Available online at http://scik.org
Commun. Math. Biol. Neurosci. 2017, 2017:21
ISSN: 2052-2541
DIRECTION AND STABILITY OF HOPF BIFURCATION
D. PANDIARAJA1, N. ARUN NAGENDRAN2, D. MURUGESWARI3, VISHNU NARAYAN MISHRA4,5,∗
1National Centre of Excellence, Statistical and Mathematical Modelling on Bio-Resources Management,
Thiagarajar College, Madurai, India
2Department of Mathematics, Thiagarajar College, Madurai, India 3Department of Zoology, Thiagarajar College, Madurai, India
4L. 1627 Awadh Puri Colony Beniganj, Phase - III, Opposite-Industrial Training Institute (I.T.I.), Faizabad-224
001, Uttar Pradesh, India
5Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak-484887, Madhya
Pradesh, India
Communicated by L. Chen
Copyright c2017 Pandiaraja, Nagendran, Murugeswari and Mishra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract.The dynamics of the spatial competition mathematical model for the invasion, removal of Kappaphycus
Algae (KA) in Gulf of Mannar (GoM) with propagation delays is investigated by applying the normal form theory
and the center manifold theorem.
Keywords:delay differential equation; Hopf bifurcation.
2010 AMS Subject Classification:34K13, 34K18, 92D40.
1. Introduction
∗Corresponding author.
E-mail address: vishnunarayanmishra@gmail.com
Received May 8, 2017
In recent years we have witnessed an increasing interest in dynamical systems with time
delays, especially in applied mathematics. Stability and direction of the Hopf bifurcation for
the predator-prey system have been discussed by using normal form theory and center manifold
theory [5,6,10,13,15,16,17]. Direction and stability of the equilibrium for a neural network
model with two delays have been investigated [7,12]. Bifurcation analysis of the predtor-prey
model has been detailed [8]. Direction and stability of the equilibrium involving various fields
have been discussed [4,9,11,14]. We reported the shifting of algal dominated reef ecosystem due
to the invasion of KA in Gulf of Mannar [1]. Subsequently, the dominance of KA over NA and
corals in competing for space has also been reported. KA sexual reproduction by spores in the
Gulf of Mannar Marine Biosphere Reserve (GoM) in future, when environmental conditions
unanimously favor this alga has been deliberated [2]. To simulate the three way competition
among corals, KA and NA, we proposed the following system of non-linear ODE’s [3].
dx
dt
=
rx
−
rx
2
−
rxy
−
rxz
−
a
1
xy
−
a
2xz
+
dy
dy
dt
=
a
1yx
+
a
3yz
+
ν
y
(
t
−
τ
1)
−
ν
yx
−
ν
y
2−
ν
yz
−
dy
dz
dt
=
a
2zx
+
hz
(
t
−
τ
2)
−
hxz
−
hzy
−
hz
2−
a
3
zy
(
1
.
1
)
2. Direction and Stability of Hopf bifurcation
We assume that the system undergoes a Hobf bifurcation at the positive equilibrium
E
(
0
,
y
∗,
0
)
for
τ
1=
τ
1∗and then
±
i
ω
denotes the corresponding purely imaginary roots of the characteristic
equation at the positive equilibrium
E
(
0
,
y
∗,
0
)
.
Without loss of generality, we assume that
τ
2∗<
τ
1∗where
τ
2∗∈
(
0
,
τ
2∗0)
and
τ
1=
τ
1∗+
µ
. Let
x
11=
x
−
x
∗,
x
21=
y
1−
y
∗1,
x
31=
y
2−
y
∗2, ¯
x
i1=
µ
i(
τ
t
)
, i=1,2,3... Here
µ
=
0 is the bifurcation
parameter and dropping the bars, the system becomes a functional differential equation in
C
=
C
([
−
1
,
0
]
,
R
3)
as
dX
where
x
(
t
) = (
x
11,
x
21,
x
31)
∈
R
3and
L
µ:
C
→
R
3,
f
:
R
×
C
→
R
3are respectively given by
L
µ(
φ
) = (
τ
1∗+
µ
)
B
φ
1(
0
)
φ
2(
0
)
φ
3(
0
)
+ (
τ
1∗+
µ
)
C
φ
1(
−τ2∗
τ1
)
φ
2(
−τ2∗
τ1
)
φ
3(
−τ2∗
τ1
)
+ (
τ
1∗+
µ
)
D
φ
1(
−
1
)
φ
2(
−
1
)
φ
3(
−
1
)
(
2
.
2
)
and
f
(
µ
,
φ
) = (
τ
1∗+
µ
)
Q
(
2
.
3
)
where Q=
(r−rφ2(0)−a1φ2(0))φ1(0) +dφ2(0)
(a1φ2(0)−ν φ2(0))φ1(0)−2ν φ22(0)−dφ2(0) + (a3φ2(0)−ν φ2(0))φ3(0) +νe−λ τ1φ2(−1) (−hφ2(0)−a3φ2(0))φ3(0) +he−λ τ2φ3(−1)
respectively where
φ
(
θ
) = (
φ
1(
θ
)
,
φ
2(
θ
)
,
φ
3(
θ
))
T∈
C
,
B
=
r
−
ry
∗−
a
1y
∗d
0
a
1y
∗−
ν
y
∗−
2
ν
y
∗−
d
a
3y
∗−
ν
y
∗0
0
−
hy
∗−
a
3y
∗
,
C
=
0 0
0
0 0
0
0 0
he
−λ τ2
,
D
=
0
0
0
0
ν
e
−λ τ10
0
0
0
.
By the Riesz representation theorem, we claim about the existence of a function
η
(
θ
,
µ
)
of
bounded variation for
θ
∈
[
−
1
,
0
)
such that
L
µ(
φ
) =
Z
0−1
d
η
(
θ
,
µ
)
φ
(
θ
)
f or
φ
∈
C
(
2
.
4
)
Now let us choose ,
η
(
θ
,
µ
) =
(
τ
1∗+
µ
)(
B
+
C
+
D
)
,
θ
=
0
(
τ
1∗+
µ
)(
C
+
D
)
,
θ
∈
[
−τ ∗ 2 τ1,
0
)
(
τ
1∗+
µ
)(
D
)
,
θ
∈
(
−
1
,
−τ ∗ 2 τ1)
For
φ
∈
C
([
−
1
,
0
]
,
R
3)
, we define
A
(
µ
)
φ
=
dφ(θ)
dθ
,
θ
∈
[
−
1
,
0
)
R
0−1
d
η
(
s
,
µ
)
φ
(
s
)
,
θ
=
0
and
R
(
µ
)
φ
=
0
,
θ
∈
[
−
1
,
0
)
f
(
µ
,
φ
)
,
θ
=
0
Then the system is equivalent to
dX
dt
=
A
(
µ
)
X
t+
R
(
µ
)
X
t,
(
2
.
5
)
where
X
t(
θ
) =
X
(
t
+
θ
)
for
θ
∈
[
−
1
,
0
]
.
Now for
ψ
∈
C
([
−
1
,
0
]
,
(
R
3)
∗)
, we define
A
∗ψ
(
s
) =
−dψ(s)
ds
,
s
∈
(
0
,
1
]
R
0−1
d
η
T(
t
,
0
)
ψ
(
−
t
)
,
s
=
0
Further we define a bilinear inner product
<
ψ
(
s
)
,
φ
(
0
)
>
=
ψ
¯
(
0
)
φ
(
0
)
−
Z
0−1
Z
θζ=0
¯
ψ
(
ζ
−
θ
)
d
η
(
θ
)
φ
(
ζ
)
d
ζ
.
(
2
.
6
)
where
η
(
θ
) =
η
(
θ
,
0
)
.
Clearly here
A
and
A
∗are adjoint operators and
±
i
ω
∗τ
0∗are eigen
values of
A
(
0
)
and so they are also eigen values
A
∗. Let
q
(
θ
) = (
1
α β
)
Te
iω∗τ0∗θbe the eigen
vector of
A
(
0
)
corresponding to
i
ω
∗τ
0∗where
α
=
−[r−ry∗−a 1y∗−iw]
d
,
β
=
(r−ry∗−a1y∗−iw)(νe−iw0−2νy∗−d−iw)−d(a1y∗−νy∗)
d(a3y∗−νy∗)
Similarly if
q
∗(
s
) =
M
(
1
α
∗β
∗)
e
iω ∗τ0∗s
be the eigen vector of
A
∗where
α
∗=
−(r−ry ∗−a1y∗−iw)
a1y∗−νy∗
,
β
∗=
(a3y∗−
νy∗)(r−ry∗−a1y∗−iw)
Then we have to determine M from
<
q
∗(
s
)
,
q
(
θ
)
>
=
1.
Thus we can take
¯
M
=
1
1
+
α
α
¯
∗+
β
β
¯
∗+
τ
1∗e
iω0∗τ0∗(
α α
∗ν
+
β β
∗h
)
(
2
.
7
)
We first compute the coordinate to describe the center manifold
C
0at
µ
=
0. Let
X
tbe the
solution of the system
(
2
.
5
)
when
µ
=
0. Define
z
(
t
) =
<
q
∗,
X
t>
W
(
t
,
θ
) =
X
t(
θ
)
−
2
Rez
(
t
)
q
(
θ
)
(
2
.
8
)
On the center manifold
C
0, we have
W
(
t
,
θ
) =
W
(
z
(
t
)
,
z
¯
(
t
)
,
θ
)
where
W
(
z
,
z
¯
,
θ
) =
W
20(
θ
)
z
22
+
W
11(
θ
)
z
z
¯
+
W
02(
θ
)
¯
z
22
+
....
(
2
.
9
)
and
z
and ¯
z
are local coordinates for center manifold
C
0in the direction of
q
∗and ¯
q
∗.
Note that W is real if
X
tis real. We consider only real solutions. For solution
X
t∈
C
0of Eq.
(
2
.
1
)
, since
µ
=
0 we have
˙
z
(
t
) =
i
ω
∗τ
0∗z
+
<
q
¯
∗(
0
)
f
(
0
,
W
(
z
,
z
¯
,
0
) +
2
Rezq
(
θ
))
>
∼
=
i
ω
∗τ
0∗z
+
q
¯
∗(
0
)
f
0(
z
,
z
¯
)
=
i
ω
∗τ
0∗z
+
g
(
z
,
z
¯
)
(
2
.
10
)
where
g
(
z
,
z
¯
) =
q
¯
∗(
0
)
f
0(
z
,
z
¯
)
=
g
20z
22
+
g
11z
z
¯
+
g
02z
22
+
g
21z
2z
¯
2
+
....
(
2
.
11
)
From
(
2
.
8
)
and
(
2
.
9
)
, we get
X
t(
θ
) =
W
(
t
,
θ
) +
2
Rez
(
t
)
q
(
θ
)
=
W
20(
θ
)
z
2
2
+
W
11(
θ
)
z
z
¯
+
W
02(
θ
)
¯
z
22
+
zq
+
z
¯
q
¯
+
....
=
W
20(
θ
)
z
22
+
W
11(
θ
)
z
z
¯
+
W
02(
θ
)
¯
z
22
+ (
1
α β
)
T
e
iω∗τ0∗+ (
1 ¯
α
β
¯
)
Te
iω∗τ0∗z
¯
+
....
Hence we have
g
(
z
,
z
¯
) =
q
¯
∗(
0
)
f
0(
z
,
z
¯
)
=
q
¯
∗(
0
)
f
(
0
,
Xt
)
=
τ
0∗M
¯
(
1 ¯
α
∗β
¯
∗)
T
=
τ
0∗M
¯
(
p
1z
2+
2
p
2z
z
¯
+
p
3z
¯
2+
p
4z
2z
¯
) +
H
.
O
.
T
(
2
.
13
)
where
T
=
(r−rx2t(0)−a1x2t(0))x1t(0) +dx2t(0)
(a1x2t(0)−νx2t(0))x1t(0)−2νx2
2t(0)−dx2t+ (a3x2t−νx2t)x3t(0) +νe−λ τ1x2t(−1)
(−hx2t(0)−a3x2t(0))x3t(0) +he−λ τ2x3t(−1)
p
1,
p
2,
p
3and
p
4values can be calculaed by using the formula.
Comparing (2.11) and (2.13)
g
20=
2
τ
0∗M p
¯
1g
11=
2
τ
0∗M p
¯
2g
02=
2
τ
0∗M p
¯
3g
21=
2
τ
0∗M p
¯
4For unknown
W
20(i)(
θ
)
,
W
11(i)(
θ
)
, i=1,2 in
g
21, we still have to compute them. From (2.5) and
(2.8)
˙
W
=
X
˙
t−
zq
˙
−
z
˙¯
q¯=
AW
−
2
Re
{
q
¯
∗(
0
)
f
0
q
(
θ
)
}
,
−
1
≤
θ
≤
0
,
AW
−
2
Re
{
q
¯
∗(
0
)
f
0q
(
θ
)
}
+
f
0,
θ
=
0
,
˙
W
=
AW
+
H
(
z
,
z
¯
,
θ
)
(
2
.
14
)
where
H
(
z
,
z
¯
,
θ
) =
H
20(
θ
)
z
2
2
+
H
11(
θ
)
z
z
¯
+
H
02(
θ
)
¯
z
22
+
...
(
2
.
15
)
From
(
2
.
14
)
and
(
2
.
15
)
[
A
(
0
)
−
2
i
ω
∗τ
0∗I
]
W
20(
θ
) =
−
H
20(
θ
)
From
(
2
.
14
)
we have for
θ
∈
[
−
1
,
0
)
H
(
z
,
z
¯
,
θ
) =
−
g
(
z
,
z
¯
)
q
(
θ
)
−
g
¯
(
z
,
z
¯
)
q
¯
(
θ
)
(
2
.
18
)
Comparing
(
2
.
15
)
and
(
2
.
18
)
H
20(
θ
) =
−
g
20q
(
θ
)
−
g
¯
02q
¯
(
θ
)
(
2
.
19
)
H
11(
θ
) =
−
g
11q
(
θ
)
−
g
¯
11q
¯
(
θ
)
(
2
.
20
)
By the definition of
A
(
θ
)
and from the above equations
W
20(
θ
) =
ig
20ω
∗τ
0∗q
(
0
)
e
iω∗τ0∗θ
+
i
g
¯
023
ω
∗τ
0∗q
¯
(
0
)
e
−iω∗τ0∗θ
+
E
1e
2iω∗τ∗
0θ
.
(
2
.
21
)
and
W
11(
θ
) =
−
ig
11ω
∗τ
0∗q
(
0
)
e
iω∗τ0∗θ
+
i
g
¯
11ω
∗τ
0∗q
¯
(
0
)
e
−iω∗τ0∗θ
+
E
2
.
(
2
.
22
)
where
q
(
θ
) = (
1
α β
)
Te
iω∗τ0∗θ,
E
1= (
E
(1) 1,
E
(2)
1
,
E
(3)
1
)
∈
R
3and
E
2= (
E
(1)
2
,
E
(2)
2
,
E
(3)
2
)
∈
R
3are
constant vectors. From
(
2
.
14
)
and
(
2
.
15
)
H
20(
0
) =
−
g
20q
(
0
)
−
g
¯
02q
¯
(
0
) +
2
τ
0∗(
c
1c
2c
3)
TH
11(
0
) =
−
g
11q
(
0
)
−
g
¯
11q
¯
(
0
) +
2
τ
0∗(
d
1d
2d
3)
T(
2
.
23
)
where
(
c
1c
2c
3)
T=
C
1,
(
d
1d
2d
3)
T=
D
1are respective coefficients of
z
2and
z
z
¯
of
f
0(
z
z
¯
)
and
they are
C
1=
c
1c
2c
3
=
−
(
a
1+
r
)
α
(
a
1−
ν
)
α
−
2
ν α
2−
dW(2)
20 (0)
2
+ (
a
3−
ν
)
α β
+
νe−λ τ12
W
(2)
20
(
−
1
)
(
−
h
−
a
3)
α β
+
he−λ τ2
2
W
(3)
20
(
−
1
)
and
D
1=
d
1d
2d
3
=
2
rW11(1)(0)
2
−
rRe
(
α
)
−
a
1Re
(
α
) +
dW11(2)(0)
2
(
a
1−
ν
)
Re
(
α
)
−
4
ν α
−
dW(2)
11 (0)
2
+ (
a
3−
ν
)
Re
(
α β
¯
) +
νe−λ τ1W
(2)
11 (−1) 2
(
−
h
−
a
3)
Re
(
α β
¯
) +
he
−λ τ2W(3)
Finally we have
(
2
i
ω
∗τ
0∗I
−
R
−01e
2iω∗τ0∗θd
η
(
θ
))
E
1=
2
τ
∗0
C
1or
C
∗E
1=
2
C
1where
C
∗=
2rx+ry+rz+a1y+a2z−r+2iω rx+a1x−d rx+a2x
νy−a1y νx+2νy+νz+d−a1x−a3z−νe−2iω∗τ1+2iω νy−a3y
hz−a2z hz+a3z −a2x−he−2iω τ2+hx+hy+2hz+a3y+2iω
, (2.24)
Thus
E
1i=
2∆i∆
where
∆
=
Det
(
C
∗
)
and
∆
ibe the value of the determinant
U
i, where
U
iformed by replacing
i
thcolumn vector of
C
∗by another column vector
(
c
1c
2c
3)
T, i =1, 2, 3.
Similarly
D
∗E
2=
2
D
1, where
D
∗=
2rx+ry+rz+a1y+a2z−r rx+a1x−d rx+a2x
νy−a1y νx+2νy+νz+d−a1x−a3z−ν νy−a3y
hz−a2z hz+a3z −a2x−h+hx+hy+2hz+a3y
, (2.25)
Thus
E
2i=
2 ¯∆i¯
∆
where ¯
∆
=
Det
(
D
∗
)
and ¯
∆
i
be the value of the determinant
V
i, where
V
iformed
by replacing
i
thcolumn vector of
D
∗by another column vector
(
d
1d
2d
3)
T, i =1,2, 3. Thus we
can determine
W
20(
θ
)
and
W
11(
θ
)
from
(
2
.
12
)
and
(
2
.
13
)
. Furthermore using them we can
compute
g
21and derive the following values.
C
1(
0
) =
2ωi∗τ∗0
(
g
20g
11−
2
|
g
11|
2
−
|g02|23
) +
g21 2
µ
2=
−Re{C1(0)} Re
dλ(τ0∗)
dτ
β
2=
2
Re
{
C
1(
0
)
}
T
2=
−Im
C1(0)+µ2Im
dλ(τ0∗)
dτ
ω∗τ0∗
These formulae give a description of the Hopf bifurcation periodic solutions of system
(
1
.
1
)
at
τ
=
τ
0∗on the center manifold. Hence we have the following result.
Theorem 2.1.
The periodic solutions is supercritical (resp.subcritical) if
µ
2>
0
(resp.
µ
2<
phase (resp.unstable) if
β
2<
0
(resp.
β
2>
0
). The period of bifurcating periodic solutions
increases(resp.decreases) if T
2>
0
(resp.T
2<
0
).
4. Conclusion
We have derived the bifurcating periodic solutions are orbitally asymptotically stable with
an asymptotical phase if
β
2<
0 and unstable if
β
2>
0 and the period of bifurcating periodic
solutions increases if
T
2>
0 and decreases if
T
2<
0.
Conflict of Interests
The authors declare that there is no conflict of interests.
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