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Mathematical Analysis of a Prey Predator and Ammensal Model with
Time Delay
Paparao. A. V
Department of Mathematics, JNTU-GV College of Engineering, Vizianagaram(A)Viziangarm-5325003
A. P, India
ABSTRACT:
We study a three-species ecological model in which the first species (x) preys on the second species(y) and the
second species exerts an amensal effect on the third species (z). Interactions between predator and prey include
an explicit time delay(τ) and the system is formulated as a set of delay differential equations. We identify the
positive coexistence equilibrium and perform a local stability analysis about this steady state. we derive sufficient
conditions for a Hopf bifurcation driven by the delay parameter (τ). By treating (τ) as the bifurcation parameter,
we determine critical delay values at which the coexistence equilibrium loses stability and periodic oscillations
emerge. Numerical simulations implemented in MATLAB confirm the analytical predictions and illustrate the
instability regimes and bifurcating limit cycles.
Keywords: Prey, Predator Co-existing state, local stability, Hopf bifurcation
AMS classification: 34 DXX
1. INTRODUCTION
Differential equations are most popular in explaining the mathematical models I ecology. The stability analysis
concept is explained in detail by Braun [9] and Simon’s [10]. The ecological models are initiated by studied
Lokta [1] and Volterra [2]. The Mathematical models and its stability analysis discussed by Kapur [3, 4].
qualitative analysis plays a big role in analysing these models due to the difficulty in finding analytical solutions
due to the non-linearity of the models arise ecology. The qualitative analyses of ecological models are widely
studied by authors [5-7]. The stability of analysis of delay-differential equations are significant in ecology. The
time delays are influence the dynamics of the system and tend to destabilize or stabilizes the system. The systems
with delay arguments and the qualitative analysis are widely discussed by the authors [11-13]. The nature of the
delay argument cause unbounded growth and extinction of populations leads to instability tendency of models.
The delay argument may classify in to continuous, discrete, distributed etc. The time lags can be discrete or
continuous. These lags will change the stable equilibrium to unstable or vice-versa. The delay models in
population dynamics are widely studied by paparao [14-21]. In this paper we take a logistic growth model of
three species for investigation. In this model first species is preying on second and second is ammensal to third
species. A discrete time lag is incorporated in the interaction of first and second species. The model is studied
by a couple of delay-differential equations. The co-existing equilibrium point is identified and discussed the
dynamics at this point. Numerical simulation is carried out carried out in support of stability analysis. It is shown
that the system exhibits instability trendies leads to Hopf bifurcation.
1. Formation of Mathematical Model:
The proposed ecological system can be modelled into the following system of equations given by
(2.1)
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2.1 Nomenclature:
Parameter
Description
x,y,z
predator, prey and amensal population density
Natural growth rates of predator, prey and amensal
Interaction rate of first and second species (positive value)
Interaction rate of second and first species (negative value)
Interaction rate of third (ammensal) and second species (negative value)
k
1
, k
2
, k
3
Carrying capacities of first, second and third species populations respectively.
3.Equilibrium Points
Equating the system of equations (2.1) to zero, and derive the co-existing state is given by
(3.1)
Co-existing state exist if (i)
(ii)
(3.2)
are Satisfied
4. Local Stability Analysis at Co-existing State
Theorem4.1: The co-existing state is locally asymptotically stable
Proof: The variational matrix for the system (2.1) is
(4.1.1)
Characteristic equation of the (4.1.1) is given by
3 2 2
1 2 3 1 2 3
( , ) ( ) 0p p p e q q q
−
= + + + + + + =
(4.1.2)
Where
3
12
1 1 2 3
1 2 3
2
22
()
az
a x a y
P a a a
k k k
= + + − + +
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2
23
2 3 1 32 2 32
11
2 1 2 1 3 2 3
1 2 1 3 2 2 1 2
3
23
2 3 3
1
2
2 1 1 1
32 1 32 2
2 1 3 3 1 2
4 2 2
44
22
2
2 2 2
a a yz a a x y a a y
a a x y a a xz
P a a a a a a
k k k k k k k k
a a z a a y
a a y
a a x a a z a a x
a a y a a y
k k k k k k
= + + + + + + +
− − − − − − − −
2
1 2 3 1 2 3 1 2 3 1 2 32 1 2 3
3 1 2 32
2 1 3 1 2 1 2 3
2
1 2 3 1 2 3 1 3 32 1 2 32 1 2 3
1 2 3
2 3 2 1 1 2
2 2 2 4 8
4 4 2 2 4
a a a y a a a x a a a z a a a x y a a a x yz
P a a a y
k k k k k k k k
a a a yz a a a zx a a a y a a a x y a a a xy
a a a
k k k k k k
= + + + + +
− − − − − −
1 21 12
q a x a y=−
2
2
3 21
2 12
2 1 12 2 12 12 3 21 32 12 32 3 21 1 21
32
2
2
a a xz
a a y
q a a x a a y a a y a a x y a a y a a x a a x
kk
= + + + + − − − −
2
2
1 21 32 12 3 2 1 21 3 2 12 3
3 1 21 3 12 32 2
1 2 1 3 3
23
1 3 21 2 3 12 2 32 12 12 3 1 21 3
21 32 12 2 3
3 2 2 3 1
2 2 4 2
2
2 4 2 2 2
a a a x y a a a y a a a xz a a a y z
q a a a x a a a y
k k k k k
a a a xz a a a y a a a y a a y z a a a x
a a x y a a a y
k k k k k
= + + + + +
− − − − − − −
Which can be written as
Case (i) For
The characteristic equation obtained from (4.1.2) by putting given by the following equation
2
3
1 2 1 2
12 21
3 1 2 1 2
( ,0) 0
az
a x a y a a x y
a a x y
k k k k k
= − + + + + + =
2
3
1 2 1 2
12 21
3 1 2 1 2
00
az
a x a y a a x y
or a a x y
k k k k k
+ = + + + + =
(4.1.3)
One of the roots is negative i.e.,
From the equation (4.1.3) find the remaining two roots. if the two roots have negative real roots if the trace of
the equation
b
a
−
is negative and the determinant
c
a
is positive.
The trace and determinant from the equation (4.1.3) are given as follows
Here the trace is =
( )
1 2 2 1
12
0
a xk a yk
b
a k k
−+
−
=
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Determinant=
( )
1 2 12 21 1 2
12
0
a a a a k k x y
c
a k k
+
=
Therefore, the system (2.1) is locally asymptotically stable at co-existing state.
Therefore, the co-existing state is locally asymptotically stable.
Case (ii) Let
0
: Suppose there is a positive
0
such that the equation (4.1.2) has pair of purely imaginary
root, let the roots be
, 0,i
therefore
i
satisfies the equation (4.1.2)
3 2 2
1 2 3 1 2 3
( ) ( ) ( ) ( ( ) ( ) ) 0
i
i p i p i p e q i q i q
−
+ + + + + + =
22
1 3 1 3 2
32
2 2 1 3
cos cos sin
[ cos sin sin ] 0
p p q q q
i w wp q q q
− + − + + +
− + + + − =
Separating real and imaginary parts, we get
22
3 1 2 1 3
( ) cos sinq q q p p
− + = −
(4.1.4)
23
2 3 1 2
cos ( )sinq q q p
− − = −
(4.1.5)
On adding, the two equations after squaring, we get
From the above equations we get the following equation (by squaring and add the two results)
2 2 2 2 3 2
3 1 2 1 3 2
( ) ( ) ( ) ( )q q q p p p
− + = − + −
6 4 2 2 2 2 2 2 2
1 2 1 2 1 3 2 1 3 3 3
( 2 ) ( 2 2 ) 0p p q p p p q q q q p
+ − − + − − + + + =
Let
32
1 2 3
( ) 0p p p N pN N
= + + + =
(4.1.6)
Where
22
1 1 1 1
22
2 2 1 2 2 1 3
33
3 3 3
2
2
22
N p p q
N p p p q q q
N q p
p
= − −
= − − +
=+
=
()po
=
If we assume that
1 2 3
0, 0, 0N N N
then equation (4.1.2) has no positive real roots. Therefore, the equation
(4.2) admits negative real roots. Hence, we can derive the conditions for existences of stability at equilibrium
point.
Theorem 4.2 The system (2.1) is locally asymptotically stable at co-existing state for all
, if the following
conditions hold.
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1 1 2 2 3 3
1 2 3
( ). ( ) 0, ( ) 0 , ( ) 0
( ). 0, 0, 0
i p q p q p q
ii N N N
+ + +
Proof: Any one of
1 2 3
,,N N N
is negative. Then equation (4.1.2) has a positive Root
0
Eliminating
cos
, from the equations (4.1.4) & (4.1.5), we have
2
1 3 2
32
2 3 1
2
3 1 2
2
2 3 1
()
cos
()
p p q
p q q
q q q
q q q
−
− − −
=
−
−−
(
by using Cramer’s rule in determinants)
2 4 2 4 2
1 3 3 3 1 1 1 3 3 2 3
2 2 4 2 2 2
3 1 1 3 2
cos
2
p q p q p q q p q w p q
q q q q w q
− − + + −
=
+ − +
42
1
0 3 1 1 0 1 3 1 3 2 3 3 3
2 4 2 2 2
0 1 0 0 2 1 3 3 0
( ) ( )
12
cos
( 2 )
k
q p q p q q p p q p q
k
q q q q q
−
− + + − −
=+
+ − +
0,1,2,3...............where k =
5. Hopf Bifurcation
Theorem 5.1: The sufficient condition for the system (4.1.1) admits bifurcation along the co-existing state E if
0
and locally asymptotically stable If
0
0
Proof: Hopf bifurcation occurs when the real part of
()t
become positive when
0
and the steady state
become unstable moreover, when
passes through the critical value
0
.
To check this result, differentiating the equation (4.1.2) With respect to
,
we get
Put in the above we get
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Real part of
=
=
Re( ) 0
d
d
By using this condition N
1
>0, N
2
> 0, N
3
>0 we have
Therefore, the Hopf bifurcation occurs at
6. Numerical Simulation
We study the Hopf bifurcations for the system (2.1) with the tolerance parameter (τ). For the system of equations,
the parameters are identified as shown in the example 1. For different values of τ the graphs are shown below.
Example: 6.1 let us choose the following parameters for examination
Fig. 6.1 (A) Fig. 6.1 (B)
The unbounded periodic solutions for the system (2.1) when τ = 0.069
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Fig. 6.1 (C) Fig. 6.1 (D)
The bounded solutions for the system (2.1) when τ = 0.068
Example: 6.2 let us choose the following parameters for examination
Fig. 6.2 (A) Fig. 6.2 (B)
The unbounded periodic solutions for the system (2.1) when τ = 0.065
Fig. 6.2 (C) Fig. 6.2 (D)
The bounded solutions for the system (2.1) when τ = 0.064
Example: 6.3 Let us choose the following parameters for examination
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x=5,y=3,z=6.
Fig. 6.3 (A) Fig. 6.3 (B)
The unbounded periodic solutions for the system (2.1) when τ = 0.072
Fig. 6.3 (C) Fig. 6.3 (D)
The bounded solutions for the system (2.1) when τ = 0.071
Example: 6.4 Let us choose the following parameters for examination
Fig. 6.4 (A) Fig. 6.4 (B)
The unbounded periodic solutions for the system (2.1) when τ = 0.65
Fig. 6.4 (C) Fig. 6.4 (D)
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The bounded solutions for the system (2.1) when τ = 0.6
CONCLUSION
A logistic growth model involving three interacting species is considered for investigation. The species are
denoted by x, y, and z. A time-delay parameter (τ) is incorporated into the interaction between the first species
(x) and the second species (y). The coexistence equilibrium of the system is shown to be locally asymptotically
stable in the absence of delay. Numerical simulations are performed for different values of (τ) the corresponding
dynamical behaviours are illustrated through suitable examples. In this study, we examine the Hopf bifurcation
that arises when the delay parameter(τ) is used as a bifurcation parameter. Four sets of numerical examples are
considered to analyze the bifurcation characteristics. For each example, the system dynamics are observed for
varying values of (τ). Based on these observations, it is found that Hopf bifurcation occurs in three out of the
four examples, as summarized in the following table.
S. No
Example
Critical value
1
Example 6.1
τ > 0.068
2
Example 6.2
τ > 0.064
3
Example 6.3
τ > 0.071
4
Example 6.4
τ > 0.6
Table (7.1)
Hence the delay parameter τ stabilizes the system.
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