Fishing effort is defined as the product of fishing activity and fishing power. The fishing effort deployed by a fleet is the sum of these products on all fishing units in the fleet, while fishing power is the ability of a fishing unit to catch fish. However, it is interesting to note that, according to the literature, effort depends on several variables, for example, ship, number of hours spent fishing, number of days spent fishing, number of stolen sorties, technology, fishing equipment, and crew.However, in this paper, effort is treated as a variable that combines all of these factors.The expression of biomass as a function of fishing effort at biological equilibrium is the solution of the system(20)r11-x=αy+βz+q1E1,r21-y=-α¯x+δz+q2E2,r31-z=-β¯x-δ¯y+q3E3.This solution of this system (20) is given by (21)x=a11E1+a12E2+a13E3+x∗,y=a21E1+a22E2+a23E3+y∗,z=a31E1+a32E2+a33E3+z∗, where (22)a11=-δδ¯q1-r2r3q1Δ,a12=βδ¯q2+αr3q2Δ,a13=-δαq3+βr2q3Δ,a21=δβ¯q1-α¯r3q1Δ,a22=-ββ¯q2-r1r3q2Δ,a23=βα¯q3+δr1q3Δ,a31=-α¯δ¯q1-β¯r2q1Δ,a32=αβ¯q2-δ¯r1q2Δ,a33=-αα¯q3-r1r2q3Δ.In matrix form, this solution can be written as X=-AE+X∗, where A=-aij1≤i,j≤3 and X∗=x∗,y∗,z∗T with aii<0 for all i=1,2,3.


### 4.1. The Net Economic Revenue

Simultaneously, an algebraic equation is also included due to the consideration of the economic profit of harvesting. According to Gordon’s economic theory(i)
Πi(E)=TotalRevenue(TR)i-TotalCostTCi,where the total revenueTRi and total cost TCi are given by(a)
TRi=Pricepi×CatchesHij.We note thatHij=qjEijXj represent the catches of species j by the fisherman i (X1=x, X2=y, and X3=z), and Eij is the effort of the fisherman i to exploit the species j. It is clear that Hj=∑i=12Hij is the total catches of species j by all fishermen.According to the above notations we have(23)TRi=p1Hi1+p2Hi2+p3Hi3=Ei,-pqAEi+pqX∗-pqAEj.(b)
TCi=ci,Ei,whereci is a constant cost per unit of harvesting effort of the fisherman i.The profit for each fisherman is represented by the functionΠi(E), so that the profit of fisherman i is given by (24)ΠiE=Ei,-pqAEi-pqX∗-ci-pqAEj.To maintain the biodiversity of the three fish populations, it is natural to assume that all biomasses remain positive; thereforeX=-AE+X∗≥0. In other words, for the fisherman i we must have AEi≤AEj-X∗.


## 4.1. The Net Economic Revenue

Simultaneously, an algebraic equation is also included due to the consideration of the economic profit of harvesting. According to Gordon’s economic theory(i)
Πi(E)=TotalRevenue(TR)i-TotalCostTCi,where the total revenueTRi and total cost TCi are given by(a)
TRi=Pricepi×CatchesHij.We note thatHij=qjEijXj represent the catches of species j by the fisherman i (X1=x, X2=y, and X3=z), and Eij is the effort of the fisherman i to exploit the species j. It is clear that Hj=∑i=12Hij is the total catches of species j by all fishermen.According to the above notations we have(23)TRi=p1Hi1+p2Hi2+p3Hi3=Ei,-pqAEi+pqX∗-pqAEj.(b)
TCi=ci,Ei,whereci is a constant cost per unit of harvesting effort of the fisherman i.The profit for each fisherman is represented by the functionΠi(E), so that the profit of fisherman i is given by (24)ΠiE=Ei,-pqAEi-pqX∗-ci-pqAEj.To maintain the biodiversity of the three fish populations, it is natural to assume that all biomasses remain positive; thereforeX=-AE+X∗≥0. In other words, for the fisherman i we must have AEi≤AEj-X∗.


## 5. Nash Equilibrium

Each of the two fishermen tries to maximize their profits and reach a fishing effort that is an optimal response to the effort of the other fisherman. Therefore, we have a Nash equilibrium situation where each fisherman’s strategy is optimal, taking into account the strategy of the second fisherman. This problem can be translated into the two following mathematical problems.The first fisherman must solve this problemp1: p1max∏E1=E1,-pqAE1+pqX∗-c1-pqAE2subjecttoAE1≤-AE2+X∗E1≥0E2given.The second fisherman must solve this problemp2:p2max∏E2=E2,-pqAE2+pqX∗-c2-pqAE1subjecttoAE2≤-AE1+X∗E2≥0E1given.By definition, the point(E1,E2) is called Nash equilibrium point if and only if E1 is a solution of problem p1 for given E2, and E2 is a solution of problem p2 for given E1.By applying the essential conditions of Karush-Kuhn-Tucker to the problemp1 we will ensure the existence of constants u1∈R+3, v1∈R+3, and λ1∈R+3 such that (25)2pqAE1+c1-pqX∗+pqAE2-u1+ATλ1=0,AE1+v1=AE2+X∗,u1,E1=λ1,v1=0.In the same way, by applying the essential conditions of Karush-Kuhn-Tucker to the problemp2 we will ensure the existence of constants u2∈R+3, v2∈R+3, and λ2∈R+3 such that (26)2pqAE2+c2-pqX∗+pqAE1-u2+ATλ2=0,AE2+v2=AE1+X∗,u2,E2=λ2,v2=0.We can notice that the two preceding problems can be written in the form of a single problem which is the following:(27)u1=2pqAE1+c1-pqX∗+pqAE2+ATλ1,u2=2pqAE2+c2-pqX∗+pqAE1+ATλ2,v1=-AE1-AE2+X∗,v2=-AE2-AE1+X∗,ui,Ei=λi,vi=0∀i=1,2,ui,Ei,λi,vi≥0∀i=1,2.To maintain the biodiversity of fish populations, it is natural to assume that all biomasses remain strictly positive; that is,X∗>0; therefore v1=v2>0. As the scalar product of λii=1,2 and vii=1,2 is zero, λi=0 for all i=1,2. In what follows in this paper, we denote v≔v1=v2. So we have the following expressions: (28)u1=2pqAE1+pqAE2+c1-pqX∗,u2=2pqAE2+pqAE1+c2-pqX∗,v=-AE1-AE2+X∗,ui,Ei=λi,vi=0∀i=1,2,ui,Ei,λi,vi≥0∀i=1,2;thus, (29)u1u2v=2pqApqAATpqA2pqA1-A-A1E1E20+c1-pqX∗c2-pqX∗X∗.Let us denote (30)z=E1E20,w=u1u2v,M=2pqApqAATpqA2pqA1-A-A1,b=c-pqX∗c-pqX∗X∗.


## 6. Linear Complementarity Problem

The Nash equilibrium problem is equivalent to the following linear complementarity problemLCP(M,b).Find vectorsz,w∈R9 such that w=Mz+b≥0, z,w≥0, and  zTw=0.Using the following theorem we can prove that this linear complementarity problemLCP(M,b) has a unique solution.Theorem 1.
L
C
P
(
M
,
b
) has a unique solution for every b if and only if M is aP-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(31)det⁡M1=-2p01q1a11>0,det⁡M2=4p1q1p2q2q2q1r3q2K2Δ>0,det⁡M3=8p1q12p2q22p3q32Δ2>0,det⁡M4=-12a11p12q13p2q22p3q32Δ2>0,det⁡M5=18p12q14p22q24p3r3q32Δ3>0,det⁡M6=27p12q14p22q24p32q34Δ4>0,det⁡M7=-9p1q13p22q24p32q34a11Δ4>0,det⁡M8=3p1q14p2q24p32q34r3Δ5>0,det⁡M=p1q12p2q22p3q32Δ4>0.
Remember that a matrixM is calledP-matrix if the determinant of every principal submatrix of M is positive (see Murty [9] and Cottle et al. [10]). The class ofP-matrices generalizes many important classes of matrices, such as positive definite matrices,M-matrices, and inverseM-matrices, and arises in applications. Note that each symmetric positive definite matrix isP-matrix, but the reverse is not always true. Since the matrix M of our problem isP-matrix, we can conclude that the linear complementarity problem LCP(M,b) admits one and only one solution. This solution is given by (32)E1=13A-1X∗-c1pq,E2=13A-1X∗-c2pq.
Then, the fishing effort that maximizes the profit of the first fisherman for caching the prey population is(33)E11=13r1q1x∗-c1p1q1+αq1y∗-c1p2q2+βq1z∗-c1p3q3;the fishing effort that maximizes the profit of the first fisherman for caching the middle predator population is (34)E12=13r2q2y∗-c1p2q2-α¯q2x∗-c1p1q1+δq2z∗-c1p3q3;the fishing effort that maximizes the profit of the first fisherman for caching the top predator population is (35)E13=13r3q3z∗-c1p3q3-β¯q3x∗-c1p1q1-δ¯q3y∗-c1p2q2.
Then, the fishing effort that maximizes the profit of the second fisherman for caching the prey population is(36)E21=13r1q1x∗-c2p1q1+αq1y∗-c2p2q2+βq1z∗-c2p3q3;the fishing effort that maximizes the profit of the second fisherman for caching the middle predator population is (37)E22=13r2q2y∗-c2p2q2-α¯q2x∗-c2p1q1+δq2z∗-c2p3q3;the fishing effort that maximizes the profit of the second fisherman for caching the top predator population is (38)E23=13r3q3z∗-c2p3q3-β¯q3x∗-c2p1q1-δ¯q3y∗-c2p2q2.


## 7. Numerical Simulations

In this section, we complement the mathematical study undertaken previously on the model by numerical simulations in order to discover the effect of the variation of the price on the profits of the fishermen. The parameters of model system (1) are considered as shown in Table 1.The economic parameters are considered as shown in Table2.Table 2
Economic parameters of the model.
Prey
Middle predator
Top predator
p1=1
p
2
=
3
p
3
=
5
q1=0,05
q
2
=
0
,
02
q
3
=
0
,
01
c1=0,01
c
1
=
0
,
01
c
1
=
0
,
01
c2=0,015
c
2
=
0
,
015
c
2
=
0
,
015Using the parameters cited in Tables1 and 2, thereafter we will see how changes in the price can affect effort fishing, catches, and profits.According to Table3, one can remark that an increase in the price level of the three fish populations leads to an increase in the fishing effort which must be provided by each fishermen to exploit them. But on arriving at a certain rank the fishing effort becomes constant. More precisely, if the price is greater than 10300, the fishing effort in this case becomes constant and it does not exceed 399,9781. This means that the fishermen must not exceed a fishing effort equal to 399,9781 to expect maximum benefit by making more reasonable catches that take into account the preservation of marine species.Table 3
The influence of the price on the fishing effort.
p1
p2
p3
E1
E2
0,5
0,75
1
182,2202
173,3303
5
7,5
10
198,2224
197,3330
20
30
60
199,4130
199,1188
50
70
100
199,7626
199,6433
90
120
250
199,8282
199,7416
220
400
510
199,9113
199,8662
2200
3000
5100
199,9113
199,9915
20000
30000
60000
199,9113
199,9915According to Table4, the catch level decreases as the price increases. If the price is equal to 4000, the catches that maximize the profit of the fishermen are equal to 2,66684; in this case the catches that allow the fisherman to have a maximum profit do not exceed 2,66684. Contrariwise, the profit is always increased even if the level of catches decreases, which is justified by the increase in the level price of fish populations (Table 5).Table 4
The influence of the price on the catches.
p1
p2
p
3
H1
H
2
0,5
0,75
1
1,52469
1,38889
5
7,5
10
1,36024
1,35055
20
30
60
1,34079
1,33824
50
70
100
1,33632
1,33531
90
120
250
1,33554
1,33479
220
400
510
1,33446
1,33408
2200
3000
5100
1,33344
1,33340
20000
30000
60000
1,33344
1,33340Table 5
The influence of the price on the profits.
p1
p2
p
3
π
1
π
2
0,5
0,75
1
2
1
5
7,5
10
34
32
20
30
60
96
95
50
70
100
229
228
90
120
250
302
301
220
400
510
611
610
2200
3000
5100
6131
6130
20000
30000
60000
71331
71300Consequently, it can be deduced that this model meets the objective of the work since it allows the fishermen to maximize their profit taking into account the preservation of marine recourses.


## 8. Conclusion

In the present paper, we have studied a bioeconomic tritrophic prey-predator model. We have maximized the profit of two fishermen exploiting the prey, middle predator, and top predator fish populations. The existence and stability of equilibrium point are studied using eigenvalue analysis and Routh-Hurwitz criterion. Using the Nash equilibrium problem and linear complementarity problem we have determined the equilibrium point that maximizes the profits of each fisherman. We have closed this paper by some numerical simulations in order to show the influence of the price on the profits of fishermen.



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*Source: 2381483-2018-01-01.xml*