ABSTRACT: This paper investigates an eco-system with two prey species and a predator. The model equations constitute a set of three first order non-linear coupled difference equations. All possible positive equilibrium points of the model are computed and criteria for the stability of all equilibrium states are established. Simulations are performed in order to illustrate the theoretical results and gain further insight into the behaviour of the system

A George

Maria Selvam

P Rathinavel

R Janagaraj

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          ISSN(Online): 2319-8753 
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International Journal of Innovative Research in Science, 
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Vol. 4, Issue 1, January 2015 
 
DOI: 10.15680/IJIRSET.2015.0401023                                         www.ijirset.com                                                                 18576 
 
A Discrete Model of Three Species Prey-
Predator System 
A.George Maria Selvam
1
, R.Janagaraj
2
 and P. Rathinavel
3
 
1
Associate Professor, Sacred Heart College, Tirupattur, India 
2
Assistant Professor, Government College of Engineering, Bargur, India 
3
M.Phil Research Scholar, Sacred Heart College, Tirupattur, India 
 
ABSTRACT: This paper investigates an eco-system with two prey species and a predator. The model equations 
constitute a set of three first order non-linear coupled difference equations. All possible positive equilibrium points of 
the model are computed and criteria for the stability of all equilibrium states are established. Simulations are performed 
in order to illustrate the theoretical results and gain further insight into the behaviour of the system. 
 
KEYWORDS: predator-prey interactions, difference equations, equilibria, stability. 
. 
I. INTRODUCTION 
Ecosystems are characterized by the interaction between different species and natural environment. Some important 
ecological interactions are Competition, Mutualism and Predation. The Lotka - Volterra predator prey equations were 
discovered independently by Alfred Lotka and by Vito Volterra in 1925-26. In 1926, Volterra devised a model to 
describe the evolution of predator and prey fish populations in the Adriatic Sea after the First World War (when fishing 
was curtailed). Population ecology typically focuses on particular species or pairs of species within webs of interacting 
species to understand variations in their abundance. Several papers appeared on 2-D models of predator - prey 
dynamics. Since the analysis of non linear systems when n > 2 (number of species) is much more difficult, only a 
limited amount of work is available on ‘Three Species Models’ [7,8]. 
II. RELATED WORK 
There exists a vast amount of literature on prey predator models, see [3,5]. The classical Lotka - Volterra predator 
prey equations predicted neutral cycles with a period determined by parameters and amplitude set by initial conditions. 
Later, many authors studied the effect of prey predator interactions by introducing logistic growth expression, 
functional responses and Allee effect [2]. It was demonstrated that some of the simplest nonlinear difference equations 
describing the growth of biological populations with nonoverlapping generations can exhibit a remarkable complex and 
rich dynamical behaviour involving stability, existence of limit cycles, bifurcation and chaos [1,2,4,8,9]. In this paper, 
we investigate the stability nature of equilibria of a system describing the interaction among three species using a 
discrete model.  
III. MODEL EQUATIONS 
 
We are concerned with the following set of first order linear difference equations which models the interaction 
between two preys and one predator. 
 
        1 1 ( )x n ax n x n bx n z n     
        1 1 ( )y n cy n y n dy n z n       (1) 
          1 1 ( )z n z n e bx n z n dy n z n      
 
 
  
          ISSN(Online): 2319-8753 
           ISSN (Print):  2347-6710 
 
International Journal of Innovative Research in Science, 
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Vol. 4, Issue 1, January 2015 
 
DOI: 10.15680/IJIRSET.2015.0401023                                         www.ijirset.com                                                                 18577 
 
where the parameters a, b, c, d and e are non-negative. The population of preys (x(n), y(n)) follow logistic growth 
model. The rate of predation is proportional to the rate at which the predator and the preys meet. This system of 
equations cannot be solved analytically, but some information about the behaviour of its solutions can be obtained via 
qualitative analysis. The mathematical analysis of the model is often used for the assessment of stability of the 
equilibrium. 
IV. EXISTENCE OF EQUILIBRIUM OF THE SYSTEM AND A LEMMA 
We are interested in the analysis of the equilibrium points as they help us to understand the dynamics of the system 
under investigation. The system (1) has the following equilibrium points namely 𝐸0(0,0,0)(extinction of all species), 
𝐸1(0,1−
1
𝐶
, 0) (survival of a prey), 𝐸2  1 −
1
𝑎
, 0,0 , 𝐸3  1−
1
𝑎
, 1−
1
𝐶
, 0  (both prey populations survive),  
𝐸4  0,
𝑒
𝑑
,
𝑐−1
𝑑
−
𝑐𝑒
𝑑2
 , 𝐸5  
𝑒
𝑏
, 0,
𝑎−1
𝑏
−
𝑎𝑒
𝑏2
  and 𝐸6  
𝑑2 𝑎−1 −𝑏𝑑  𝑐−1 +𝑏𝑐𝑒
𝑐𝑏2+𝑎𝑑2
,
𝑏2 𝑐−1 −𝑏𝑑  𝑎−1 +𝑎𝑑𝑒
𝑐𝑏2+𝑎𝑑2
,
𝑏𝑐  𝑎−1 +𝑎𝑑  𝑐−1 −𝑎𝑐𝑒
𝑐𝑏2+𝑎𝑑2
 (existence 
of all species). 
 The first equilibrium point E0 corresponds to the extinction of species in the eco-system where as E1 and E2 
are axial equilibrium points corresponding to the situation in which two species become extinct. The equilibrium 
points E3, E4 and E5 correspond to the extinction of a species. The equilibrium points E6 is an interior equilibrium point 
which establishes the coexistence of all species in the system. The following Lemma is useful in the discussion of the 
stability properties of the equilibrium points of (1) 
Lemma 1. Let [which can be found in (3)] 
3 2
1 2 3
( ) 0p A A A            (2) 
be the characteristic equation for a matrix. Then we have: 
 • If every root of equation (2) has absolute value less than one, then the equilibrium point of the system (1) is 
locally asymptotically stable and equilibrium point is called a sink. 
V. STABILITY OF EQUILIBRIUM POINTS AND NUMERICAL STUDY 
In this section, we provide numerical examples and simulations which help us to analyse the stability properties of 
the system (1) by graphing time plots for x(n), y(n), z(n) and phase diagrams. We consider ranges of the parameters 
provided by the analysis presented in the following propositions. The examples explain the complex dynamics of the 
system. The local stability analysis of the model can be carried out by computing the Jacobian corresponding to each 
equilibrium point and finding eigen values from the Jacobian matrix. The Jacobian matrix J for the system (1) has the 
form 
 
(1 2 ) 0
( , , ) 0 (1 2 )
1
a x bz bx
J x y z c y dz dy
bz dz e bx dy
   
 
   
 
    
    (3) 
The characteristic equation of (3) can be put in the form 
3 2
( ) [ ( )] [ ( )] ( ) 0p Tr J Det of Dia J Det J      
.
 
where Tr(J)= (1 2 ) (1 2 ) 1a x bz c y dz e bx dy        
 
is the trace and  
Det (J) =[ (1 2 )][ (1 2 ) ][1 ]a x c y dz e bx dy     
 
is the determinant of the Jocobian matrix J.  
 For the system (1), we have the following properties accompanied by numerical simulations. 
 
 
 
 
 
 
 
  
          ISSN(Online): 2319-8753 
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International Journal of Innovative Research in Science, 
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Vol. 4, Issue 1, January 2015 
 
DOI: 10.15680/IJIRSET.2015.0401023                                         www.ijirset.com                                                                 18578 
 
Proposition 1. The equilibrium point E0 is sink if 1a  , 1c  and 0 2e  . 
Proof: Jacobian matrix for the trivial equilibrium E0 is 0
0 0
( ) 0 0
0 0 1
a
J E c
e


 
 
 
  
 
 
The Eigen values of the matrix J(E0) are 1 a  , 2 c  , 3 1 e   . In view of Lemma 1, we see that E0 is a 
sink if 1a  , 1c  . 
 
Figure 1. Time plot for trivial equilibrium point E0 
Proposition 2. The equilibrium point E1 is a sink if 1a  ,1 3c  and 
1
ce
d
c


. 
Proof: From (3), Jacobian matrix for E1 is given by 
1
0 0
1
( ) 0 2 1
1
0 0 1 1
a
J E c d
c
e d
c
  
  
 
 
 
  
   
 
  
    
 
The eigen values of the matrix J(E1) are 1 a  , 2 2 c    and 3
1
1 1e d
c
    
 
 
 
.  
The axial equilibrium point E1 is a sink if 1a  ,1 3c   and
1
ce
d
c


. 
 
  
          ISSN(Online): 2319-8753 
           ISSN (Print):  2347-6710 
 
International Journal of Innovative Research in Science, 
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(An ISO 3297: 2007 Certified Organization) 
Vol. 4, Issue 1, January 2015 
 
DOI: 10.15680/IJIRSET.2015.0401023                                         www.ijirset.com                                                                 18579 
 
 
Figure 2. Time plot for axial equilibrium point E1 
 
Proposition 3. The equilibrium point E2 is a sink if,1 3a  , 1c   and 
1
ae
b
a


. 
Proof: Jacobian matrix for E2 is obtained from (3) as  2
1
2 0 1
( ) 0 0
1
0 0 1 1
a b
a
J E c
e b
a
 

  
   
   
 
 
  
  
  
 
The eigen values of the matrix J(E2) are 1 2 c   , 2 a   and 3
1
1 1e b
a
    
 
 
 
 
The axial equilibrium point E2 is a sink if1 3a  , 1c   and
1
ae
b
a


. 
 
 
Figure 3. Time plot for Axial equilibrium point E2 
 
 
  
          ISSN(Online): 2319-8753 
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International Journal of Innovative Research in Science, 
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Vol. 4, Issue 1, January 2015 
 
DOI: 10.15680/IJIRSET.2015.0401023                                         www.ijirset.com                                                                 18580 
 
Proposition 4. The equilibrium point E3 is a sink if 1 3a  , 1 3c  and 
( 1) ( 1)bc a ad c
e
ac
  
  
Proof:  Jacobian matrix for E3 is 3
1
2 0 1
1
( ) 0 2 1
1 1
0 0 1 1 1
a b
a
J E c d
c
e b d
a c
 
  
    
   
   
 
  
   
 
    
   
       
The eigen values of the matrix J (E3) are 1 2 a   , 2 2 c    and 3
1 1
1 1 1e b d
a c

   
        
   
 
From Lemma 1, we see that E3 is a sink if1 3a  , 1 3c  and
( 1) ( 1)bc a ad c
e
ac
  
 . 
 
 
Figure 4. Time plot for Axial equilibrium point E3
 
Proposition 5. The equilibrium point E4 is a sink if 
2
(1 ) 1
1
d a
e d
bc c

  
 
 
 
and
1
1e d
c
 
 
 
 
. 
Proof: From (3) Jacobian matrix for E4 is given by 
2
4
2
( 1)
0 0
( ) 0 1
( 1)
1 1
b c bce
a
d d
ce
J E e
d
b c bce ce
c
d dd
 
  
 
   
 
 
   
  
 
The eigen values of the matrix J(E4) are 
2
1 2
bd ad bcd bce
d

  
  and 
2 2 2 2 2
2,3
2 1
4 4 4
2 2
d ce
c e cd e cde d e
d d


     . In view of Lemma 1, we see that E4 is a 
sink if 
2 (1 ) 1
1
d a
e d
bc c
  
   
 
and
1
1e d
c
 
  
 
. 
 
  
          ISSN(Online): 2319-8753 
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International Journal of Innovative Research in Science, 
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Vol. 4, Issue 1, January 2015 
 
DOI: 10.15680/IJIRSET.2015.0401023                                         www.ijirset.com                                                                 18581 
 
 
 
Figure 5. Time plot for axial equilibrium point E4
 
Proposition 6. The equilibrium point E5 is a sink if 
2 (1 ) (1 )b c bd b
e
ad
  
 and
1
1e b
a
 
  
 
.  
Proof:  Computation of Jacobian matrix for E5 yields   
5 2
2
1 0
1
( ) 0 1 0
( 1)
1 1
ae
e
b
a ade
J E c d
b b
ae d a ade
a
d b b
 
  
 
  
      
 
 
    
 
The eigen values of the matrix J(E5) are
2
1 2
bd cb add ade
b

  
  and 
2 2 2 2 2
2,3
2 1
4 4 4
2 2
b ae
a e ab e abe b e
b b


     . In view of Lemma 1, we see that E5 is a 
sink if 
2 (1 ) (1 )b c bd b
e
ad
  
 and
1
1e b
a
 
  
 
. 
 
  
          ISSN(Online): 2319-8753 
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International Journal of Innovative Research in Science, 
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Vol. 4, Issue 1, January 2015 
 
DOI: 10.15680/IJIRSET.2015.0401023                                         www.ijirset.com                                                                 18582 
 
 
 
Figure 6. Time plot for Axial equilibrium point E5
 
 
Jacobian matrix for the interior equilibrium point of the system (1) is  
 2 2 2 2
2 2 2 2
2 2 2 2 2
6 2 2 2 2
2 2
2 2 2 2
(1 ) ( 1) ( 1) ( 1)
1 0
( 1) ( 1) ( 1) ( 1)
( ) 0 1
( 1) ( 1) ( 1) ( 1)
1
ad a ab d c ce b d c bd a b ce
cb ad cb ad
cb a ad c adce bd a b d c ad e
J E
cb ad cb ad
cb a ad c adce bcd a ad c adce
cb ad cb ad
        
 
  
        
  
  
        

  


and 
 
 
2
6 1 22 2
(1 ) ( 1)
[ ( )]1
ad a ab d c ce
Det J E p p
cb ad
   
 
  where  
2 2
1 2 2
( 1) ( 1)
1
cb a ad c adce
p
cb ad
   
 

 
2 2 2
2 2 2 2 2
( 1) ( 1) ( 1) ( 1)bcd a ad c adce bd a b d c ad e
p
cb ad cb ad
       

 
  
  
    
 
Example 1. In this example, we take a = 2.95, b = 1.9, c = 2.99, d = 1.15 and e = 0.999. Computations yield E6 = (0.37, 
0.43, 0.45). The eigen values are 1 0.9976   and 2,3 0.5531 0.7409i   such that 1,2,3 1  . Hence the criteria 
for stability are satisfied. The time plot and phase portrait in Figure - 7 shows a sink and the trajectory spirals towards 
the fixed point E6. 
 
  
          ISSN(Online): 2319-8753 
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International Journal of Innovative Research in Science, 
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DOI: 10.15680/IJIRSET.2015.0401023                                         www.ijirset.com                                                                 18583 
 
 
 
Figure 7. Stability of time plot and Phase Portrait for interior equilibrium point 
 
Example 2. Taking a = 2.95, b = 1.88, c = 2.99, d = 1.299 and e = 0.999, we obtain the eigen values 1 0.9876   and 
2,3 1.0531 0.7409i    such that 2,3 1  . Hence the criteria for instability are satisfied. The time plot and phase 
portrait are shown in Figure - 8. 
 
Figure 8. Instability of time plot and Phase Portrait for interior equilibrium point 
 
 
 
 
  
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DOI: 10.15680/IJIRSET.2015.0401023                                         www.ijirset.com                                                                 18584 
 
VI. BIFURCATION DIAGRAMS 
Bifurcation diagrams provide information about abrupt changes in the dynamics, see Figure-8. The parametric 
values at which these changes occur are called bifurcation points. They provide information about the dependence of 
the dynamics on a certain parameter. If the qualitative change occurs in a neighbourhood of an equilibrium point or 
periodic solution, it is called a local bifurcation. Any other qualitative change that occurs is considered as a global 
bifurcation. 
 
Figure 9. Bifurcation for Prey - Predator Populations 
 
Example 3. The parameters are assigned the values b = 1.88, c = 2.99, d = 0.99, e = 0.999 and the bifurcation 
diagram is plotted for the growth parameter in the range 3.0001 – 4.5. Both prey and predator population exhibit chaos. 
REFERENCES 
[1] Abd-Elalim A. Elsadany, H. A. EL-Metwally, E. M. Elabbasy, H. N. Agiza, Chaos and  bifurcation of a nonlinear discrete prey-predator system, 
Computational Ecology and Software, 2012,2(3):169-180. 
[2] CananCelik, OktayDuman, Allee effect in a discrete-time predator prey system, Chaos, Solitons and Fractals, 40, (2009), 1956-1962. 
[3] Leah Edelstein-Keshet, Mathematical Models in Biology, SIAM, Random House, New York, 2005. 
[4] Marius Danca, StelianaCodreanu and Botond Bako, Detailed Analysis of a Nonlinear Prey-predator Model, Journal of Biological Physics 23: 
1120, 1997. 
[5] J.D.Murray, Mathematical Biology I: An Introduction, 3-e, Springer International Edition, 2004. 
[6] Saber Elaydi, An Introduction to Difference Equations, Third Edition, Springer International Edition, First Indian Reprint, 2008. 
[7] Temple H. Fay, Johanna C. Greeff, Lion, wildebeest and zebra: A predator–prey model, ecological modelling, 196(2006)237–244. 
[8] M.ReniSagaya Raj, A.George Maria Selvam, R.Janagaraj, Dynamical Behaviour of Discrete Prey-Predator system with Scavenger, 
International Journal of Emerging Research in Management & Technology ISSN: 2278-9359 (Volume-2, Issue-11), November 2013, pp: 27 - 
33. 
[9] Robert M.May, Biological Populations with Non overlapping Generations: Stable Points, Stable Cycles and Chaos, Science, New Series, 
Vol.186, No.4164 (Nov. 15, 1974), 645 - 647. 


A Discrete Model of Three Species Prey- Predator System

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A Discrete Model of Three Species Prey- Predator System

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