# Solving Multispecies Lotka–Volterra Equations by the Daftardar-Gejji and Jafari Method

Authors: Belal Batiha; Firas Ghanim; O. Alayed; Ra’ed Hatamleh; Ahmed Salem Heilat; Hamzeh Zureigat; Omar Bazighifan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this article, we apply the Daftardar-Gejji and Jafari method (DJM) to solve the multispecies Lotka–Volterra equation. A comparison between the DJM, differential transformation method (DTM), the variational iteration method (VIM), and Adomian decomposition method (ADM) shows that the DJM is a reliable and powerful method for solving nonlinear equations. The efficiency and applicability of this method are confirmed by considering some examples. The proposed procedure provides better results in comparison to some existing methods.


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## Body

## 1. Introduction

The area of mathematics called numerical analysis is in charge of coming up with practical methods for calculating answers to difficult computational calculations. The majority of mathematical issues in engineering and science are very challenging, and sometimes, there is no straightforward solution. To make a difficult mathematical problem simpler to solve, measurement is thus crucial. As a contemporary tool for scientists and engineers, numeracy has grown in popularity as a result of the tremendous developments in computing technology. As a consequence, a variety of software packages, including MATLAB, Mathematica, Maple, and others, are being created to solve even the most challenging issues quickly and simply. These programs provide features that make use of conventional numerical techniques, allowing the user to run a single command without entering any parameters and obtain the desired results. The creation, analysis, and application of algorithms for solving numerical problems in continuous mathematics are all made using the numerical analysis approach, which is mostly utilized in mathematics and computer science. These kinds of issues often come up in the actual world when algebra, geometry, and calculus are applied, and they also include continuous variables. These issues arise in all areas of study, including the scientific and social sciences, engineering, health care, and business [1–9]. Numerical analysis introduced realistic mathematical models which have become more prevalent in science and engineering over the last 50 years as a result of the expansion in the power and accessibility of digital computers. We shall learn more about numerical approaches and their analysis here. PDE solutions may be solved using the same numerical techniques used for ODEs. Many difficulties may be solved using the techniques mentioned for handling initial value concerns, for example, see references [6, 10–14].The Lotka–Volterra equations describe the time history of a biological system [15]. The Lotka–Volterra equations are applied in a number of engineering areas. The one-species Lotka–Volterra equation is used to demonstrate a simple nonlinear control system [16].The Lotka–Volterra equations were solved by many numerical methods like hybrid deep network [17], Gröbner bases elimination method [18], generalized backstepping control method [19], the differential transformation method (DTM) [20], the Adomian decomposition method (ADM), and variational iteration method (VIM) [21].The differential transformation method (DTM) was first proposed by Zhou [22] (also check [23, 24]). The DTM is an iterative method that obtains the Taylor series solutions of different kinds of differential equations (see [20, 25–27]). The DTM can be applied directly to different kinds of DEs without requiring linearization, discretization, or perturbation, and it is a very accurate method with less computational work [28].The Adomian decomposition method (ADM) was introduced by Adomian [29] to solve nonlinear differential equations and physical problems [30–33].The VIM was first proposed by He [34] (see also [35–37]). The VIM has successfully been used for many ordinary and partial differential equations [21, 38–40].In 2006, the DJM was first proposed by Daftardar-Gejji and Jafari [41]; the method can solve many nonlinear differential equations and physical problems [42–52]. Recently, the DJM was applied to create a quite new predictor-corrector method [53, 54]. Noor et al. [55–59] used the DJM to create numerical techniques to solve algebraic equations.In this paper, we apply the DJM to solve the multispecies Lotka–Volterra equation and compare the results obtained with DTM, VIM, ADM, and exact solution to show the simplicity and accuracy of this method. The efficiency and applicability of this method are confirmed by considering some examples. The proposed procedure provides better results in comparison to some existing methods. The DJM method will be implemented in a direct way without any linearization, perturbation, or restrictive assumptions.


## 2. The Daftardar-Gejji and Jafari Method (DJM)

Here, the DJM ([41]) will be described, which was successfully applied to solve nonlinear DEs of the following form:(1)v=f+Lv+Nv,where f is a function given, L is the linear operator and N is the nonlinear operator. The solution of equation (1) will be as follow:(2)v=∑i=0∞vi.Suppose(3)H0=Nv0,Hm=N∑i=0mvi−N∑i=0m−1vi.So,(4)H0=Nv0,H1=Nv0+v1−Nv0,H2=Nv0+v1+v2−Nv0+v1,H3=Nv0+v1+v2+v3−Nv0+v1+v2.Thus,Nv is decomposed as follows:(5)N∑i=0∞vi=Nv0+Nv0+v1−Nv0+Nv0+v1+v2−Nv0+v1+Nv0+v1+v2+v3−Nv0+v1+v2+⋯.So,(6)v0=f,v1=Lv0+H0,vm+1=Lvm+Hm,m=1,2,….SinceL is linear, then(7)∑i=0mLvi=L∑i=0mvi.Then,(8)∑i=1m+1vi=∑i=0mLvi+N∑i=0mvi=L∑i=0mvi+N∑i=0mvi,m=1,2,….Thus,(9)∑i=0∞vi=f+L∑i=0∞vi+N∑i=0∞vi.Thek− term approximate solution is given as follows:(10)v=∑i=0k−1vi.


## 3. Convergence of the DJM

Theorem 1.
“For anyn and for some real L>0 and ui≤M<1/e,i=1,2,…, if ℕ is C∞ in the neighborhood of u0 and ℕnu0≤L, then ∑n=0∞Hn is absolutely convergent and Hn≤LMnen−1e−1,n=1,2,….”Proof.
Please see reference [46] for full details of the proof.Theorem 2.
“The series∑n=0∞Hn is absolutely convergent if ℕ is C∞ and ℕnu0≤M≤e−1,∀n.”Proof.
Please see reference [46] for full details of the proof.


## 4. Analysis of Multispecies Lotka–Volterra Equations

In this section, we will study thenth general Lotka–Volterra system in the form as follows:(11)dyidt=yiβi+∑j=1nαijyj,i=1,2,…,n.To solve equation (11) with the initial condition y0=y0 by the Daftardar-Gejji and Jafari method (DJM), we write it in the following integral equation:(12)yt=y0+∫0tyiβi+∑j=1nαijyjdt,i=1,2,…,n.Then, we will apply the DJM as in the previous section.


### 4.1. One Species

In this section, equation (11) is reduced to one species:(13)dydt=yβ+αy,β>0,α<0,y0>0,where α and β are constants. With exact solution,(14)yt=βeβtβ+αy0/y0−αeβtforβ≠0,yt=y01−αy0t,forβ=0.To solve equation (13) with the initial condition y0=0.1 by the Daftardar-Gejji and Jafari method (DJM), we write it in the following integral equation:(15)yt=0⋅1+∫0tyβ+αydt.By applying DJM, we obtain the following:(16)y0=0.1,y1=0.07t,y2=−0.0049t2t−2.857143,⋮.The four-term solution is as follows:(17)yt=0.1+0.07t−0.0049t2t−2.857143−0.00001029t+5.587558t3t−1.234763t−3.493845t−7.525617−2.117682×10−11t+5.877433t+5.005629t4t−4.090877⋅t−6.871677t−7.828486t2+6.991252t+21.95022⋅t2−2.520506t+1.605527t2−10.84848t+38.63197.


### 4.2. Two Species

In this section, equation (11) is reduced to two species:(18)dy1dt=y1β1+α11y1+α12y2,(19)dy2dt=y2β2+α21y1+α22y2,where α11, α12, α21, α22, β1, and β2 are constants.To solve equations (18) and (19) by DJM with initial conditions y10=4 and y20=10, we write it in the following integral equation:(20)y1,1=4+∫0ty1β1+α11y1+α12y2dt,y2,1=10+∫0ty2β2+α21y1+α22y2dt.The rest components of the formulas (20) and (20) can be obtained using the computer algebra package Maple.


### 4.3. Three Species

In this section, equation (11) is reduced to three species:(21)dy1dt=y11−y1−αy2−βy3,(22)dy2dt=y21−βy1−y2−αy3,(23)dy3dt=y31−αy1−βy2−y3,where α and β are constants.To solve equations (21)–(23) by DJM with initial conditions y10=0.2, y20=0.3, and y30=0.5, we write it in the following integral equation:(24)y1,1=0.2+∫0ty11−y1−αy2−βy3dt,(25)y2,1=0.3+∫0ty21−βy1−y2−αy3dt,(26)y3,1=0.5+∫0ty31−αy1−βy2−y3dt.Again, the rest components of the formulas (24)–(26) can be obtained using the computer algebra package Maple.


## 4.1. One Species

In this section, equation (11) is reduced to one species:(13)dydt=yβ+αy,β>0,α<0,y0>0,where α and β are constants. With exact solution,(14)yt=βeβtβ+αy0/y0−αeβtforβ≠0,yt=y01−αy0t,forβ=0.To solve equation (13) with the initial condition y0=0.1 by the Daftardar-Gejji and Jafari method (DJM), we write it in the following integral equation:(15)yt=0⋅1+∫0tyβ+αydt.By applying DJM, we obtain the following:(16)y0=0.1,y1=0.07t,y2=−0.0049t2t−2.857143,⋮.The four-term solution is as follows:(17)yt=0.1+0.07t−0.0049t2t−2.857143−0.00001029t+5.587558t3t−1.234763t−3.493845t−7.525617−2.117682×10−11t+5.877433t+5.005629t4t−4.090877⋅t−6.871677t−7.828486t2+6.991252t+21.95022⋅t2−2.520506t+1.605527t2−10.84848t+38.63197.


## 4.2. Two Species

In this section, equation (11) is reduced to two species:(18)dy1dt=y1β1+α11y1+α12y2,(19)dy2dt=y2β2+α21y1+α22y2,where α11, α12, α21, α22, β1, and β2 are constants.To solve equations (18) and (19) by DJM with initial conditions y10=4 and y20=10, we write it in the following integral equation:(20)y1,1=4+∫0ty1β1+α11y1+α12y2dt,y2,1=10+∫0ty2β2+α21y1+α22y2dt.The rest components of the formulas (20) and (20) can be obtained using the computer algebra package Maple.


## 4.3. Three Species

In this section, equation (11) is reduced to three species:(21)dy1dt=y11−y1−αy2−βy3,(22)dy2dt=y21−βy1−y2−αy3,(23)dy3dt=y31−αy1−βy2−y3,where α and β are constants.To solve equations (21)–(23) by DJM with initial conditions y10=0.2, y20=0.3, and y30=0.5, we write it in the following integral equation:(24)y1,1=0.2+∫0ty11−y1−αy2−βy3dt,(25)y2,1=0.3+∫0ty21−βy1−y2−αy3dt,(26)y3,1=0.5+∫0ty31−αy1−βy2−y3dt.Again, the rest components of the formulas (24)–(26) can be obtained using the computer algebra package Maple.


## 5. Discussion

We used Maple to code the DJM algorithm. Maple environment variable digits is set to 16 in all calculations done in this paper.The numerical solutions obtained by using the DJM are compared with the exact solution and those obtained by ADM [21], DTM [20], and VIM [40]. Table 1 shows a comparison between the exact solution, the four iterations DJM with the DTM of order 6, two iterations of VIM, and 3-term ADM in the case b=1,a=−3, and y0=0.1 for t∈0,1; we can see the method is efficient to solve the one-species Lotka–Volterra equation. In Table 2, we compare four iteration DJM with DTM, VIM, ADM, and the exact solution where b=1,a=−3, and y0=0.1 for t∈0,3; in this table, we can prove the stability of DJM for large t. In Table 3, we perform the numerical comparison when β1=0.1,α11=−0.0014,α12=−0.0012,β2=0.08,α21=−0.0009,α22=−0.001,y10=4,y20=10, and h=0.001. In Table 4, we compare DJM with 4-Iterate VIM and RK4 when α=0.1,β=0.1,y10=0.2,y20=0.3,y30=0.5, and h=0.001Table 1
Comparison study whenβ=1,α=−3,y0=0.1.
tExactDJM4ADM,ϕ3 [21]VIM2 [21]DTM6 [20]0.00.10000000.10000000.10000000.10000000.10000000.20.11453290.11453290.11456000.11455450.11453290.40.13000110.13000110.13024000.13025900.13000040.60.14616290.14616270.14704000.14744450.14615460.80.16272590.16272560.16496000.16712630.16267901.00.17936720.17936690.18400000.19152490.1791887Table 2
Comparison study whenβ=1, α=−3, y0=0.1.
tExactDJM4VIM2 [40]ADM,ϕ3 [21]DTM9 [20]0.00.100000.100000.100000.100000.100000.50.138010.138010.138620.138500.138011.00.179360.179370.191520.184000.179371.50.219210.219210.298770.236500.219322.00.253330.253330.302860.296000.254422.50.279750.27969−4.48990.362500.285193.00.298640.29824−69.3170.436000.31533Table 3
Numerical comparison whenβ1=0.1,α11=−0.0014,α12=−0.0012,β2=0.08,α21=−0.0009,α22=−0.001,y10=4,y20=10, and h=0.001.
DJM2-iterate VIM [21]RK4ty1y2y1y2y1y20.04.0000010.000004.0000010.000004.0000010.000000.14.0330810.066574.0330710.066574.03307010.06650.24.0663910.133484.0663610.133494.06636310.13340.34.0998310.200754.0998710.200754.09987810.20070.44.1336510.268354.1336110.268364.1336110.26830.54.1675310.336314.1675810.336324.16758010.33630.64.2017610.404604.2017710.404624.20176710.40460.74.2361710.473244.2361810.473284.23618010.47320.84.2707310.542234.2708210.542284.27081810.54220.94.3056210.611564.3056910.611634.30568310.61161.04.3406710.681244.3407910.681334.34077510.6813Table 4
Numerical comparison whenα=0.1,β=0.1,y10=0.2,y20=0.3,y30=0.5, and h=0.001.
DJM4-Iterate VIM [21]RK4ty1y2y3y1y2y3y1y2y30.00.200000.300000.500000.200000.300000.500000.200000.300000.500000.10.214750.319150.522330.214730.319140.522340.214730.319140.522340.20.230190.338860.544420.230100.338730.544290.230100.338730.544280.30.246230.359030.566270.246090.358670.565730.246090.358670.565720.40.262980.379770.587780.262650.378890.586620.262640.378880.586550.50.280340.400990.609020.279750.399310.606930.279730.399270.606670.60.298380.422640.629930.297340.419870.626790.297290.419740.626010.70.317140.444840.650620.315400.440540.646530.315270.440200.644490.80.336530.467630.671090.333930.461380.666830.333610.460540.662080.90.356580.490820.691220.352940.482540.688870.352220.480680.678711.00.377320.514670.711060.372560.504380.714550.371050.500530.69438


## 6. Conclusions

In this article, the DJM is used for solving the multispecies Lotka–Volterra equation. The Daftardar-Gejji and Jafari method was implemented in a direct way without any linearization, perturbation, or restrictive assumptions. Comparisons with the VIM, DTM, and ADM show that the DJM is a better method for solving nonlinear equations. We proved that DJM is a precise and efficient method to solve the multispecies Lotka–Volterra equation.



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*Source: 1839796-2022-12-26.xml*