DYNAMICAL BEHAVIOURS OF PREY-PREDATOR FISHERY MODEL WITH TWO RESERVED AREA FOR PREY IN THE PRESENCE OF TOXICITY AND RESPONSE FUNCTION HOLLING TYPE IV

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. This paper proposes a model with two type of preys and one predator of fishery with Holling type IV function response. The effect of harvesting was incorporated to both populations and thoroughly analysed. We study the dynamics as the prey-predator system of fishing in two fishing zones: a free zone and a reserved zone. The equilibrium points are calculated and the local and global stability conditions of the system are obtained. The local stability conditions were obtained by the Routh-Hurwitz criterion. In addition, the global stability of the coexistence equilibrium point is proved by defining an appropriate Lyapunov function. The optimal harvesting policy is discussed by using the Pontryagin’s Maximal Principle. Finally, numerical simulations are carried to verify the analytical results.


INTRODUCTION
Currently, the relationship between prey and predator has become a very important topic to discuss in ecology. The prey-predator system has attracted many researchers to study the interaction between species (see [4,11,14,24,28,34]). For this, we use the aspect of mathematical ecology to study the interacting species. The current study considers fishing between two areas: a free area and another area reserved for fishing as a case study. However, current trends in species dynamics between the two areas have been obtained by reading various publications such as [13,15,18,23,25]. This literature clearly explains the current trend on this kind of models and fisheries management.
Dubey et al. [8] proposed and analyzed the dynamics of a predator-prey model where one prey species is reserved and the other free for fishing. In addition, it assumes that the predator is not authorized to enter the reserved zone, however it is authorized to access the free zone. Kar et al. [13] they proposed and analyzed a mathematical prey-predator model made up of two prey in two areas (free fishing area, reserved area) and a predator authorized to enter the free area.
The dynamics follow the response of the type I Holling function. Yang et al. [31] proposed and studied a predator-prey model with harvest and reserve area for prey in the presence of toxicity where the dynamics follow a Holling type II response. They studied the boundedness of solutions and the existence of the equilibria of this system. The optimal harvest policy was also discussed. Thus, the Holling II type functional response is fairly representative in the case of the model presented by Yang et al. [31].
In theoretical ecology, the mathematicians and ecologists study more often the interactions between the model prey-predator using the response functions of Holling type (I, II and III) [10]. The functional response of Holling type IV comes in the form : Sokol et al. [27] proposed this function in this form : ax b+cx+x 2 . However, in this paper, the prey-predator interaction modeling using a simplified type IV Holing functional response is expressed by : ax b+x 2 .
When the prey population is low, predators who follow a type IV functional response consume very little. For such predators, the highest consumption of prey occurs when the size of the prey is intermediate. When their numbers is low, the prey therefore undergoes a relaxation of the predation pressure. Therefore, they are opportunistic predators, capable of diversifying the prey they hunt.
The specific study aims to analyze the impact of the type IV functional response on fishing activities between the two zones on the model proposed by Yang et al. [31]. Thus, we are studying a prey-predator system of two prey and a single predator in a two-zone environment : one zone accessible to fishing and the other is prohibited. We assumed that the prey migrated randomly between the two plots. The growth of prey in each plot in the absence of a predator is assumed to be logistic and linear respectively. The Holling response to predation from predators is type IV.
Considering the hypotheses below, we formulate the system of equations of the model as : (1) with initial conditions x(0) > 0, y(0) > 0, z(0) > 0. Here, x ≡ x(t), y ≡ y(t), z ≡ z(t) are the population densities of prey and predator respectively. Model system (1) is defined on the set Further, the biological meanings of parameters are described in Table   1. The prey x in an unreserved area, reproduces in logistic terms of carrying capacity K and intrinsic growth rate r 1 according to the term r 1 x 1 − x K . For the prey y in the reserve zone, it increases linearly according to the term r 2 y. It is assumed that there is a migration from the unreserved area to the reserved area of the prey species x according to the term σ 1 x and vice versa for the prey y following the term σ 2 y. The Interaction between prey x and predator z is expressed by Holling type-IV functional response (see [29,30] If there is no migration of fish population from reserved area to unreserved area (σ 2 = 0) and (r 1 −σ 1 −q 1 E < 0), we find thatẋ < 0. Similarly, if there is no migration of fish population from unreserved area to reserved area (σ 1 = 0) and So, from our analysis, we suppose that : What follows is organized in the following way. In the next section, we show the bournitude system solutions (1). In section 3-4, we study the existence and stability of all the equilibria of our model. Then, we discuss the optimal harvesting policy of the system (1) in section 5.
Finally, we present the numerical simulations to study the stability of the equilibria.

BASIC RESULTS AND EXISTENCE OF EQUILIBRIA
In the first part of this section, we demonstrate the boundedness of the solutions of our system (1). From the point of view of biology, we are only interested in the dynamics of system (1) in the first octant R 3 + .
Lemma 2.1. The set Ω = (x, y, z) ∈ R 3 + : x + y + 1 β z ≤ H d+w+q 2 E is a region of attraction for all solutions initiating in the interior of the positive octant, where Proof. Let us consider X(t) = x(t) + y(t) + 1 β z(t), then the time derivative along the solutions of the system (1) is given by Applying the theory of differential inequality [3,17], we get and when t → ∞, 0 < X(t) ≤ H d+w+q 2 E , proving the Lemma.
In this section, we discuss the existence of positive equilibria of system (1).
Equilibria of model (1) is obtained by solving dx/dt = dy/dt = dz/dt = 0. It can be checked that model (1) has four positive equilibria.
(2) The equilibrium pointP(x,ȳ, 0), where (x,ȳ) is the positive solution of the following equations : After the calculations, x is satisfied by the following cubic equation, where The above equation (4) had a unique positive solution if the following inequalities hold.
We find that a 1 < 0 and, according to the criteria of Descartes [7] it is necessary to impose that : Then, we obtain Thus, from the first equation of (3), we deducē y =x σ 2 (3) For the interior equilibrium P * (x * , y * , z * ), (x * , y * , z * ) is the positive solution of the following equations : From the last equation of (7), we get two positive solutions : Substitute x * i for i = 1, 2 in the second equation of (7), we get : Using the third equation of (7), we obtain : We therefore state the existence of the positives equilibrium points in the following theorem.

LOCAL STABILITY OF EQUILIBRIUM POINTS
In this section, we will study local stability by finding the eigenvalues of the Jacobian matrix Theorem 3.1. The equilibrium P 0 (0, 0, 0) of the system (1) is unstable.
Proof. The characteristic equation of P 0 (0, 0, 0) is : Let λ 2 and λ 3 be the two other eigenvalues. Evidently Therefore λ 2 and λ 3 have one positive value. Hence, Theorem 3.2. The equilibrium pointP(x,ȳ, 0) of the system (1) is locally asymptotically stable Proof. The characteristic equation atP is written as : . Let λ 2 and λ 3 be two other eigenvalues. Then λ 2 and λ 3 are the root of the equation : (13) and (14) are realized, which are given in the proof.
Using the Routh-Hurwitz criteria, it is to check that all roots of the equation had a negative real parts if (13) and (14) realized, then P * 1 is locally asymptotically stable.
Proof. The characteristic equation of the system (1) at P * 2 is : Using the Routh-Hurwitz criteria, based on (9) it is easy to check that b 0 < 0. Then P * 2 is unstable.

GLOBAL STABILITY OF THE EQUILIBRIUM POINTSP AND
To prove the global stability of the interiors equilibrium pointsP and P * 1 , we state and prove the following theorems. Proof. We consider the following positive definite function aboutP : where l is a positive constant, to be chosen later on.
Differentiating V with respect to time along the solutions of model (1), After the calculations we obtain : Therefore, the equilibriumP is globally asymptotically stable.
Proof. We utilize the geometric approach [16] to prove a global stability of P 1 (x * 1 , y * 1 , z * 1 ) The variational matrix J(x, y, z) of the system (1) is : and its second additive compound matrix A [2] is : . We set B as the following diagonal matrix : So we get : [2] B −1 can be written in the following block form : We consider the norm in R 3 as : Where |(u, v, w)| is a vector in R 3 and denote by η(B) the Lozinskiǐ measure with respect to this norm is given by : ; r 2 − 2vy Then, g 1 and g 2 are expressed in this form : Hence, from (15) we obtain η(C) ≤ sup{g 1 , g 2 } =ẋ x + max{ξ 1 , ξ 2 } where : where ε = in f (x, y, z) is the constant of uniform persistence. Now, we impose that (16) aε(1−β ) Then, from the above calculation we get The model (1) is globally asymptotically stable if (16) realized.

OPTIMAL HARVESTING POLICY
Recently, the optimal harvesting policy has played an important role in the field of biomathematics. Increasingly the inclusion of economic factors in population models, optimal harvesting policies is one of the most important problems taking into account both biological and economic. In this section, we wish to evaluate the economic performance of reserve-based management with the employment of the Pontryagin's Principle. The objective is to select a harvest method which can maximize the net profit. The present value I of a continuous time-stream of revenues is given by : where D be the constant Harvesting cost per unit effort, p 1 is the constant price per unit biomass of the prey in the unreserved zone, p 2 is the constant price per unit biomass of the predator and δ is the instantaneous discount rate.
Thus, our objective is to maximize I subject to state equations (1) and to the control constraints : To solve this optimization problem, we utilize the Pontryagin's maximal Principle (see [6,26,33]). The associated Hamiltonian of the problem is given by : where λ 1 , λ 2 and λ 3 are the adjoint variables and Now, the adjoint equation are : And then since H is linear in the control variable E. The optimal control E which maximizes H must satisfy the following conditions : The function λ i e δt , (i = 1, 2, 3) is the usual shadow price and p 1 q 1 x + p 2 q 2 z − D is the net economic revenue on a unit harvest. Economically, if the first condition of (18) is satisfied, after the payment of all expenditure, we obtains a positive profit. According to the second condition of (18), we have a negative profits that is to say to a loss, then the fisherman will not exert any effort.
When ∂ H ∂ E = 0, i.e. when the shadow price equals the net economic revenue on a unit harvest, the Hamiltonian H becomes independent of the variable E. This is necessary condition for the singular control E * to be optimal over the control set 0 < E * < E max .
Thus, the optimal harvesting policy is : When ∂ H ∂ E = 0, it follows that : This involves that the user's cost of harvest per unit of effort equals the discounted value of the future marginal profit of the effort at the steady-state level. In the interior of equilibria P * 1 . The third equation of (17) in the following form : where : . By calculation, we get λ 3 (t) = M 2 M 1 +δ e −δt . Similarly, where : Then λ 2 (t) = N 2 N 1 +δ e −δt . The first equation of (17) can be written as where It easy to verify that : λ 1 = T 2 T 1 +δ e −δt . According to the above calculation, we have : Thus, we denote the function by : Hence (19) can be written as G(x * 1 ) = 0. After the calculations we get G(0) > 0 There exists a unique positive root x * 1 = x δ of G(x * 1 ) = 0 in the interval 0 < x * 1 < K if the following inequalities hold : (20) G(K) < 0,Ġ(x) < 0 f or 0 < x < K Therefore, we summarize the analysis above by the following theorem.  (20) is satisfied, then the optimal harvesting control E δ and the corresponding solutions x δ , y δ , z δ exist that maximize. Where : Furthermore, the adjoint functions λ 1 , λ 2 and λ 3 exist that satisfy equations (17) with the the conditions λ i (t f ) = 0, i = 1, 2, 3. from equation (19), we have Thus, the net economic revenue π(x, y, z, E,t) = 0.
So, if the discount rate δ is infinite, the economic gain is zero, which leads to the closure of the fishery. Consequently, the interest rate rises, which leads to an increase in the rate of inflation.
Consequently, the real value of the resource is reduced.

NUMERICAL SIMULATIONS
It is difficult to find realistic data in order to validate the results of our model (1). For this reason, we take some hypothetical data in order to illustrate the results that we have already established in the previous sections.
(1) In order to ensure the local stability of the equilibriumP, we consider the following parameters : The system (1) with the initial conditions (x(0), y(0), z(0)) = (20,20,20). For this set of parameters (21), the conditions of existence (2) and (6)  Although, the predator populations rapidly decrease to reach equilibrium. (see Figure   1) (2) We consider the following parameters to prove the local stability of the equilibrium P * 1 :  (2) and (12) (1) approaches to the equilibrium point P * 1 which is locally asymptotically stable. We remark that there are a few oscillations of prey populations x and y, then they decreases to converge towards the equilibrium. So, the predators population z can attack the prey x which implies the diminution of prey. After that, it converges to the equilibrium state. (see Figure 2)

CONCLUSION AND DISCUSSION
In recent years, researchers have proposed various prey-predator models to describe the interaction between the prey and predator population using the holling II function response. In this paper, we have proposed a mathematical model analyze and study the dynamics of preypredator fishing using the function response holling type IV and toxic substances. Interactions between species are based on the following assumptions : • The prey can move between the two fishing zones (free zone and prohibited fishing zone).
• The populations of predators are prohibited in the prohibited fishing area.
• The interaction between species of prois and predators is linked by the Holling type IV functional response.
To describe the behavior of our system (1), we proved the boundedness of the solutions. Then, we studied the existence of equilibria as well as their local and global stability. We studied the global stability of the equilibriumP using the Lyapunov function, and we used the geometric approach to prove the global stability of the equilibrium P * 1 . To study the economic side, we formulated an optimal control problem which is solved by the Pontryagin's Maximum Principle.
Finally, in numerical simulations, we choose the parameters to ensure the overall stability of the system.
In this article, there are factors that are not taken into account. As predator species can access the prohibited fishing area, predators can be divided into several types (Infected and susceptible), and competition factors, etc. So, we will try to take those factors into consideration in a future research.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.