X. R. CHEN

Copyright © 2013 SciRes. ENG

2 22

2212122

()(1 )(1 )ythhAhhtFh tFh tF=+++++

(19.2)

2 22

231 2122

()(1 )(1 )zthhAhhtHh tHh tH=+++++

(19.3)

where

2

111 11212313

ErAaAa AAaAA=−+ ++

;

2

122 22112323

F rAbAbAAbAA=−+ ++

;

;

2

21 1111212313

22 2ErAaAa AAaAA=−+ ++

;

22111112 2111

31 131

2( )

()

ErEaAEa AEAF

a AHAE

=−+ +−

++

;

2

21 2222112323

22 2FrAbAbAAbAA=−+ ++

;

22212 2111121

32131

()

FrFbAFbAFAE

b 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 [5]. 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 31231

2 31

0.5,0.4, 0.4,0.1,0.69,0.49

0.1,0.1, 0.8.

a abbbd

d 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. 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 pro blems.

REFERENCES

[1] Y. N. Xiao and F. Van Den Bosch, “The Dynam ics of an

Eco-Epidemic Mod el with Biologieal Eontrol,” Eeol Mo-

deling, Vol. 168, 2003, pp. 203-214.

[2] I. Hashim, M. S. M. Noorani and B. Batiha, “A Note on

the Adomian Decomposition Method for the Generalized

Huxley Equation,” Applied Mathematics and Computa-

tion, Vol. 2, 2006, pp. 1439-1445.

http://dx.doi.org/10.1016/j.amc.2006.03.011

[3] J. H. He, “Application of Homotopy Perturbation Method

to Nonlinear Wave Equations,” Chaos Solitons Fractals,

Vol. 26, 2005, pp. 695-700.

http://dx.doi.org/10.1016/j.chaos.2005.03.006

[4] I. Hashim, M. S. M. Noorani and B. Batiha, “A Note on

the Adomian Decomposition Method for the Generalized

Huxley Equation,” Applied Mathematics and Computa-

tion, Vol. 2, 2006, pp. 1439-1445.

http://dx.doi.org/10.1016/j.amc.2006.03.011

[5] S. J. Liao, “Homotopy Analysis Method and Its Applica-

tion,” PhD Dissertation, Shanghai Jiao Tong University,

1992.

[6] S. J. Liao, “On the Homotopy Analysis Method for Non-

linear Problems,” Applied Mathematics and Computation,

Vol. 147, 2004, pp. 499-513.

http://dx.doi.org/10.1016/S0096-3003(02)00790-7

[7] W. Wu. and S. J. Liao, R. N. Mohapatra and K. Vajravelu,

“Solving Solitary Waves with Discontinuity by Means of

the Homotopy Analysis Method,” Journal of Mathemati-

cal Analysis and Applications, Vol. 26, 2005, pp. 177-

185.

[8] J. Cheng, S. J. Liao, R. N. Mohapatra and K. Vajravelu,

“Series Solutions of Nano Boundary Layer Flows by

Means of the Homotopy Analysis Method,” Journal of

Mathematical Analysis and Applications, Vol. 343, 2008,

pp. 233-245.

http://dx.doi.org/10.1016/j.jmaa.2008.01.050

[9] F. Talay Akyildiz, K. Vajravelu and S. J. Liao, “A New

Method for Homoclinic Solutions of Ordinary Differen-

tial Equations,” Chaos, Solitons Fractals, Vol. 39, 2009,

pp. 1073-1082.

-2 -1.5 -1 -0.5h

0.795

0.805

0.81

0.815

0.82

0.825

x

-2 -1.5-1 -0.50.5 h

0.12

0.14

0.16

y

-2 -1.5-1 -0.5 h

0.10345

0.1035

0.10355

0.1036

0.10365

z