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

. In this article, we apply the Daftardar-Gejji and Jafari method (DJM) to solve the multispecies Lotka–Volterra equation. A comparison between the DJM, diferential 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. Te efciency and applicability of this method are confrmed by considering some examples. Te proposed procedure provides better results in comparison to some existing methods.


Introduction
Te area of mathematics called numerical analysis is in charge of coming up with practical methods for calculating answers to difcult computational calculations. Te majority of mathematical issues in engineering and science are very challenging, and sometimes, there is no straightforward solution. To make a difcult 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. Tese 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. Te 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. Tese kinds of issues often come up in the actual world when algebra, geometry, and calculus are applied, and they also include continuous variables. Tese issues arise in all areas of study, including the scientifc and social sciences, engineering, health care, and business [1][2][3][4][5][6][7][8][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 difculties may be solved using the techniques mentioned for handling initial value concerns, for example, see references [6,[10][11][12][13][14].
Te Lotka-Volterra equations describe the time history of a biological system [15]. Te Lotka-Volterra equations are applied in a number of engineering areas. Te one-species Lotka-Volterra equation is used to demonstrate a simple nonlinear control system [16].
Te diferential transformation method (DTM) was frst proposed by Zhou [22] (also check [23,24]). Te DTM is an iterative method that obtains the Taylor series solutions of diferent kinds of diferential equations (see [20,[25][26][27]). Te DTM can be applied directly to diferent kinds of DEs without requiring linearization, discretization, or perturbation, and it is a very accurate method with less computational work [28].
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. Te efciency and applicability of this method are confrmed by considering some examples. Te proposed procedure provides better results in comparison to some existing methods. Te DJM method will be implemented in a direct way without any linearization, perturbation, or restrictive assumptions.

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: where f is a function given, L is the linear operator and N is the nonlinear operator. Te solution of equation (1) will be as follow: Suppose So, Tus, N(v) is decomposed as follows: Since L is linear, then Ten, Te k− term approximate solution is given as follows: 3. Convergence of the DJM Theorem 1. "For any n and for some real L > 0 and H n is absolutely convergent and ‖H n ‖ ≤ LM n e n− 1 (e − 1), n � 1, 2, . . .." Proof. Please see reference [46] for full details of the proof.
Proof. Please see reference [46] for full details of the proof.

□ . Analysis of Multispecies Lotka-Volterra Equations
In this section, we will study the n th general Lotka-Volterra system in the form as follows: To solve equation (11) with the initial condition y(0) � y(0) by the Daftardar-Gejji and Jafari method (DJM), we write it in the following integral equation: Ten, we will apply the DJM as in the previous section.

One Species.
In this section, equation (11) is reduced to one species: where α and β are constants. With exact solution, To solve equation (13) with the initial condition y(0) � 0.1 by the Daftardar-Gejji and Jafari method (DJM), we write it in the following integral equation: By applying DJM, we obtain the following: Te four-term solution is as follows:
To solve equations (18) and (19) by DJM with initial conditions y 1 (0) � 4 and y 2 (0) � 10, we write it in the following integral equation:       (20) and (20) can be obtained using the computer algebra package Maple. (11) is reduced to three species:

Tree Species. In this section, equation
where α and β are constants.

Discussion
We used Maple to code the DJM algorithm. Maple environment variable digits is set to 16 in all calculations done in this paper.

Conclusions
In this article, the DJM is used for solving the multispecies Lotka-Volterra equation. Te 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 efcient method to solve the multispecies Lotka-Volterra equation.

Data Availability
No data were used to support this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.