Dynamics and optimal Harvesting strategy for biological models with Beverton –Holt growth

In this work, the dynamic behavior of discrete models is analyzed with BevertonHolt function growth . All equilibria are found . The existence and local stability are investigated of all its equilibria.. The optimal harvest strategy is done for the system by using Pontryagin’s maximum principle to solve the optimality problem. Finally numerical simulations are used to solve the optimality problem and to enhance the results of mathematical analysis .


1-Introduction
For many organisms births occur in reguler times each month or year or each circle. Discrete time function is well used to describe the life of them. Many researchers have analyzed models that described by system of difference eqautions [1][2][3][4]. For one dimension model,the general form is governed by first order difference eqaution ( ) where denotes the size of population at year or period t. For two or more dimantions model, the well known model is Lotka -Voltera ,that was first introduced by Lotka and Voltera [5].After their work, more realistic models were introduced and modified by many authors ,we refere to reader for more detials see [6][7][8][9][10].
It is well known that harvesting plays an important role in managing the renew resources , so one should consider useful strategies in order to decrease the risks of extinction as well as to increase the net gains .Scientists and researchers used different harvesting strategies in their models [11][12][13] . For example, Sanchez and Braner analyzed and investigated the effect of periodic harvesting in periodic environments [14,15].A great deal of attention was given to the discrete as well as continuous logistic models in [16]. Other models are considered in the literatures, for example Ricker model, with constant depletion rate [17] . It was shown numerically that the population exhibits chaotic ISSN: 0067-2904 oscillations, which are not necessarily lead to extinction [18,19]. In this paper we will investigate the dynamic and an optimal harvesting of biological models with Beverton-Holt model in one and two dimensions models.
This work is organized as follows: In section 2 we study the dynamics behaviour of a single species with Beverton-Holt growth function with and without harvesting . In section 3, Stability analysis of the prey-predator model is analyzed. In section 4 the model is extended to an optimal harvesting problem. We used the extension of Pontryagin's maximum principle to find the optimal solutions . In section 5, numerical simulations are used to solve the optimality problem and to enhance the results of mathematical analysis. Finally conculsion is provided.

2) Single species
In this section we will study and investigate the dynamic behaviour of the classical Beverton Holt models with and without constant rate harvesting for single population .
The model without harvesting is given by (1) Where is the inherent growth rate, and is the population carrying capacity. The dynamics of (1) are well known ,For any point , <1 the extinction equilibrium point is globally asymptotically stable ,while if >1 then is unstable and the survival equilibrium is sink point. The last case when the equilibrium point is nonhyperbolic point . Now if one puts a constant rate harvesting then the model will be as the following : ( Where is a constant representing the intensity of harvesting due to fishing or hunting , here q depends on the density of the population , so that we cannot harvest more than the population density. Thus our mathematical analysis is concerned with 0<q < . The model (2) has also two equilibria ,the exinction equilibrium always exists,and the unique positive equilibrium if and only if So that is always sink point.

3-Stability analysis of the prey-predator model.
In this section we will study the dynamics of two -dimension model , prey-predator system, with Beverton-Holt growth in prey species . The system is given by , are the population carrying capacity , the growth rate of the prey and the predator respectively ,while the positive parameters and represent the maximum per capita killing rate and conversion rate of predator respectively. All equilibrium points of the system (3) can be determined by solving the following algebraic equation : (4) After simple calculation, we have the following lemma: Lemma 2: The system (3) has the following equilibira for all parameters values 1 -( ) the trivial equilibrium always exists without any restriction .

-
) ) the unique positive equilibrium which exists if ( ) and In order to discuss the local stability analysis of system(3) around the equilibrium points , we have to compute the general Jacobian matrix of the system (3) at point ( ). This is given by : where ( ) The characteristic polynomial of (5)   { }

4-
Proof: We will apply lemma (3) ,the Jacobian matrix at is given by :

4-one can easily note that F( -1) = 0 if and only if = Now
And if p = 2 then by the same way one get r = So that is nonhyperbolic point when

4) Optimal harvesting strategy
We will study an optimal control strategy by using discrete version of PMP Pontryagin maximum principle to solve the optimality problem [21,22,23,24]. In this optimal control problem the state equations are : The and are the prey population density and the predator density at period time t respectivety . The parameters r ,b and d are defined as before while the parameter refers to the cotrol variable ,which represents the harvesting amount at period time t, with where A is the maximum harvesting one can get .
The objective functional that we have to maximize is : is the amount of money that one can to earn and is the associated with the cost of catching and supporting the animals . To solve the problem one has to form the Hamilitonian function for t = 0,1,2,………,T-1 ,this is given by : Now finding the optimal harvesting with corrosponding optimal state solution at time t will be obtained numerically .

5-Numerical results:
we will use different sets of the values to show the local stability of and for the system (30 , we also choose other values for the optimal harvesting solution. We choose the values of the parameters for , as follows r =0.9 ,a=1,b = 2,c =0.3 and d = 1 with initial condition ( 1.1 ,0.15 ) . Therefore the ( 1) condition in theorem(1) is satisfied. Figure-1 Figure-3 . For solving the optimal problem we use an iterative method to compute the optimality [ ] . Starting by an initial solution of the control with initial of state variable ,then we solve the state system (6 )forward while the adjoint system ( 8) is solve backward and combine the new control with previous one to update the control . This procedure continuous until getting the optimal solutions with corresponding state variables. We choose the set of values of parameter the total optimal harvesting is . Figures-5and 6 show the effect of optimal harvesting on the prey and optimal control variable as function of time respectively. Table-1 compares the total optimal harvesting and other total harvesting starategies using the same values of the parameters.

6-Conclusions
Discrete time models with Beverton-Holt function growth has been studied and analyzed. All equilibria are found. The local stability for all equilibria is investigated, then the model has been extended to an optimal control problem. The Pontryagin's maximum principle used to solve the optimality problem. All theoretical results confirmed by numerical simulations.