In this paper, a mathematical model of one-predator two-prey system is discussed. This model is derived from predator-prey Lotka-Volterra model by adding another population of prey into the system. The model derived is a nonlinear system of ODEs. So the approach to this model is different from the linear system of ODEs. With reference to that, Adomian Decomposition Method (ADM) is one of the semi-analytical approaches being applied in this paper to solve the system. The approximate solution is made until four terms. The solution obtained is analyzed graphically

Farahanie, Fauzi

Muhammad Saufi Firdaus, Azmi

Norlaila, Md Nor

Wan Khairiyah Hulaini, Wan Ramli

English

Universiti Teknologi MARA Institutional Repository


 
JOURNAL OF MATHEMATICS AND COMPUTING SCIENCE, 2018,  
Volume 1, Number 1, 1 - 8 
 
1 
 
Solving One-Predator Two-Prey System by using Adomian Decomposition Method  
Wan Khairiyah Hulaini Wan Ramli1*, Muhammad Saufi Firdaus Azmi2, Farahanie Fauzi 3 and Norlaila Md Nor4 
1,2,3,4Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Kelantan, Bukit Ilmu, Machang, 
Kelantan, Malaysia 
wkriyah@kelantan.uitm.edu.my 
*Corresponding author 
Abstract: In this paper, a mathematical model of one-predator two-prey system is discussed. This model is 
derived from predator-prey Lotka-Volterra model by adding another population of prey into the system. The 
model derived is a nonlinear system of ODEs. So the approach to this model is different from the linear system 
of ODEs. With reference to that, Adomian Decomposition Method (ADM) is one of the semi-analytical 
approaches being applied in this paper to solve the system. The approximate solution is made until four terms. 
The solution obtained is analyzed graphically. 
 
Keywords: Adomian Decomposition Method, Adomian Polynomials, Lotka-Volterra, 
Predator Prey 
 
1 Introduction 
 
Population is a group of organisms of the same species which live in the same region at the same 
time. A population consists of all the individuals of a given species in a specific region or area at a 
certain time. It contains genetic variation between themselves. From one individual to another, there 
might be a slight difference among them such as in terms of size, hair color, and so forth. Most 
importantly, not all individuals in the population have the same ability to survive and reproduce. 
All living things within an ecosystem are dependent on each other. A change in the size of one 
population affects all other organisms within the ecosystem. There are many types of interaction that 
happen among the population and one of them is predation. According to Obaid and Hamoud [1], 
predation is the species interaction when one species, the predator eats another species, the prey as a 
source of food. 
In the 1920's, a mathematical model was created independently by Alfred Lotka and Vito Volterra, 
and it is characterized by oscillations in the population size of both predator and prey [2]. This model 
was proposed to explain the dynamic, natural populations and describe the relationship of predators 
and prey. It is an application of a nonlinear system of differential equations in mathematics as it 
involves more than 1 population. This model is known as Lotka-Volterra predator-prey model. 
From the model, a new system is derived by adding another prey population into the system. In 
formulating the system, there are a few assumptions made. The assumptions are : 
 
1. Consider there are only one predator and two prey species in the system. 
2. Both preys do not compete with each other. 
3. Food sources for both preys are different and unlimited. 
4. The system is free from any disease that can increase the death rate. 
5. The same species are competing in nature. 
 
In [3], many topics under the system of Lotka-Volterra were discussed, one of them was two species 
Lotka-Volterra systems. The author has reviewed about the evolution of predator and prey fish 
populations in Adriatic Sea. He stated that there are several points of criticism worth noting for the 
Lotka-Volterra model. The changes of birth and death rates do nothing but the changes in the period 
of oscillation - i.e. no population - can dominate, and it is impossible for either population being 
driven to extinction. 
Wan Khairiyah Hulaini 
2 
 
Another study was conducted by Schrum [4] where the author discussed the two-dimensional Lotka-
Volterra systems where the interaction of two species was assumed as a closed system. The two given 
species are only affected to each other and no other external factors can influence the system. Schrum 
stated that the main weakness of this model is the exponential growth assumption. It is biologically 
unrealistic for the unbounded growth of the prey species without predators. He extended the 
discussion by stating that the system exhibits stable periodic behavior for all non-zero initial 
conditions. 
Obaid and Hamoud [1] discussed two types of interaction in their article, which are one-prey one-
predator system and one-predator two-prey system. The authors showed how both interaction was 
being modeled. Numerical simulation is also used in the verification of the results. After both 
mathematical model were examined, predator and prey populations seemed to be cyclical endless 
without settling down quickly. The same result is also stated in [5].  
Kar and Batabyal [6] emphasized on the persistence and stability of the one-predator two-prey system. 
The authors used Holling type I and Holling type II response function to represent the interaction 
between the predator and prey. Kar and Chaudhuri [7] considered a two-prey one-predator harvesting 
model with interference. The model was based on Lotka-Volterra dynamics with two competing 
species which were affected not only by harvesting but also by the presence of a predator, the third 
species.  
Biazar et al. [8]  discussed solving the system of ordinary differential equations. It is known that 
among the many, Lotka-Volterra is a type of first order nonlinear system of differential equations. The 
author showed that Adomian Decomposition Method (ADM) was able to solve both linear and 
nonlinear system of differential equations. According to Babolian et al. [9], ADM was initially 
proposed by George Adomian in the early of 1980s. The authors stated that a solution of functional 
equation was assumed as the summation of infinite series that usually converges to the solution. There 
were two topics discussed which are system of linear equation and system of Volterra integral 
equation. The authors presented numerical examples to show a comparison between ADM and Jacobi 
iterative method for solving system of linear equations. The results indicate both methods are the 
same which means if Jacobi Iterative method diverges, so will ADM. 
The main part in solving a system of differential equation using ADM is computing the Adomian 
polynomial. Many authors suggested different algorithms for computing Adomian polynomials [10-
12]. This paper uses an alternate algorithm for computing Adomian polynomials [11]. 
2 Problem Formulation 
 
The derivation of the system was discussed in Obaid and Hamoud [1]. The system was given by Eq. 
(1). 
  
   
   
   
   
033312322
3
022212
2
0111322
1
0,
0,
0,
yyyyeydc
dt
dy
yyyeydyc
dt
dy
yyyybbya
dt
dy



       (1)   
  
where )(1 ty represents the predator’s populations at time t, )(2 ty  represents the first prey’s 
populations at time t and )(3 ty  represents the second prey’s populations at time t. The parameter a 
represents per capita death rate of predator in the absence of prey, parameter c represents per capita 
birth rate of first prey in the absence of predator and parameter 2c  represents per capita birth rate of 
second prey in the absence of predator . Parameter b and 2b  represent interaction parameter of 
predator with the first and second prey respectively. Parameter d and 2d  represent the interaction 
 Solving one predator two prey system by using Adomian decomposition method 
3 
 
parameter of first and second prey populations respectively. Lastly, parameter e and 2e  respresent the 
first and second prey death rate killed by predator respectively. Eq. (1) is called one-predator two-prey 
system. This nonlinear system will be solved by using Adomian decomposition method. 
 
3 Adomian Decomposition Method 
 
The Adomian Decomposition Method is a semi analytical method for solving Ordinary Differential 
Equations (ODE) and non-linear Partial Differential Equations (PDE). The purpose of this method is 
towards a unified theory for the solution of PDE, an aim which has been superseded by a more 
general theory of the homotopy analysis method. 
The important aspect of ADM is employment of the Adomian Polynomials which allows for solutions 
to convergence of the nonlinear portion of the equation, without help to linearizing the system. These 
polynomials are mathematically generalized to Maclaurin series about an arbitrary external parameter 
that give more flexibility compared to the Taylor series. 
 
Consider the one-predator two-prey system with the initial condition given by Eq. (1). The system can 
be represented in the form of L operator : 
 
 
 
  3123223
2122
13221
yyeydcLy
yeydycLy
yybbyaLy



       (2) 
 
where L is linear operator 
dt
d
 with inverse,  dtL
t
 

0
1
. By applying the 1L on both sides of each 
Eq. (2), the equation becomes : 
 
     
     
     





t
t
t
dtyyeydcyty
dtyeydycyty
dtyybbyayty
0
31232223
0
21222
0
132211
0
0
0
     (3) 
 
 
Then, according to Babolian et. al [9], the solution is considered as the sum of infinite series with 
parameter λ . 
 












0
,33
0
,22
0
,11
n
n
n
n
n
n
n
n
n
yy
yy
yy



         (4) 
 
 
 
 
 
Wan Khairiyah Hulaini 
4 
 
 
Let each integrand in Eq. (3) as the sum of the following series : 
 
   



0
,30,3,20,2,10,1,13211
...,,,...,,,...,,,,
n
n
nnnn yyyyyyAyyyf   
   



0
,30,3,20,2,10,1,23212
...,,,...,,,...,,,,
n
n
nnnn yyyyyyAyyyf      (5) 
   



0
,30,3,20,2,10,1,33213
...,,,...,,,...,,,,
n
n
nnnn yyyyyyAyyyf   
 
where nnn AAA ,3,2,1 ,,  are called Adomian Polynomials. Then, substitute Eq. (4) and Eq. (5) into Eq. 
(3): 
 
    





0
,30,3,20,2,10,1,1
0
1
0
,1 ...,,,...,,,...,,0
n
nnnn
t
n
n
n
n dtyyyyyyAyy   
    





0
,30,3,20,2,10,1,2
0
2
0
,2 ...,,,...,,,...,,0
n
nnnn
t
n
n
n
n dtyyyyyyAyy     (6) 
    





0
,30,3,20,2,10,1,3
0
3
0
,3 ...,,,...,,,...,,0
n
nnnn
t
n
n
n
n dtyyyyyyAyy   
 
Then, define the following scheme : 
 
    dtyyyyyyAyyy nnnn
t
n ,30,3,20,2,10,1,1
0
110,1 ...,,,...,,,...,,,0    
    dtyyyyyyAyyy nnnn
t
n ,30,3,20,2,10,1,2
0
120,2 ...,,,...,,,...,,,0      (7) 
    dtyyyyyyAyyy nnnn
t
n ,30,3,20,2,10,1,3
0
130,3 ...,,,...,,,...,,,0    
 
In computing all the Adomian Polynomials, an alternate algorithm method is used [11]. From the 
series in Eq. (4), if we derive  321 ,, yyyf  as a power series of λ , then the coefficient of 
nλ will be 
 nnnni yyyyyyA ,30,3,20,2,10,1, ...,,,...,,,...,, . Then, the expansion will be as follows : 
 
 
Predator equation : 
 
0,30,120,20,10,10,1 yybybyayA   

















 



 rr
r
rr
r
yybyybayA 1,3,1
1
0
21,2,1
1
0
1,11,1  

















 



 rr
r
rr
r
yybyybayA 2,3,1
2
0
22,2,1
2
0
2,12,1      (8) 

















 



 rr
r
rr
r
yybyybayA 3,3,1
3
0
23,2,1
3
0
3,13,1  
 
 Solving one predator two prey system by using Adomian decomposition method 
5 
 
 
 
1st  Prey equation : 
 
   0,20,1
2
0,20,20,2 yeyydcyA   
 

















 



 rr
r
rr
r
yyeyydcyA 1,2,1
1
0
1,2,2
1
0
1,21,2  

















 



 rr
r
rr
r
yyeyydcyA 2,2,1
2
0
2,2,2
2
0
2,22,2      (9) 

















 



 rr
r
rr
r
yyeyydcyA 3,2,1
3
0
3,2,2
3
0
3,23,2  
  
 
2nd Prey equation : 
 
   0,30,1220,320,320,3 yyeydycA   
 

















 



 rr
r
rr
r
yyeyydycA 1,3,1
1
0
21,3,3
1
0
21,321,3  

















 



 rr
r
rr
r
yyeyydycA 2,3,1
2
0
22,3,3
2
0
22,322,3      (10) 

















 



 rr
r
rr
r
yyeyydycA 3,3,1
3
0
23,3,3
3
0
23,323,3  
  
 
So, in general : 
 

















 



 rnr
n
r
rnr
n
r
nn yybyybayA ,3,1
0
2,2,1
0
,1,1  

















 



 rnr
n
r
rnr
n
r
nn yyeyydcyA ,2,1
0
,2,2
0
,2,2      (11) 

















 



 rnr
n
r
rnr
n
r
nn yyeyydycA ,3,1
0
2,3,3
0
2,32,3  
 
In this paper, the solution is approximated until four terms. So, all the Adomian Polynomials will be 
generated for n = 0, 1, 2. The four terms approximation to the solution are considered as 
 
 ( ) 3,2,1,0, +++≈ iiiii yyyyty         (12) 
 
The three populations for 21, yy  and 3y  are fox, rabbit and turkey populations respectively where 
their initial population are 100, 90 and 80. The parameters a = 0.5, b = 0.01, 02.02 b , c = 1, d = 
0.01, e = 0.01, 2.32 c , 04.02 d  and 03.02 e are used for solving the model [1]. 
 
Wan Khairiyah Hulaini 
6 
 
4 Results and Discussion 
 
The approximations to the solutions with four terms for ( ) ( )tyty 21 ,  and ( )ty3 are as follow: 
   321 36667.975.80200100 tttty   
   322 54.661.178190 tttty   
   323 2.126550424080 tttty   
From the solutions obtained, some of the approximated values are shown below : 
Table 1 : Numerical results obtained from the solutions using ADM 
t ( )ty1  ( )ty2  ( )ty3  
0.1 119.0976333 81.79554 59.7748 
0.2 136.0010666 73.64832 42.0384 
0.3 150.1260999 65.95758 19.1996 
0.4 160.8885331 59.12256  
0.5 167.7041662 53.54250  
0.6 169.9887993 49.61664  
0.7 167.1582322 47.74422  
0.8 158.6282650 48.32448  
0.9 143.8146976 51.75666  
1.0 122.1333300 58.44000  
1.1 92.9999622 68.77374  
1.2 55.8303942 83.15712  
1.3 10.0404260 101.98938  
1.4  125.66976  
 
Table 1 shows some numerical results after using Adomian Decomposition Method in approximating 
the predator and both preys population for some value of t. The negative value means that the 
population extincts. This result is analyzed further by a graph that has been plotted using MAPLE 
software. 
 
Figure 1: Population of foxes and rabbits versus time 
 Solving one predator two prey system by using Adomian decomposition method 
7 
 
Figure 1 shows the population of foxes and rabbit. This graph agrees with the graph plotted in Biazar 
and Montazeri [13]. 
 
Figure 2: Population of foxes, rabbits and turkey versus time 
 
In Figure 2, all the three populations are plotted. We can summarize that there are patterns we can see. 
First, populations go up and down, this refers to a graph where shape of the graph is oscillated up and 
down. Second, the number of predator population depends upon the number of prey population.  As 
can be seen in the table and graph, the number of foxes increases gradually over the time, t at the 
beginning. On the other hand, the number of rabbits and turkeys decreases as we can say the 
population is in the process of adapting to the environment. 
Foxes are known as the predator that readily adapt to new environments and so are very successful 
invasive species. Foxes grow quickly and occupy new area and generally prey on native species, 
which are easily caught. So, it is relevant to say that number of foxes increases at the beginning. 
Yet, the number of rabbits decreases at first and then going up back when foxes turn to decrease. It is 
logically realistic for the rabbits population to show an increase when there are less foxes. For turkeys 
population, we can see the values continuously go down until they reach 0 and drop to a negative. It 
means that the turkeys population is experiencing an extinction. The graph agrees with Vlastmil and 
Eisner [14] who said, when resources grow exponentially, handling times are zero and apparent 
competition always leads to extinction of the weaker resource. 
5 Conclusion 
 
This paper discusses solving one-predator two-prey system by using Adomian Decomposition 
Method. This method is claimed to be a powerful device for solving nonlinear problems. From the 
two preys’ population, one prey goes to extinction while the other prey survives. The predator 
population is also heading towards extinction. All predators and preys can neither leave these 
populations nor enter them from the outside system. This means that there is no possible immigration 
into the system. This model also assumes that the predator has only two types of prey as food sources. 
This is rarely the case in nature and the real world. As for the recommendation, this method should be 
applied to an improved model which considers realistic assumptions and an open system. 
 
 
 
 
 
Wan Khairiyah Hulaini 
8 
 
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[3]  Baigent, S. Lotka –Volterra Dynamical Systems. Dynamical and Complex Systems: 2017 
 
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Solving One-Predator Two-Prey System by using Adomian Decomposition Method / Wan Khairiyah Hulaini Wan Ramli... [et. al]

http://ir.uitm.edu.my/id/eprint/24202/1/AJ_WAN%20KHAIRIYAH%20HULAINI%20WAN%20RAMLI%20JMCS%20D%2016.pdf

Solving One-Predator Two-Prey System by using Adomian Decomposition Method / Wan Khairiyah Hulaini Wan Ramli... [et. al]

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