Optimization of Sampling of Small Pelagic Fishes in the Exclusive Economic Zone of Senegal under the Climate Impact

Senegal is in a very favourable geographical position for sea fishing. Its coast has an upwelling favouring a good development of phytoplankton very appreciated by the various fish families that populate its Exclusive Economic Zone (EEZ). The little pelagic fish make up the majority of landings. The dynamics of this family of resources is very complex while its perfect mastery is essential for a fishing well controlled maritime. The mathematical models that exist in the literature have not address the different issues related to maritime fisheries and climate change in the Senegalese fishing areas. The linear programming model in integer numbers has been developed after calculation of equilibrium biomass, catches at equilibrium catchability by the application of Schaefer and Freon models in the Senegalese Economic Exclusive Zone. Two proposals have been developed to better explain the tools used in the writing of the mathematical model. The objective is to maximize the biomass of this family of fish resources on the Petite Côte, Grande Côte and Cape Verde depending on samples and climate change. In the application of the model, real data made it possible to test the Linear program in integer numbers obtained. This optimization study allowed us to find an effective way to maximize recruitment within this resource family. This consists in setting up several less expensive marine refuges to build in the fishing zones targeted by the study. The simulation computer program of the model is presented in the appendix.


Introduction
A marine ecosystem is a natural unit made up of all its components (animals, plants, microorganisms) and physicochemical (abiotic) factors with which they interact.It includes three elements which are the biotope1 ,biocenosis2 and the relationships that can exist and develop within this system.Many scientific teams are studying the impact of anthropogenic pressures and global warming on marine biodiversity, laying and recruitment.Small coastal pelagic fish are all small fish (anchovy, ethmos/bonga, horse mackerel, mackerel, sardine and sardinella) which pass most or almost all of their adult phase on the surface or in full water.These resources constitute the major part of landings in the North-West African sub-region, with annual catches of about 2 million tonnes out of a total of 2.8 million tonnes (Eichelsheim, 2014).These types of fish have a very complex organization and dynamics (Lepage et al. 2012).In 1975, Nihoul used a coupled physical model (salinity, temperature,...)/biology (biomass, energy, ...) three-dimensional to express the change in time and space of a fish resource biomass B. Models such as the so-called global models (Schaefer, 1954), (Fox, 1970) and (Pella and Tomlinson, 1969) have been developed in the same period to meet a pressing need for resource management fisheries.These models differ only in their production function but generally they make a stock assessment over time.The climate aspect is not at the base considers in the parameters of these models but scientists in the field of marine fisheries as (Griffin et al. 1976) and(Freon et al. 1988) introduced a climatic variable (the Eupwelling index) based on these global models.Matrix models making a growth assessment Structured populations in length were developed by Drouineau et al. in 2008.The latter thus presented a model of population dynamics of Merlu under constraints.They constructed a matrix containing abundances in each area and each length class in time steps, a matrix containing survival rates in a time step, a matrix containing migration rates for each length class between different areas, a growth matrix and a recruitment matrix in time steps.The fishing mortality caused by a fishing subunit on the size of a population length class at a time step is the product of the catchability of the fishing subunit.In order to limit the parameters (Froysa et al., 2002) and(Deriso et al., 1985), have modelled catch-ability by a sigmoidalmathematical function representing an increasing rate of retention for trawlers and a gamma law for long-liners and gillnetters.The study of renewable resource management can lead to a series of optimal control problems (Lavigne, 2008).These mathematical questions have been studied by (Clark, Clarke, Munro, 1979) and have given rise to several scientific publications on the same topic.They have based this approach of optimal control in order to find a fishing effort rate so that the current value of the fishery is maximized.Brochier and al. (2015) modelled the importance of establishing an artificial housing area in a marine protected area by the system (predator/prey) 3 The importance of establishing an artificial housing area in a protected marine area.

Presentation of the Senegalese Pelagic Zone
The Senegalese maritime zone is rich in two kinds of pelagic resources.On the high seas, of shore pelagic resources (tropical and small coastal tunas), coastal pelagic resources (sardinella, horse mackerel, mackerel, etc.), which constitute more than 70% of catches made in the exclusive economic zone of Senegal, mainly via artisanal fishing , for the most important of the local consumption (Department of Marine Fisheries, 2016).This distribution of pelagic resources is shown in Fig. 1.

Introduce the Problem
The different models converge towards the same objective which is to make an estimation and an optimal distribution of stock.Climate change is not insignificant nowadays, its consideration in mathematical or environmental models is more than a challenge in the research world.Senegal is involved in several programmes to protect small pelagic fish.Thus, the DPM4 and the CRODT5 work together for a participative management because this resource constitutes the main catch of our artisanal fishermen.These small pelagic fish are migrating towards the Mauritanian coast.In 2011, 29% of marine fish stocks were overfished (FAO, 2014).The purpose of our study is to model the effect of overfishing and the impact of climate change on the dynamics of these small pelagic fish in Senegal, then to propose optimal management solutions.

Method
There are many mathematical models developed according to different parameters.These models make it possible to make projections on the evolution of the fishery and stocks of natural populations.The dynamics of these populations depend strongly on biological factors, such as the rate of population growth, which includes birth and death as well as the movements of individuals (imigration and emigration).Human activity is also an important factor influencing population dynamics.Variables are the rate of growth, the effort that corresponds to the number and duration of exploitation, the costs related to effort and catchability.
The parameters and indices in Table 1 will be used throughout.migration coefficient between zone i and zone j b i constant of the rate of climatic variation in a zone i Alim Diet of pelagic resource R recruitment

Proposition 1
The biomass variation B i with B i ∈ R * + , a pelagic resource exploited and impacted by natural variations in a zone i ∈ N follows the following ordinary dynamic equation and

Proof
The population density of a stock follows the following equation: The models of linear synthetic production are based on the description of the evolution of the relative instantaneous growth rate of the biomass by the logistic curve (Freon et al., 1988).The graphic description of the state the stock in each fishing zone corresponds to the following discrete state representation in discrete time: B t+1,i = B t,i + ∂B ∂t .The knowledge of ∂B ∂t is crucial in the evaluation and optimization of the abstractions performed on a fish stock.In the absence of exploitation in each fishing zone and in modifying the model of Shaefer and Freon by them a variable zone i, is obtained the following relation in a known time: ∂B ∂t 1 Result from the combination of natural variations and fishing levies, is obtained the basic equation of the so-called Schaefer model which, applied to a zone i of Senegal, give the relation (1): = cte, modelise the fact that the habitat change affects both B ∞ i and k i .This relationship applied to fishing areas, allows to identify the highly threatened areas and potentially rich areas.The hydroclimatic phenomena, according to this formulation, can intervene only at two levels, on q i if the catchability varies, or on the pair k i − B ∞ i (the ratio of these two quantities being constant), if These are natural variations of abundance.The relation (1) to equilibrium makes it possible to estimate the current recruitment6 in each fishing zone of Senegal from the data of the Oceanographic Research Center of Dakar Thiaroy (CRODT).If the stock has the equilibrium state and the increase of the biomass is zero, this leads from equation (1): Replacing U e i and Y e i respectively by q i .B e i and f i .U e i , we obtain: is obtained by looking for the value of f i which cancels the derivative of equation ( 3), ie: For obtain the maximum values of U opt i and Y max i , we replace them in q i .B e i and f i .U e i , f i is replaced by the formula of f opt i in these equations.We obtain the following result: 10, No. 4;2018

Proposition 2
Suppose catchability q i , with q i 0 varies from area to the other and that B i ∈ R * + and h i ∈ R * + , the average catches per zone U e i is obtained for the following relation: We obtain f opt i , U opt i and Y max i from the relation (8).
Proof Catchability can affect hydroclimatic phenomena at either of its two components, which are accessibility and vulnerability (Freon et al. 1988).To give just a few examples, in a given fishery, movements of water masses may induce coastal parallel or perpendicular migrations, which will affect the accessibility of the stock.Consider a stock in equilibrium conditions not only with the fishery, but also with the environment.Assuming that catchability varies from one zone i to the other and that g(vi) = B ∞ i .vi , the mean catches per zone U e i are obtained by the following relation: We obtain f opt i , Y max i and U opt i from this last relation.f opt i is obtained by canceling the derivative with respect to q of U e i :

Data Analysis
Table (2) is a result of analysis of real data over a period of thirty years relating to the fishing effort of Petite Cote, Grande Cote and Cape Verde.Verde.
Table (4) shows the results of the analyzes and calculations above.The CRODT data allowed us to obtain answers reflecting the reality of the catches.The data were classified according to the three target areas of the study.There are slight differences in some physical parameters.A simulation of the linear regression function ( 12) on matlab give the figure 2 Despite a large evolution in the number of trips at sea over the years, catches have increased slightly for this species and the higher the Eupwelling index, the more fish there is.These results led to the design of the optimization model below, making it possible to locate the areas in which spatial management measures can be applied.

Location of Marine Refuges
This part is a consequence of part (2) The aim is to locate highly vulnerable areas and to introduce management policies such as the creation of marine refuges, the establishment of artificial reefs.

Hypotheses
• The number of sub-zones for developing marine refuges is fixed at 6 in each region.

Model
In this model we will maximize the function This represents the recruitment of the species in the three fishing zones of the study to maximize its value in these areas.
X i, j = { 1 if the sub-zone j of zone i meets the criteria for the establishment of a marine refuge 0 otherwise

Mathematical Formulation
The objective is to maximize the biomass of pelagic resources in the Senegalese coast by optimizing the recruitment function (13). maximize Under the constraints: The objective (a) is to maximize the recruitment of this resource in the different fishing zones.
The constraint (b) shows that a marine refuge has a unique location.
The constraint (c) informs us that marine refuge development areas should be areas where catches exceed a certain threshold.The constraint (d) shows that areas favorable to marine refuges must exceed a certain amount per unit catch.
Stress (e) tells us that a selected area must have a bathymetry not very favorable to the development of these small pelagic fish with the aim of introducing artificial reefs7 .

Figure 2 .
Figure 2. Catch's evolution by output area

Table 2 .
Table (3) is a result of analysis of actual catch data for the Petite Cote, Grande Cote and Cape Materials and Fishing Gear

Table 3 .
Fishing Efforts and Catches