- Email: [email protected]
Application of perturbation–iteration method to Lotka–Volterra equations
Alexandria Engineering Journal (2016) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
Application of perturbation–iteration method to Lotka–Volterra equations _ Yig˘it Aksoy a,*, U¨nal Go¨ktasß b, Mehmet Pakdemirli a, Ihsan Timuc¸in Dolapc¸ı a a b
Department of Mechanical Engineering, Celal Bayar University, 45140 Muradiye, Manisa, Turkey Department of Computer Engineering, Turgut O¨zal University, 06010 Kec¸io¨ren, Ankara, Turkey
Received 2 November 2015; revised 8 February 2016; accepted 20 February 2016
KEYWORDS Perturbation method; Perturbation–iteration method; Symbolic computation; Lotka–Volterra equations; Systems of first order differential equations
Abstract Perturbation–iteration method is generalized for systems of first order differential equations. Approximate solutions of Lotka–Volterra systems are obtained using the method. Comparisons of our results with each other and with numerical solutions are given. The method is implemented in Mathematica, a major computer algebra system. The package PerturbationIteration.m automatically carries out the tedious calculations of the method. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The study of methods for approximate solutions of nonlinear models in real life has always been a growing branch of applied mathematical sciences. Many methods with different capabilities and limitations have been developed. The well-known approximate analytical method, the perturbation technique [1] can deal with weakly nonlinear systems due to the small parameter assumption. To overcome this limitation, methods such as the linearized perturbation method [2], the Lindstedt–Poincare´ method with modified frequency expansion [3], the multiple-scale Lindstedt–Poincare´ method [4] and the parameter expanding method [5] were developed. Also methods such as the Adomian decomposition method [6], the variational iteration method [7], and the homotopy analysis method [8] were among the non-perturbative methods that were applied to many interesting mathematical problems. * Corresponding author. E-mail address: [email protected] (Y. Aksoy). Peer review under responsibility of Faculty of Engineering, Alexandria University.
Other attempts to treat both weakly and strongly nonlinear problems were through iteration procedures which used preformed alternative equations to obtain approximate solutions. Just to list a few, He [9] linearized the nonlinear terms by substitution of iterative solution functions from previous iteration results, Mickens’ iteration procedure [10] was for specific problems, and variational iteration method [11] was used to solve boundary value problems. Recently, the perturbation–iteration method, which gives valuable solutions for strongly nonlinear problems [12,13] has been developed. In [14], approximate solutions of some nonlinear heat transfer problems were obtained, and the comparison of the results showed that perturbation–iteration method fits better than the variational iteration method as the parameter measuring the nonlinearity takes larger values. The aim of this study was to develop perturbation–iteration algorithms for systems of first order differential equations and to obtain accurate solutions of Lotka–Volterra differential equations. Many analytical methods such as Adomian decomposition [15], variational iteration method [16], homotopy analysis method [17], and optimal parametric iteration method [18] were successfully applied to this type of problems.
http://dx.doi.org/10.1016/j.aej.2016.02.015 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Y. Aksoy et al., Application of perturbation–iteration method to Lotka–Volterra equations, Alexandria Eng. J. (2016), http://dx. doi.org/10.1016/j.aej.2016.02.015
2
Y. Aksoy et al.
The paper is organized as follows. In Section 2, the perturbation–iteration algorithm for systems is explained. The method is applied to Lotka–Volterra equations in Section 3. In Section 4, comparisons of our results with other methods are given. The package PerturbationIteration.m is described in Section 5. Concluding remarks are given in Section 6. 2. Perturbation–iteration method for systems of first order differential equations In this section, the perturbation–iteration method, described in [12,13], is generalized toward systems of first order differential equations. The generalization is developed by taking arbitrary number of terms in Taylor series expansion and one correction term in perturbation expansion PIA(1,M) i.e. the first number expresses the correction terms in the perturbation expansion and the second number expresses the derivative orders in the Taylor expansion. Consider the following system of first order differential equations: Fk ðu_ k ; uj ; e; tÞ ¼ 0;
j ¼ 1; 2; . . . ; K; k ¼ 1; 2; . . . ; K;
ð1Þ
where uj are the dependent variables, t is the independent variable, K is the number of dependent variables, e is the artificially introduced small parameter and the dot stands for the derivative. Reconsider Eq. (1) in the following iterative form: Fk ðu_ k;nþ1 ; uj;nþ1 ; e; tÞ ¼ 0;
ð2Þ
where the subscript n expresses the number of iterations completed in the procedure. More clearly, the system of equations is given below F1 ¼ F1 ðu_ 1 ; u1 ; u2 ; . . . ; uK ; e; tÞ F2 ¼ F2 ðu_ 2 ; u1 ; u2 ; . . . ; uK ; e; tÞ .. . FK ¼ FK ðu_ K ; u1 ; u2 ; . . . ; uK ; e; tÞ:
ð3Þ
ð4Þ
Taylor series expansion of equation (1) is given as follows: m M X 1 d Fk ¼ Fk em ; k ¼ 1; 2 . . . . . . :K ð5Þ m! de e¼0 m¼0
The Lotka–Volterra equations, also known as the predator– prey equations, are frequently used by mathematicians to describe the time evolution of the species in the dynamics of biological systems [15–17]. In the more general framework one can study the following multidimensional Lotka–Volterra system: ! K X duk ¼ uk bk þ akj uj ; uk ð0Þ ¼ ck ; k ¼ 1; 2; . . . ; K; ð8Þ dt j¼1 where ck represent the populations of the species at the beginning of the evolution. In the following subsections, solutions of one dimensional and multidimensional systems such as K = 1, 2, 3,. . . are investigated by the perturbation–iteration method. 3.1. One dimensional system (K = 1) One dimensional Lotka–Volterra equation, known as the Verhulst equation [19], describes the behavior of population in time of one species competing for a given finite source of food: du ¼ uðb þ auÞ; dt
K X d @ u_ k;nþ1 @ @uj;nþ1 @ @ ¼ þ þ : de @e @e @ u_ k;nþ1 j¼1 @e @uj;nþ1
b > 0;
a < 0;
uð0Þ > 0:
The exact solution of (9) is 8 bebt ; b–0 < bþauð0Þ aebt uð0Þ uðtÞ ¼ : uð0Þ ; b ¼ 0: 1auð0Þt
ð9Þ
ð10Þ
3.1.1. Perturbation iteration method with first order derivatives in the Taylor series expansion PIA(1,1) First rewrite Eqs. (9) and (3) in the following form: F1 ¼ u_ 1;nþ1 bu1;nþ1 eau21;nþ1 ¼ 0;
ð11Þ
euc1;n ;
ð12Þ
u1;nþ1 ¼ u1;n þ
with the derivative operator defined as follows: ð6Þ
Calculating derivatives at e = 0 and substituting Eq. (6) into Eq. (5) yields ( !m ) M K X X 1 @ @ @ Fk ¼ þ ucj;n þ Fk em ¼ 0; u_ ck;n _ m! @ u @u @e k;nþ1 j;nþ1 m¼0 j¼1 e¼0
k ¼ 1; 2 . . . . . . :K
3. Application to Lotka–Volterra problems
Approximate solutions of Eq. (9) will be obtained using two different perturbation–iteration algorithms by taking M = 1 and M = 2.
Next, define the following iterative perturbation series: uj;nþ1 ¼ uj;n þ eucj;n ;
are taken, the algorithm is PIA(1,m). A more general algorithm with PIA(n,m) can be constructed but would cause too much complexity in the applications.
ð7Þ
Note that, by taking larger values of M, i.e., increasing the number of terms in the Taylor series expansion, one can develop different algorithms for the specific problem with the help of Eq. (7). Since one correction term in the perturbation expansion and m’th order derivatives in the Taylor expansion
where e is artificially introduced as a small parameter. Reorganizing Eq. (7) for Eq. (9) yields duc1;n @F1 =@unþ1 c eF1 @F1 =@e þ u ¼ ; dt @F1 =@ u_ nþ1 1;n @F1 =@ u_ nþ1 @F1 =@ u_ nþ1 where the derivatives are F1 je¼0 ¼ u_ n bun ; @F1 ¼ a u2n : @e e¼0
@F1 ¼ b; @unþ1 e¼0
ð13Þ
@F1 ¼ 1; @ u_ nþ1 e¼0 ð14Þ
Finally, the iteration equation is obtained by substituting (14) into (13) and setting e = 1: duc1;n buc1;n ¼ u_ 1;n þ bu1;n þ au21;n : dt
ð15Þ
Please cite this article in press as: Y. Aksoy et al., Application of perturbation–iteration method to Lotka–Volterra equations, Alexandria Eng. J. (2016), http://dx. doi.org/10.1016/j.aej.2016.02.015
Application of perturbation–iteration method
3
One can take the initially assumed function as follows: u1;0 ¼ ae ;
ð16Þ
bt
which satisfy the initial condition exactly. Starting with the above initial function, first uc1;0 is calculated from Eq. (15) and then substituted into Eq. (12) to obtain u1,1 as the solution of the first iteration. The iteration process is repeated using the previous solution as an initial guess until satisfactory result is obtained. This way, one can obtain the first and second iteration solutions as follows: a2 a bt bt e ðe 1Þ; b 2 2 a aebt ðebt 1Þ a3 a2 ebt ðebt 1Þ þ ¼ aebt þ b b2
u1;1 ¼ aebt þ u1;2
a4 a3 ebt ðebt 1Þ : 3b3
ð18Þ
3.1.2. Perturbation iteration method with second order derivatives in the Taylor series expansion PIA(1,2) Taking up to second order derivative terms in the Taylor series expansion leads to the iteration formula: duc1;n ðb þ 2au1;n Þuc1;n ¼ u_ 1;n þ bu1;n þ au21;n : dt
The second and third iteration solutions are as follows: 2aa 2aað2ebt 1Þ aebt b 4aa 2aaebt bt 2aa 2aaebt u1;2 ¼ ð21Þ 3e b þ 4e b þ e b e b ðe 1Þ ; 8 b 2að2þebt Þa 2að1þ2ebt Þa 2að1þ4ebt Þa 6aa 6aebt a 1 bt6aa u1;3 ¼ 384 e b a 117e b þ 15e b þ 161e b þ 90e b e b 2aebt a b
4aa
4aebt a
2að1þebt Þa
ð1 þ ebt Þð20e b þ e b þ 12e b Þa bt Þa 2að1þe 2 2 2aa 2aebt a 2 : þ 24ab2 a e b ð1 þ ebt Þ 5e b þ 2e b
ð22Þ
3.2. Two dimensional system (system with two species, K = 2) Two dimensional Lotka–Volterra system was first proposed by Lotka [20] and Volterra [21] without knowing each other’s work. The following two dimensional competitive Lotka– Volterra equation models a pair of species competing for a common resource: ¼ u1 ðb1 þ a11 u1 þ a12 u2 Þ;
u1 ð0Þ ¼ a;
¼ u2 ðb2 þ a21 u1 þ a22 u2 Þ;
u2 ð0Þ ¼ b;
e¼0
ð25Þ 1 2 F1 je¼0 ¼ u_ 1;n b1 u1;n ; F2 je¼0 ¼ u_ 2;n b2 u2;n ; @ u@F ¼ 1; @ u@F ¼ 1; _ 1;nþ1 _ 2;nþ1 e¼0 e¼0 @F1 1 1 ¼ b1 ; @u@F ¼ a11 u21;n a12 u1;n u2;n ; ¼ 0; @F @u1;nþ1 @e e¼0 2;nþ1 e¼0 e¼0 @F2 2 2 ¼ a22 u22;n a21 u1;n u2;n ; @u@F ¼ b2 ; @u@F ¼ 0: @e e¼0 2;nþ1 1;nþ1 e¼0
e¼0
ð26Þ
Substituting (26) into (25) leads to the final form of the iteration equation: u_ c1;n b1 uc1;n ¼ a11 u21;n þ a12 u1;n u2;n u_ 1;n þ b1 u1;n ; ð27Þ u_ c2;n b2 uc2;n ¼ a22 u22;n þ a21 u1;n u2;n u_ 2;n þ b2 u2;n : u1;nþ1 ¼ u1;n þ uc1;n ;
ð23Þ
where a11, a12, a21, a22 are parameters representing the interaction of the species and the self-interaction, and a, b are the initial size of populations. Perturbation–iteration method with first order derivatives in the Taylor series PIA(1,1) will be applied to Eq. (23). First, re-write Eq. (23) as follows:
ð28Þ
u2;nþ1 ¼ u2;n þ uc2;n ; taking u1;0 ¼ aeb1 t ;
u2;0 ¼ beb2 t ;
ð29Þ
as an initial approximation satisfying the initial conditions leads to the following: uc1;0 ¼ a uc2;0 ¼
2a
tb tb 11 e 1 ð1þe 1 Þ
b1
aba21 etb2 ð1þetb1 Þ b1
þ aba12 e þ
tb1 ð1þetb2 Þ
b2
;
ð30Þ
b2 a22 etb2 ð1þetb2 Þ ; b2
as correction terms. Combining (29) and (30) in (28) yields the first iteration solutions: u1;1 ¼ aetb1 þ a
2a
In second and third iteration solutions, as the calculations get more involved, the second term in (19), b+2au1,n is approximated as b+2au1,0 for simplicity and help of a computer algebra system such as Mathematica is essential.
du1 dt du2 dt
Setting K = 2 and M = 1, transforms Eq. (7) into the following: nh i o @ @ @ F1 ffi F1 je¼0 þ u_ c1;n @ u_ 1;nþ1 þ uc1;n @u1;nþ1 þ uc2;n @u2;nþ1 þ @e@ F1 e¼0 nh i oe¼0 @ @ @ F2 ffi F2 je¼0 þ u_ c2;n @ u_ 2;nþ1 þ uc2;n @u2;nþ1 þ uc1;n @u1;nþ1 þ @e@ F2 e¼0
As Eq. (4) become: ð19Þ
Using the same initial guess function u1;0 ¼ aebt ; one can obtain the first iteration solution: a 2aa bt ð20Þ u1;1 ¼ ebt 1 þ e b ðe 1Þ ; 2
12a e b
ð24Þ
with ð17Þ
3
þ
F1 ¼ u_ 1;nþ1 u1;nþ1 ðb1 þ ea11 u1;nþ1 þ ea12 u2;nþ1 Þ ¼ 0 F2 ¼ u_ 2;nþ1 u2;nþ1 ðb2 þ ea21 u1;nþ1 þ ea22 u2;nþ1 Þ ¼ 0:
u2;1 ¼ betb2 þ
tb tb 11 e 1 ð1þe 1 Þ
b1
aba21 etb2 ð1þetb1 Þ b1
þ aba12 e 2
þb
a22
tb1 ð1þetb2 Þ
b2
etb2 ð1þetb2 Þ b2
; :
ð31Þ
For briefness, higher iteration results are not given here, but numerical results of those are given in numerical comparison tables. 3.3. Three dimensional system (system with three species, K = 3) The three dimensional Lotka–Volterra system models population dynamics of three competitive species in an ecosystem [22]: du1 dt du2 dt du3 dt
¼ u1 ðb1 þ a11 u1 þ a12 u2 þ a13 u3 Þ; u1 ð0Þ ¼ a; ¼ u2 ðb2 þ a21 u1 þ a22 u2 þ a23 u3 Þ; u2 ð0Þ ¼ b;
ð32Þ
¼ u3 ðb3 þ a31 u1 þ a32 u2 þ a33 u3 Þ; u3 ð0Þ ¼ c:
Using u1;0 ¼ aeb1 t ; u2;0 ¼ beb2 t ; u3;0 ¼ ceb3 t as initial functions, approximate solutions after the first iteration are as follows:
Please cite this article in press as: Y. Aksoy et al., Application of perturbation–iteration method to Lotka–Volterra equations, Alexandria Eng. J. (2016), http://dx. doi.org/10.1016/j.aej.2016.02.015
4
Y. Aksoy et al. a2 a11 etb1 ð1 þ etb1 Þ aba12 etb1 ð1 þ etb2 Þ þ b1 b2 aca13 etb1 ð1 þ etb3 Þ þ ; b3
u1;1 ¼ aetb1 þ
aba21 etb2 ð1 þ etb1 Þ b2 a22 etb2 ð1 þ etb2 Þ þ b1 b2 bca23 etb2 ð1 þ etb3 Þ þ ; b3 aca31 etb3 ð1 þ etb1 Þ bca32 etb3 ð1 þ etb2 Þ ¼ cetb3 þ þ b1 b2 c2 a33 etb3 ð1 þ etb3 Þ þ : b3
ð33Þ
u2;1 ¼ betb2 þ
u3;1
ð34Þ
ð35Þ
For second and higher iteration solutions, the parameters have to be set to values.
from PIA(1,2) are given in the last three columns of Table 1. It is easy to see that both iteration algorithms rapidly converge to the exact solution. For the PIA(1,2) case, third iteration solutions acquire almost the same accuracy with the fifth iteration solutions for the PIA(1,1) case. Representative values of known numerical and perturbation–iteration solutions of (23) are given in Table 2. Only in two iterations, very high accuracy has been achieved as expected. Comparisons for approximate solutions of three dimensional Lotka–Volterra system (32) are given in Table 3. Satisfactory results on the whole domain are obtained in four or five iterations. The results indicate that solutions obtained with the perturbation–iteration method rapidly converge to the known numerical solutions. 5. The mathematica package
3.4. Multi-dimensional Lotka–Volterra Equations In this subsection, we will obtain perturbation–iteration solutions of multi-dimensional Lotka–Volterra equations having arbitrary number of competitive species. We seek solutions with first order derivatives in the Taylor series expansion. First, rewrite Eq. (8) as follows: ! K X ð36Þ Fk ¼ u_ k;nþ1 uk;nþ1 bk þ e akj uj;nþ1 ¼ 0:
To use the code, first load the Mathematica package PerturbationIteration.m [23] using the command In[2]:=Get[‘‘PerturbationIteration.m”];
Proceeding with the two dimensional Lotka–Volterra system (23) as an example, call the function PerturbationIterationAl gorithm (which is part of the package):
j¼1
For m = 1, the terms included in Eq. (7) are as follows: @Fk @Fk ¼ 1; ¼ bk ; Fk je¼0 ¼ u_ k;n bk uk;n ; @ u_ k;nþ1 e¼0 @uk;nþ1 e¼0 K X @Fk ¼ akj uk;n uj;n : ð37Þ @e e¼0
j¼1
Hence substituting these terms in Eq. (7) yields u_ ck;n bk uck;n ¼ u_ k;n þ bk uk;n þ
K X akj uk;n uj;n :
ð38Þ
j¼1
Integrating Eq. (38) gives the following: Z Z uck;n ¼ ebk t ebk t u_ k;n dt þ ebk t bk uk;n ! Z K X bk t akj uk;n uj;n dt þ cn ; þ e
ð39Þ
j¼1
where cn is integration constant to be determined by the initial conditions. Then n’ th iteration solution for multidimensional Lotka–Volterra equations becomes the following: Z Z uk;nþ1 ¼ uk;n þ ebk t ebk t u_ k;n dt þ ebk t bk uk;n ! Z K X bk t akj uk;n uj;n dt þ cn : ð40Þ þ e j¼1
4. Numerical results and comparisons Comparisons of approximate solutions of Eq. (9) with the exact solutions are presented in Table 1. Solutions up to five iterations are computed using PIA(1,1). Solutions resulted
In[3]: = PerturbationIterationAlgorithm[ {u0 [t] == u[t] (b1 + a11 u[t] + a12 v[t]), v0 [t] == v[t] (b2 + a21 u[t] + a22 v[t])}, {u[0] == a, v[0] == b}, {u, v}, t, 1, 1, 2]
which will produce the solutions in (31) and go one more iteration. Similarly, results given in subsections 3.1–3.3 of this paper can be reproduced by using our package. The usage of the main function PerturbationIterationAlgo rithm is as follows: PerturbationIterationAlgorithm[,
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
Application of perturbation–iteration method to Lotka–Volterra equations _ Yig˘it Aksoy a,*, U¨nal Go¨ktasß b, Mehmet Pakdemirli a, Ihsan Timuc¸in Dolapc¸ı a a b
Department of Mechanical Engineering, Celal Bayar University, 45140 Muradiye, Manisa, Turkey Department of Computer Engineering, Turgut O¨zal University, 06010 Kec¸io¨ren, Ankara, Turkey
Received 2 November 2015; revised 8 February 2016; accepted 20 February 2016
KEYWORDS Perturbation method; Perturbation–iteration method; Symbolic computation; Lotka–Volterra equations; Systems of first order differential equations
Abstract Perturbation–iteration method is generalized for systems of first order differential equations. Approximate solutions of Lotka–Volterra systems are obtained using the method. Comparisons of our results with each other and with numerical solutions are given. The method is implemented in Mathematica, a major computer algebra system. The package PerturbationIteration.m automatically carries out the tedious calculations of the method. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The study of methods for approximate solutions of nonlinear models in real life has always been a growing branch of applied mathematical sciences. Many methods with different capabilities and limitations have been developed. The well-known approximate analytical method, the perturbation technique [1] can deal with weakly nonlinear systems due to the small parameter assumption. To overcome this limitation, methods such as the linearized perturbation method [2], the Lindstedt–Poincare´ method with modified frequency expansion [3], the multiple-scale Lindstedt–Poincare´ method [4] and the parameter expanding method [5] were developed. Also methods such as the Adomian decomposition method [6], the variational iteration method [7], and the homotopy analysis method [8] were among the non-perturbative methods that were applied to many interesting mathematical problems. * Corresponding author. E-mail address: [email protected] (Y. Aksoy). Peer review under responsibility of Faculty of Engineering, Alexandria University.
Other attempts to treat both weakly and strongly nonlinear problems were through iteration procedures which used preformed alternative equations to obtain approximate solutions. Just to list a few, He [9] linearized the nonlinear terms by substitution of iterative solution functions from previous iteration results, Mickens’ iteration procedure [10] was for specific problems, and variational iteration method [11] was used to solve boundary value problems. Recently, the perturbation–iteration method, which gives valuable solutions for strongly nonlinear problems [12,13] has been developed. In [14], approximate solutions of some nonlinear heat transfer problems were obtained, and the comparison of the results showed that perturbation–iteration method fits better than the variational iteration method as the parameter measuring the nonlinearity takes larger values. The aim of this study was to develop perturbation–iteration algorithms for systems of first order differential equations and to obtain accurate solutions of Lotka–Volterra differential equations. Many analytical methods such as Adomian decomposition [15], variational iteration method [16], homotopy analysis method [17], and optimal parametric iteration method [18] were successfully applied to this type of problems.
http://dx.doi.org/10.1016/j.aej.2016.02.015 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Y. Aksoy et al., Application of perturbation–iteration method to Lotka–Volterra equations, Alexandria Eng. J. (2016), http://dx. doi.org/10.1016/j.aej.2016.02.015
2
Y. Aksoy et al.
The paper is organized as follows. In Section 2, the perturbation–iteration algorithm for systems is explained. The method is applied to Lotka–Volterra equations in Section 3. In Section 4, comparisons of our results with other methods are given. The package PerturbationIteration.m is described in Section 5. Concluding remarks are given in Section 6. 2. Perturbation–iteration method for systems of first order differential equations In this section, the perturbation–iteration method, described in [12,13], is generalized toward systems of first order differential equations. The generalization is developed by taking arbitrary number of terms in Taylor series expansion and one correction term in perturbation expansion PIA(1,M) i.e. the first number expresses the correction terms in the perturbation expansion and the second number expresses the derivative orders in the Taylor expansion. Consider the following system of first order differential equations: Fk ðu_ k ; uj ; e; tÞ ¼ 0;
j ¼ 1; 2; . . . ; K; k ¼ 1; 2; . . . ; K;
ð1Þ
where uj are the dependent variables, t is the independent variable, K is the number of dependent variables, e is the artificially introduced small parameter and the dot stands for the derivative. Reconsider Eq. (1) in the following iterative form: Fk ðu_ k;nþ1 ; uj;nþ1 ; e; tÞ ¼ 0;
ð2Þ
where the subscript n expresses the number of iterations completed in the procedure. More clearly, the system of equations is given below F1 ¼ F1 ðu_ 1 ; u1 ; u2 ; . . . ; uK ; e; tÞ F2 ¼ F2 ðu_ 2 ; u1 ; u2 ; . . . ; uK ; e; tÞ .. . FK ¼ FK ðu_ K ; u1 ; u2 ; . . . ; uK ; e; tÞ:
ð3Þ
ð4Þ
Taylor series expansion of equation (1) is given as follows: m M X 1 d Fk ¼ Fk em ; k ¼ 1; 2 . . . . . . :K ð5Þ m! de e¼0 m¼0
The Lotka–Volterra equations, also known as the predator– prey equations, are frequently used by mathematicians to describe the time evolution of the species in the dynamics of biological systems [15–17]. In the more general framework one can study the following multidimensional Lotka–Volterra system: ! K X duk ¼ uk bk þ akj uj ; uk ð0Þ ¼ ck ; k ¼ 1; 2; . . . ; K; ð8Þ dt j¼1 where ck represent the populations of the species at the beginning of the evolution. In the following subsections, solutions of one dimensional and multidimensional systems such as K = 1, 2, 3,. . . are investigated by the perturbation–iteration method. 3.1. One dimensional system (K = 1) One dimensional Lotka–Volterra equation, known as the Verhulst equation [19], describes the behavior of population in time of one species competing for a given finite source of food: du ¼ uðb þ auÞ; dt
K X d @ u_ k;nþ1 @ @uj;nþ1 @ @ ¼ þ þ : de @e @e @ u_ k;nþ1 j¼1 @e @uj;nþ1
b > 0;
a < 0;
uð0Þ > 0:
The exact solution of (9) is 8 bebt ; b–0 < bþauð0Þ aebt uð0Þ uðtÞ ¼ : uð0Þ ; b ¼ 0: 1auð0Þt
ð9Þ
ð10Þ
3.1.1. Perturbation iteration method with first order derivatives in the Taylor series expansion PIA(1,1) First rewrite Eqs. (9) and (3) in the following form: F1 ¼ u_ 1;nþ1 bu1;nþ1 eau21;nþ1 ¼ 0;
ð11Þ
euc1;n ;
ð12Þ
u1;nþ1 ¼ u1;n þ
with the derivative operator defined as follows: ð6Þ
Calculating derivatives at e = 0 and substituting Eq. (6) into Eq. (5) yields ( !m ) M K X X 1 @ @ @ Fk ¼ þ ucj;n þ Fk em ¼ 0; u_ ck;n _ m! @ u @u @e k;nþ1 j;nþ1 m¼0 j¼1 e¼0
k ¼ 1; 2 . . . . . . :K
3. Application to Lotka–Volterra problems
Approximate solutions of Eq. (9) will be obtained using two different perturbation–iteration algorithms by taking M = 1 and M = 2.
Next, define the following iterative perturbation series: uj;nþ1 ¼ uj;n þ eucj;n ;
are taken, the algorithm is PIA(1,m). A more general algorithm with PIA(n,m) can be constructed but would cause too much complexity in the applications.
ð7Þ
Note that, by taking larger values of M, i.e., increasing the number of terms in the Taylor series expansion, one can develop different algorithms for the specific problem with the help of Eq. (7). Since one correction term in the perturbation expansion and m’th order derivatives in the Taylor expansion
where e is artificially introduced as a small parameter. Reorganizing Eq. (7) for Eq. (9) yields duc1;n @F1 =@unþ1 c eF1 @F1 =@e þ u ¼ ; dt @F1 =@ u_ nþ1 1;n @F1 =@ u_ nþ1 @F1 =@ u_ nþ1 where the derivatives are F1 je¼0 ¼ u_ n bun ; @F1 ¼ a u2n : @e e¼0
@F1 ¼ b; @unþ1 e¼0
ð13Þ
@F1 ¼ 1; @ u_ nþ1 e¼0 ð14Þ
Finally, the iteration equation is obtained by substituting (14) into (13) and setting e = 1: duc1;n buc1;n ¼ u_ 1;n þ bu1;n þ au21;n : dt
ð15Þ
Please cite this article in press as: Y. Aksoy et al., Application of perturbation–iteration method to Lotka–Volterra equations, Alexandria Eng. J. (2016), http://dx. doi.org/10.1016/j.aej.2016.02.015
Application of perturbation–iteration method
3
One can take the initially assumed function as follows: u1;0 ¼ ae ;
ð16Þ
bt
which satisfy the initial condition exactly. Starting with the above initial function, first uc1;0 is calculated from Eq. (15) and then substituted into Eq. (12) to obtain u1,1 as the solution of the first iteration. The iteration process is repeated using the previous solution as an initial guess until satisfactory result is obtained. This way, one can obtain the first and second iteration solutions as follows: a2 a bt bt e ðe 1Þ; b 2 2 a aebt ðebt 1Þ a3 a2 ebt ðebt 1Þ þ ¼ aebt þ b b2
u1;1 ¼ aebt þ u1;2
a4 a3 ebt ðebt 1Þ : 3b3
ð18Þ
3.1.2. Perturbation iteration method with second order derivatives in the Taylor series expansion PIA(1,2) Taking up to second order derivative terms in the Taylor series expansion leads to the iteration formula: duc1;n ðb þ 2au1;n Þuc1;n ¼ u_ 1;n þ bu1;n þ au21;n : dt
The second and third iteration solutions are as follows: 2aa 2aað2ebt 1Þ aebt b 4aa 2aaebt bt 2aa 2aaebt u1;2 ¼ ð21Þ 3e b þ 4e b þ e b e b ðe 1Þ ; 8 b 2að2þebt Þa 2að1þ2ebt Þa 2að1þ4ebt Þa 6aa 6aebt a 1 bt6aa u1;3 ¼ 384 e b a 117e b þ 15e b þ 161e b þ 90e b e b 2aebt a b
4aa
4aebt a
2að1þebt Þa
ð1 þ ebt Þð20e b þ e b þ 12e b Þa bt Þa 2að1þe 2 2 2aa 2aebt a 2 : þ 24ab2 a e b ð1 þ ebt Þ 5e b þ 2e b
ð22Þ
3.2. Two dimensional system (system with two species, K = 2) Two dimensional Lotka–Volterra system was first proposed by Lotka [20] and Volterra [21] without knowing each other’s work. The following two dimensional competitive Lotka– Volterra equation models a pair of species competing for a common resource: ¼ u1 ðb1 þ a11 u1 þ a12 u2 Þ;
u1 ð0Þ ¼ a;
¼ u2 ðb2 þ a21 u1 þ a22 u2 Þ;
u2 ð0Þ ¼ b;
e¼0
ð25Þ 1 2 F1 je¼0 ¼ u_ 1;n b1 u1;n ; F2 je¼0 ¼ u_ 2;n b2 u2;n ; @ u@F ¼ 1; @ u@F ¼ 1; _ 1;nþ1 _ 2;nþ1 e¼0 e¼0 @F1 1 1 ¼ b1 ; @u@F ¼ a11 u21;n a12 u1;n u2;n ; ¼ 0; @F @u1;nþ1 @e e¼0 2;nþ1 e¼0 e¼0 @F2 2 2 ¼ a22 u22;n a21 u1;n u2;n ; @u@F ¼ b2 ; @u@F ¼ 0: @e e¼0 2;nþ1 1;nþ1 e¼0
e¼0
ð26Þ
Substituting (26) into (25) leads to the final form of the iteration equation: u_ c1;n b1 uc1;n ¼ a11 u21;n þ a12 u1;n u2;n u_ 1;n þ b1 u1;n ; ð27Þ u_ c2;n b2 uc2;n ¼ a22 u22;n þ a21 u1;n u2;n u_ 2;n þ b2 u2;n : u1;nþ1 ¼ u1;n þ uc1;n ;
ð23Þ
where a11, a12, a21, a22 are parameters representing the interaction of the species and the self-interaction, and a, b are the initial size of populations. Perturbation–iteration method with first order derivatives in the Taylor series PIA(1,1) will be applied to Eq. (23). First, re-write Eq. (23) as follows:
ð28Þ
u2;nþ1 ¼ u2;n þ uc2;n ; taking u1;0 ¼ aeb1 t ;
u2;0 ¼ beb2 t ;
ð29Þ
as an initial approximation satisfying the initial conditions leads to the following: uc1;0 ¼ a uc2;0 ¼
2a
tb tb 11 e 1 ð1þe 1 Þ
b1
aba21 etb2 ð1þetb1 Þ b1
þ aba12 e þ
tb1 ð1þetb2 Þ
b2
;
ð30Þ
b2 a22 etb2 ð1þetb2 Þ ; b2
as correction terms. Combining (29) and (30) in (28) yields the first iteration solutions: u1;1 ¼ aetb1 þ a
2a
In second and third iteration solutions, as the calculations get more involved, the second term in (19), b+2au1,n is approximated as b+2au1,0 for simplicity and help of a computer algebra system such as Mathematica is essential.
du1 dt du2 dt
Setting K = 2 and M = 1, transforms Eq. (7) into the following: nh i o @ @ @ F1 ffi F1 je¼0 þ u_ c1;n @ u_ 1;nþ1 þ uc1;n @u1;nþ1 þ uc2;n @u2;nþ1 þ @e@ F1 e¼0 nh i oe¼0 @ @ @ F2 ffi F2 je¼0 þ u_ c2;n @ u_ 2;nþ1 þ uc2;n @u2;nþ1 þ uc1;n @u1;nþ1 þ @e@ F2 e¼0
As Eq. (4) become: ð19Þ
Using the same initial guess function u1;0 ¼ aebt ; one can obtain the first iteration solution: a 2aa bt ð20Þ u1;1 ¼ ebt 1 þ e b ðe 1Þ ; 2
12a e b
ð24Þ
with ð17Þ
3
þ
F1 ¼ u_ 1;nþ1 u1;nþ1 ðb1 þ ea11 u1;nþ1 þ ea12 u2;nþ1 Þ ¼ 0 F2 ¼ u_ 2;nþ1 u2;nþ1 ðb2 þ ea21 u1;nþ1 þ ea22 u2;nþ1 Þ ¼ 0:
u2;1 ¼ betb2 þ
tb tb 11 e 1 ð1þe 1 Þ
b1
aba21 etb2 ð1þetb1 Þ b1
þ aba12 e 2
þb
a22
tb1 ð1þetb2 Þ
b2
etb2 ð1þetb2 Þ b2
; :
ð31Þ
For briefness, higher iteration results are not given here, but numerical results of those are given in numerical comparison tables. 3.3. Three dimensional system (system with three species, K = 3) The three dimensional Lotka–Volterra system models population dynamics of three competitive species in an ecosystem [22]: du1 dt du2 dt du3 dt
¼ u1 ðb1 þ a11 u1 þ a12 u2 þ a13 u3 Þ; u1 ð0Þ ¼ a; ¼ u2 ðb2 þ a21 u1 þ a22 u2 þ a23 u3 Þ; u2 ð0Þ ¼ b;
ð32Þ
¼ u3 ðb3 þ a31 u1 þ a32 u2 þ a33 u3 Þ; u3 ð0Þ ¼ c:
Using u1;0 ¼ aeb1 t ; u2;0 ¼ beb2 t ; u3;0 ¼ ceb3 t as initial functions, approximate solutions after the first iteration are as follows:
Please cite this article in press as: Y. Aksoy et al., Application of perturbation–iteration method to Lotka–Volterra equations, Alexandria Eng. J. (2016), http://dx. doi.org/10.1016/j.aej.2016.02.015
4
Y. Aksoy et al. a2 a11 etb1 ð1 þ etb1 Þ aba12 etb1 ð1 þ etb2 Þ þ b1 b2 aca13 etb1 ð1 þ etb3 Þ þ ; b3
u1;1 ¼ aetb1 þ
aba21 etb2 ð1 þ etb1 Þ b2 a22 etb2 ð1 þ etb2 Þ þ b1 b2 bca23 etb2 ð1 þ etb3 Þ þ ; b3 aca31 etb3 ð1 þ etb1 Þ bca32 etb3 ð1 þ etb2 Þ ¼ cetb3 þ þ b1 b2 c2 a33 etb3 ð1 þ etb3 Þ þ : b3
ð33Þ
u2;1 ¼ betb2 þ
u3;1
ð34Þ
ð35Þ
For second and higher iteration solutions, the parameters have to be set to values.
from PIA(1,2) are given in the last three columns of Table 1. It is easy to see that both iteration algorithms rapidly converge to the exact solution. For the PIA(1,2) case, third iteration solutions acquire almost the same accuracy with the fifth iteration solutions for the PIA(1,1) case. Representative values of known numerical and perturbation–iteration solutions of (23) are given in Table 2. Only in two iterations, very high accuracy has been achieved as expected. Comparisons for approximate solutions of three dimensional Lotka–Volterra system (32) are given in Table 3. Satisfactory results on the whole domain are obtained in four or five iterations. The results indicate that solutions obtained with the perturbation–iteration method rapidly converge to the known numerical solutions. 5. The mathematica package
3.4. Multi-dimensional Lotka–Volterra Equations In this subsection, we will obtain perturbation–iteration solutions of multi-dimensional Lotka–Volterra equations having arbitrary number of competitive species. We seek solutions with first order derivatives in the Taylor series expansion. First, rewrite Eq. (8) as follows: ! K X ð36Þ Fk ¼ u_ k;nþ1 uk;nþ1 bk þ e akj uj;nþ1 ¼ 0:
To use the code, first load the Mathematica package PerturbationIteration.m [23] using the command In[2]:=Get[‘‘PerturbationIteration.m”];
Proceeding with the two dimensional Lotka–Volterra system (23) as an example, call the function PerturbationIterationAl gorithm (which is part of the package):
j¼1
For m = 1, the terms included in Eq. (7) are as follows: @Fk @Fk ¼ 1; ¼ bk ; Fk je¼0 ¼ u_ k;n bk uk;n ; @ u_ k;nþ1 e¼0 @uk;nþ1 e¼0 K X @Fk ¼ akj uk;n uj;n : ð37Þ @e e¼0
j¼1
Hence substituting these terms in Eq. (7) yields u_ ck;n bk uck;n ¼ u_ k;n þ bk uk;n þ
K X akj uk;n uj;n :
ð38Þ
j¼1
Integrating Eq. (38) gives the following: Z Z uck;n ¼ ebk t ebk t u_ k;n dt þ ebk t bk uk;n ! Z K X bk t akj uk;n uj;n dt þ cn ; þ e
ð39Þ
j¼1
where cn is integration constant to be determined by the initial conditions. Then n’ th iteration solution for multidimensional Lotka–Volterra equations becomes the following: Z Z uk;nþ1 ¼ uk;n þ ebk t ebk t u_ k;n dt þ ebk t bk uk;n ! Z K X bk t akj uk;n uj;n dt þ cn : ð40Þ þ e j¼1
4. Numerical results and comparisons Comparisons of approximate solutions of Eq. (9) with the exact solutions are presented in Table 1. Solutions up to five iterations are computed using PIA(1,1). Solutions resulted
In[3]: = PerturbationIterationAlgorithm[ {u0 [t] == u[t] (b1 + a11 u[t] + a12 v[t]), v0 [t] == v[t] (b2 + a21 u[t] + a22 v[t])}, {u[0] == a, v[0] == b}, {u, v}, t, 1, 1, 2]
which will produce the solutions in (31) and go one more iteration. Similarly, results given in subsections 3.1–3.3 of this paper can be reproduced by using our package. The usage of the main function PerturbationIterationAlgo rithm is as follows: PerturbationIterationAlgorithm[
- ,
where
- . One should keep in mind that n has to be smaller than or equal to m. The default value of PerturbationParameter is e. If set to another parameter, then this new parameter is used as the perturbation parameter. IntroduceEpsilon option is used for controlling the introduction of e. If the system has the perturbation parameter already in place, then setting IntroduceEpsilon to False will skip introduction of the perturbation parameter. The default value of GivenUZero is set to Automatic, in which case the function automatically computes
Please cite this article in press as: Y. Aksoy et al., Application of perturbation–iteration method to Lotka–Volterra equations, Alexandria Eng. J. (2016), http://dx. doi.org/10.1016/j.aej.2016.02.015
Application of perturbation–iteration method Table 1
5
Comparisons of perturbation–iteration solutions of (9) with exact solution [16] (a = 3, b = 1 and a = 0.1).
Exact solution
PIA(1,1)
t
u1
u1,1
u1,2
u1,3
u1,4
u1,5
PIA(1,2) u1,1
u1,2
u1,3
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.100000 0.107137 0.114533 0.122164 0.130001 0.138013 0.146163 0.154414 0.162726 0.171057 0.179367
0.100000 0.107030 0.114028 0.120818 0.127171 0.132785 0.137272 0.140132 0.140729 0.138259 0.131705
0.100000 0.107139 0.114555 0.122253 0.130259 0.138625 0.147445 0.156869 0.167126 0.178536 0.191525
0.100000 0.107137 0.114532 0.122159 0.129983 0.137958 0.146021 0.154084 0.162023 0.169658 0.176725
0.100000 0.107137 0.114533 0.122164 0.130002 0.138017 0.146175 0.154449 0.162814 0.171261 0.179808
0.100000 0.107137 0.114533 0.122164 0.130001 0.138012 0.146162 0.154411 0.162717 0.171033 0.179306
0.100000 0.107138 0.114543 0.122206 0.130121 0.138293 0.146738 0.155492 0.164618 0.174207 0.184389
0.100000 0.107137 0.114533 0.122164 0.130002 0.138016 0.146174 0.154445 0.162802 0.171228 0.179721
0.100000 0.107137 0.114533 0.122164 0.130001 0.138013 0.146163 0.154415 0.162728 0.171064 0.179386
Table 2 Comparisons of perturbation–iteration solutions of (23) with numerical solutions [16] (b1 = 0.1, a11 = 0.0014, a12 = 0.0012, b2 = 0.08, a21 = 0.0009, a22 = 0.001, a = 4 and b = 10). Numerical solution
PIA(1,1)
t
u1
u2
u1,1
u2,1
u1,2
u2,2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4.000000 4.033071 4.066363 4.099879 4.133617 4.167580 4.201767 4.236180 4.270819 4.305684 4.340776
10.000000 10.066572 10.133490 10.200753 10.268363 10.336320 10.404623 10.473273 10.542271 10.611618 10.681312
4.000000 4.033059 4.066316 4.099771 4.133422 4.167269 4.201312 4.235549 4.269980 4.304604 4.339420
10.000000 10.066553 10.133411 10.200573 10.268037 10.335802 10.403865 10.472226 10.540883 10.609834 10.679076
4.000000 4.033071 4.066364 4.099879 4.133618 4.167582 4.201770 4.236185 4.270826 4.305694 4.340796
10.000000 10.066573 10.133490 10.200754 10.268365 10.336322 10.404627 10.473281 10.542282 10.611634 10.681334
Table 3 Comparisons of perturbation–iteration solutions of (32) with numerical solutions [16] (b1 = 1, a11 = 1, a12 = 0.1, a13 = 0.1, b2 = 1, a21 = 0.1, a22 = 1, a23 = 0.1, b3 = 1, a31 = 0.1, a32 = 0.1, a33 = 1, a = 0.2, b = 0.3, c = 0.5). Numerical solutions
PIA(1,1)
t
u1
u2
u3
u1,4
u2,4
u3,4
u1,5
u2,5
u3,5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.20000 0.21473 0.23010 0.24609 0.26264 0.27973 0.29729 0.31527 0.33361 0.35222 0.37105
0.30000 0.31914 0.33873 0.35867 0.37888 0.39927 0.41974 0.44020 0.46054 0.48068 0.50053
0.50000 0.52234 0.54428 0.56572 0.58655 0.60667 0.62601 0.64449 0.66208 0.67871 0.69438
0.20000 0.21473 0.23010 0.24609 0.26265 0.27975 0.29734 0.31540 0.33393 0.35294 0.37256
0.30000 0.31915 0.33873 0.35867 0.37889 0.39931 0.41987 0.44054 0.46138 0.48254 0.50438
0.50000 0.52234 0.54429 0.56573 0.58662 0.60693 0.62679 0.64654 0.66683 0.68887 0.71455
0.20000 0.21473 0.23010 0.24609 0.26264 0.27973 0.29729 0.31526 0.33356 0.35210 0.37075
0.30000 0.31915 0.33873 0.35867 0.37888 0.39927 0.41973 0.44015 0.46042 0.48037 0.49979
0.50000 0.52234 0.54428 0.56572 0.58654 0.60664 0.62591 0.64420 0.66128 0.67675 0.68991
uj;0 . If GivenUZero is set to a user defined expression as GivenUZero ->{u1;0 ,u2;0 ;. . .} then computations continue using the initial guess values u1;0 ,u2;0 ;. . .. Further documentation and more examples about the package can be found in [23].
6. Concluding remarks In this study, the perturbation–iteration method [12,13] is extended to the systems of first order differential equations and applied to the Lotka–Volterra equations. Contrary to
Please cite this article in press as: Y. Aksoy et al., Application of perturbation–iteration method to Lotka–Volterra equations, Alexandria Eng. J. (2016), http://dx. doi.org/10.1016/j.aej.2016.02.015
6 the classical perturbation methods, small parameter assumption is not needed for the new perturbation–iteration method. In the study of one dimensional Lotka–Volterra equations, solutions arising from PIA(1,2) have better accuracy in comparison with those of PIA(1,1). Hence increasing the number of terms in Taylor series expansion results in a higher accuracy as in [24]. Also solutions of general multi-dimensional Lotka–Volterra equations without any small parameter are obtained using PIA(1,1). It is easy to see that these results cover the one-two and three dimensional cases. References [1] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley and Sons, New York, 1981. [2] J.H. He, Linearized perturbation technique and its applications to strongly nonlinear oscillators, Comput. Math. Appl. 45 (2003) 1–8. [3] H. Hu, A classical perturbation technique which is valid for large parameters, J. Sound Vib. 269 (2004) 409–412. [4] M. Pakdemirli, M.M.F. Karahan, H. Boyacı, A new perturbation algorithm with better convergence properties: multiple scales Lindstedt Poincare´ method, Math. Comput. Appl. 14 (2009) 31–44. [5] L. Xu, Determination of limit cycle by He’ s parameter expanding method for strongly nonlinear oscillators, J. Sound Vib. 302 (2007) 178–184. [6] A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput. 166 (2005) 652–663. [7] J.H. He, Variational iteration method — a kind of non-linear analytical technique: some examples, Int. J. Non-Linear Mech. 34 (1999) 699–708. [8] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004) 499–513. [9] J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B 20 (2006) 1141–1199. [10] R.E. Mickens, A generalized iteration procedure for calculating approximations to periodic solutions of ‘‘truly nonlinear oscillators’’, J. Sound Vib. 287 (2005) 1045–1051.
Y. Aksoy et al. [11] H. Tari, Modified variational iteration method, Phys. Lett. A 369 (2007) 290–293. [12] Y. Aksoy, M. Pakdemirli, New perturbation–iteration solutions for Bratu-type equations, Comput. Math. Appl. 59 (2010) 2802– 2808. [13] M. Pakdemirli, Y. Aksoy, H. Boyacı, A new perturbation iteration approach for first order differential equations, Math. Comput. Appl. 16 (2011) 890–899. [14] Y. Aksoy, M. Pakdemirli, S. Abbasbandy, H. Boyacı, New perturbation–iteration solutions for nonlinear heat transfer equations, Int. J. Numer. Meth. Heat Fluid Flow 22 (2012) 814–828. [15] S. Olek, An accurate solution to the multispecies Lotka– Volterra equations, Soc. Ind. Appl. Math. 36 (3) (1994) 480–488. [16] B. Batiha, M.S.M. Noorani, I. Hashim, Variational iteration method for solving multispecies Lotka-Volterra equations, Comput. Math. Appl. 54 (2007) 903–909. [17] A.S. Bataineh, M.S.M. Noorani, I. Hashim, Series solution of the multispecies Lotka–Volterra equations by means of the homotopy analysis method, Diff. Equat. Nonlinear Mech. 2008 (2008) 14, http://dx.doi.org/10.1155/2008/816787 816787. [18] Vasile Marinca, Nicolae Herisßanu, Optimal parametric iteration method for solving multispecies Lotka–Volterra Equations, Discrete Dyn. Nat. Soc. 2012 (2012) 10, http://dx.doi.org/ 10.1155/2012/842121 842121. [19] P.H. Verhulst, Notice sur la loi que la population suit dans son accroissement, Corresp. Mathe´matique et Physique 10 (1838) 113–121. [20] V. Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Memorie della Reale Accademia dei Lincei 2 (1926) 31–113. [21] A.J. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, 1925. [22] R.M. May, W.J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975) 243–253. [23] U¨. Go¨ktasß, 2011. Software is available on
Please cite this article in press as: Y. Aksoy et al., Application of perturbation–iteration method to Lotka–Volterra equations, Alexandria Eng. J. (2016), http://dx. doi.org/10.1016/j.aej.2016.02.015