IMPACT OF POLLUTION ON SARDINE, SARDINELLA, AND MACKEREL FISHERY: A BIOECONOMIC APPROACH

. This paper studies a bioeconomic model of three species of small pelagic marine species in Moroccan coastal areas: Sardine, Sardinella and shark. The model combines competition and predation. Two areas are proposed, one is polluted and the other is not. The model combines a biological part describing the evolution of the biomass of stocks subjected to ﬁshing mortality and an economic part explaining the mortality rate. We study the existence and stability of equilibrium states through eigenvalue analysis and the Routh-Hirwitz criterion, then introduce economic approaches to determine the effort needed to maximize the ﬁshermen’s income. Numerical simulations are performed. The objective of this paper is to study the impact of pollution on the existence, evolution of biomass and predation, ﬁshing effort, catches, and proﬁts


INTRODUCTION
Marine pollution is defined as the direct or indirect introduction of wastes, substances, or energy, including underwater sound sources of human origin, which results or is likely to result in adverse effects on living resources and marine ecosystems [1]. The consequences of marine pollution include a loss of biodiversity [2], risks to human health [3], impediments to marine activities such as fishing, tourism, and recreation [4], and reductions in the amenity value of the marine environment.
This pollution can stem from human activities in the catchment area, including industrial, agricultural, urban, or port origins, that reach the marine environment directly through discharges into the sea or indirectly through rivers [5]. However, the true impact of marine pollution on the environment is difficult to determine as the precise quantities of pollutants that reach the sea are not fully understood [6]. Further research is needed to deepen our knowledge on the transfer of pollutants within watersheds and their fate in transition zones.
Our study focuses on the Moroccan coasts, which, due to its geographical location between Europe and Africa, and between the Mediterranean and the Atlantic, offers a diverse range of ecosystems and marine species. The Mediterranean Sea is a highly productive area for fish, with numerous commercially important species caught in the region [7]. Fishing in the Mediterranean Sea is largely based on the exploitation of pelagic fish species, such as Sardine and Sardinella, and 450 species of fish in total are related to oceanic species found on the coasts of Portugal or Morocco.
We examine two different areas in the Mediterranean Sea: a polluted area (Area A) and a nonpolluted area (Area B). Our focus is on the predator shark, and its relationship with small pelagic species such as Sardine and Sardinella, considering the negative impact of pollution on their existence, evolution, and exploitation. Within the framework of a differential equation-based prey-predator and competition model, our results demonstrate the importance of continuous monitoring of the marine environment to assess its health.
In this context, we can cite these works to demonstrate the importance of incorporating the effects of pollution in bioeconomic models in order to understand the impacts of human activities on marine ecosystems. In this work [8], the authors developed a bioeconomic model to examine the impacts of marine pollution on a fishery system. The model incorporated both biological and economic components, and the authors found that pollution can have significant impacts on the long-term sustainability of the fishery.
In [9], The authors developed a bioeconomic model to assess the impact of pollution on marine biodiversity conservation. The model incorporated economic and ecological variables, including the effects of fishing effort, pollution, and the interplay between predator and prey species.
In this work [10], the athors used a bioeconomic model to evaluate the impacts of marine pollution on fishing activities. The model considers the effects of pollution on fishing costs, revenue, and effort, and assesses the implications for fishing communities. We can cite also [11], where the authors study the predation interaction between phytoplankton and zooplankton under their exploitation in multi-fishing zones using a bioeconomic spatiotemporal discrete model.
The entire domain is represented by a grid of colored cells, with two harvesting control strategies used to guarantee the survival of the organisms.
This work will study a bioeconomic model of three fish populations, Sardine, Sardinella, and shark, combining competition and predation. Our model is based on hypotheses that the three fish populations grow according to a logistic equation and that predators compete with each other for space and food. The bioeconomic model considers the negative effect of pollution on fishing effort, catches, fishermen's profits, and biomasses. In the first part of the work, we will determine the equilibrium points of the biological system and study their stabilities. In a subsequent part, we will introduce economic approaches to determine the effort necessary to maximize each fisherman's net economic income and perform numerical simulations to see the impact of pollution on fishing effort, catches, and profits.

BIOLOGICAL MODEL DESCRIPTION, FORMULATION, AND ANALYSIS
2.1. The mathematical model and the hypotheses. In this study we are interested in the study of three marine populations which are of the prey-predator type. The preys are distributed in two different zones: the first zone is a unpolluted zone A and the second a polluted zone B.
These prey are the preferred prey of predators.
In the unpolluted zone A, the evolution of the biomass of prey in this zone is defined by x A .They grow according to a logistic equation with growth rate r 1 and carrying capacity K 1 .This population is preyed with the response rate α 1 .
In the second area; the polluted area B, the evolution of the biomass of prey in this zone is defined by x B . They grow according to a logistic equation with growth rate r 2 and carrying capacity K 2 . This population is preyed with the response rate α 2 . These preys die by the pollution of this zone by the rate δ . So it is clear that r 2 > δ .
The evolution of the biomass of predators is defined by y.These predators feed on prey from both polluted and unpolluted areas where β 1 , β 2 represent the rate of conversion to predators of prey in non polluted zone and polluted zone respectively. The coefficient d represents the natural mortality coefficient of the predator population. The parameters γ denote the coefficients of toxicity mortality by feeding on prey from the polluted area. So it is clear that α 2 β 2 > γ, d < α 1 β 1 k 1 and δ < α 2 β 2 .
Based on these given assumptions, we find the system that describes the evolution of the biomass of these three marine populations Subject to initial conditions: All the parametres used in this model are assumed to be positive and all variables are not negative. Then the model proposed of the three populations can be rewritten as The system (1) is defined in the field Proof. According to the system of equations (2.1) and the initial conditions we have So, All the solutions are positive.
2) We consider its derivative with respect to time is given by For all η > 0, we have Then, there exists ε > 0, with dϕ dt +ηϕ(t) < ε. By applying the theory of differential inequality, we obtain Hence, all the solutions of the system with initial value in R 3 + are included in the following domain

THE STEADY STATES OF THE SYSTEM
We propose to study the existence of equilibrium states and the stability of the interior equilibrium point of our model [12].
3.1. Existence of different equilibrium points. The equilibrium states of the system are solutions of the following system This system of equations has eight solutions i: The trivial equilibrium point P 1 (0, 0, 0) and the axial equilibrium points ii: The equilibrium points in the plane (x A , y) is P 5 x iii: The equilibrium points in the plane (x B , y) is P 6 0, x The equilibrium points P 7 x A , x B , y (7) , where The system of (3.1) has several solutions, but only one of them can give the coexistence of the biomass of the three species; this solution is the point

THE STABILITY OF THE STEADY STATES
The variational matrix of system (2.1) is as follow The variational matrix of system at the steady state P 1 (0, 0, 0) is Proposition 2. The steady state P 2 (K 1 , 0, 0) is unstable as shown in Figure 1.
The variational matrix of system at the steady state P 2 (K 1 , 0, 0) is Dynamical behaviour and Phase portraits of the three populations for , 0 is unstable, see Figure 2.
The variational matrix of system at the steady state The eigenvalues of P 3 are , 0 is stable, as shown in Figure 3.
The variational matrix of system at the steady state The eigenvalues of P 4 are Proposition 5. The steady state P 5 x is unstable, see Figure 4.
The variational matrix of system at the steady state P 5 x nothing can be concluded, see Figure 5.
The variational matrix of system at the steady state P 6 0, x is: A , x B , y (7) , where is unstable, see Figure 6.
The variational matrix of system at the steady state P 7 x A , x B , y (7) , where is: The eigenvalues of P 7 are The variational matrix of system at the steady state P 8 x ( * ) In this cace, the characteristic polynomial is given by: P(λ ) = a 0 λ 3 + a 1 λ 2 + a 2 λ + a 3 where By using the conditions of stability of Routh-Hurwitz, one can proof that a 0 , a 1 , a 2 , a 3 and a 0 a − a 0 a 3 are positive.
Then the interior equilibrium point P 8 x

PROFIT MAXIMISIZATION
The main objective of this part is to maximize the profits of the fishing fleets which exploit these marine species from their fishing effort (see [13]). For it, the three species proposed in this model are assumed to be caught by three fishing fleets. So the model becomes as follows where q i represents the catchability coefficient and E i j 1≤i, j≤3 represents the fishing effort deployed by fishing fleets to capture the species, it is defined as the product of fishing activity and fishing power.
The solution of the system (2) at bioeconomic equilibrium is given by , j≤3 with b ii < 0 for i = 1, 2, 3. Then we can write the solution of the system(2) in the following matrix form X = −BE + X * We want to maximize the profits of the fishing fleets that exploit these marine species. According to Gordon, the profit formula is defined by Therefore the final formula of Total Revenue We thus obtain the final formula of the profit of fisherman i is given by In order to maximize the profits of the fishermen, we must first of all take into consideration the maintenance of the biodiversity of the three marine species, so we will assume that all the biomasses remain positive Y = −BY + Y * ≥ 0 i.e. for the fisherman i we must have Each of the three fleets tries to maximize their profits and achieve a fishing effort that is an optimal response to the effort of the other fishing fleets. And so, we have a Nash equilibrium situation where the strategy of each fishing fleet is optimal, taking into account the strategy of the other fishing fleets (see [14] and [15]). This problem can be translated mathematically into the following three problems: The first fishing fleet must solve this problem (P) 1 The seconde fishing fleet must solve this problem (P) 2 The third fishing fleet must solve this problem (P) 3 The point E 1 , E 2 , E 3 is called the Nash equilibrium point if and only if E 1 is a solution of the problem (P) 1 for given E 3 , E 2 and E 2 is a solution of the problem (P) 2 for given E 1 , E 3 and E 3 is a solution of the problem (P) 3 for given E 1 , E 2 .
In order to find our Nash equilibrium point we will use the essential conditions of Karush-Kuhn-Tucker (KKT). By applying these conditions to the first problem (P) 1 this will give us the existence of constants u 1 ∈ R 3 + , v 1 ∈ R 3 + and µ 1 ∈ R 3 + such that By applying the conditions of (KKT) to the second problem (P) 2 , this will give us the existence In the same way, by applying the essential conditions of (KKT) to (P) 3 , this will give us the existence of constants u 3 ∈ R 3 + , v 3 ∈ R 3 + and µ 3 ∈ R 3 + such that From the previous problems we get the following expressions We have the scalar product of µ i and v i is zero and to maintain the biodiversity of the three marine species, it is natural to assume that all biomasses remain strictly positive, i.e. Y * > 0 then v i > 0 and µ i = 0, ∀i = 1, 2, 3. We note v : We can also write it in the following matrix form w = NL + q where The generalized Nash equilibrium problem is equivalent to the following Linear Complementarity Problem LCP(N, q) : find vectors w, L ∈ R 16 such that The LCP(N, q) has a unique solution for every q if and only if N is a P−matrix.
Since the matrix N of our problem is P−matrix, we can deduce that the linear complementarity problem LCP(N, q) admits one and only one solution. The solution is given by Finally, we obtain the fishing effort that maximizes the profit of the first fisherman for caching the prey population in non-polluted zone as follow the fishing effort that maximizes the profit of the first fisherman for caching the prey population in polluted zone as follow the fishing effort that maximizes the profit of the first fisherman for caching the predator population as follow the fishing effort that maximizes the profit of the second fisherman for caching the prey population in non-polluted zone as follow the fishing effort that maximizes the profit of the second fisherman for caching the prey population in polluted zone as follow the fishing effort that maximizes the profit of the second fisherman for caching the predator population as follow the fishing effort that maximizes the profit of the third fisherman for caching the prey population in non-polluted zone as follow the fishing effort that maximizes the profit of the third fisherman for caching the prey population in polluted zone as follow the fishing effort that maximizes the profit of the third fisherman for caching the predator population as follow

NUMERICAL SIMULATION
In this part, we will see the impact of pollution on the profits of fishermen, their fishing effort as well as on the catches made by these fishermen. For the catches of the first species is higher followed by the catches of the second species then the catches of the third species. For the catches of the first species, we note that the catches are almost stable with a small variation which is almost negligible even if the pollution rate increases.

FIGURE 8. Evolution of catches in relation to pollution rate
For the catches of the second species, we notice that from the value 0.1 to the value 0.4 of the pollution rate, the catches are almost stable with a very small decrease and from the pollution rate equal to 0.5 up to the value 1 we notice a large reduction in catches.
For the captures of the third species, we note that from the value 0.1 to the value 0.4 of the pollution rate, the captures are almost stable and from the pollution rate equal to 0.5 to the value 1 we notice a large decrease in the level of catches of this species. Thus, the total catches of these three marine species are also decreasing.  We now move on to the fishing effort deployed by fishermen to capture these three marine species where the orange bars represent the fishing effort deployed to capture the first species, the yellow bars represent the fishing effort deployed to capture the second species and the green bars represent the fishing effort deployed to catch the first species while the blue curve represents the total fishing effort as shown in Figure 10. We note that the fishing effort deployed to catch the first species is almost the same for all values of the pollution rate. For the fishing effort deployed to catch the second species, we notice that from the pollution rate equal to 0.1 up to the value 0.4 the effort was almost stable then from the value 0.5 up to the value 1 we notice a large decrease in fishing effort. Similarly for the fishing effort deployed to capture the third species, we note that the effort was stable from the value 0.1 up to the value 0.4 and after this value we notice a large decrease in the level of the fishing effort. The explanation that can be given for the results obtained in Figures 8, 9 and 10 is that for the first species which is found in an unpolluted environment, the fishing effort, the catches and the profits have not changed since this area is not polluted and this species has no direct interaction with other species found in a polluted area. For the second species which is in a polluted area and which is the prey of the third species, where the latter feeds on the two species the first which is found in an unpolluted area and the second in a polluted area, the second species after a certain higher value this species begins to be reduced by this pollution and even to die which directly affects the predator which is gradually reduced and its biomass too. This reduction in the level of the second and third species thus implies a reduction in the level of fishing effort because pollution makes marine species more vulnerable to capture. This reduction in the level of fishing effort thus implies a reduction in the level of captures and therefore a reduction in the level of captures since this-Pollution has a negative effect on the seabed and the entire ecosystem because it can kill these species or cause diseases that can be transmitted to humans.

CONCLUSION
In this work, we conclude that the effect of pollution on fishery resources is significant, particularly for both polluted and unpolluted marine areas. Our bioeconomic model, which considers the interplay between competition and predation among three fish populations (Sardine, Sardinella, and Shark) and the impact of pollution on fishing effort, catches, fishermen's profits, and biomasses, is proposed as a tool to study this issue. Our results showed the importance of controlling the exploitation of this marine population in ensuring their sustainability, and highlighted the critical role that pollution plays in affecting the mortality rates of fishery resources in both polluted and unpolluted marine areas also has a significant impact on fishing effort, catches, and profits in the fishery industry.