TIME DELAY MODEL FOR A PREDATOR AND TWO SPECIES
WITH MUTUALISM INTERACTION
Y. Suresh Kumar
1, N. Seshagiri Rao
2and B. V. Appa Rao
31
Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur, Andhra Pradesh, India
2Department of Applied Mathematics, School of Applied Natural Science, Adama Science and Technology University, Adama, Ethiopia 3
Department of Mathematics, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur, Andhra Pradesh, India E-Mail: [email protected]
ABSTRACT
The present paper focus on to know the dynamical behavior of a three species system made up of a predator together with mutualism interaction between two species in the limited resources, whereas the predator is depending on both the mutual species. In this model the time delay is proposed to the predator and the first mutual species to recognize the sustainability of the system in long run. Local asymptotic stability of diverse existing positive equilibrium solutions is investigated to understand the dynamics of the system. Further the global stability is established using appropriate Lyapunov functional at positive interior equilibrium solution. Finally numerical simulation is execute to examine the delay impact that can lead to transformation from stable to unstable or unstable to stable culminate hopf bifurcation.
Keywords: mutualism, predator, asymptotic stability, global stability, laypunov function, hopf bifurcation.
1. INTRODUCTION
Mathematical developments in population biology have been extensively growing in all areas of ecology and it has long history generated by mathematics studying the dynamical properties of population developments. The investigations on laboratory controlled organisms and small mammals assist much attentiveness in mathematical formulation of the models easily. The theory of population biology has started and taken a formal shape after the great work from Lotka and Volterra. During last two decades the variety of ecological processes were done mathematically on dynamical modelling. We need some systematic methodologies in the case of difficult or complex dynamical models which carry out some physical conditions to understand an ecosystem appropriately. The construction of the model over the population ecosystem is very important to know the system behavior under certain assumed conditions. If not at all have satisfactory model with respect to theoretical observations and practical observations at first time re-modify the model till to get a satisfactory model, so that we can have the future information approximately. Many authors have designed mathematical models on the relationship between predator and prey based on their existence and importance in the nature. It seems to be simple problem mathematically at first sight but is very complex and changeling one when we go through involving different conditions around them. Still there are good numbers of unsolved open problems between the prey and predator relationship even though we receive considerable amount of progress in last forty years.
Apart from delay differential equations plays much interest of complex real world problems such as infectious diseases, biotic population, neuronal networks and even finance and economics. Latest developments related to delay differential equations have been growing rapidly but still there are many challenging open questions remain. To solve delay differential equations practically is
2. BASIC MATHEMATICAL MODEL
The ecological setup of multi-system involving three species is specified under:
2 1
1 1 11 1 12 1 2 13 1 1
2 2
2 2 22 2 21 1 2 23 2
2
3 33 32 2 31 2 1
( ) ( ) 1 ( ) ( ) t t dx
a x x x x x K t u y u du dt
dx
a x x x x x y dt
dy
a y y x y y K t u x u du dt
where
x x y
1,
2,
represents population densities of mutual and predator species.a a a
1,
2,
3 are the natural growth rates of three species,
ii, (
i
1, 2,3)
representsthe decreasing rates of the species because of its insufficient resources,
12,
21are mutual coefficients,13
,
23
are the rate of decrease of the mutual species owing to predator,
31,
32 are the consumption coefficients of the predator over the mutual species respectively with the following initial conditions:1
(0)
0,
2(0)
0, (0)
0
x
x
y
.Normalizing the kernals
K
1andK
2with the following conditions1 2 1 2
0 0 0 0
( ) 1, ( ) 1, ( ) , ( ) (2)
K z dz K z dz zK z dz and zK z dz
The above system (1) now with the delay kernel conditions becomes
2 1
1 1 11 1 12 1 2 13 1 1 0 2
2
2 2 22 2 21 1 2 23 2
2
3 33 32 2 31 2 1
0
( ) ( )
(3)
( ) ( ) dx
a x x x x x K z y t z dz dt
dx
a x x x x x y dt
dy
a y y x y y K z x t z dz dt
3. STEADY STATES AND THEIR EXISTENCE
The system (3) own the following eight equilibria (i). The washed out equilibrium:
1
:
10
;
20
;
0
E
x
x
y
(ii). Both the first mutual species and predator free equilibrium
2
2 1 2
22
;
:
0
a
;
0
E
x
x
y
(iii). Both the second mutual species and predator free equilibrium
1
3 1 2
11
:
a
;
0
;
0
E
x
x
y
(iv). Both mutual species washed out equilibrium: 3
4 1 2
33
:
0
;
0
;
a
E
x
x
y
(v). The predator free equilibrium
2 12 1 22 2 11 1 21
5 1 2
11 22 12 21 11 22 12 21
0
;
:
a
a
a
a
;
E
x
x
y
This case exists only when
11 22
12 21. (vi). Second mutual species washed out equilibrium1 33 3 13 1 31 3 11
6 1 2
11 33 13 31 11 33 13 31
;
;
:
a
a
0
a
a
E
x
x
y
This case exists only when
a
1
33
a
3
13. (vii). First mutual species washed out equilibrium2 33 3 23 2 32 3 22
7 1 2
22 33 23 32 22 33 23 32
:
0
;
a
a
;
a
a
E
x
x
y
This case exists only when
a
2
33
a
3
23. (viii). The coexistence state12 23 3 3 13 22 2 12 33 2 13 32 1 33 22 1 23 32 8 1
12 21 33 12 31 23 11 22 33 31 22 13 32 21 13 23 11 32
:
a
a
a
a
a
a
;
E
x
13 21 3 2 13 31 3 11 23 1 31 23 1 33 21 2 11 33 2
12 21 33 12 31 23 11 22 33 31 22 13 32 21 13 23 11 32
;
a
a
a
a
a
a
x
12 21 3 3 11 22 2 12 31 1 31 22 1 32 21 2 11 32 12 21 33 12 31 23 11 22 33 31 22 13 32 21 13 23 11 32
a a a a a a
y
This case exists only when
2 12 33 1 33 22 1 23 32 12 23 3 3 13 22 2 13 32
a a a a a a
2 13 31 1 33 21 2 11 33 3 11 23 1 31 23 13 21 3
a a a a a a
3 11 22 2 12 31 1 31 22 1 32 21 2 11 32 12 21 3
a a a a a a
12 31 23 11 22 33 31 22 13 32 21 13 23 11 32 12 21 33
3.1. Dynamical behavior: Stability analysis
The stability nature of the system can observe depending on the Eigen values of the Jacobian matrix at
each steady state. The associated Jacobian matrix to the system (3) is as follows:
*
1 11 1 12 2 13 12 1 13 1 1
21 2 2 22 2 21 1 23 23 2
*
31 2 32 3 33 31 1 32 2
2
( )
2
(4)
( )
2
a
x
x
y
x
x k
J
x
a
x
x
y
x
E
yk
y
a
y
x
x
by taking
u
1
A e
1 t,
u
2
A e
2 t,
u
3
A e
3 t, the system of equations (3) becomes* 1
1 11 1 12 2 13 1 12 1 2 13 1 1 3
2
21 2 1 2 22 2 21 1 23 2 23 2 3
* 3
31 2 1 32 2 3 33 31 1 32 2 3
( 2 ) ( )
( 2 )
( ) ( 2 )
du
a x x y u x u x k u dt
du
x u a x x y u x u dt
du
yk u yu a y x x u
dt
(5)
where 1* 1 0
( )
( ) exp(
)
k
k z
z dz
and*
2 2
0
( )
( ) exp(
)
k
k z
z dz
are the LaplaceTransforms of
k z
1( )
andk z
2( )
. The local stability nature of each equilibrium point can be discussed in the following prepositions.Proposition 1. The equilibrium point
E
1 is always unstable.The Jacobian matrix of the equilibrium point
E
1 is 1 1 2 30
0
0
0
0
0
Ea
J
a
a
Hence
E
1is unstable because of all three eigen values are positive.Proposition 2. The equilibrium point
E
2 is a saddle point.The Jacobian matrix of the equilibrium point
E
2is 2 2 12 1 22 2 23 2 21 2 22 22 2 32 3 22
0
0
0
0
Ea
a
a
a
J
a
a
a
The characteristic values of the system are 2 32
2 12
1 2 3
22 22
,
,
a
.
a
a
a a
Hence the steady state is a saddle point. In this case if it holds the conditions2 12 1 22
a
a
and 3 2 32 22a
a
thenE
2is stable otherwise unstable.Proposition 3. The equilibrium point
E
3 is always unstable.The Jacobian matrix of the equilibrium point
E
3is
3
* 1 13 1 1 12 1 11 11 1 21 2 11 1 31 3 11
( )
0
0
0
0
Ea
K
a
a
a
J
a
a
a
The characteristic values of the above matrix are 1 31
1 21
1 2 3
11 11
,
a
,
a
a a
a
. So one can observe thatthe present steady state is always is unstable.
Proposition 4. The equilibrium point
E
4 is a saddle point.The relevant Jacobian matrix of this stateis
3 13 1 33 3 23 4 2 33 *
3 31 2 3 32
3 33 33
0
0
0
0
( )
a
a
a
J
a
a
K
a
a
The Eigen values of this matrix are 3 13 3 23
1 2 3
33 33
,
,
.
a
a
a
a
a
conditions 1 3 13 33
a
a
and 2 3 23 33a
a
thenE
4is stable.Proposition 5. The equilibrium point
E
5 is always unstable.The Jacobian matrix of the equilibrium point
E
5is
*
1 11 1 12 2 12 1 13 1 1
5 21 2 2 22 2 21 1 23 2
3 32 2 31 1
2 ( )
2
0 0
a x x x x K
J x a x x x
a x x
The characteristic equation of the system is
2
1 2 12 2 21 1 11 1 22 2 3 32 2 31 1 1 2 1 21 1 2 11 1 1 22 2 2 12 2
2 2
11 22 1 2 11 1 12 22 2
2
2
[
(
)]
2
2
0
4
2
2
a
a
x
x
x
x
a
x
x
a a
a
x
a
x
a
x
a
x
x x
x
x
It is also clear that this steady state is always unstable.
Proposition 6. The equilibrium point
E
6 is stable.The corresponding Jocabian matrix for this state is
6
*
1 11 1 13 12 1 13 1 1
2 21 1 23
*
31 2 32 3 33 31 1
2 ( )
0 0
( ) 2
E
a x y x x K
J a x y
yK y a y x
The secular equation of the system at this state is
2
1 3 31 1 13 11 1 33
2 2 21 1 23 1 3 1 31 1 1 33 3 11 11 33 1 11 31 1
2 * *
3 13 13 33 13 31 1 13 31 1 1 2
2
2
[
(
)]
2
2
4
2
0
2
( )
( )
a
a
x
y
x
y
a
x
y
a a
a
x
a
y
a
x
x y
x
a
y
y
x y
x yK
K
The steady state is stable if satisfies the condition
2 21 1 23
0
a
x
y
, 1 3 31 1 21 1 11 1 33 132
2
a
a
x
x
x
y
y
,
2 2
1 3 1 31 1 1 33 3 11 11 33 1 11 31 1 3 13 13 33
* *
13 31 1 13 31 1 1 2
2
2
4
2
2
( )
( )
0
a a
a
x
a
y
a
x
x y
x
a
y
y
x y
x yK
K
Proposition 7. The equilibrium point
E
7 is stable.The related Jocabian matrix for this state is
7
1 12 2 13
21 2 2 22 2 33 23 2
*
31 2 32 3 33 32 2
0 0
2
( ) 2
E
a x y
J x a x y x
yK y a y x
The characteristic equation of the system is
2
2 3 32 2 22 2 33
2 1 12 2 13 2 3 2 32 2 2 33 3 22 2 22 33 2 22 32 2
2 2
3 33 33 33 33 32 32 23 2
2
3
[
(
)]
2
2
4
2
0
2
a
a
x
x
y
a
x
y
a a
a
x
a
y
a
x
x y
x
a
y
y
y
x y
It is also stable if holds the condition 1 12 2 13
0
a
x
y
2 3 32 2
2
22 23
33a
a
x
x
y
and
2 2 3 2 32 2 2 33 3 22 2 22 33 2 22 32 2
2 2
3 33 33 33 33 32 32 23 2
2 2 4 2
2 0
a a a x a y a x x y x
a y y y x y
Proposition 8. The coexistence equilibrium point 8
E
is locally asymptotically stable. The Jacobian matrixfor this state is
*
1 11 1 12 2 13 12 1 13 1 1
8 21 2 2 22 2 21 1 23 23 2
*
31 2 32 3 33 32 2 31 1
2
( )
2
( )
2
a
x
x
y
x
x K
J
x
a
x
x
y
x
yK
y
a
y
x
x
The corresponding characteristic equation of the above matrix is
3 2
1 2 3
0
b
b
b
, where1
(
1 2 3)
b
b
2
1 2
2 3
3 1
12 21 1x x
2
23 32x y
2
13 31 1x yK
1*( )
K
2*( )
* *
3 3 12 21 1 2 12 23 31 1 2 2 13 21 32 1 2 1 1 23 32 2
* *
2 13 31 1 1 2 1 2 3
( )
( )
( )
( )
b
x x
x x yK
x x yK
x y
x yK
K
1
a
12
11 1x
12 2x
13y
2
a
22
22 2x
21 1x
23y
3
a
32
33y
32 2x
31 1x
Now2 2 2 * *
1 2 3 1 2 3 2 1 3 3 1 2 1 2 3 1 3 13 31 1 1 2
* *
2 3 23 32 2 1 2 12 21 1 2 12 23 31 2 13 21 32 1 1 2
(
)
(
)
(
) 2
(
)
( )
( )
(
)
(
)
(
( )
( ))
0
b b
b
x yK
K
x y
x x
K
K
x x y
By Routh-Hurwitz criterion, the steady state is always stable.
4. GLOBAL STABILITY
In this sub section, we shall establish the global asymptotic stability of the co-existing state
E
8by a suitable Lyapunov's function.Let us define a Lipunov function for the system (3) as follows:
1 2
1 2 1 1 1 2 2 2
1 2
2 2
13 1 31 2 1 1
0 0
( ,
, )
log
log
log
1
1
( )
(
)
( )
(
)
6
2
2
t t
t z t z
x
x
y
V x x y
x
x
x
x
x
x
y
y
y
x
x
y
k z
y
y
dudz
k z
x
x
dudz
The above function
V x x y
( ,
1 2, )
is zero at the equilibrium points and is positive for all other positivevalues of
x x y
1,
2,
. Now the derivative of (6) with respect to time along the solution of (3) is
2
1 1 2 2
1 2 13 1
1 2 0
2 2
13 1 31 2 1 1
0 0
2
31 2 1 1
0
1
( )
( )
2
1
1
( )
(
)
( )
( )
(7)
2
2
1
( )
(
)
2
x
x
x
x
dV
y
y
x
x
y
k z
y t
y dz
dt
x
x
y
k z
y t
z
y dz
k z
x t
x
dz
k z
x t
z
x
dz
1 1 1 11 1 12 2 13 1 2 2 2 22 2 21 1 23 0
2
3 33 32 2 31 2 1 13 1
0 0
2 2
13 1 31 2 1 1
0 0
( ) ( ) ( ) ( )
1
( ) ( ) ( ) ( ) ( ) (8)
2
1 1
( ) ( ) ( ) ( )
2 2
dV
x x a x x K z y t z dz x x a x x y
dt
y y a y x K z x t z dz k z y t y dz
k z y t z y dz k z x t x d
231 2 1 1
0
1
( ) ( ) 2
z
k z x t z x dz
By proper choice of the parameter
a a a
1,
2,
3as follows the equation (8) becomes1 11 1 12 2 13 1 0
( ) (
)
a
x
x
K z y t
z dz
2 22 2 21 1 23
a
x
x
y
3 33 32 2 31 2 1
0
( ) (
)
a
y
x
K z x t
z dz
2 2
11 1 1 12 1 1 2 2 22 2 2 21 1 1 2 2 2
2 2
33 32 2 2 13 13 1
0
2 2
31 1 1 31 2 1 1
0
(
)
(
)(
)
(
)
(
)(
)
1
1
(
)
(
)(
)
(
)
( )
(
)
(9)
2
2
1
1
( )
(
)
2
2
dV
x
x
x
x
x
x
x
x
x
x
x
x
dt
y
y
x
x
y
y
y
y
k z
y t
z
y dz
x
x
k z
x t
z
x
dz
By the following inequalities 2 2
2
1 1
0 0
,
( )[ (
)
]
( )
1,
2
a
b
ab
k z y t
z
y dz
k z dz
The equation (9) becomes
2 2 2 2 2
11 1 1 12 2 2 12 1 1 22 2 2 21 1 1
2 2 2 2 2
21 2 2 33 32 32 2 2 13
2 2 2
13 31 1 1 31 1 1
1
1
1
(
)
(
)
(
)
(
)
(
)
2
2
2
1
1
1
1
(
)
(
)
(
)
(
)
(
)
(10)
2
2
2
2
1
1
1
(
)
(
)
2
2
2
dV
x
x
x
x
x
x
x
x
x
x
dt
x
x
y
y
y
y
x
x
y
y
y t
z
y
x
x
x t
z
x
2 211 12 21 31 1 1 22 12 21 32 2 2
2
33 32 13 13 31
1 1 1 1 1 1
( ) ( )
2 2 2 2 2 2
1 1 1
( ) (11)
2 2 2
dV
x x x x
dt y y
2 2 2
1 1 2 2
(
)
(
)
(
)
dV
l x
x
l x
x
l y
y
dt
(12)where
11 22 33 12 21 23 32 13 31
1 1 1 1 1 1
min
2 2 2 2 2 2
l
Thus,
dV
0
dt
strictly for all positive values ofx x y
1,
2,
.Hence
V x x y
( ,
1 2, )
satisfies Lyapunov's asymptoticstability theorem and hence the interior equilibrium point 8
E
of system (3) is globally asymptotically stable.5. NUMERICAL SIMULATION
2 1
1 1 11 1 12 1 2 13 1 1 2
2
2 2 22 2 21 1 2 23 2
2
3 33 32 2 31 2
1
3 13 1
2
31 2
(13) dx
a x x x x x w dt
dx
a x x x x x y dt
dy
a y y x y yw dt
dw
x w
dt dw
y w dt
By defining the kernels as
1 1
(
) ( )
,
2 2(
) ( )
1, ( )
1, ( )
2,
0,
0
t t
u u
w
k t
u y u du w
k t
u x u du k u
e
k u
e
5.1. Simulation in the absence of a time delay
In this sub section, we verified the existence of the results which were already proved analytically in [25], where the author was considered a system without time delay in (3) (as the system (14) below) by assigning suitable values to parameters in the model using numerical solutions with the same package.
2 1
1 1 11 1 12 1 2 13 1 2
2
2 2 22 2 21 1 2 23 2 2
3 33 32 2 31 1
dx
a x
x
x x
x y
dt
dx
a x
x
x x
x y
dt
dy
a y
y
x y
yx
dt
(14)
(a) (b)
Figure-1. 1 11 12 13 2 21 22 23
3 31 32 33 1 2
12.52; 12.48; 22.96; 16.96; 18.56; 12.48; 9.84; 21.44; 3.24; 2.64; 2.98; 14.8; (0) 10; (0) 10; (0) 5;
E(7.861,5.786,2.786)
a a
a x x y
(a) (b)
Figure-2. Let 1 11 12 13 2 21 22 23
3 31 32 33 1 2
12.52; 12.48; 22.96; 16.96; 18.56; 12.48; 9.52; 21.44; 2.88; 2.64; 2.98; 14.8; (0) 10; (0) 10; (0) 5;
E(9.009,6.677,3.147)
a a
a x x y
Note: One can easily identify that the system of equations (14) which is free from the time delay terms is always stable at the coexisting equilibrium points E(7.861, 5.786, 2.786), E(9.009,6.677, 3.147) by observing the trajectories of the species with respect to time t in Figure-1(a) and Figure-2(a). Initially the species having rich growth rates and then tending to the above asymptotic values later and also in the phase portrait diagrams 1(b), Figure-2(b) shows the stability of the system in terms of taking any trajectory with initial value nearby coexisting equilibrium points E(7.861, 5.786,2.786), E(9.009, 6.677, 3.147) is converging spirally in.
5.2. Simulation in the presence of a time delay
In this sub section, the dynamical behavior of the model (3), which is involving the delay terms is identified numerically by fixing all parameter values are same as in Example: 2 and taking different kernel values. The following Figures with (a) shows the species nature with respect to the time t and Figures of (b) gives the information about stability to instability or instability to stability at partially washed out state and coexistence states of the system taking the trajectories with initial values in the neighborhoods of the above said equilibrium points.
(a) (b)
Figure-3.
0.01;
0.01; E(0,1.0398,0.4039)
(a)
(b)
Figure-4.
0.2;
0.2; E(0.0235,1.0285,0.4226)
.
(a)
(b)
Figure-5.
0.6;
0.6; E(1.8833,1.8705,1.1310)
.(a)
(b)
Figure-6.
0.7;
0.7; E(2.8047,2.4624,1.4051)
(a) (b)
Figure-7.
0.8;
0.8; E(4.0874,3.3158,1.7738)
.(a)
(b)
Figure-8.
0.85;
0.85; E(4.9338,3.8862,2.0125)
.(a)
(b)
(a) (b)
Figure-10.
0.95;
0.95; E(7.2953,5.4975,2.6713)
.(a) (b)
Figure-11.
1.01;
1.01; E(9.1868,6.7889,3.2031)
.(a) (b)
(a) (b)
Figure-13.
1.55;
0.97; E(24.0085,16.2801,7.3244)
.
(a) (b)
Figure-14.
1.78;
0.96; E(10.9427,6.9798,4.3108)
.
6. CONCLUSIONS
In this paper, we have taken and discussed a dynamical system consisting of a predator and two species with mutual interaction among them. This work can be looked upon as an extension of the work in [25], the main modification here is that imposing delay terms to both the first mutual species and the predator. The objectives were the analysis of the dynamical properties of different equilibrium points of the system. Later by well known theorem (stability theory of ordinary differential equations), we have shown that the system is stable globally at the interior equilibrium of the reduced model under certain conditions. On the other hand, it is important to note that the presence of delay terms
and
can impact the existence and the behavior of the system around the positive equilibrium. Using numerical simulations the following important remarks we observed.a) The values of
and
various from 0.01 to 0.2, there is no considerable growth rates of all three species. Since the delay term doesn't effect on the second mutual species and so is sustainable for long time. But because the consumption of the predator over the first mutual species it is extinct from the beginning. From thephase portrait diagrams Figure-3(b) and Figure-4(b), we can observe that the system is unstable.
b) When the delay terms
and
increasing from0.6 to1.01, all three species gradually increasing and then reaching their asymptotic values later. Further, the kernal values in the above range, the impact of the predator over the mutual species doesn't trouble in long time. Hence the dynamical system is stable as the trajectories coming to the equilibrium points spirally can see from Figure-5(b) to Figure-11(b).c) As soon as the kernel terms crossing the point 1.01, all three species having oscillatory behavior and no considerable growth in the predator species but the other two mutual species sustaining long time with rich growth rates.
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