 Engineering, 2013, 5, 446-449 http://dx.doi.org/10.4236/eng.2013.510B091 Published Online October 2013 (http://www.scirp.org/journal/eng) Copyright © 2013 SciRes. ENG The Solutions for the Eco-Epidemic Model with Homotopy Analysis Method Xiurong Chen Department of Science and Information, Qingdao Agricultural University, Qingdao, China Email: xrchen_100@163.com Received 2013 ABSTRACT In this paper, the Homotopy Analysis Method (HAM) has been used to solve an eco-epidemic model equation. The al-gorithm 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. Keywords: Nonlinear Partial Differential Equations; Homotopy Analy s i s Method; Eco-Epidemic Equations 1. 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 , Adomian decompo-sition method , homotopy perturbation method , and so on. The HAM first envisioned by Liao , is another po-werful analytical method for nonlinear problem. Many authors have applied the HAM for solving nonlinear eq-uations in , for solving solitary waves with discontinu-ity in , for s eri e s s olutions of nano boundary layer flows in , for nonlinear equations in  and many other subjects. The technique employed here is very powerful and has been already successfully applied to various com-plicated problems [9-13]. 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 pro-vides us with a convenient way to control the conver-gence of approximation series, which is a fundamental qualitative difference in analysis between HAM and oth-er methods. The purpose of this paper is to implement HAM to eco-epidemic model equations : 21 12212()()xr xaxbxybxzyry cxy dyyzyzzyzd zyzββ′=−− −′= +−+−′=−+ (1) where (),(), ()xtyt zt are the density of a wild p lant spe- cies, a susceptible pest and infected pest which live on the crop. All parameters are positive cons tants. 2. The Solution by Ham Note 12 13 2121, ,,,,aaab abbcbd===== 31bdβ= + 12ddβ= −, this leads to the dimensionless equations 211 23221 23212xr xaxaxyaxzyryb xybybyzzd yzdz′=−−−′=+− −′= − (2) with the initial conditions 123(0),(0), (0)x Ay Az A= = = Due to the governing Equation (2), w e choose the aux-iliary linear operators as follows: 1(; )[(;)]XtpLXtp t∂=∂ (3.1) 2(; )[( ;)]YtpLYtp t∂=∂ (3.2) 3(; )[( ;)]ZtpL Ztpt∂=∂ (3.3) which satisfy 11[]0LC =, 12[]0LC =, 3[]0LC = (4) where 1C, 2C and 3C are integral constants, and the (; ),(; ),(; )XtpYtp Ztp are real functions. Furthermore, due to (2), we define the non-linear operators X. R. CHEN Copyright © 2013 SciRes. ENG 447 2111 23[,,]NXYZXrXaXaXYaXZ′=−+ ++ (5.1) 222123[,,]NXYZYrYbXYbYb YZ′=−−+ + (5.2) 23 12[,,]NXYZZd YZdZ′=−+ (5.3) Then, introducing a non-zero auxiliary h, we construct the zero-order deformation equations 10111(1)[(; )()]()[,, ]pLXtpxthHtpNXYY− −= (6.1) 2022 2(1)[(;)()]()[,,]pLYtpy thHtpNXYY− −= (6.2) 3033 3(1)[ (;)()]()[, ,]pLZtpz thHtpNXYY− −= (6.3) Obviously, when 0p= and 1p=, we have the so-lutions 000( ;0)( ),(;0)(),(;0)( )XtxtYtytZtzt= ==, (7) ( ;1)( ),( ;1)( ),( ;1)( )XtxtYtytZtzt= == (8) Therefore, as the embedding parameter p increases from 0 to 1, (; ),(; )XtpYtp and (; )Ztp vary from the initials 00(), ()xtyt and 0()zt to the exa c t s o lu ti on ()xt, ()yt and ()zt governed by (2). This is the basic idea of the homotopy and this kind of variation is called de-formations in topology. Thus, by Taylor’s theorem and (7), we can express 01( ,,)()()kkkXxtpx tx tp∞== +∑ (9.1 ) 01( ,,)()()kkkYxtpy ty tp∞==+∑ (9.2 ) 01( ,,)()()kkkZxtpz tz tp∞== +∑ (9.3 ) where 01(;)() !kkpkXtpxt kp=∂=∂ (10.1) 01(; )(,)!kkpkYtpy xtkp=∂=∂ (10.2) 01(; )(,)!kkpkZtpz xtkp=∂=∂ (10.3) If the auxiliary linear parameter, the initial conditions, and the auxiliary parameters 123,,hhh, 12( ),(),HtHt3()Ht are chosen, the above series converge at 1p=, and one has 01() ()()kkxtx txt∞==+ ∑ (11 .1 ) 01() ()()kkyty ty t∞== +∑ (11 .2 ) 01() ()()kkztz tzt∞== +∑ (11 .3 ) Let123hhhh== = and 123()()()1HtHtHt== =. Differentiating the zero-order deformation Equations (6) m times with resp ect to p and then dividing them by !m and finally setting 0p=, the mth-order deformation equations read 111 111[ ()()][,,]mmmm mmmL xtxthRxyzχ−− −−−= (12.1) 212 111[ ()()][,,]mmmm mmmL ytythRxyzχ−− −−−= (12.2) 313 111[ ()()][,,]mmmm mmmL ztzthRxyzχ−− −−−= (12.3) with the initial conditions (1m≥) (0) 0,(0) 0,(0) 0mmmxyy=== (13) where 111111 1121111311()mmm m mjmjjmjjjmjmjjRtxrxax xaxya xz−−− −−−−−= =−−−=′=−+ ++∑∑∑ (14.1) 112111121111311()mmmmrmjm jjm jjjmjmjjR tyrybxybyyb xz−−− −−−−−= =−−−=′=−− ++∑∑∑ (14.2) 113111 2111()mmmmjm jjmjjjR tzdyzdzz−−− −−−−= =′=−+∑∑ (14.3) and 1101mmmχ>=≤ (15) Now, the solutions of the mth-order deformation Equ-ations (12) for 1m≥ become 110()tm mmmxxhR dχ ττ−= +∫ (16.1) 120()tm mmmyy hR dχ ττ−=+ ∫ (16.2) 130()tm mmmzz hRdχ ττ−=+ ∫ (16.3 ) Mathematica software is used to solve the linear Equa-tions (16) under the initial conditions up to first few or-der of approxim a t ions. We ha ve : 01()xt A=; 02()yt A=; 03()zt A= (17) 1 11()xthAhtE= + (1 8.1) 1 21()y thAhtF=+ (18 .2 ) 1 31()z thAhtH= + (18 .3 ) 2 22211 2122()(1 )(1 )xthhAhhtEh tEh tE=+++++ (19.1) X. R. CHEN Copyright © 2013 SciRes. ENG 448 2 222212122()(1 )(1 )ythhAhhtFh tFh tF=+++++ (19.2) 2 22231 2122()(1 )(1 )zthhAhhtHh tHh tH=+++++ (19.3)  where 2111 11212313ErAaAa AAaAA=−+ ++; 2122 22112323F rAbAbAAbAA=−+ ++; 2112323HdAA dA=−+; 221 111121231322 2ErAaAa AAaAA=−+ ++; 22111112 211131 1312( )()ErEaAEa AEAFa AHAE=−+ +−++; 221 222211232322 2FrAbAbAAbAA=−+ ++; 22212 2111121321312( )()FrFbAFbAFAEb AHAF=−++ +++ ; etc. Therefore, 012()() ()()xtxtxtxt=+++ (20.1) 012()() ()()ytytytyt= +++ (20.2 ) 012()()()()ztztztzt= +++ (20.3 ) 3. 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 . The valid region of h is a horizontal line segment. Thus the valid region of h is shown in Figure 1 when A1 = 0.85, A2 = 0.15, r2 = 0.1, a1 = 1, (a) (b) (c) 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. 2 312312 310.5,0.4, 0.4,0.1,0.69,0.490.1,0.1, 0.8.a abbbdd Ar== == ==== = 4. Conclusion The model equations which arise from problems are usu- ally 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 equ-ations. HAM contains the auxiliary parameter which pro- vides us with a convenient way to adjust and control con- vergence region and rate of solution series. 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