APPLICATION OF HOMOTOPY PERTURBATION METHOD FOR ECOSYSTEMS MODELLING Zaid Odibat(1) and Cyrille Bertelle(2) (1)
Prince Abdullah Bin Ghazi Faculty of Science and IT Al-Balqa’Applied University Salt, Jordan email:
[email protected] (2)
LITIS, University of Le Havre 25 rue Ph. Lebon, BP 540 76058 Le Havre cedex, France email:
[email protected] Keywords: Homotopy perturbation method; ecosystems; Adomian decomposition method; variational iteration method.
way, we need both to build automatic equation and identify the parameters value but we need also to solve automatically the generated equations. The method proposed here is the Homotopy Perturbation Method (HPM) which is applied to nonlinear ecosystems.
ABSTRACT The HPM method can be considered as one of the new methods belonging to the general classification of perturbation methods. These methods deals with exact solvers for linear differential equations and approximative solvers for non linear equations. In this paper, we focus our attention on the generation of the decomposition steps to build a solver using the HPM method. We present how this method can be used in ecosystem modelling. We develop some solvers for prey-predator systems involving 2 or 3 populations.
1
Usually, perturbation methods need some kind of small parameter to be used. In the HPM method, which doesn’t require a small parameter in an equation, a homotopy with an imbedding parameter p ∈ [0, 1] is constructed. The method provides analytical approximate solutions for different types of nonlinear ecosystems. The results reveal that the method is very effective and simple for obtaining approximate solutions of nonlinear systems of differential equations. The HPM, proposed first by He (He 1999; He 2000), for solving differential and integral equations, linear and nonlinear, has been the subject of extensive analytical and numerical studies. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which is easily solved. This method has a significant advantage in that it provides an analytical approximate solution to a wide range of nonlinear problems in applied sciences. The HPM is applied to Volterra’s integro-differential equation (El-Shahed 2005), to nonlinear oscillators (He 2004a), bifurcation of nonlinear problems (He 2005a), bifurcation of delay-differential equations (He 2005b), nonlinear wave equations (He 2005c), boundary value problems (He 2006) , quadratic Riccati differential equation of fractional order (Odibat and Momani 2006), and to other fields. This HPM yields a very rapid convergence of the solution series in most
INTRODUCTION
Ecosystems modelling can be approach in many ways. Global methods are based on differential systems and Individual-based methods allow to represent local phenomena. The first methods are efficient to formulate general behavior of the whole system by using global parameters. The seconds deal with a better understanding on which local phenomenon are hidden inside these global parameters but they lead to high consuming computing. Innovative models are today based on hybrid approaches which manage during computing different way to express the ecosystems behavior. When and where some regularities are observed, we can change from individual-based model to differential ones. When automatic processes are developped in that 1
2
cases, usually only a few iterations leading to very accurate solutions.
2
ANALYSIS OF HPM
fi (t, x01 + px11 + p2 x21 + . . . , . . . , x0n + px1n + p2 x2n + . . .)
The HPM which provides an analytical approximate solution is applied to various nonlinear problems (see the references). In this section, we introduce a reliable algorithm to handle in a realistic and efficient way the nonlinear ecosystems. The proposed algorithm will then be used to investigate the system Dx1 (t) =
Pn
a1j (t)xj + f1 (t, x1 , x2 , . . . , xn ),
Dx2 (t) =
Pn
a2j (t)xj + f2 (t, x1 , x2 , . . . , xn ),
j=1
j=1
(2.1)
.. . Dxn (t) =
where the functions fi1 , fi2 , fi3 , . . . satisfy the following equation
= fi1 (t, x01 , x02 , . . . , x0n ) + pfi2 (t, x01 , x02 , . . . , x0n , x11 , x12 , . . . , x1n ) +p2 fi3 (t, x01 , x02 , . . . , x0n , x11 , x12 , . . . , x1n , x21 , x22 , . . . , x2n ) + . . . .
Setting p = 1 in the Eq. (??) yields the solution of the system (??). It obvious that the above linear equations are easy to solve, and the components xki , k ≥ 0 of the homotopy perturbation solution can be completely determined, thus the series solution is entirely determined. P k Finally, we approximate the solution xi (t) = ∞ k=0 xi (t) by the truncated series
Pn
φi (t) =
j=1 anj (t)xj + fn (t, x1 , x2 , . . . , xn ),
(2.2)
where fi is a nonlinear function for i = 1, 2, . . . , n. In view of the homotopy perturbation technique, we can construct, for i = 1, 2, . . . , n, the following homotopy aij (t)xj = pfi (t, x1 , x2 , . . . , xn ),
(2.3)
j=1
where p ∈ [0, 1]. The homotopy parameter p always changes from zero to unity. In case of p = 0, Eq. (??) becomes the linearized equation Dxi (t) =
n X
aij (t)xj ,
(2.4)
and when it is one, Eq. (??) turns out to be the original equation given in the system (??). The basic assumption is that the solution of the system (??) can be written as a power series in p: xi = x0i + px1i + p2 x2i + p3 x3i + . . .
Pn
aij (t)x0j , x0i (0) = ci ,
=
Pn
fi1 (t, x0 ),
p2 : Dx2i (t) =
Pn
aij (t)x2j + fi2 (t, x0 , x1 ), x2i (0) = 0,
p3 : Dx3i (t) = .. .
Pn
aij (t)x3j + fi3 (t, x0 , x1 , x2 ), x3i (0) = 0,
1
p :
Dx1i (t)
1 j=1 aij (t)xj
j=1
j=1
+
n X
aij (t)xj = pfi (t, x1 , x2 , . . . , xn ),
(2.7)
j=1
P 0 where p ∈ [0, 1]. In this case, the term n j=1 aij (t)xj is comPn 1 1 bined with the component xi and the term j=1 aij (t)xj 2 is combined with the component xi and so on. This variation reduces the number of terms in each component and may minimize the size of calculations. Substituting (??) into (??), we obtain the following series of linear equations
x1i (0)
p1 : Dx1i (t) =
Pn
aij (t)x0j + fi1 (t, x0 ), x1i (0) = 0,
p2 : Dx2i (t) =
Pn
aij (t)x1j + fi2 (t, x0 , x1 ), x2i (0) = 0,
p3 : Dx3i (t) = .. .
Pn
aij (t)x2j + fi3 (t, x0 , x1 , x2 ), x3i (0) = 0.
j=1
j=1
j=1
(2.5)
.
Substituting Eq. (??) into Eq. (??), and equating the terms with identical powers of p, we can obtain a series of linear equations of the form j=1
Dxi (t) − p
p0 : Dx0i (t) = 0, x0i (0) = ci ,
j=1
p0 : Dx0i (t) =
(2.6)
It is also useful, for the system (??), to construct the homotopy, for i = 1, 2, . . . , n,
x1 (0) = c1 , x2 (0) = c2 , . . . , xn (0) = cn ,
n X
xki (t).
k=0
subject to the initial condition
Dxi (t) −
N−1 X
= 0,
3
NUMERICAL TATION
IMPLEMEN-
To demonstrate the effectiveness of the HPM algorithm discussed above, several examples of nonlinear systems will be studied. In the first example we choose a linear system to show the features of HPM and the convergence of the homotopy perturbation solution. Example 3.1 Consider the linear system (Momani and Odibat 2006)
3
x′ (t) = y(t), (3.1)
′
y (t) = 2x(t) − y(t),
“ = − 43 1 − 2t +
subject to the initial conditions x(0) = 1
y(0) = −1.
,
y(t) = −1 + 3t − 52 t2 +
(3.2)
According to the homotopy given in Eq. (??), Substituting (??) and the initial conditions (??) into the homotopy (??) and equating the terms with identical powers of p, we obtain the following two sets of linear equations:
+ 13
“
1+t+
t2 2!
11 3 t 6
(−2t)2 2!
+
t3 3!
t4 4!
+
+
21 4 t 24
−
(−2t)3 3!
+
t5 5!
+
+
43 5 t 120
(−2t)4 4!
+ ...,
+ ...
”
(3.4)
”
+ ... ,
which converges to the exact solution x(t) = 23 e−2t + 13 et ,
0
0
0
p :
Dx = 0,
x (0) = 1,
p1 :
Dx1 = y 0 ,
x1 (0) = 0,
p2 :
Dx2 = y 1 ,
x2 (0) = 0,
p3 :
Dx3 = y 2 , .. .
x3 (0) = 0,
Example 3.2 Consider the predator-prey system (Momani and Odibat 2006)
Dx(t) = x(t) − x(t)y(t),
p0 :
Dy 0 = 0,
p1 :
Dy 1 = 2x0 − y 0 ,
y 1 (0) = 0,
p2 :
Dy 2 = 2x1 − y 1 ,
y 2 (0) = 0,
p3 :
Dy 3 = 2x2 − y 2 , .. .
y 3 (0) = 0,
y 0 (0) = −1,
,
x1 = −t
subject to the initial conditions x(0) = 1
x2 = 32 t2
Dx0 = 0,
y 1 = 3t,
p1 :
Dx1 = x0 − x0 y 0 ,
p2 :
Dx2 = x1 − x0 y 1 − x1 y 0 ,
p3 :
Dx3 = x2 − x0 y 2 − x1 y 1 − x2 y 0 , .. .
p0 :
Dy 0 = 0,
p1 :
Dy 1 = −y 0 + x0 y 0 ,
p2 :
Dy 2 = −y 1 + x0 y 1 + x1 y 0 ,
p3 :
Dy 3 = −y 2 + x0 y 2 + x1 y 1 + x2 y 0 , .. .
y 2 = t − 52 t2 ,
,
11 4 t 24
y3 =
,
11 3 t , 6
21 4 y 4 = − 24 t ,
,
21 5 x5 = − 120 t .. .
y5 =
,
43 5 t , 120
and so on, in this manner the rest of components of the homotopy perturbation solution for the system (??) can be obtained. The solution in series form is given by x(t) = 1 − t + 32 t2 − 56 t3 + =
2 3
+ 31
“
“
1 − 2t +
1+t+
t2 2!
(−2t)2 2!
+
t3 3!
+ +
11 4 t 24
(−2t)3 3! t4 4!
+
t5 5!
+
−
21 5 t 120
(−2t)4 4!
”
+... ,
+ ...,
+
(−2t)5 5!
+ ...
”
(3.3)
(3.7)
According to the homotopy given in Eq. (??), Substituting (??) and the initial conditions (??) into the homotopy (??) and equating the terms with identical powers of p, we obtain the following two sets of linear equations: p0 :
x3 = − 65 t3 x4 =
y(0) = 0.5.
,
y 0 = −1, ,
(3.6)
Dy(t) = −y(t) + x(t)y(t),
Consequently, solving the above equations, the first few components of the homotopy perturbation solution for the system (??) are derived as follows x0 = 1
(3.5)
y(t) = − 43 e−2t + 13 et .
x0 (0) = 1, x1 (0) = 0, x2 (0) = 0, x3 (0) = 0,
y 0 (0) = 0.5, y 1 (0) = 0, y 2 (0) = 0, y 3 (0) = 0,
Consequently, solving the above equations, the first few components of the homotopy perturbation solution for the system (??) are derived as follows
4
0
0
x = 1,
,
y =
x1 = 12 t
,
y 1 = 0,
x2 = 18 t2
,
1 3 t x3 = − 48
where a, b, c, d, e, f, g, h and i are constants, subject to the initial conditions
1 , 2
x(0) = c1
y3 =
1 3 t , 48
.. . and so on, in this manner the rest of components of the homotopy perturbation solution can be obtained. The fourthterm approximate solution for the system (??) is given by x(t) = 1 + 12 t + 18 t2 − y(t) =
1 2
+ 18 t2 +
1 3 t , 48
1 3 t , 48
(3.8)
which is the same solution for the system (??) obtained in (Momani and Odibat 2006) using Adomian decomposition method and variational iteration method. Example 3.3 Consider the predator-prey system
(3.9)
where a, b, c and d are constants, subject to the initial conditions x(0) = c1
,
y(0) = c2 .
(3.10)
This system is a generalization of the system (??). Using the homotopy given in Eq. (??), the third-term approximate solution for the system (??) is given by
(3.13)
p0 : p1 : p2 :
Dy 0 = −dy 0 , y 0 (0) = c2 , 1 1 0 0 Dy = −dy + ex y − f y 0 z 0 , y 1 (0) = 0, 2 2 0 1 1 0 Dy = −dy + e(x y + x y ) − f (y 0 z 1 + y 1 z 0 ), .. .
p0 : p1 : p2 :
Dz 0 = −gz 0 , z 0 (0) = c3 , 1 1 0 0 Dz = −gz + hx z + iy 0 z 0 , z 1 (0) = 0, 2 2 0 1 1 0 Dz = −gz + h(x z + x z ) + i(y 0 z 1 + y 1 z 0 ), .. .
Consequently, solving the above equations, the first few components of the homotopy perturbation solution for the system (??) are derived as follows x0
=
c1 exp(at),
y0
=
c2 exp(−dt),
0
=
c3 exp(−gt),
x1
=
y
y1
= −
“ ” (3.11) y(t) = c2 exp(−ct) + dca1 c2 exp((a − c)t) − exp(−ct) “ ” d2 c 2 c + a1 2 exp((2a−c)t) − exp((a−c)t) 2a a “ ” bdc c2 − exp((a−c)t) + c1 2 exp((a−2c)t) a−c a “ 2 2 h i” d c c bdc c2 1 + 2a12 2 − c1 2 a−c − a1 exp(−ct).
z1
= −
cc1 c3 bc1 c2 exp((a − d)t) + exp((a − g)t) d g ” “ bc c cc1 c3 1 2 exp(at), + d g f c2 c3 ec1 c2 exp((a − d)t) + exp(−(d + g)t) a g ” “ ec c f cc c3 1 2 exp(−dt), + a g hc1 c3 ic2 c3 exp((a − g)t) − exp(−(d + g)t) a d ” “ hc c ic2 c3 1 3 exp(−gt). − a d
.. .
Example 3.5 Consider the system (Aziz-Alaoui 2006)
Example 3.4 Consider the predator-prey system
Dx(t) = a0 x(t) − b0 x2 (t) −
Dx(t) = ax(t) − bx(t)y(t) − cx(t)z(t),
Dy(t) = −a1 y(t) + (3.12) Dz(t) = a2 z(t) −
υ0 x(t)y(t) , d0 +x(t)
υ1 x(t)y(t) d1 +x(t) 2
Dz(t) = −gz(t) + hx(t)z(t) + iy(t)z(t),
z(0) = c3 .
Dx0 = ax0 , x0 (0) = c1 , Dx1 = ax1 − bx0 y 0 − cx0 z 0 , x1 (0) = 0, 2 2 0 1 1 0 Dx = ax − b(x y + x y ) − c(x0 z 1 + x1 z 0 ), .. .
−
“ ” x(t) = c1 exp(at) + bc1cc2 exp((a − c)t) − exp(at) “ ” bdc2 c − a1 2 exp((2a−c)t) + exp((a−c)t) a−c c “ ” b2 c c2 + exp((a−c)t) − c1 2 − exp((a−2c)t) 2c c “ 2 h i ” bdc1 c2 b2 c c2 1 + + 1c + 2c12 2 exp(at), a a−c
Dy(t) = −dy(t) + ex(t)y(t) − f y(t)z(t),
,
p0 : p1 : p2 :
Dx(t) = ax(t) − bx(t)y(t), Dy(t) = −cy(t) + dx(t)y(t),
y(0) = c2
According to the homotopy given in Eq. (??), Substituting (??) and the initial conditions (??) into the homotopy (??) and equating the terms with identical powers of p, we obtain the following two sets of linear equations:
y 2 = 81 t2 , ,
,
υ3 z (t) , d3 +y(t)
−
υ2 y(t)z(t) , d2 +y(t)
(3.14)
5
where a0 , b0 , υ0 , d0 , a1 , υ1 , d1 , υ2 , d2 , a2 , υ3 and d3 are model parameters assuming only positive values, subject to the initial conditions x(0) = c1
y(0) = c2
,
,
z(0) = c3 .
(3.15)
Multiplying the first equation by the factor d0 + x(t), the second equation by the factor (d1 + x(t))(d2 + y(t)) and the third equation by the factor d3 + y(t). According to the homotopy given in Eq. (??), the first few components of the homotopy perturbation solution for the system (??) are derived as follows: x0
=
c1 ,
0
=
c2 ,
y0
=
c3 ,
x1
=
“
y1
=
z1
=
y
υ0 c1 c2 ” t, d0 + c1 “ υ1 c1 c2 υ2 c2 c3 ” − a1 c2 + − t, d1 + c1 d2 + c2 “ 2 ” υ3 c3 a2 c3 − t. d3 + c2 a0 c1 − b0 c21 −
.. .
4
CONCLUSION
In this work, the HPM has been successfully applied to construct approximate solutions for nonlinear systems of differential equations. The method were used in a direct way to study nonlinear ecosystems. There are some points to make here. First, the HPM doesn’t require a small parameter in an equation and the perturbation equation can be easily constructed by a homotopy in topology. Second, the HPM provides the solution in terms of convergent series with easily computable components. Third, the results show that the homotopy perturbation solution in example 1 converges to the exact solution and the approximate solution in example 2 is the same approximate solution obtained using Adomian decomposition method and variational iteration method. Fourth, it is clear and remarkable, from example 3, 4 and 5, that the HPM is effective and simple to solve nonlinear systems, specifically predator-prey systems with 2 or 3 populations. It can be easily generalized to any finite populations number.
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