Application of perturbation–iteration method to Lotka–Volterra 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.

Keywords

Perturbation method; Perturbation–iteration method; Symbolic computation; Lotka–Volterra equations; Systems of first order differential equations

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–Poincaré method with modified frequency expansion [3], the multiple-scale Lindstedt–Poincaré 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.

Other attempts to treat both weakly and strongly nonlinear problems were through iteration procedures which used pre-formed 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] and [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.

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] and [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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_k({\overset{\cdot}{u}}_k\mbox{,}u_j\mbox{,}\epsilon \mbox{,}t)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad j=1\mbox{,}2\mbox{,}\ldots \mbox{,}K\mbox{,}\quad k= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1\mbox{,}2\mbox{,}\ldots \mbox{,}K\mbox{,}

(1)

where u_j are the dependent variables, t is the independent variable, K   is the number of dependent variables, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon }

is the artificially introduced small parameter and the dot stands for the derivative. Reconsider Eq. (1) in the following iterative form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_k({\overset{\cdot}{u}}_{k\mbox{,}n+1}\mbox{,}u_{j\mbox{,}n+1}\mbox{,}\epsilon \mbox{,}t)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}

(2)

where the subscript n expresses the number of iterations completed in the procedure. More clearly, the system of equations is given below

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_1=F_1({\overset{\cdot}{u}}_1\mbox{,}u_1\mbox{,}u_2\mbox{,}\ldots \mbox{,}u_K\mbox{,}\epsilon \mbox{,}t) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_2=F_2({\overset{\cdot}{u}}_2\mbox{,}u_1\mbox{,}u_2\mbox{,}\ldots \mbox{,}u_K\mbox{,}\epsilon \mbox{,}t) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \vdots Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_K=F_K({\overset{\cdot}{u}}_K\mbox{,}u_1\mbox{,}u_2\mbox{,}\ldots \mbox{,}u_K\mbox{,}\epsilon \mbox{,}t)\mbox{.}

(3)

Next, define the following iterative perturbation series:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{j\mbox{,}n+1}=u_{j\mbox{,}n}+\epsilon u_{j\mbox{,}n}^c\mbox{,}

(4)

Taylor series expansion of equation (1) is given as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_k=\sum_{m=0}^M\frac{1}{m!}{\left[{\left(\frac{d}{d\epsilon }\right)}^mF_k\right]}_{\epsilon =0}{\epsilon }^m\mbox{,}\quad k= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1\mbox{,}2\ldots \ldots .K

(5)

with the derivative operator defined as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{d}{d\epsilon }=\frac{\partial {\overset{\cdot}{u}}_{k\mbox{,}n+1}}{\partial \epsilon }\frac{\partial }{\partial {\overset{\cdot}{u}}_{k\mbox{,}n+1}}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \sum_{j=1}^K\left(\frac{\partial u_{j\mbox{,}n+1}}{\partial \epsilon }\frac{\partial }{\partial u_{j\mbox{,}n+1}}\right)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\partial }{\partial \epsilon }\mbox{.}

(6)

Calculating derivatives at ε = 0 and substituting Eq. (6) into Eq. (5) yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_k=\sum_{m=0}^M\frac{1}{m!}{\left\{{\left({\overset{\cdot}{u}}_{k\mbox{,}n}^c\frac{\partial }{\partial {\overset{\cdot}{u}}_{k\mbox{,}n+1}}+\sum_{j=1}^Ku_{j\mbox{,}n}^c\frac{\partial }{\partial u_{j\mbox{,}n+1}}+\frac{\partial }{\partial \epsilon }\right)}^mF_k\right\}}_{\epsilon =0}{\epsilon }^m= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad k=1\mbox{,}2\ldots \ldots .K

(7)

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 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.

3. Application to Lotka–Volterra problems

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], [16] and [17]. In the more general framework one can study the following multidimensional Lotka–Volterra system:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{du}_k}{dt}=u_k\left(b_k+\sum_{j=1}^Ka_{kj}u_j\right)\mbox{,}\quad \quad u_k(0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): c_k\mbox{,}\quad k=1\mbox{,}2\mbox{,}\ldots \mbox{,}K\mbox{,}

(8)

where c_k 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{du}{dt}=u(b+au)\mbox{,}\quad b>0\mbox{,}\quad a<0\mbox{,}\quad u(0)>0\mbox{.}

(9)

The exact solution of (9) is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u(t)=\begin{array}{ll} \frac{{be}^{bt}}{\frac{b+au(0)}{u(0)}-{ae}^{bt}}\mbox{,} & b\quad \not =\quad 0\\ \frac{u(0)}{1-au(0)t}\mbox{,} & b=0\mbox{.} \end{array}

(10)

Approximate solutions of Eq. (9) will be obtained using two different perturbation–iteration algorithms by taking M = 1 and M = 2.

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_1={\overset{\cdot}{u}}_{1\mbox{,}n+1}-{bu}_{1\mbox{,}n+1}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \epsilon {au}_{1\mbox{,}n+1}^2=0\mbox{,}

(11)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}n+1}=u_{1\mbox{,}n}+\epsilon u_{1\mbox{,}n}^c\mbox{,}

(12)

where ε is artificially introduced as a small parameter. Reorganizing Eq. (7) for Eq. (9) yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{du}_{1\mbox{,}n}^c}{dt}+\frac{\partial F_1/\partial u_{n+1}}{\partial F_1/\partial {\overset{\cdot}{u}}_{n+1}}u_{1\mbox{,}n}^c= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\frac{\epsilon F_1}{\partial F_1/\partial {\overset{\cdot}{u}}_{n+1}}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\partial F_1/\partial \epsilon }{\partial F_1/\partial {\overset{\cdot}{u}}_{n+1}}\mbox{,}

(13)

where the derivatives are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {F_1}_{\epsilon =0}={\overset{\cdot}{u}}_n-{bu}_n\mbox{,}\quad {\frac{\partial F_1}{\partial u_{n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -b\mbox{,}\quad {\frac{\partial F_1}{\partial {\overset{\cdot}{u}}_{n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1\mbox{,}\quad {\frac{\partial F_1}{\partial \epsilon }}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -a\quad u_n^2\mbox{.}

(14)

Finally, the iteration equation is obtained by substituting (14) into (13) and setting ε = 1:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{du}_{1\mbox{,}n}^c}{dt}-{bu}_{1\mbox{,}n}^c= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -{\overset{\cdot}{u}}_{1\mbox{,}n}+{bu}_{1\mbox{,}n}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {au}_{1\mbox{,}n}^2\mbox{.}

(15)

One can take the initially assumed function as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}0}=\alpha e^{bt}\mbox{,}

(16)

which satisfy the initial condition exactly.

Starting with the above initial function, first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{1\mbox{,}0}^c}

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}1}=\alpha \mbox{e}^{bt}+\frac{{\alpha }^2a}{b}\mbox{e}^{bt}(e^{bt}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1)\mbox{,}

(17)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}2}=\alpha \mbox{e}^{bt}+\frac{{\alpha }^2a\mbox{e}^{bt}(\mbox{e}^{bt}-1)}{b}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\alpha }^3a^2\mbox{e}^{bt}{\left(\mbox{e}^{bt}-1\right)}^2}{b^2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\alpha }^4a^3\mbox{e}^{bt}{\left(\mbox{e}^{bt}-1\right)}^3}{3b^3}\mbox{.}

(18)

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{du}_{1\mbox{,}n}^c}{dt}-(b+2{au}_{1\mbox{,}n})u_{1\mbox{,}n}^c= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -{\overset{\cdot}{u}}_{1\mbox{,}n}+{bu}_{1\mbox{,}n}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {au}_{1\mbox{,}n}^2\mbox{.}

(19)

Using the same initial guess function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{1\mbox{,}0}=\alpha e^{bt}\mbox{,}}

one can obtain the first iteration solution:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}1}=\frac{\alpha }{2}e^{bt}\left(1+e^{\frac{2\alpha a}{b}(e^{bt}-1)}\right)\mbox{,}

(20)

The second and third iteration solutions are as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}2}=\frac{\alpha \mbox{e}^{bt-\frac{2a\alpha }{b}}}{8}\left(3\mbox{e}^{\frac{2a\alpha }{b}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. 4\mbox{e}^{\frac{2a\alpha \mbox{e}^{bt}}{b}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \mbox{e}^{\frac{2\alpha a(2\mbox{e}^{bt}-1)}{b}}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \frac{4\alpha a}{b}\mbox{e}^{\frac{2\alpha a\mbox{e}^{bt}}{b}}(\mbox{e}^{bt}-\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. 1)\right)\mbox{,}

(21)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}3}=\frac{1}{384}\mbox{e}^{bt-\frac{6a\alpha }{b}}\alpha 117\mbox{e}^{\frac{6a\alpha }{b}}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 15\mbox{e}^{\frac{6a\mbox{e}^{bt}\alpha }{b}}+161\mbox{e}^{\frac{2a(2+\mbox{e}^{bt})\alpha }{b}}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 90\mbox{e}^{\frac{2a(1+2\mbox{e}^{bt})\alpha }{b}}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{e}^{\frac{2a(-1+4\mbox{e}^{bt})\alpha }{b}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\frac{12a}{b}\mbox{e}^{\frac{2a\mbox{e}^{bt}\alpha }{b}}(- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1+\mbox{e}^{bt})(20\mbox{e}^{\frac{4a\alpha }{b}}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mbox{e}^{\frac{4a\mbox{e}^{bt}\alpha }{b}}+12\mbox{e}^{\frac{2a(1+\mbox{e}^{bt})\alpha }{b}})\alpha Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): +\frac{24{\alpha }^2a^2}{b^2}\mbox{e}^{\frac{2a(1+\mbox{e}^{bt})\alpha }{b}}{\left(-1+\mbox{e}^{bt}\right)}^2\left(5\mbox{e}^{\frac{2a\alpha }{b}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. 2\mbox{e}^{\frac{2a\mbox{e}^{bt}\alpha }{b}}\right)\mbox{.}

(22)

In second and third iteration solutions, as the calculations get more involved, the second term in (19), b+2au_1,n is approximated as b+2au_1,0 for simplicity and help of a computer algebra system such as Mathematica is essential.

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{du}_1}{dt}=u_1(b_1+a_{11}u_1+a_{12}u_2)\mbox{,}u_1(0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \alpha \mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{du}_2}{dt}=u_2(b_2+a_{21}u_1+a_{22}u_2)\mbox{,}u_2(0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \beta \mbox{,}

(23)

where a₁₁, a₁₂, a₂₁, a₂₂ are parameters representing the interaction of the species and the self-interaction, and α, β 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_1={\overset{\cdot}{u}}_{1\mbox{,}n+1}-u_{1\mbox{,}n+1}(b_1+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \epsilon a_{11}u_{1\mbox{,}n+1}+\epsilon a_{12}u_{2\mbox{,}n+1})= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_2={\overset{\cdot}{u}}_{2\mbox{,}n+1}-u_{2\mbox{,}n+1}(b_2+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \epsilon a_{21}u_{1\mbox{,}n+1}+\epsilon a_{22}u_{2\mbox{,}n+1})= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{.}

(24)

Setting K = 2 and M = 1, transforms Eq. (7) into the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_1\cong {F_1}_{\epsilon =0}+{\left\{\left[{\overset{\cdot}{u}}_{1\mbox{,}n}^c\frac{\partial }{\partial {\overset{\cdot}{u}}_{1\mbox{,}n+1}}+u_{1\mbox{,}n}^c\frac{\partial }{\partial u_{1\mbox{,}n+1}}+u_{2\mbox{,}n}^c\frac{\partial }{\partial u_{2\mbox{,}n+1}}+\frac{\partial }{\partial \epsilon }\right]F_1\right\}}_{\epsilon =0}\epsilon = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_2\cong {F_2}_{\epsilon =0}+{\left\{\left[{\overset{\cdot}{u}}_{2\mbox{,}n}^c\frac{\partial }{\partial {\overset{\cdot}{u}}_{2\mbox{,}n+1}}+u_{2\mbox{,}n}^c\frac{\partial }{\partial u_{2\mbox{,}n+1}}+u_{1\mbox{,}n}^c\frac{\partial }{\partial u_{1\mbox{,}n+1}}+\frac{\partial }{\partial \epsilon }\right]F_2\right\}}_{\epsilon =0}\epsilon = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0

(25)

with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {F_1}_{\epsilon =0}={\overset{\cdot}{u}}_{1\mbox{,}n}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): b_1u_{1\mbox{,}n}\mbox{,}\quad {F_2}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\overset{\cdot}{u}}_{2\mbox{,}n}-b_2u_{2\mbox{,}n}\mbox{,}\quad {\frac{\partial F_1}{\partial {\overset{\cdot}{u}}_{1\mbox{,}n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1\mbox{,}\quad {\frac{\partial F_2}{\partial {\overset{\cdot}{u}}_{2\mbox{,}n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1\mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\frac{\partial F_1}{\partial u_{1\mbox{,}n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -b_1\mbox{,}\quad {\frac{\partial F_1}{\partial u_{2\mbox{,}n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{,}\quad {\frac{\partial F_1}{\partial \epsilon }}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -a_{11}u_{1\mbox{,}n}^2-a_{12}u_{1\mbox{,}n}u_{2\mbox{,}n}\mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\frac{\partial F_2}{\partial \epsilon }}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -a_{22}u_{2\mbox{,}n}^2-a_{21}u_{1\mbox{,}n}u_{2\mbox{,}n}\mbox{,}\quad {\frac{\partial F_2}{\partial u_{2\mbox{,}n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -b_2\mbox{,}\quad {\frac{\partial F_2}{\partial u_{1\mbox{,}n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{.}

(26)

Substituting (26) into (25) leads to the final form of the iteration equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\overset{\cdot}{u}}_{1\mbox{,}n}^c-b_1u_{1\mbox{,}n}^c= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): a_{11}u_{1\mbox{,}n}^2+a_{12}u_{1\mbox{,}n}u_{2\mbox{,}n}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\overset{\cdot}{u}}_{1\mbox{,}n}+b_1u_{1\mbox{,}n}\mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\overset{\cdot}{u}}_{2\mbox{,}n}^c-b_2u_{2\mbox{,}n}^c= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): a_{22}u_{2\mbox{,}n}^2+a_{21}u_{1\mbox{,}n}u_{2\mbox{,}n}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\overset{\cdot}{u}}_{2\mbox{,}n}+b_2u_{2\mbox{,}n}\mbox{.}

(27)

As Eq. (4) become:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}n+1}=u_{1\mbox{,}n}+u_{1\mbox{,}n}^c\mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{2\mbox{,}n+1}=u_{2\mbox{,}n}+u_{2\mbox{,}n}^c\mbox{,}

(28)

taking

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}0}=\alpha e^{b_1t}\mbox{,}\quad u_{2\mbox{,}0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \beta e^{b_2t}\mbox{,}

(29)

as an initial approximation satisfying the initial conditions leads to the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}0}^c=\frac{{\alpha }^2a_{11}\mbox{e}^{{tb}_1}(-1+\mbox{e}^{{tb}_1})}{b_1}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\alpha \beta a_{12}\mbox{e}^{{tb}_1}(-1+\mbox{e}^{{tb}_2})}{b_2}\mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{2\mbox{,}0}^c=\frac{\alpha \beta a_{21}\mbox{e}^{{tb}_2}(-1+\mbox{e}^{{tb}_1})}{b_1}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\beta }^2a_{22}\mbox{e}^{{tb}_2}(-1+\mbox{e}^{{tb}_2})}{b_2}\mbox{,}

(30)

as correction terms. Combining (29) and (30) in (28) yields the first iteration solutions:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}1}=\alpha \mbox{e}^{{tb}_1}+\frac{{\alpha }^2a_{11}\mbox{e}^{{tb}_1}(-1+\mbox{e}^{{tb}_1})}{b_1}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\alpha \beta a_{12}\mbox{e}^{{tb}_1}(-1+\mbox{e}^{{tb}_2})}{b_2}\mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{2\mbox{,}1}=\beta \mbox{e}^{{tb}_2}+\frac{\alpha \beta a_{21}\mbox{e}^{{tb}_2}(-1+\mbox{e}^{{tb}_1})}{b_1}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\beta }^2a_{22}\mbox{e}^{{tb}_2}(-1+\mbox{e}^{{tb}_2})}{b_2}\mbox{.}

(31)

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]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{du}_1}{dt}=u_1(b_1+a_{11}u_1+a_{12}u_2+a_{13}u_3)\mbox{,}\quad u_1(0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \alpha \mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{du}_2}{dt}=u_2(b_2+a_{21}u_1+a_{22}u_2+a_{23}u_3)\mbox{,}\quad u_2(0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \beta \mbox{,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{du}_3}{dt}=u_3(b_3+a_{31}u_1+a_{32}u_2+a_{33}u_3)\mbox{,}\quad u_3(0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \gamma \mbox{.}

(32)

Using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{1\mbox{,}0}=\alpha \mbox{e}^{b_1t}\mbox{,}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{2\mbox{,}0}=\beta \mbox{e}^{b_2t}\mbox{,}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{3\mbox{,}0}=\gamma \mbox{e}^{b_3t}}

as initial functions, approximate solutions after the first iteration are as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{1\mbox{,}1}=\alpha \mbox{e}^{{tb}_1}+\frac{{\alpha }^2a_{11}\mbox{e}^{{tb}_1}(-1+\mbox{e}^{{tb}_1})}{b_1}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\alpha \beta a_{12}\mbox{e}^{{tb}_1}(-1+\mbox{e}^{{tb}_2})}{b_2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\alpha \gamma a_{13}\mbox{e}^{{tb}_1}(-1+\mbox{e}^{{tb}_3})}{b_3}\mbox{,}

(33)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{2\mbox{,}1}=\beta \mbox{e}^{{tb}_2}+\frac{\alpha \beta a_{21}\mbox{e}^{{tb}_2}(-1+\mbox{e}^{{tb}_1})}{b_1}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\beta }^2a_{22}\mbox{e}^{{tb}_2}(-1+\mbox{e}^{{tb}_2})}{b_2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\beta \gamma a_{23}\mbox{e}^{{tb}_2}(-1+\mbox{e}^{{tb}_3})}{b_3}\mbox{,}

(34)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{3\mbox{,}1}=\gamma \mbox{e}^{{tb}_3}+\frac{\alpha \gamma a_{31}\mbox{e}^{{tb}_3}(-1+\mbox{e}^{{tb}_1})}{b_1}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\beta \gamma a_{32}\mbox{e}^{{tb}_3}(-1+\mbox{e}^{{tb}_2})}{b_2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\gamma }^2a_{33}\mbox{e}^{{tb}_3}(-1+\mbox{e}^{{tb}_3})}{b_3}\mbox{.}

(35)

For second and higher iteration solutions, the parameters have to be set to values.

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): F_k={\overset{\cdot}{u}}_{k\mbox{,}n+1}-u_{k\mbox{,}n+1}\left(b_k+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \epsilon \sum_{j=1}^Ka_{kj}u_{j\mbox{,}n+1}\right)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\mbox{.}

(36)

For m = 1, the terms included in Eq. (7) are as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {F_k}_{\epsilon =0}={\overset{\cdot}{u}}_{k\mbox{,}n}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): b_ku_{k\mbox{,}n}\mbox{,}{\frac{\partial F_k}{\partial {\overset{\cdot}{u}}_{k\mbox{,}n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1\mbox{,}\quad {\frac{\partial F_k}{\partial u_{k\mbox{,}n+1}}}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -b_k\mbox{,}\quad {\frac{\partial F_k}{\partial \epsilon }}_{\epsilon =0}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\sum_{j=1}^Ka_{kj}u_{k\mbox{,}n}u_{j\mbox{,}n}\mbox{.}

(37)

Hence substituting these terms in Eq. (7) yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\overset{\cdot}{u}}_{k\mbox{,}n}^c-b_ku_{k\mbox{,}n}^c= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -{\overset{\cdot}{u}}_{k\mbox{,}n}+b_ku_{k\mbox{,}n}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \sum_{j=1}^Ka_{kj}u_{k\mbox{,}n}u_{j\mbox{,}n}\mbox{.}

(38)

Integrating Eq. (38) gives the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{k\mbox{,}n}^c=e^{b_kt}\left(-\int e^{-b_kt}{\overset{\cdot}{u}}_{k\mbox{,}n}dt+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \int e^{-b_kt}b_ku_{k\mbox{,}n}+\int e^{-b_kt}\sum_{j=1}^Ka_{kj}u_{k\mbox{,}n}u_{j\mbox{,}n}dt+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. c_n\right)\mbox{,}

(39)

where c_n is integration constant to be determined by the initial conditions. Then n’ th iteration solution for multidimensional Lotka–Volterra equations becomes the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_{k\mbox{,}n+1}=u_{k\mbox{,}n}+e^{b_kt}\left(-\int e^{-b_kt}{\overset{\cdot}{u}}_{k\mbox{,}n}dt+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \int e^{-b_kt}b_ku_{k\mbox{,}n}+\int e^{-b_kt}\sum_{j=1}^Ka_{kj}u_{k\mbox{,}n}u_{j\mbox{,}n}dt+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. c_n\right)\mbox{.}

(40)

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 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.

Table 1. Comparisons of perturbation–iteration solutions of (9) with exact solution [16] (a = −3, b = 1 and α = 0.1).

Exact solution		PIA(1,1)					PIA(1,2)
t	u1	u1,1	u1,2	u1,3	u1,4	u1,5	u1,1	u1,2	u1,3
0.0	0.100000	0.100000	0.100000	0.100000	0.100000	0.100000	0.100000	0.100000	0.100000
0.1	0.107137	0.107030	0.107139	0.107137	0.107137	0.107137	0.107138	0.107137	0.107137
0.2	0.114533	0.114028	0.114555	0.114532	0.114533	0.114533	0.114543	0.114533	0.114533
0.3	0.122164	0.120818	0.122253	0.122159	0.122164	0.122164	0.122206	0.122164	0.122164
0.4	0.130001	0.127171	0.130259	0.129983	0.130002	0.130001	0.130121	0.130002	0.130001
0.5	0.138013	0.132785	0.138625	0.137958	0.138017	0.138012	0.138293	0.138016	0.138013
0.6	0.146163	0.137272	0.147445	0.146021	0.146175	0.146162	0.146738	0.146174	0.146163
0.7	0.154414	0.140132	0.156869	0.154084	0.154449	0.154411	0.155492	0.154445	0.154415
0.8	0.162726	0.140729	0.167126	0.162023	0.162814	0.162717	0.164618	0.162802	0.162728
0.9	0.171057	0.138259	0.178536	0.169658	0.171261	0.171033	0.174207	0.171228	0.171064
1.0	0.179367	0.131705	0.191525	0.176725	0.179808	0.179306	0.184389	0.179721	0.179386

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.

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, α = 4 and β = 10).

Numerical solution			PIA(1,1)
t	u1	u2	u1,1	u2,1	u1,2	u2,2
0.0	4.000000	10.000000	4.000000	10.000000	4.000000	10.000000
0.1	4.033071	10.066572	4.033059	10.066553	4.033071	10.066573
0.2	4.066363	10.133490	4.066316	10.133411	4.066364	10.133490
0.3	4.099879	10.200753	4.099771	10.200573	4.099879	10.200754
0.4	4.133617	10.268363	4.133422	10.268037	4.133618	10.268365
0.5	4.167580	10.336320	4.167269	10.335802	4.167582	10.336322
0.6	4.201767	10.404623	4.201312	10.403865	4.201770	10.404627
0.7	4.236180	10.473273	4.235549	10.472226	4.236185	10.473281
0.8	4.270819	10.542271	4.269980	10.540883	4.270826	10.542282
0.9	4.305684	10.611618	4.304604	10.609834	4.305694	10.611634
1.0	4.340776	10.681312	4.339420	10.679076	4.340796	10.681334

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.

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, α = 0.2, β = 0.3, γ = 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	0.20000	0.30000	0.50000	0.20000	0.30000	0.50000	0.20000	0.30000	0.50000
0.1	0.21473	0.31914	0.52234	0.21473	0.31915	0.52234	0.21473	0.31915	0.52234
0.2	0.23010	0.33873	0.54428	0.23010	0.33873	0.54429	0.23010	0.33873	0.54428
0.3	0.24609	0.35867	0.56572	0.24609	0.35867	0.56573	0.24609	0.35867	0.56572
0.4	0.26264	0.37888	0.58655	0.26265	0.37889	0.58662	0.26264	0.37888	0.58654
0.5	0.27973	0.39927	0.60667	0.27975	0.39931	0.60693	0.27973	0.39927	0.60664
0.6	0.29729	0.41974	0.62601	0.29734	0.41987	0.62679	0.29729	0.41973	0.62591
0.7	0.31527	0.44020	0.64449	0.31540	0.44054	0.64654	0.31526	0.44015	0.64420
0.8	0.33361	0.46054	0.66208	0.33393	0.46138	0.66683	0.33356	0.46042	0.66128
0.9	0.35222	0.48068	0.67871	0.35294	0.48254	0.68887	0.35210	0.48037	0.67675
1.0	0.37105	0.50053	0.69438	0.37256	0.50438	0.71455	0.37075	0.49979	0.68991

5. The mathematica package

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 PerturbationIterationAlgorithm (which is part of the package):

In[3]: = PerturbationIterationAlgorithm[

{u′[t] == u[t] (b1 + a11 u[t] + a12 v[t]),

v′[t] == v[t] (b2 + a21 u[t] + a22 v[t])}, {u[0] == α,

v[0] == β}, {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, 3.2 and 3.3 of this paper can be reproduced by using our package.

The usage of the main function PerturbationIterationAlgorithm is as follows:

PerturbationIterationAlgorithm[

<equations>,

<list of initial conditions>,

<functions>,

<independent variable>,

<n: number of correction terms (integer) >,

<m: number of terms included in the Taylor series expansion (integer) >,

<number of iterations to be carried out (integer)>,

<options>]

where <options> can be Perturbation Parameter → <parameter>, IntroduceEpsilon → <True | False >, or GivenUZero → <list of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{j\mbox{,}0}}

initial guess values>. One should keep in mind that n has to be smaller than or equal to m.

The default value of PerturbationParameter is ε. If set to another parameter, then this new parameter is used as the perturbation parameter. IntroduceEpsilon option is used for controlling the introduction of ε. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{j\mbox{,}0}} . If GivenUZero is set to a user defined expression as GivenUZero ->{Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{1\mbox{,}0}} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{2\mbox{,}0}\mbox{,}} …} then computations continue using the initial guess values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{1\mbox{,}0}} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{2\mbox{,}0}\mbox{,}} ….

Further documentation and more examples about the package can be found in [23].

6. Concluding remarks

In this study, the perturbation–iteration method [12] and [13] is extended to the systems of first order differential equations and applied to the Lotka–Volterra equations. Contrary to 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. [1] A.H. Nayfeh; Introduction to Perturbation Techniques; Wiley and Sons, New York (1981)
2. [2] J.H. He; Linearized perturbation technique and its applications to strongly nonlinear oscillators; Comput. Math. Appl., 45 (2003), pp. 1–8
3. [3] H. Hu; A classical perturbation technique which is valid for large parameters; J. Sound Vib., 269 (2004), pp. 409–412
4. [4] M. Pakdemirli, M.M.F. Karahan, H. Boyacı; A new perturbation algorithm with better convergence properties: multiple scales Lindstedt Poincaré method; Math. Comput. Appl., 14 (2009), pp. 31–44
5. [5] L. Xu; Determination of limit cycle by He’ s parameter expanding method for strongly nonlinear oscillators; J. Sound Vib., 302 (2007), pp. 178–184
6. [6] A.M. Wazwaz; Adomian decomposition method for a reliable treatment of the Bratu-type equations; Appl. Math. Comput., 166 (2005), pp. 652–663
7. [7] J.H. He; Variational iteration method — a kind of non-linear analytical technique: some examples; Int. J. Non-Linear Mech., 34 (1999), pp. 699–708
8. [8] S.J. Liao; On the homotopy analysis method for nonlinear problems; Appl. Math. Comput., 147 (2004), pp. 499–513
9. [9] J.H. He; Some asymptotic methods for strongly nonlinear equations; Int. J. Mod. Phys. B, 20 (2006), pp. 1141–1199
10. [10] R.E. Mickens; A generalized iteration procedure for calculating approximations to periodic solutions of ‘‘truly nonlinear oscillators’’; J. Sound Vib., 287 (2005), pp. 1045–1051
11. [11] H. Tari; Modified variational iteration method; Phys. Lett. A, 369 (2007), pp. 290–293
12. [12] Y. Aksoy, M. Pakdemirli; New perturbation–iteration solutions for Bratu-type equations; Comput. Math. Appl., 59 (2010), pp. 2802–2808
13. [13] M. Pakdemirli, Y. Aksoy, H. Boyacı; A new perturbation iteration approach for first order differential equations; Math. Comput. Appl., 16 (2011), pp. 890–899
14. [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), pp. 814–828
15. [15] S. Olek; An accurate solution to the multispecies Lotka–Volterra equations; Soc. Ind. Appl. Math., 36 (3) (1994), pp. 480–488
16. [16] B. Batiha, M.S.M. Noorani, I. Hashim; Variational iteration method for solving multispecies Lotka-Volterra equations; Comput. Math. Appl., 54 (2007), pp. 903–909
17. [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), Article 816787 http://dx.doi.org/10.1155/2008/816787
18. [18] Vasile Marinca, Nicolae Herişanu; Optimal parametric iteration method for solving multispecies Lotka–Volterra Equations; Discrete Dyn. Nat. Soc., 2012 (2012), Article 842121 http://dx.doi.org/10.1155/2012/842121
19. [19] P.H. Verhulst; Notice sur la loi que la population suit dans son accroissement; Corresp. Mathématique et Physique, 10 (1838), pp. 113–121
20. [20] V. Volterra; Variazioni e fluttuazioni del numero d’individui in specie animali conviventi; Memorie della Reale Accademia dei Lincei, 2 (1926), pp. 31–113
21. [21] A.J. Lotka; Elements of Physical Biology; Williams & Wilkins, Baltimore (1925)
22. [22] R.M. May, W.J. Leonard; Nonlinear aspects of competition between three species; SIAM J. Appl. Math., 29 (1975), pp. 243–253
23. [23] Ü. Göktaş, 2011. Software is available on <http://www.turgutozal.edu.tr/ugoktas/>.
24. [24] Ü. Göktaş, M. Pakdemirli and Y. Aksoy, Symbolic Computation of Perturbation–Iteration Solutions for Differential Equations, in: Proc. 2nd International Symposium on Computing in Science and Engineering (ISCSE 2011), June 1–4, 2011, Kuşadası, Aydın, Turkey, 60–66, ed. M. Güneş, İzmir (Turkey), Gediz University Publications, 2011.