Optimal Harvesting Strategy of a Discretization Fractional-Order Biological Model

Optimal control methods are used to get an optimal policy for harvesting renewable resources. In particular, we investigate a discretization fractional-order biological model, as well as its behavior through its fixed points, is analyzed. We also employ the maximal Pontryagin principle to obtain the optimal solutions. Finally, numerical results confirm our theoretical outcomes

Fractional-order derivative provides a precise description of the dynamics of biological or epidemiological models due to in consideration of information about a population memory compared to the other descriptions for that many researchers prefer to model their systems by fractional-order derivative. For more details about the fractional-order derivative we refer to these references [6,11,12,[17][18][19].
A general system of two dimensional prey-predator without harvesting is described by ordinary differential equations is as following: ( ) = ( ) − ( , )y Here the variables ( ), and ( ) denote to the size of prey and predator population at time , respectively. Parameter is the conversion rate. Parameter denotes the death natural rate of predator species. The function represents growth rate of prey , while the function ( , y) is called the functional response of predator to prey density. This work is organized as follows: The fractional-order derivative model is described in section 2, as well as its discretization is analyzed and investigated through its fixed points. Then we extend the discretization system to an optimal control problem, this is done in section 3. In section 4 numerical results are presented to clarify the theoretical analysis. A discussion follows in section 5.
Where ( ), and ( ) denote the densities of prey, and predator species at time , respectively. In this system the prey growths logistically. The parameter represents the conversion rate part from the prey species to the predator species. The parameter denotes the death rate of predator species. The functional response is the ratio-dependent predatorprey. a, b are the half saturation constants. h 1 , h 2 are the rate harvesting or the removal rate of prey and predator, respectively. Throughout this article we assume that h 2 = 0,and h 1 = ℎ.
Applying discretization method to the fractional-order system (2). For more details we refer to [6,10]. The system (2) is reduced to Definition 2 [22]: Let be a discrete time system the point e * is called a fixed point of equation (4) if e * = f(e * ). If |λ i | < 1 for i = 1,2, … . . , λ i are the eigenvalues of the Jacobian matrix J at e * then it is called local stable point. Otherwise e * is called unstable point. While if |λ i | = 1 for some 1≤ i ≤ n then e * is called a non-hyperbolic point. To discuss the dynamic behavior of the system (3) we have to compute the Jacobian matrix of (3). The Jacobian matrix at (x, y) is as follows : J(x, y) = [ j 11 j 12 j 21 j 22 ] For the local stability of the fixed points e 0 , and e 1 of system (3) we have the following theorem.

Theorem 1 1-
The e 0 is never to be locally stable point.

Theorem 2
The point e 2 is locally stable if h ∈ ( then the condition3 in lemma 1 holds . Therefore the point e 2 is local stable point.

3-Optimal harvesting approach.
This part of the article deals with the optimal harvesting amounts so that the system (3) becomes as follows The all parameter are the same previous interpolation, while the parameter h n represents the control variable. We form the objective functional as follows: Subject to the considered system (5) the parameters c 1 and c 2 are positive constants. Now we have to find out the optimal solution h n * that satisfies J(h n * ) = Max J(h n ) for all 0 ≤ h n ≤ h Max , h Max represents the maximum harvesting. We apply the Pontryagin's Maximum Principle [1,3,[23][24][25] to get the necessary conditions for the optimal variable control and corresponding states.

Proof:
The Hamiltonian function is ] . Now the optimal variable will be h n * = 1 − +1

4-Numerical results
This section verifies the effectiveness of our theoretical results, so that some numerical simulations are given. To confirm the behavior of the system (3)   Hence the Theorem 2 is verified, and the point is stable. This is displayed in Figure 2. Trajectories of the prey species and the predator species as a function of time which Indicates that the point e 2 is local stability. This is done in Figure 3.  We use and employ iterative method to find the optimal control solution. We use an iterative algorithm. For more details we refer to [5,14,25]. The values of parameters as follows: = 0.3; = 0.8 ; = 0.45; = 0.5; = 0.98, c 1 = 0.2, . and c 2 = 0.2 with initial guess 0 = 0.4, 0 =0.5 for prey , and predator , respectively. We obtain the total net optimal harvesting is (h n * ) = 0.1090. Figure 4 shows the optimal solution variable as function of time, while Figures 5-6 indicate the effect of optimal solution and the fixed harvesting amount on the prey, predator, respectively.    (6) is plotted with control, without control, and with fixed harvest amount.

5-Discussions and Conclusions
In this paper, a discretization of fractional-order prey-predator system with ratiodependent predator-prey functional response has been presented and analyzed. The local stability of its fixed point is studied. Our analysis shows the considered system has three fixed points as well as the trivial fixed point is never to be stable point, while the other points are locally stable under certain conditions. We also conclude that the equilibrium harvesting amount as well as any constant harvesting amount cannot be the optimal solution. We can see in Figures 4 and 5 that the level of prey species density, predator density with optimal control are lower than their equilibrium level. It is also seen that the heavily harvesting will lead to increase the possibility of extinction.