Comparative Numerical Study of SBA (Somé Blaise-Abbo) Method and Homotopy Perturbation Method (HPM) on Biomathematical Models Type Lotka-Volterra

In this work the Homotopy Perturbation Method (HPM) is used to find an exact or approximate solutions of Lotka-Volterra models. Then we compare the HPM solution with the solution given by SBA (Somé Blaise Abbo) method.


Introduction
We are interested in the resolution of nonlinear systems of general form : in a suitable Hilbert space, where f (u, v) et ϕ (u, v) are the linear functions and ϕ(u, v) and ψ(u, v) are nonlinear fonction.
These systems model a large number of phenomena in biomathematics. They are also Lotka-Volterra equations type of the form: The paper is organized as follows: in the following section the homotopy perturbation method is explained. In Section 3 we solve two problems. Numerical results are reported in Section 4. Finally, the paper is concluded in Section 5.

HPM for System of ODEs
To illustrate the basic idea of the HPM for system of PDEs, we consider the following non-homogeneous, non-linear system of ODEs ( M.S.H. Chowdhury & et al., 2010;M.S.H. Chowdhury & et al., 2015).

Example 1
Consider a Lotka-Volterra model which describe predators-prey interactions (ABBO, B., 2007) : with the initial conditions The exact solution of (10) obtained by SBA method is (ABBO, B., 2007) : In order to apply the homotopy perturbation method, we construct the following homotopy equations : We obtain the following homotopy system: Assume the solution of (10) to be in the form : Substituting (12) into (11) and equating the coefficients of like p, we get the following set of differential equations : p 2 : . . .
Solving the above equations, we obtain For n ≥ 1, we have u n (t) = v n (t) = 0 Hence the solution of (10) by HPM is given by

Numerical Results and Discussion
For the example 1, we get its exact solution with both methods (HPM and SBA method). For example 2, in oder to verify the efficiency of the proposed method in comparison with the exact solution, we calculate the values of these solutions for different values of t.

Conclusion
In this paper, the homotopy perturbation method was used for finding exact or approximate solutions of Lotka-Volterra models.
Through the examples studied, we have shown that we obtain practically the same solutions with HPM and the SBA method.