FISHING ACTIVITY IN THE ATLANTIC MOROCCAN OCEAN: MATHEMATICAL MODELING AND OPTIMAL CONTROL

. As it spans the Mediterranean sea and the Atlantic Ocean on the north and west respectively, Morocco is characterized by its marine sources’ diversity. Fishing Activity is considered to be the backbone of the country’s economy. the process of structuring the exploited population, and the marine environment are two major ﬁelds that the kingdom must understand to implement management measures to sustain the equilibrium of national ﬁsheries and marine sources. Here, we have built a discrete-time model to describe the dynamic of ﬁshing of two small pelagic species on the Atlantic Coast. The equilibrium between marine population and ﬁshing effort, and maximizing proﬁt are what the control parameter proposed have to achieve. We use a discrete version of Pontryagin’s principal maximum to calculate the optimal system. The numerical simulation is carried out using Matlab.


INTRODUCTION
According to [1], in Morocco "The most important stocks remains too small pelagics whose production exceeds 80% of the weight landed at the national level. 2017 was characterized by  In 2006, the catch of anchovies (Engraulis encrasicolus) amounted to around 10,000 tonnes, which accounts for about 1 percent of the total small pelagic catch.
According to [1], 1.46 million tonnes of small pelagics, composed of Sardine 73%, followed by Chub Mackerel 17%, Chinchard 7% and Sardinelle 2% and Anchors 1%, the important capture are realized in the Sud and Centre Zone.
In the North, Centre, and Sud Zones, Sardine Pilchardus and Chub Mackerel distribution continue all along the coast of this area, they grow in abundance along the Atlantic Ocean.   According to [4], Colin Clark published a book entitled Mathematical Bioeconomics: The optimal management of renewable resources in 1976, this book influenced many researchers to work on problems of management of natural resources. This book shows the importance of biological and economic aspects when studying natural resources. Among these works we can refer to [5] this paper deals with the problem of combined harvesting of two competing fish species, it analyses the dynamic behavior of the exploited system and studies some aspects of the optimal harvest policy. In [6] Chaudhuri studies the problem of dynamic optimization of the exploitation policy connected with the combined harvesting of two competing fish species, and discusses the possibilities of the existence of bioeconomic equilibrium in a system of combined harvesting of the prey-predator system in [7]. Also we can refer to ([8] [9] [10] [11] [12]).
All works take problem of management of nature maritime resources in the point of view of stability.
In this work, we propose to study a specific prey-predator model, witch the prey is two marine species ( Sarcina Pilchardus and Scomber colias) the predator are fishing efforts of both (Sardina P and Scomber C), plus two equations describing the dynamics of the market price of Sardine and Chub Mackerel. We assume that Sardine and Chub Mackerel compete for food and space. The model in question is a discrete multi-region system since the distribution of Sardine and Chub Mackerel continues along the Atlantic sea. Using a discrete version of Pontryagin's Maximum Principle, we built the optimality system using two controls u 1 u 2 (Conserving an equilibrium state between Sardine(Sardina Pilcardus) and its harvesting and Conserving an equilibrium state between Chub Mackerel (Scomber Colias) and its harvesting, also maximizing the profit). The optimal result is tested using an iterative function compiled on Matlab.

PRESENTATION OF MODEL
The model that we will study is a discrete multi-region model, which describes the fishing of two competing pelagic species x (Sardina Pilchardus) and y (Scomber Colias); Essentially these two species (Sardine and Chub Mackerel are found all along the Atlantic (North Zone, Center Zone and finally South Zone). We focus the study on the Atlantic since statistics in the Mediterranean are rare.
The model describing the dynamic of fishing of Sardine and Chub Mackerel is: ,0 > 0. Our model contains six equations that we can divide into three parts. The first part contains the first and second equations describing the evolution of the biomass of sardine (Sardina pilchardus) and Chub Mackerel (Scomber Colias) in presence of harvesting and competitions between these two species. The second part contains the third and fourth equations describing the evolution of the fishing efforts for both Sardine and Chub Mackerel, whereas the third part describes the evolution of the market price of these species.
We assume in this model that the evolution of Sardine x and Chub Mackerel y obey to a discrete version of the law of logistic growth. r ). The biomass of these species is decreasing by harvesting. We choose a general harvesting function ( [13]) for , and the same general harvesting function is used for decreasing the biomass of Chub mackerel y q The biomass x and y are also decreasing by competition for food and spaces between this two species θ x is the intrinsic growth rate of stock x and y respictively. The variation of the price market for x and y is the difference between the demand

AN OPTIMAL CONTROL PROBLEM
On the first hand, if fishing is carried on at specific periods in the year when maritime reserves are ecologically low in density, there is a risk of the extinction of some marine species.
On other hand, if vessels' activity lowered fishing activity will disappear. u 1 , u 2 will be restricting vessels to use certain fishing techniques, to limit the catch especially when the number of biomass decrease for some natural raisins. The tables below reflect the evolution of the biomass of Sardine and Chub mackerel in Morocco in 2015 and 2007.
Therefore, the balance must be struck between maritime resources density and the frequency of fishing activity.
According to the report of 2015 In this work, we introduce two controls parameter. The first one is called u 1 which is the parameter control that leads to conserving state equilibrium between Sardine and its harvesting. The second parameter control is u 2 which leads to conserving stable equilibrium between Chub Mackerel and its harvesting. u 1 , u 2 are real constant choosing also to bring much profit.
Our goal is to maximize the objective functional presented as follow, for j ∈ {1, 2, 3} Were the parametre ρ 1 > 0, ρ 2 > 0, ξ 1 > 0 and ξ 2 are the cost coefficients, they are selected to weigh the relative importance u 1,i , u 2,i at time i. t f is the final time.

Existence theorem.
To demonstrate the existence of the optimal pair for J, we use a result of Dabbs and present in ( [16], [17], [18], [19]). Theorem 1. There exists an optimal control (u Proof. For all j ∈ {1, 2, 3}, since the coefficients of the state equations are bounded and there are a finite number of time steps, and corresponding sequences of states Since there is a finite number of uniformly bounded sequences, there exist (u Finally, due to the finite dimensional structure of system (4) In ordre to calculate the adjoints functions, we apply the Pontryagin's principle ( [20], [[21], [22]). ).

Numerical Simulation.
Our aim goal in this section is to test how successful the controls u 1 , u 2 are. So, we solve the optimality system numerically using an iterative method. Note that we calculate the state system with an initial guess using a progressive schema in time, while the adjoint functions with transversality conditions are obtained using a regressive schema in time.
We concentrate the calculus to the Centre and Sud Zones in the Atlantic ocean since we have statistics in just these two zones.
The statistics for Sardine (Sardina P) [3]. The statistics for Chub Mackerel ( Scomber C) Intrinsic growth rate r 1.55 Carring Capacity K (thousand of tonnes) 602 Centre Zone: For the centre Zone, the data used in centre zone are, In the center zone, we observe that for α 1 = α 2 = 0.44 β 1 = β 2 = 0.48 the effort decrease in general since the biomass x and y decrease, but the effort provided to fish with control is less than that provided to fish without control in both biomass Sardine and Chub Mackerel.
Furthermore, the price market of Sardine and Chub Mackerel with control is greater than the price market of Sardine and Chub Mackerel than the price with no control, which will reflect positively on the profit and we see the fishing activity is not stopped at the end of the simulation.
For the second case α 1 = α 2 = 0.66 β 1 = β 2 = 0.60 we observe that the effort decrease in general but the effort with control provided to fish with control is less than that one provided For α 1 = α 2 = 0.44 β 1 = β 2 = 0.48 the effort decrease in general, but the effort provided to fish with control is less than that provided to fish without control for both Sardine and Chub Mackerel. Furthermore, the price market of Sardine and Chub Mackerel with control is greater than the price market of Sardine and Chub Mackerel than the price with no control. For the second and third cases α 1 = α 2 = 0.66 β 1 = β 2 = 0.60 α 1 = α 2 = 0.44 β 1 = β 2 = 0.48 , we observe that the effort decrease in general but the effort provided to fish with control is less than that one provided to fish without control for both Sardine et Chub Mackerel. Figure 5, show the development of E 1 , E 2 , P 1 , andP 2 in the presence, absence of controls. We notice on day 4 and day 3 respectively, that E 1 andE 2 without control respectively start to rise, as we notice that values reach by E 1 without control are greater than values reaches by E 2 without control. While, values reach by E 1 with control remaining constant, and on the fifth day, these values increase very slightly. On the contrary, E 2 with control remains constant until the last day of hunting, when a slight decrease in its value is noted.

Discussions.
Regarding P 1 without control, the value keeps increasing until the penultimate day when this value decreases. As for, P 2 without control increases until the seventh day of hunting. On the other hand, we note that the value of each P 1 , P 2 with control respectively rise but on the fourth, and fifth days respectively these values begin to decrease.
And from it, we conclude, in the case of development of the system without control with high values of E 1 andE 2 and low values of P 1 andP 2 this will make losses higher than profits or that will make benefits very low. The opposite occurs when the system develops with control with a low value of E 1 E 2 and a high value of P 1 , P 2 , which will make the profits bigger or losses will be lower than profits.  Simulation 7, the case of (α 1 = α 2 = 0.44 and β 1 = β 2 = 0.48), values of E 1 without-with decrease from the first day, the difference between them begins to become clear from the fourth day, but remains very small. Same thing for E 2 with the presence and absence of controls.
For the center Zone, with the Schaefer parameter the development of the system with control gives good results since efforts (E 1 , E 2 ) decrease and the price (P 1 , P 2 ) increase with these parameter. We conclude that there is a difference with without control in the system but it remains very small.
Simulation 8, case of (α 1 = α 2 = 1 and β 1 = β 2 = 1), we note that the value of E 1 without control increase, same for E 2 without control. On the other, we note that the value of P 1 without control decreases same for P 2 without control. And on the contrary E 1 , E 2 with control decrease while, P 1 , P 2 with control increase. As here as before the system with control makes benefits big than losses.
Simulation 9, case of (α 1 = α 2 = 0.65 and β 1 = β 2 = 0.60) in general both E 1 , E 2 in the presence-absence of control decrease but, values of E 1 , E 2 with control respectively are lower than value of E 1 , E 2 without control. Simulation 9, the difference between E 1 , E 2 withoutwith control is very small.

CONCLUSION
Fishing is the backbone of Morocco's economy. As a result of this, several studies have been devoted to researching the biological properties of marine resources and so on, to find ways to catch these resources without compromising the biodiversity that characterizes Morocco.
In this paper, we built a discrete multi-region model to describe the fishing activity for both Sardine and Chub Mackerel. We introduce lately two controls to weigh the fishing activity and the biomass of that two species. To calculate the optimality system, we use a discrete version of Pontryagin's maximum principle to calculate the adjoint function and deduce the optimality system. At the end of the simulation, we observe that the fishing activity is not stopped and also the effort provides to fish Sardine and Chub Mackerel with control is less than that one provided to fish with control, and for some cases, we observe that the price market for both Sardine and Chub mackerel is higher than the price with no controls which reflect positively to the profit.

DECLARATIONS
No funds, grants, or other support were received.
This work is original and not have been published elsewhere in any form or language.
The datasets generated during and/or analyzed during the current study are available in Laboratoires centraux-Casablanca,Département pêche, Etat des stocks et des pêcheries Marocaines