Journal of Applied Mathematics

Journal of Applied Mathematics / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 8892388 | https://doi.org/10.1155/2021/8892388

Nossaiba Baba, Imane Agmour, Yousef El Foutayeni, Naceur Achtaich, "Bioeconomic-Epidemiological Model of Scomber colias Population in the Moroccan Coasts", Journal of Applied Mathematics, vol. 2021, Article ID 8892388, 22 pages, 2021. https://doi.org/10.1155/2021/8892388

Bioeconomic-Epidemiological Model of Scomber colias Population in the Moroccan Coasts

Academic Editor: Fernando Sim es
Received25 Aug 2020
Revised28 Jan 2021
Accepted22 Feb 2021
Published20 Mar 2021

Abstract

In this paper, we develop and study a mathematical model for the dynamics of Scomber colias and Thunnus thynnus prey-predator with parasitic helminths. We search to analyze a bioeconomic model in which both susceptible and infected prey populations Scomber colias are exposed to the predator Thunnus thynnus, with varying degrees of exposure. However, the predator feeds preferentially on the most numerous prey types. This implies a kind of switching from the susceptible class to the infected class, and vice versa, as these two types of prey change in numerical superiority. So, the positivity, boundedness, equilibria, stability, and bioeconomic equilibrium are studied. Some numerical simulation of stability is cited. For giving a high yield and keeping the Scomber colias and Thunnus thynnus populations away from extension, we use the Maximum Principle of Pontryagin.

1. Introduction

Morocco is one of the largest fish producers in the world, according to the 2018 report from the Food and Agriculture Organization of the United Nations (FAO). In 2018, national fishery production totaled 1371683 tons for a turnover of 11579544 thousand MAD (Moroccan Dirhams). The export volume reached more than 722921 tons for a turnover of 22,531 million MAD (DPM, 2018) [1]. A study published in December 2019 reveals that the main reason for the repression of Moroccan fish is the presence of parasites. The study was carried out by the Hassan II Agronomic and Veterinary Institute (HAVI) and the center specializing in the pathology of aquatic animals at the National Fisheries Research Institute (NFRI). The parasites are the source of major problems with natural fish stocks. Indeed, the parasites determine pathologies slowing the growth and increasing the mortality of their hosts and constitute, consequently, a limiting factor of the success of the productivity in aquaculture. Following the analysis concerning 1678 pieces of fish of different species, 537 of which come from the Atlantic (port of Essaouira and wholesale market of Casablanca) and 1141 pieces from the Mediterranean, the extent of parasitism is almost the same on the Mediterranean coast (31.1%) and the Atlantic (32%). Nematodes (anisakis + acanthocephalans) occupy most of the parasites in the Atlantic and the Mediterranean, with respective prevalence of 21.4% and 24.9%. The plerocercoid larva of the cestode was found in silver saber, with a prevalence of 8.3% at the Atlantic level, as is shown in Figure 1. It should be noted that to be contaminated, the fish must feed on plants or animals carrying parasites. In fact, several animal species can carry parasites and contaminate the fish’s environment through their excrement, which sometimes contains parasite eggs. To develop, these eggs will infect plankton, small crustaceans, snails, etc. If fish eat these organisms, they in turn can carry parasites. These reasons encouraged us to study this phenomenon and to see all the interactions that occur between populations of fish. In this context, many mathematical models have been developed to describe the dynamics of fisheries, and we can refer, for example, [27]. Moreover, we can cite [8], in this work, the authors have defined a bioeconomic equilibrium model for Parapeaneus longirostris and small pelagic fish populations in two different areas; the first one is protected against fishing and the second is a free access zone.

They studied the influence of the predator mortality rate variation on the evolution of prey biomass and the profit of coastal trawlers. In [9], the authors have sought to highlight that the increase of the carrying capacity of marine species does not always lead to an increase on the catch levels and on the incomes. They considered a bioeconomic model of Sardina pilchardus, Engraulis encrasicolus, and Xiphias gladius marine species that are exploited by several seiners in the Moroccan Atlantic zone based on the parameters given by NFRI. In [10], the authors studied a model consisting of susceptible and infected prey populations and a predator population. More precisely, they assume that the likelihood of a predator catching a susceptible prey or an infected prey is proportional to the numbers of these two different types of prey species. They assume that the predator will eventually die as a result of eating infected prey. In previous works, in particular, in the works [1123], the authors did not take into account parasitic helminths which have a negative impact on the evolution and behavior of populations. Therefore, the purpose of this paper is to identify parasitic Helminths in Scomber colias and Thunnus thynnus of great economic importance, which occupy a prominent place in the trophodynamic chain. We search to study a bioeconomic model in which both susceptible and infected prey populations (Scomber colias) are exposed to the predator (Thunnus thynnus), with varying degrees of exposure. However, the predator feeds preferentially on the most numerous prey types. This implies a kind of switching from the susceptible class to the infected class, and vice versa, as these two types of prey change in numerical superiority. Switching may simply come about due to individual behavior changing with varying abundance of that class of prey. The paper is organized as follows. In the next section, we present the biological model of susceptible and infected Scomber colias with the presence of the predators Thunnus thynnus; in other words, we resolve a system of three ordinary differential equations, the first equation describes the natural growth of the susceptible Scomber colias fish population a prey of the Thunnus thynnus fish population, the second equation describes the natural growth of infected Scomber colias fish population a prey of the Thunnus thynnus fish population, and the third equation describes the natural growth of the Thunnus thynnus fish population as a predator of the susceptible and infected Scomber colias. The basic reproduction number, positivity, and boundedness are studied in the first part. The existence of the steady states of this system and their stability are studied using eigenvalue analysis, and we define a bioeconomic equilibrium model for these fish populations exploited by a fishing fleet. In Section 3, we compute some numerical simulations to determine the optimal conditions under which the biological steady state can be attained. In Section 4, we give a numerical simulation of the mathematical model and discussion of the results. Finally, we give a conclusion and some potential perspectives in Section 5.

2. Biological Model

Our study is based on an epidemiological model which describes the interaction between the susceptible and infected fish population Scomber colias and their predators Thunnus thynnus. We assume that the disease is transmitted by direct contact with the prey. Due to the presence of the disease, the prey population is divided into two disjoint classes, the susceptible fish population denoted by per unit designated area and the infected fish population denoted by per unit designated area. Therefore, at time , the total Scomber colias population is . The predator population per unit designated area is denoted by . If there is no predation and no infection, then the fish population follows a logistic growth model. Both susceptible and infected Scomber colias are preyed upon by Thunnus thynnus, but the Thunnus thynnus preferentially eats the infected fish, because the infection causes the fish to be more vulnerable to predation (as shown in Figure 2).

Let be the density of susceptible Scomber colias at time . This population grows according to a logistic equation with growth rate and carrying capacity in the absence of the disease. The parameter represents the rate of transmission multiplied by the area under consideration, measured in units of area. The susceptible Scomber colias is a prey of the Thunnus thynnus with the response rate as follows

Let be the density of infected Scomber colias with the rate of transmission of the disease . This population is preyed by the Thunnus thynnus with the response rate . The infected Scomber colias populations died by the death rate as follows

The Thunnus thynnus is a predator of susceptible Scomber colias and infected Scomber colias where represents the rate of conversion to Thunnus thynnus of susceptible mackerel, is the rate of conversion to Thunnus thynnus of infected Scomber colias, and is the death rate of the Thunnus thynnus as follows

Following the previous assumptions, the biological model is formulated as follows

We assume that all parameters are positive, . The system of equations (4) has a unique solution in since the right-hand sides of them are Lipschitz continuous in .

At the term is not defined. As for and are small, and so it is natural to interpret as zero at For we have the same interpretation at

3. Biological Model Analysis

3.1. Basic Reproduction Number

We assume that we have the following SI model

By dividing the second equation by , we obtain Then, for , each infected individual will contaminate more than one susceptible individual, and the disease will spread to an ever-increasing number of individuals. This will remain the case until the number of susceptible is such that The ratio can then be interpreted as the number of contacts that can transmit the disease by infected individuals throughout their period of contagion. If , there will necessarily be an epidemic, whereas in the opposite case, if , only a few individuals will be infected before the spread of the disease does not stop by itself. Let where This threshold is the basic reproduction rate.

3.2. Positivity

We will denote by

Then, the system (4) becomes subject to the initial condition MS (0) > 0,MI (0) > 0 and T (0) > 0. The system (8) is defined on the set

Let be the solution of the system (4) at the biological equilibrium. Then, all the solutions of the system (4) are nonnegative. By system of equations (8) with initial condition, we have

Therefore, all solutions starting from an interior of the first octant remain in it for all future time.

3.3. Boundedness

We define the function , and we take the time derivative along the solution of the system For any positive constant , we have

We choose as a minimum of . On the one hand, we have

Consequently, Then, and That improves that the susceptible and infected preys are always bounded. On the other hand, there exists such that , thus, for

Which gives, for

Hence, that independently of the initial conditions. Finally, we conclude that the trajectories of the system are bounded.

3.4. Equilibrium Points

For the system (4), we can see that the equilibrium points satisfy the following equations

After calculation, it is obvious that system (15) has five equilibrium points as follows

For the last equilibrium point is given by where is given in the following calculation.

Since so we obtain . Hence, there is no economic equilibrium when

Consider the function defined on an interval by where

If i.e., , then, equation (18) admits two different solutions: the first one is trivial and the other solution exists if and When , we have with and

Equation (18) admits at least one solution which implies that system (1) has at least one capital-labor equilibrium with

To determine the uniqueness of the solution , we discuss the following cases (i)If then the equation (18) has a unique positive solution given by(ii)If and then equation (18) has a unique positive solution given by (iii)If by applying the rule signs of Descartes, equation (18) has a unique positive root if any of the following three conditions is satisfied

To depress the cubic equation, we substitute and we solve . Then, we get

Then, we obtain where

By using Cardano’s formula, we obtain then

3.5. Local Stability of Equilibrium Points

To ensure the positivity of the population’s biomass, we assure that

We consider the matrix that represents the variational matrix of the system at the equilibrium point where (1)At the equilibrium point , the variational matrix is

It has three eigenvalues: , and Consequently, the point is unstable. (2)At the variational matrix is

It has these eigenvalues:

Hence, the point is unstable. (3)In the same manner, at , the variational matrix is

It has these eigenvalues: where

Then, following the conditions and , we can deduce that the point is unstable. (4)The variational matrix at is given by

Since one of the eigenvalues is positive Then, the point is unstable. (5)The local stability of is investigated by considering the variational matrix based on the system (1) withwhere (i)If thenwhere

In this case, the characteristic polynomial where where

By considering the previous characteristic polynomial, if one of the coefficients is zero or negative while at least one other coefficient is positive, then there is one or more imaginary roots or roots with a positive real part. If all the coefficients are positive, we calculate the Routh table. The Routh Hurwitz tabular is given by with So, according to the Routh Hurwitz stability criterion, we can conclude that the equilibrium point in this case is stable. (ii)If and then

In this case, the characteristic polynomial is as follows: , where where

The Routh Hurwitz tabular is given by with Then, according to the Routh Hurwitz stability criterion, we can conclude that the equilibrium point in this case is stable. (iii)If , thenwhere and

In this case, the characteristic polynomial is as follows: where where , and