The Solutions for the Eco-Epidemic Model with Homotopy Analysis Method

In this paper, the Homotopy Analysis Method (HAM) has been used to solve an eco-epidemic model equation. The algorithm of approximate analytical solution is obtained. HAM contains the auxiliary parameter h which provides us with a convenient way to adjust and control convergence region and rate of solution series. The results obtained show that these algorithms are accurate and efficient for the model.


Introduction
There has recently been much attention devoted to the search for better and more efficient solution methods for determining a solution, approximate or exact, analytical or numerical, to nonlinear models.Finding approximate solutions of these nonlinear equations is interesting and important.Many methods have been developed to solve nonlinear partial differential equations (NPDEs) such as biologically-based technologies [1], Adomian decomposition method [2], homotopy perturbation method [3], and so on.
The HAM first envisioned by Liao [4], is another powerful analytical method for nonlinear problem.Many authors have applied the HAM for solving nonlinear equations in [5], for solving solitary waves with discontinuity in [6], for series solutions of nano boundary layer flows in [7], for nonlinear equations in [8] and many other subjects.The technique employed here is very powerful and has been already successfully applied to various complicated problems [9][10][11][12][13].Thus, through HAM, explicit analytic solutions of nonlinear problems are possible.
The application of the HAM in engineering problems in highly considered by scientists, because HAM provides us with a convenient way to control the convergence of approximation series, which is a fundamental qualitative difference in analysis between HAM and other methods.The purpose of this paper is to implement HAM to eco-epidemic model equations [1]: where ( ), ( ), ( ) x t y t z t are the density of a wild plant species, a susceptible pest and infected pest which live on the crop.All parameters are positive constants.

The Solution by Ham
with the initial conditions Due to the governing Equation (2), we choose the auxiliary linear operators as follows: where 1 C , 2 C and 3 C are integral constants, and the ( ; ), ( ; ), ( ; ) X t p Y t p Z t p are real functions.Furthermore, due to (2), we define the non-linear operators Then, introducing a non-zero auxiliary h, we construct the zero-order deformation equations Obviously, when 0 p = and 1 p = , we have the solutions  2).This is the basic idea of the homotopy and this kind of variation is called deformations in topology.
Thus, by Taylor's theorem and ( 7), we can express If the auxiliary linear parameter, the initial conditions, and the auxiliary parameters 1 2 3 , , ( ), ( ), H t H t 3 ( ) H t are chosen, the above series converge at 1 p = , and one has . Differentiating the zero-order deformation Equations ( 6) m times with respect to p and then dividing them by !m and finally setting 0 p = , the mth-order deformation equations read where Now, the solutions of the mth-order deformation Equations (12) for Mathematica software is used to solve the linear Equations (16) under the initial conditions up to first few order of approximations.We have: ( ) ( )

Results and Analysis
It is important to ensure that the solution series (20) is convergent.Note that the solution series (20) contain the auxiliary parameter h, which we can choose properly by plotting the so-called h -curves to ensure solution series converge, as suggested by Liao [5].The valid region of h is a horizontal line segment.Thus the valid region of h is shown in Figure 1 when A 1 = 0.85, A 2 = 0.15, r 2 = 0.1, a 1 = 1,

Conclusion
The model equations which arise from problems are usually nonlinear and such biological equations are difficult to estimate numerically or analytically.In this paper, the HAM was applied to solve the eco-epidemic model equations.HAM contains the auxiliary parameter which provides us with a convenient way to adjust and control convergence region and rate of solution series.In this regard the HAM is found to be a very useful analytic technique to get highly accurate and purely analytic solution to such kind of problems.
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Figure 1 .
Figure 1.The h-curves of the 6-th order approximation for x, y and z when t = 0.5: (a) x − y, (b) y − h, (c) z − h.