Fractional Ecosystem Model and Its Solution by Homotopy Perturbation Method

In the present paper we propose an algorithm based on Homotopy Perturbation Method to solve the fractional Ecosystem model and show the efficiency and accuracy of the algorithm . This Ecosystem model is solved with time fractional derivatives in the sense of Caputo. The nonlinear terms can be easily handled by the using He’s Polynomials. The numerical solutions of the Ecosystem model reveal that only a few iterations are sufficient to obtained accurate approximate analytical solutions. The numerical results obtained are presented graphically. The four different cases are considered and proved that the method is extremely effective due to fractional approach and performance. Comparing the methodology (FHPM) with the some known technique (HPM) shows that the present approach is effective and powerful. The proposed scheme finds the solution with the help of Mathematica and without any restrictive assumptions. We implement it to four different problems.


Introduction
Ecosystem models p lay an important ro le to know the dy namics o f th e system. Su ch mod els are fo rmed by comb ining known eco logical relations with data gathered fro m field observations ([1]- [8]). These models are then used to make predict ions about the dynamics of the real system. However, the study of deficiencies in the model leads to the generat ion of hypoth eses about the possib le eco log ical relations that are not yet known o r well understood. Models enable researchers to simulate the conclusions of large-scale experiments that might be too expensive to perform on a real ecosystem. These also enable the simulat ion of ecological processes over very long period of t ime (may be so me process may take centuries of t ime in reality), However this can be done in a matter of minutes in a computer model. Ecosystem models have applications in a wide variety of disciplines, in particu lar, to natural resource management. An important aspect any model evaluation is the validation, that is, the process of confirming that the model behaviour corresponds to reality. The results o f the model can be validated by comparison with field data, but this technique limits the validation to a particu lar set of field conditions. We obtain a sequence of equations in which the solution at any stage is close to the solution at nearby stages. In the present paper we discuss the fractional Ecosystem model and solve it using Homotopy Perturbation Method.
We consider the following fractional ecosystem model given by fractional init ial value problem of the type:

Basic Definitions
In this section, we g ive the related definit ions and properties of the fractional calculus fro m [9].
Some of the properties of the operator J α , wh ich we need here, are as follows: Definiti on 2.3. [10] The fractional derivative D α of ( ) h t in the Caputo's sense is defined as The follo wing are the two basic properties of the Caputo's fractional derivative [10]:

Analysis of HPM
The Homotopy Perturbation Method (HPM), wh ich provides analytically an appro ximate solution, is applied to various nonlinear problems ( [11]- [14]). In this section, we introduce a reliable algorith m to handle the nonlinear ecosystems in a realistic and efficient way. The proposed algorith m will then be used to investigate the system given by: ...
and when p is 1, equation (3) turns out to be the original equation given as system (1). Assuming that the solution of the system (1) is a power series in p g iven by: Substituting equation (5) in equation (3), and equating the terms having identical powers of p, we obtain a series of linear equations in the form: ..., ... i n n n t y py p y p y g y py p y It is also useful, for the system ( )

Numerical Example
In this section we show the application o f the algorith m based on HPM given above to solve four different linear and non linear fractional Ecosystem model. Example 4.1. Consider the linear system [15] ( ) ( ), subject to the initial conditions ( ) ( ) We construct the Homotopy The solution of ( ) 8 is given by: Fro m Fig. 1 and Fig. 2 we see that the solution obtained by the proposed algorithm for 1 α = is same as the solution obtained by [16]. Also figures 1and 2 show the solution for other different values of α . Fig 1 and 2 show the solutions in red by HPM of the original system and by HPM of the fractional ecosystem model by blue thick dots.
Example 4.2. Consider the predator -prey system [15].      Fro m the Fig. 3 and Fig. 4 we see that the solution obtained by the proposed algorithm for 1 α = is same as the solution obtained by [16]. Also fig. 3 and 4 show the solution for other different values of α .
Hence the solution is given as