Solving one species Lotka–Volterra equation by the new iterative
method (NIM)
BELAL BATIHA
Mathematics Department, Faculty of Science and Information Technology
Jadara University, Irbid, JORDAN
Abstract: In this paper, we investigate the use of the new iterative method, referred to as the NIM, for solving the
one species LotkaVolterra equation. This equation, which describes the dynamics of populations in ecological
systems, has been widely studied in the field of mathematical biology. However, finding an analytical solution
to this equation can be challenging. To overcome this, we propose using the NIM as an alternative method for
solving the equation. To demonstrate the effectiveness of the NIM, we conduct a comparative study between it and
other wellestablished techniques such as the differential transformation method (DTM), the variational iteration
method (VIM), and the Adomian decomposition method (ADM). Through numerical simulations, we show that
the NIM is able to accurately and efficiently solve the one species LotkaVolterra equation, making it a promising
tool for researchers in the field of mathematical biology.
KeyWords: Lotka–Volterra equation, The new iterative method (NIM), The differential transformation method
(DTM), The variational iteration method (VIM), The Adomian decomposition method (ADM)
Received: September 14, 2022. Revised: April 5, 2023. Accepted: April 28, 2023. Published: May 10, 2023.
1 Introduction
Nonlinear phenomena play a crucial role in vari
ous areas of science and engineering. However, solv
ing reallife nonlinear models is often challenging,
both numerically and theoretically. To make these
models tractable, unnecessary assumptions are often
made, [1], [2], [3], [4].
The LotkaVolterra equations, first introduced in
the field of mathematical biology, provide a mathe
matical model for describing the time evolution of a
biological system as reported in [5]. These equations
have also been applied in various engineering fields,
such as in the simultaneous control of chemical
processes and nonlinear systems, [6]. In particular,
the onepredator oneprey LotkaVolterra equation
serves as a simple example of a nonlinear control
system.
The DTM was initially introduced by [7], and
further developed in [8], [9]. DTM is an iterative
technique that aims to obtain Taylor series solutions
of various types of differential equations, as explored
in [10], [11], [12]. One of the main benefits of DTM
is that it can be applied to a wide range of differ
ential equations, without the need for linearization,
discretization, or perturbation. This makes it an
accurate method with relatively low computational
requirements as reported in [13].
The Adomian decomposition method (ADM)
was developed by [14], as a method for solving
challenging nonlinear physical problems. Since
its introduction, it has been utilized to address a
wide variety of differential equations, as reported in
literature such as [15], [16], [17], [18].
The VIM was first introduced by [19], and further
expanded upon in subsequent publications such as
[20], [21], [22]. This method has been demonstrated
to be effective in solving a wide range of ordinary
and partial differential equations, as shown in various
studies including, [23], [24], [25], [26].
The NIM was first introduced by [27]. Since
its inception, this method has been shown to be
a powerful technique for solving a wide range of
nonlinear equations as reported in [28], [29], [30],
[31], [32], [33], [34]. Recently, NIM has been used
to develop a novel predictorcorrector method, [35].
Additionally, Noor et al. have used NIM to create
numerical methods for solving algebraic equations as
reported in [36].
In this article, we present an application of the
NIM to solve the one species LotkaVolterra equa
tion, which describes the dynamics of populations in
ecological systems. The LotkaVolterra equation is a
nonlinear differential equation that can be challeng
ing to solve analytically. To overcome this, we em
ploy the NIM as an alternative method for solving the
equation. In order to evaluate the effectiveness of the
NIM, we conduct a comparison of the results obtained
with the NIM with the exact solution, as well as other
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DOI: 10.37394/23206.2023.22.38
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Volume 22, 2023
wellestablished techniques such as the DTM, VIM,
and the ADM. Through this comparison, we aim to
show the potential of the NIM as a powerful tool for
solving the one species LotkaVolterra equation and
similar nonlinear differential equations.
2 The new iterative method (NIM)
In this section, the NIM numerical method will be
outlined as follows, [29], [30], [31], [32]:
u=f+L(u) + N(u),(1)
In the equation above, fis a known function, and
Land Nare linear and nonlinear operators, respec
tively.
The NIM solution for Eq. (1) has the form
u=
∞
i=0
ui.(2)
Since Lis linear then
L∞
i=0
ui=
∞
i=0
L(ui).(3)
The nonlinear operator Nin Eq. (1) is decomposed
as below
N∞
i=0
ui=N(u0)
+
∞
i=1
N
i
j=0
uj
−N
i−1
j=0
uj
=
∞
i=0
Ai,
where
A0=N(u0)
A1=N(u0+u1)−N(u0)
A2=N(u0+u1+u2)−N(u0+u1)
.
.
.
Ai=
N
i
j=0
uj
−N
i−1
j=0
uj
, i ≥1.
Using Eqs.(2), (3) and (4) in Eq. (1), we get
∞
i=0
ui=f+
∞
i=0
L(ui) +
∞
i=0
Ai.(4)
The solution of Eq. (1) can be expressed as
u=
∞
i=0
ui=u0+u1+u2+. . . +un+. . . , (5)
where
u0=f
u1=L(u0) + A0
u2=L(u1) + A1
.
.
.
un=L(un−1) + An−1
.
.
. (6)
Algorithm
INP UT :Read M (N umber of iterations);
Read L(u); N(u); f
Step −1 : u−1= 0, u0=f
Step −2 : F or(n= 0, n ≤M, n + +)
{
Step −3 : An=f(un)−f(un−1);
Step −4 : un+1 =f+L(un) + An;
Step −5 : u=un+1
}end
OUT P U T :u
3 The convergence of the NIM
Theorem 1: For any nand for some real L > 0
and ||ui|| ≤ M < 1
e, i = 1,2, ..., if Nis C(∞)
in the neighborhood of u0and ||N(n)(u0)|| ≤ L,
then ∞
n=0 Hnis convergent absolutely and ||Hn|| ≤
LMnen−1(e−1), n = 1,2, ....
Proof:
||Hn|| ≤ LMn
∞
in=1
∞
in−1=0
···
∞
i1=0
n
j=1
1
ij!
=LMnen−1(e−1).(7)
Thus the series ∞
n=1 ||Hn|| is dominated by the
convergent series LM(e−1) ∞
n=1(Me)n−1, where
M < 1/e. Hence, ∞
n=0 Hnis absolutely conver
gent, due to the comparison test.
As it is difficult to show boundedness of ui, for
all i, a more useful result is provedin the following
theorem, where conditions on N(k)(u0)are given
which are sufficient to guarantee convergence of the
series.
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Theorem 2: The series ∞
n=0 Hnis convergent
absolutely if Nis C(∞)and ||N(n)(u0)|| ≤ M≤
e−1,∀n.
Proof: Consider the recurrence relation
εn=ε0exp(εn−1), n = 1,2,3, ..., (8)
where ε0=M. Define ηn=εn−εn−1, n = 1,2,3,··
·. We observe that
||Hn|| ≤ ηn, n = 1,2,3,···.(9)
Let
σn=
n
i=1
ηi=εn−ε0.(10)
Not that ε0=e−1>0,ε1=ε0exp(ε0)> ε0and
ε2=ε0exp(ε1)> ε0exp(ε0) = ε1. In general, εn>
εn−1>0. Hence ηnis a series of positive real
numbers. Note that
0< ε0=M=e−1<1,
0< ε1=ε0exp(ε0)< ε0e1=e−1e1= 1,(11)
0< ε2=ε0exp(ε1)< ε0e1= 1.
In general 0< εn<1. Hence, σ=εn−ε0<
1. This implies that {σn}∞
n=1 is bounded above by 1,
and hence convergent. Therefore, Hnis absolutely
convergent by comparison test.
4 Numerical results and discussion
Here, we will focus on the study of the one species
LotkaVolterra equation in the context of competition
for a finite source of food. The LotkaVolterra equa
tion is a mathematical model that describes the dy
namics of populations in ecological systems and it is
widely used to study the interactions between differ
ent species. In this particular scenario, we will inves
tigate the behavior of one species that is competing for
a limited food supply. The analysis will involve un
derstanding the rate of change of the population of the
species and how it is affected by the food availability
and the competition with other species. Through this
analysis, we aim to gain insights into the impact of
competition and resource availability on population
dynamics and the behavior of the species. The one
species LotkaVolterra equation in the form:
du
dt =u(b+au), b > 0, a < 0, u(0) >0,(12)
where aand bare constants. With exact solution:
u(t) = bebt
b+au(0)
u(0) −aebt for b= 0,(13)
u(t) = u(0)
1−au(0)t,for b= 0.(14)
To apply the NIM to solve equation (12) with the
initial condition of u(0) = 0.1, we integrate eq. (12)
and utilize the specified initial condition to obtain:
u(t) = 0.1 + t
0
u(b+au)dt (15)
By applying NIM we get:
u0= 0.1,
u1= 0.07t
u2=−0.0049 t2(t−2.857143)
.
.
. (16)
The fourterm solution is:
u(t) = 0.1+0.07 t−0.0049 t2(t−2.857143)
−0.00001029 (t+ 5.587558)t3(t−1.234763)
(t−3.493845)(t−7.525617)
−2.117682 ×10−11 (t+ 5.877433)
(t+ 5.005629)t4(t−4.090877)
(t−6.871677)(t−7.828486)
(t2+ 6.991252 t+ 21.95022)
(t2−2.520506 t+ 1.605527)
(t2−10.84848 t+ 38.63197) (17)
4.1 Discussion
The solutions calculated using the NIM are eval
uated against the accurate solution and solutions ob
tained through other techniques like the ADM, [25],
DTM, [12], and VIM ,[26]. Table 1 presents a com
parison between the exact solution and the numerical
solutions obtained using the NIM, ADM, DTM, and
VIM for b= 1, a =−3,u(0) = 0.1and t∈[0,1].
Similarly, Table 2 compares the numerical solutions
obtained by the NIM, ADM, DTM, VIM, and the ex
act solution for b= 1, a =−3,u(0) = 0.1and
t∈[0,3].
5 Conclusions
In this article, we employ the new iterative method
(NIM) developed by DaftardarGejji and Jafari to
solve the one species LotkaVolterra equation. This
method was implemented in a direct manner, with
out the need for linearization, perturbation, or any re
strictive assumptions. The results obtained using the
NIM were compared to those obtained using other
wellestablished methods such as the variational it
eration method (VIM), the differential transforma
tion method (DTM), and the Adomian decomposition
method (ADM), and it was found that the NIM is a
more effective method for solving nonlinear equa
tions. Through our analysis, we have demonstrated
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that the NIM is a highly efficient, accurate, and cost
effective method for solving the one species Lotka
Volterra equation.
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Contribution of Individual Authors to the
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Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflicts of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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Table 1: Comparison study under the conditions b=
1,a=−3, and an initial value of u(0) = 0.1
tExact solution NIM4ADM, ϕ3[25] V IM2[25] DT M6[12]
0.0 0.1000000 0.1000000 0.1000000 0.1000000 0.1000000
0.2 0.1145329 0.1145329 0.1145600 0.1145545 0.1145329
0.4 0.1300011 0.1300011 0.1302400 0.1302590 0.1300004
0.6 0.1461629 0.1461627 0.1470400 0.1474445 0.1461546
0.8 0.1627259 0.1627256 0.1649600 0.1671263 0.1626790
1.0 0.1793672 0.1793669 0.1840000 0.1915249 0.1791887
Table 2: Comparison study under the conditions b=
1,a=−3, and an initial value of u(0) = 0.1
tExact solution NIM4V IM2[26] ADM, ϕ3[25] DT M9[12]
0.0 0.10000 0.10000 0.10000 0.10000 0.10000
0.5 0.13801 0.13801 0.13862 0.13850 0.13801
1.0 0.17936 0.17937 0.19152 0.18400 0.17937
1.5 0.21921 0.21921 0.29877 0.23650 0.21932
2.0 0.25333 0.25333 0.30286 0.29600 0.25442
2.5 0.27975 0.27969 −4.4899 0.36250 0.28519
3.0 0.29864 0.29824 −69.317 0.43600 0.31533
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