Dynamics Analysis and Optimality in Selective Harvesting Predator-Prey Model With Modi

Abstract: In this work, we consider the optimal harvesting and stability problems of a prey-predator model with modi ed Leslie-Gower and Holling-type II functional response. The model is governed by a system of three di erential equations which describe the interactions between prey, predator and harvesting e ort. Boundedness and existence of solutions for this system are showed. The existence and local stability of the possible steady states are analyzed and the conditions of global stability of the interior equilibrium are established by using the Lyapunov function, we prove also the occurrence of Hopf bifurcation at this point. By using the Pontryagin’s maximal principle, we formulate and we solve the problem of the optimal harvest policy. In the end, some numerical simulations are given to support our theoretical results.


Introduction and Mathematical model
In the last recent, the mathematical modeling of prey-predator models since Lotka ( ) and Volterra ( ) is a very interesting area of research for many ecologists, mathematicians and economists. The problem of optimal harvesting in predator-prey systems is a dominant theme in ecology and bio-economics in due to its importance. Many authors have red their models in this theme of Clark [1] who proved the optimal equilibrium policy for joint harvesting of two independent species and he suppose that each population follows a logistic growth law in the absence of harvesting and its harvest rate is proportional to both its stock level and harvesting e ort. This analysis has been studied and analyzed by several authors Braverman and Mamdani [4], Fan and Wang [2], Qiu et al [3]. There are also numerous works on the e ect of harvesting on predator-prey interaction, Kar [8], Brauer and Soudack [15], Pal et al. [9], Kalyan [11], Agarwal [12], Xiao and Ruan [13] and Myerscough et al. [14]. Many researchers concentrate their research work in the eld of optimal management of renewable resources. Costa et al. [16] considered Lotka-Volterra and Leslie-Gower predator-prey model and proved the global stability of a desired equilibrium population. Rebaza [18] studied the boundedness of solutions, existence of bionomic equilibria, positivity of equilibrium states, local and global stability of equilibrium points and peri-odic solutions of a predator-prey model with harvesting. Das et al. [23] introduced prey-predator harvesting model with toxicity and discussed optimal harvesting policy using Pontryagin's maximal principle. Fister and Lenhart [30] studied a predator-prey model, where the predator is represented by the Lotka-Volterra ordinary di erential equation. Zhang and chen [31] considered the stage structured predator prey model and optimal harvesting policy. Always in this topic, Beddington and May [25], Liu [24], Li and Wang [26] have studied the optimal harvesting strategy of stochastic population systems (species a ected by environmental natural factors) and some other authors have discussed and analyzed a prey-predator model with harvesting. The exploitation of biological resources and the harvest of population species are commonly experienced in shery, forestry and wildlife management. In the natural world and among the serious and immediate global problem for future sheries management and sustainable development of ecosystem, is over exploitation of sheries, di erent species of shes are decreased due to enhancement of shing power, high growth rate of world population and lack of knowledge of the characteristics of exploited species. Therefore several authors have studied the optimal harvest policy in shery. Palma [27] found the optimal harvest policy in an open access shery in which both prey and predator populations are exposed to non selective harvesting and the growth rate of prey species is a ected by Allee-e ect. Chen and Hsui [28] investigated the shery policy when considering the future opportunity of harvesting. Dubey and Chandra [29] investigated the dynamics of a sher resource system in an aquatic environment in two zones harvest in reserve area. In Roy and Roy [42], the authors formulated a mathematical model describing the interactions between three species: super predator-predator-prey with di erent types of functional responses. The conditions of local and global stabilities of the interior equilibrium are given. In Roy and Roy [43], a model with four compartments of prey (refuge region-predatory part) and predator (super predator-predator) and only predator population is harvested. The obtained model is studied in terms of stability and Hopf bifurcation. To optimize the utilization of the resource, the authors considered the shing e ort as a control to harvest predator population. On the other hand, in Roy et al. [45] a mathematical model is introduced to investigate the e ect on prey of two predators (predator-generalist predator) and the the e ect of the harvesting e ort for the generalist predator. The conditions of stabilities and Hopf bifurcation are established. In Roy et al. [44], the authors studied the e ect of harvesting on prey and the a ect of time delay on a formulated Holling-Tanner prey-predator model with Beddington-DeAngelis functional response. They studied the stability of the interior equilibrium and the existence of small amplitude periodic solutions and their stability via the Hopf bifurcation Theorem. Motivated by the above works, we consider the following predator-prey model which incorporates the Holling type II and a modi ed Leslie-Gower functional response (see Aziz Alaoui et al. [5]) and takes into account the harvesting in prey population and integrates the equation of variation of the e ort of harvesting: the variable u is the density of prey population and v is the density of predator population at any instant of time t subject to the non-negative initial condition u( ) > and v( ) > , the term c v u+k denotes the functional response of the predator, which is known as Holling type II response function, h(t) > is the rate of prey harvesting. According the catch per unit e ort hypothesis [41], the functional form of harvest is generally considered to de ne the assumption that the catch per unit e ort is proportional to the stock level. The harvest function depends on several factors, revenue, market demand and harvest cost. Therefore, we consider that the harvest e ort is a variable from the real point of view, which takes the following form: where w is the e ort used to harvest the population, m( < m < ) is the fraction of the stock available for harvesting and q is the catchability coe cient.
We suppose that shery e ort itself is a dynamic variable that satis es where p is the price per unit biomass of landed sh, c is the constant shing cost per unit e ort and λ is sti ness parameter.
In this paper, we have considered the following ordinary di erential equations: Our goal, in this paper is to study the dynamics of model presented in paper of Aziz Alaoui and Daher [5] but with presence of harvesting and adding a dynamic variable is shery e ort.
In the absence of harvesting, the rst model proposed in this optic is given by Aziz Aloui et al. [5] as follows: with initial conditions u( ) > and v( ) > , u and v represent population densities at time t. c , a , b , k , c , a and k are model parameters assuming only positive values. a is the growth rate of preys u. a describes the growth rate of predators v. b measures the strength of competition among individuals of species u. c is the maximum value of the per capita reduction of u due to v. c has a similar meaning to c . k measures the extent to which environment provides protection to prey u. k has a similar meaning to k relatively to the predator v. In [5,6], the authors investigated this model in terms of boundedness of solutions, existence of a positively invariant and attracting set, stability analysis of the coexisting interior equilibrium, permanence and extinction, existence and uniqueness of limit cycle. The notion of global stability, existence and uniqueness of limit cycle and Hopf bifurcation are showed by Hsu et al. [32,33,39] and Chakraborty et al. [40]. In [35,36], Ya a et al. studied a delayed version of this model, they showed the existence of periodic solutions and their stability. The global stability and persistence are proved in [37,38] by using lyapunov function. The organization of this paper is as follows. In section 2, we prove the existence of the equilibrium points and the local stability of the trivial steady state and the boundedness of solutions. Section 3 is devoted to the local and global stability of the nontrivial steady state. In section 4, we establish the conditions of the occurrence of Hopf bifurcation. In Section 5, we solve the problem of determining the optimal harvest policy by using Pontryagin's maximal principle. Numerical simulations are performed to illustrate the e ectiveness of our results in section 6.

Preliminaries . Boundedness, boundary equilibria
In this subsection, we recall some results on model (1.4) such as boundedness, existence of boundary equilibria and their stability.
ii) All solutions of (1.4) initiating in Θ are ultimately bounded with respect to R + and eventually enter the attracting set Θ.
is a positive solution of (1.4) initiating from the positive initial condition (u( ), v( ), w( )), we have From Lemma ( . ) of chen [17], we deduce that As u and v are positives and from (2.2), we get This complete the proof of the theorem.
The jacobian matrix associated with the system (1.4), At E = ( , , ) the associated jacobian matrix is Then, we nd the following eigenvalues a , a and −λc.
At E = ( a b , , ) the associated jacobian matrix is as follows: The associated characteristic equation is given by Then, we nd the corresponding eigenvalues −a , a and λ( a b pmq − c).
At E = ( , a k c , ) the jacobian matrix reads as: The characteristic equation is as follows Then, we nd the corresponding eigenvalues a − c a k k c , −a and −λc. Consequently, E = ( , a k c , ) is locally asymptotically stable if c a k > a k c .
At E = ( u , v , ) the associated jacobian matrix is The characteristic equation is If pmq u > c, then ) the jacobian matrix is Then, we nd the following eigenvalues a > and x , unstable.

Interior equilibrium and stability
In this section, we discuss the local and global stability of the positive steady state E * = (u * , v * , w * ) of system (1.4).  then E * = (u * , v * , w * ) is locally asymptotically stable.

Proof.
We denote E = (u, v, w) T and system (1.4) can be written as:  The characteristic polynomial of L E (E * ) is given by . (3.11) Then, the positive equilibrium E * = (u * , v * , w * ) is globally asymptotically stable.
Proof. Let us consider the following Lyapunov function, . The time derivative of V along the solution of system (1.4), we have T , T , and T are positive constants to be determined in the subsequent steps, using the following expressions and choosing Then, we deduce that: From the expression (3.11), the coe cients of (u − u * ) and (v − v * ) are negative. Then we deduce that dV dt < and dV dt = if and only if u = u * , v = v * and w = w * . Then, the function V satisfy the Lyapunov conditions and by LaSalle's Theorem [10], E * = (u * , v * , w * ) is globally asymptotically stable.
In the next, we will establish the existence of periodic oscillations around the interior equilibrium via Hopf bifurcation Theorem.

Bifurcation analysis
Choosing a as a parameter of bifurcation, then we have the following result on the existence of bifurcating branch of periodic solutions. Proof. The characteristic equation associated to E * = (u * , v * , w * ) is given by

Optimal Harvesting Policy
In commercial exploitation of renewable resources, the fundamental problem from the economic point of view is to determine the optimal trade o between present and future harvests. In the prey-predator model, the biological resources are most likely to be harvested and sold with the purpose of achieving the economic interest which motivates the introduction of harvesting in the prey-predator model [19]. Our objective in this section, is to determine an optimal harvesting policy and to maximize the total return produced by the exploitation of the resources. The optimal control problem is given by the value J, which presents a continuous time-stream of revenues [20].
where δ denotes the instantaneous annual rate of discount [21]. Thus, our goal is to maximize J subject to the state equations (1.4) by using Pontryagin's maximum principle [22]. The convexity of the function J with respect to m, the linearity of the di erential equations in the control and the compactness of the range values of the state variables can be combined to give the existence of the optimal control. We choose m δ is an optimal control with corresponding states u δ , v δ and w δ . We take E δ = (u δ , v δ , w δ ) as optimal equilibrium point. Here we intend to derive optimal control such that where Γ is the control set de ned by The associated Hamiltonian function is given by: Where λ , λ , andλ are the adjoint variables. The transversality conditions give λ i (t f ) = , i = , , .
The control variable m δ is subjected to the constraints m min ≤ m δ (t) ≤ mmax is the control set where mmax (resp m min ) is a feasible upper limit for the harvesting e ort (resp. feasible lower limit for the harvesting effort).
Here we treat with an optimal equilibrium solution. Since we are considering an equilibrium solution, u δ , v δ and w δ are to be used as constants in the subsequent steps. By Pontryagin's maximum principle, the adjoint equations at the point The Hamiltonian H must be maximised for m δ ∈ [m min , mmax]. With the control constraints m min ≤ m δ (t) ≤ mmax are not binding (that is, the optimal equilibrium does not occur either at m δ = m min or m δ = mmax), we have singular control given by [18]. Then ∂H ∂m = pqu δ w δ − λ qu δ w δ + λ λpqu δ w δ = at m δ (t). (5.12) Now substituting the value of λ and λ into equation (5.12), we get Therefore, we summarize the above analysis by the following theorem: Theorem 5.1. There exist an optimal control m δ and corresponding solutions u δ , v δ and w δ which maximize J(m) over Γ. Furthermore, there exists adjoint functions λ , λ , andλ satisfying equations (5.9) and (5.11) with transversality conditions λ i (t f ) = , i = , , . Moreover, the optimal control is given by:

Numerical examples
In this section, we present some numerical simulations with hypothetical set of parameters to understanding the theoretical results which have been showed in the previous sections of this work. With the following parameters: b = . , k = . , k = . , a = . , c = . , c = . , m = . , q = . , λ = . , p = . , c = .
(6.1) Firstly from the Figure (1) and (2), we observe that populations of prey-predator and harvesting e ort con-      .  verge to their steady states with the passage of time and E * = (u * , v * , w * ) is locally asymptotically stable for system (1.4). If we increase the value of the bifurcation parameter a = .
In the next, we study the variation of the prey, predator population and harvesting e ort with time for di erent price per unit biomass of landed sh p and the parameters are xed as follows: a = . , b = . , k = . , k = . , a = . , c = . , c = . , m = . , q = . , λ = . , c = . . (6.2) From Figure (7), we observe that prey and predator populations decreases with the increase of price value p. Theorem (5.1) giving existence of the optimal equilibrium and by using the parameters values in table 1 and from optimal harvesting policy, we have proved the optimal value of m δ and the corresponding optimal equilibrium E δ . Thus we deduce this table 2:   and other parameters are shown in the Table 1. and other parameters are shown in the Table 1.  .