Engineering, 2013, 5, 446-449
http://dx.doi.org/10.4236/eng.2013.510B091 Published Online October 2013 (http://www.scirp.org/journal/eng)
The Solutions for the Eco-Epidemic Model with Homotopy
Analysis Method
Xiurong Chen
Department of Science and Information, Qingdao Agricultural University, Qingdao, China
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 [1], Adomian decompo-
sition method [2], homotopy perturbation method [3], and
so on.
The HAM first envisioned by Liao [4], is another po-
werful analytical method for nonlinear problem. Many
authors have applied the HAM for solving nonlinear eq-
uations in [5], for solving solitary waves with discontinu-
ity in [6], for s eri e s s olutions 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 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 [1]:
2
1 12
21
2
()
()
xr xaxbxybxz
yry cxy dyyzyz
zyzd zyz
β
β
′=−− −
′= +−+−

′=−+

(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=====
31
bd β = +
12
dd β = −
, this leads to the dimensionless equations
2
11 23
2
21 23
2
12
xr xaxaxyaxz
yryb xybybyz
zd 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
(; )
[(;)]Xtp LXtp t
∂
=∂
(3.1)
2
(; )
[( ;)]Ytp LYtp t
∂
=∂
(3.2)
3
(; )
[( ;)]Ztp L Ztpt
∂
=∂
(3.3)
which satisfy
11
[]0LC =
,
12
[]0LC =
,
3
[]0LC =
(4)
where
1
C
,
2
C
and
3
C
are integral constants, and the
(; ),(; ),(; )XtpYtp Ztp
are real functions. Furthermore,
due to (2), we define the non-linear operators

X. R. CHEN
447
2
111 23
[,,]NXYZXrXaXaXYaXZ
′
=−+ ++
(5.1)
2
22123
[,,]NXYZYrYbXYbYb YZ
′
=−−+ +
(5.2)
2
3 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
( ,,)()()
k
k
k
Xxtpx tx tp
∞
=
= +
∑
(9.1 )
01
( ,,)()()
k
k
k
Yxtpy ty tp
∞
=
=+∑
(9.2 )
01
( ,,)()()
k
k
k
Zxtpz tz tp
∞
=
= +
∑
(9.3 )
where
0
1(;)
() !
k
kp
k
Xtp
xt kp
=
∂
=∂
(10.1)
0
1(; )
(,)!
k
kp
k
Ytp
y xtkp=
∂
=∂ (10.2)
0
1(; )
(,)!
k
kp
k
Ztp
z xtkp=
∂
=∂ (10.3)
If the auxiliary linear parameter, the initial conditions,
and the auxiliary parameters
123
,,hhh
,
12
( ),(),HtHt
3
()Ht
are chosen, the above series converge at
1p=
,
and one has
01
() ()()
k
k
xtx txt
∞
=
=+ ∑
(11 .1 )
01
() ()()
k
k
yty ty t
∞
=
= +
∑
(11 .2 )
01
() ()()
k
k
ztz tzt
∞
=
= +
∑
(11 .3 )
Let
123
hhhh== =
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 mmm
L xtxthRxyz χ
−− −−
−=
(12.1)
212 111
[ ()()][,,]
mmmm mmm
L ytythRxyz χ
−− −−
−=
(12.2)
313 111
[ ()()][,,]
mmmm mmm
L ztzthRxyz χ
−− −−
−=
(12.3)
with the initial conditions (
1m≥
)
(0) 0,(0) 0,(0) 0
mmm
xyy===
(13)
where
11
1111 1121
11
1
31
1
()
mm
m m mjmjjmj
jj
m
jmj
j
Rtxrxax xaxy
a xz
−−
− −−−−−
= =
−
−−
=
′
=−+ +
+
∑∑
∑
(14.1)
11
2111121
11
1
31
1
()
mm
mmrmjm jjm j
jj
m
jmj
j
R tyrybxybyy
b xz
−−
− −−−−−
= =
−
−−
=
′
=−− +
+
∑∑
∑
(14.2)
11
3111 21
11
()
mm
mmjm jjmj
jj
R tzdyzdzz
−−
− −−−−
= =
′ =−+
∑∑
(14.3)
and
11
01
m
m
m
χ
>

=≤

(15)
Now, the solutions of the mth-order deformation Equ-
ations (12) for
1m≥
become
11
0
()
t
m mmm
xxhR d χ ττ
−
= +
∫
(16.1)
12
0
()
t
m mmm
yy hR d χ ττ
−
=+ ∫
(16.2)
13
0()
t
m mmm
zz hRd χ 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 22
211 2122
()(1 )(1 )xthhAhhtEh tEh tE=+++++
(19.1)

X. R. CHEN
448
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
112323
HdAA dA=−+
;
2
21 1111212313
22 2ErAaAa AAaAA=−+ ++
;
22111112 2111
31 131
2( )
()
ErEaAEa AEAF
a AHAE
=−+ +−
++
;
2
21 2222112323
22 2FrAbAbAAbAA=−+ ++
;
22212 2111121
32131
2( )
()
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

X. R. CHEN
449
http://dx.doi.org/10.1016/j.chaos.2007.04.021
[10] S. J. Liao, “Series Solution of Nonlinear Eigenvalue Pro-
blems by Means of the Homotopy Analysis Method,”
Nonlinear Analysis: Real World Applications, Vol. 10,
2009, pp. 2455-2470.
http://dx.doi.org/10.1016/j.nonrwa.2008.05.003
[11] A. R. Ghotbi, Bararnia, G. Domairry and A. Barari, “In-
vestigation of a Powerful Analytical Method into Natural
Convection Boundary Layer Flow,” Communications in
Nonlinear Science and Numerical Simulation, Vol. 14,
2009, pp. 2222-2228.
http://dx.doi.org/10.1016/j.cnsns.2008.07.020
[12] M. Khan, Z. Abbas and T. L. Hayat, “Analytic Solution
for Flow of Sisko Fluid through a Porous Medium Au-
thors,” Transport in Porous Media, Vol. 71, 2008, pp. 23-
37. http://dx.doi.org/10.1007/s11242-007-9109-4
[13] S. Abbasbandy and A. Shirzadi, “The Series Solution of
Problems in Calculus of Variations via Homotopy Analy-
sis Method,” Zeitschrift für Naturforschung A, Vol. 64,
2009, pp. 30-36.