2. The Existence of Hopf Bifurcations
In this section, we study the existence of the Hopf bifurcations of system (2). Clearly, when

1
<
n
<
0
, system (2) has only one positive equilibrium, that is,
(3)
E
(
1
+
γ
ɛ

(
1

γ
ɛ
)
2

4
ɛ
γ
n
2
ɛ
,
1

γ
ɛ
+
(
1

γ
ɛ
)
2

4
ɛ
γ
n
2
)
.
Let
(4)
x
1
(
0
)
=
1
+
γ
ɛ

(
1

γ
ɛ
)
2

4
ɛ
γ
n
2
ɛ
;
x
2
(
0
)
=
1

γ
ɛ
+
(
1

γ
ɛ
)
2

4
ɛ
γ
n
2
,
then system (2) becomes
(5)
x
˙
1
=

ɛ
x
1
(
0
)
(
x
1

x
1
(
0
)
)

x
1
(
0
)
(
x
2
(
t

τ
)

x
2
(
0
)
)

(
x
1

x
1
(
0
)
)
(
x
2
(
t

τ
)

x
2
(
0
)
)

ɛ
(
x
1

x
1
(
0
)
)
2
,
x
˙
2
=
x
2
(
0
)
n
+
x
2
(
0
)
(
x
1

x
1
(
0
)
)
+
(

γ
+
2
n
x
1
(
0
)
x
2
(
0
)
+
x
1
(
0
)
(
x
2
(
0
)
)
2
(
n
+
x
2
(
0
)
)
2
)
(
x
2

x
2
(
0
)
)
+
⋯
.
By introducing the new variables
z
1
(
t
)
=
x
1
(
t
)

x
1
(
0
)
,
z
2
(
t
)
=
x
2
(
t
)

x
2
(
0
)
and denoting
f
(
x
1
,
x
2
)
=

γ
x
2
+
(
x
2
/
(
n
+
x
2
)
)
x
1
x
2
, system (5) can be rewritten in a simpler form as
(6)
z
˙
1
(
t
)
=

α
1
z
1
(
t
)

α
2
z
2
(
t

τ
)

z
1
(
t
)
z
2
(
t

τ
)

ɛ
z
1
2
(
t
)
,
z
˙
2
(
t
)
=
r
1
z
1
(
t
)
+
r
2
z
2
(
t
)
+
∑
1
i
!
j
!
c
i
j
z
1
i
(
t
)
z
2
j
(
t
)
,
where
(7)
α
1
=
ɛ
x
1
(
0
)
,
α
2
=
x
1
(
0
)
,
r
1
=
x
2
(
0
)
n
+
x
2
(
0
)
,
r
2
=

γ
+
2
n
x
1
(
0
)
x
2
(
0
)
+
x
1
(
0
)
(
x
2
(
0
)
)
2
(
n
+
x
2
(
0
)
)
2
,
and
c
i
j
=
∂
i
+
j
f
(
x
1
(
0
)
,
x
2
(
0
)
)
/
∂
x
1
i
∂
x
2
j
. Then the linearization of system (2) at
E
is
(8)
z
˙
1
(
t
)
=

α
z
1
(
t
)

α
2
z
2
(
t

τ
)
,
z
˙
2
(
t
)
=
r
1
z
1
(
t
)
+
r
2
z
2
(
t
)
.
The associated characteristic equation of (8) is given by
(9)

λ
+
α
1
α
2
e

λ
τ

r
1
λ

r
2

=
0
.
That is,
(10)
λ
2
+
(
α
1

r
2
)
λ

α
1
r
2
+
α
2
r
1
e

λ
τ
=
0
.
The equilibrium
E
is stable if all roots of (10) have negative real parts. Clearly, when
τ
=
0
, the characteristic equation (10) becomes
(11)
λ
2
+
(
α
1

r
2
)
λ

α
1
r
2
+
α
2
r
1
=
0
.
By directly computing, we known that
r
2
<
0
when

1
<
n
<
0
. Therefore all roots of (11) have negative real parts. Obviously,
λ
=
i
ω
(
ω
>
0
)
is a root of (10) if and only if
ω
satisfies
(12)

ω
2
+
i
(
α
1

r
2
)
ω

α
1
r
2
+
α
2
r
1
e

i
ω
τ
=
0
.
Separating the real and imaginary parts, we have
(13)

ω
2

α
1
r
2
+
α
2
r
1
cos
ω
τ
=
0
,
(
α
1

r
2
)
ω

α
2
r
1
sin
ω
τ
=
0
,
which leads to
(14)
ω
4
+
(
α
1
2
+
r
2
2
)
ω
2
+
α
1
2
r
2
2

α
2
2
r
1
2
=
0
.
When
α
1
2
r
2
2

α
2
2
r
1
2
<
0
, (14) has only one positive root
(15)
ω
*
=

(
α
1
2
+
r
2
2
)
+
(
α
1
2
+
r
2
2
)
2

4
(
α
1
2
r
2
2

α
2
2
r
1
2
)
2
.
Substituting (15) into (13), we obtain
(16)
τ
j
=
1
ω
*
arccos
ω
*
2
+
α
1
r
2
α
2
r
1
+
2
j
π
ω
*
,
j
=
0,1
,
2
,
…
.
Thus, when
τ
=
τ
j
, the characteristic equation (10) has a pair of purely imaginary roots
±
i
ω
*
.
Lemma 1.
Let
λ
j
(
τ
)
=
η
j
(
τ
)
+
i
ω
j
(
τ
)
be the root of (10) satisfying
(17)
η
j
(
τ
j
)
=
0
,
ω
j
(
τ
j
)
=
ω
*
,
j
=
0,1
,
2
,
…
,
and then
(18)
η
j
′
(
τ
j
)
>
0
.
Proof.
Differentiating both sides of (10) with respect to
τ
, we obtain
(19)
(
2
λ
+
α
1

r
2

α
2
r
1
τ
e

λ
τ
)
d
λ
d
τ
=
α
2
r
1
λ
e

λ
τ
.
Therefore,
(20)
sign
{
d
(
Re
(
λ
)
)
d
τ
}
τ
=
τ
j
=
sign
{
Re
(
d
λ
d
τ
)

1
}
τ
=
τ
j
=
sign
{
Re
[
2
ω
*
+
(
α
1

r
2
)
]
(
cos
λ
τ
+
i
sin
λ
τ
)

τ
j
α
2
r
1
i
ω
*
α
2
r
1
}
=
sign
{
[
[
2
ω
*
+
(
α
1

r
2
)
]
[
α
1

r
2
]
]
α
2
2
r
1
2
}
>
0
.
Thus, the lemma follows.
Therefore, from Lemma 1 and the relations between roots of (10) and (11) [17], we have the following conclusion.
Lemma 2.
When
τ
∈
[
0
,
τ
0
)
, all roots of (10) have negative real parts. When
τ
=
τ
0
, all roots of (10) have negative real parts except
±
i
ω
*
. When
τ
∈
(
τ
j
,
τ
j
+
1
]
, (10) has
2
(
j
+
1
)
roots with positive real parts.
Furthermore, from Lemma 2, the following theorem holds.
Theorem 3.
If
τ
∈
[
0
,
τ
0
)
, then the positive equilibrium
E
is asymptotically stable and unstable if
τ
>
τ
0
. If
τ
=
τ
j
, (2) undergoes a Hopf bifurcation at
E
.
3. Stability and Direction of the Hopf Bifurcation
Let
u
i
=
z
i
(
τ
t
)
and
τ
=
τ
j
+
μ
, where
μ
∈
R
. Then (2) can be written as a functional differential equation in
C
=
C
(
[

1,0
]
,
R
2
)
as
(21)
u
˙
(
t
)
=
L
μ
(
u
t
)
+
F
(
μ
,
u
t
)
,
where
u
t
(
θ
)
=
x
(
t
+
θ
)
∈
C
, and
L
μ
:
C
→
R
,
F
:
R
×
C
→
R
are given, respectively, by
(22)
L
μ
(
ϕ
)
=
(
τ
(
j
)
+
μ
)
(

α
1
0
r
1
r
2
)
(
ϕ
1
(
0
)
ϕ
2
(
0
)
)
+
(
τ
(
j
)
+
μ
)
(
0

α
2
0
0
)
(
ϕ
1
(

1
)
ϕ
2
(

1
)
)
,
(23)
F
(
μ
,
ϕ
)
=
(
τ
(
j
)
+
μ
)
(

ϕ
1
(
0
)
ϕ
2
(

1
)

ɛ
ϕ
1
2
(
0
)
+
h
.
o
.
t
Σ
1
i
!
j
!
c
i
j
ϕ
1
i
(
0
)
ϕ
2
j
(
0
)
+
h
.
o
.
t
)
,
where
h
.
o
.
t
denotes the higher order terms.
From the discussions above, we known that if
μ
=
0
, then system (21) undergoes a Hopf bifurcation at the zero equilibrium and the associated characteristic equation of system (21) has a pair of simple imaginary roots
±
i
τ
j
ω
0
.
By the Reiz representation theorem, there exists a function
η
(
θ
,
μ
)
of bounded variation for
θ
∈
[

1,0
]
, such that
(24)
L
μ
ϕ
=
∫

1
0
d
η
(
θ
,
0
)
ϕ
(
θ
)
for
ϕ
∈
C
.
In fact, we can choose
(25)
η
(
θ
,
μ
)
=
(
τ
(
j
)
+
μ
)
(

α
1
0
r
1
r
2
)
δ
(
θ
)

(
τ
(
j
)
+
μ
)
(
0

α
2
0
0
)
δ
(
θ
+
1
)
,
where
(26)
δ
(
θ
)
=
{
0
,
θ
≠
0
,
1
,
θ
=
0
.
For
ϕ
∈
C
1
(
[

1,0
]
,
R
2
)
, define
(27)
A
(
μ
)
ϕ
=
{
d
ϕ
(
θ
)
d
θ
,
θ
∈
[

1,0
]
,
∫

1
0
d
η
(
s
,
μ
)
ϕ
(
s
)
,
θ
=
0
,
R
(
μ
)
ϕ
=
{
0
,
θ
∈
[

1,0
]
,
F
(
μ
,
ϕ
)
,
θ
=
0
.
Then we can rewrite (21) as
(28)
u
˙
t
=
A
(
μ
)
u
t
+
R
(
μ
)
x
t
,
where
u
t
(
θ
)
=
u
(
t
+
θ
)
,
θ
∈
[

1,0
]
. For
ϕ
∈
C
1
(
[
0,1
]
,
R
2
)
, define
(29)
A
*
ψ
(
s
)
=
{

d
ψ
(
s
)
d
s
,
s
∈
[
0,1
]
,
∫

1
0
ψ
(

t
)
d
η
(
t
,
0
)
,
s
=
0
and a bilinear inner product
(30)
〈
ψ
(
s
)
,
ϕ
(
θ
)
〉
=
ψ
¯
(
0
)
ϕ
(
0
)

∫

1
0
∫
ξ
=
0
θ
ψ
¯
(
ξ

θ
)
d
η
(
θ
)
ϕ
(
ξ
)
d
ξ
,
where
η
(
θ
)
=
η
(
θ
,
0
)
. Then
A
*
and
A
(
0
)
are adjoint operators. By the discussion of Section 2, we known that
±
i
ω
0
τ
0
are eigenvalues of
A
(
0
)
. Thus, they are also eigenvalues of
A
*
.
Suppose that
q
*
(
s
)
=
D
(
1
,
α
*
)
e
i
s
ω
0
τ
(
j
)
is the eigenvector of
A
(
0
)
corresponding to
i
τ
(
j
)
ω
0
. Then,
A
(
0
)
q
(
θ
)
=
i
τ
(
j
)
ω
0
q
(
θ
)
. From the definition of
A
(
0
)
and (25), we obtain
(31)
τ
(
j
)
(
i
ω
+
α
1
α
2
e

i
ω
0
τ
j

r
1
i
ω
0

r
2
)
q
(
0
)
=
(
0
0
)
,
which yields
(32)
q
(
0
)
=
(
1
,
α
)
T
=
(
1
,
r
1
i
ω
0

r
2
)
T
.
Similarly, it can be verified that
q
*
(
s
)
=
D
(
1
,
α
*
)
e
i
s
ω
0
τ
(
j
)
is the eigenvector of
A
*
corresponding to

i
ω
0
τ
(
j
)
, where
(33)
α
*
=
α
1

i
ω
0
r
1
.
Let
〈
q
*
(
s
)
,
q
(
θ
)
〉
=
1
; that is,
(34)
〈
q
*
(
s
)
,
q
(
θ
)
〉
=
D
¯
(
1
,
α
*
¯
)
(
1
,
α
)
T

∫

1
0
∫
ξ
0
D
¯
(
1
,
α
*
¯
)
e

i
(
ξ

θ
)
ω
0
τ
(
j
)
d
η
(
θ
)
(
1
,
α
)
T
e
i
ξ
ω
0
τ
(
j
)
d
ξ
=
D
¯
{
1
+
α
α
*
¯

∫

1
0
(
1
,
α
*
¯
)
θ
e
i
θ
ω
0
τ
(
j
)
d
η
(
θ
)
(
1
,
α
)
T
}
=
D
¯
{
1
+
α
α
*
¯

α
2
α
τ
(
j
)
e

i
ω
0
τ
(
j
)
}
=
1
.
Thus, we can choose
(35)
D
=
1
1
+
α
¯
α
*

α
2
α
¯
τ
(
j
)
e
i
ω
0
τ
(
j
)
such that
〈
q
*
(
s
)
,
q
(
θ
)
〉
=
1
,
〈
q
*
(
s
)
,
q
¯
(
θ
)
〉
=
0
.
Using the same notations as in Hassard et al. [18] and Song et al. [19], we first compute the center manifold
C
0
at
μ
=
0
. Let
x
t
be the solution of (21) when
μ
=
0
. Define
(36)
z
(
t
)
=
〈
q
*
,
x
t
〉
,
W
(
t
,
θ
)
=
x
t
(
θ
)

(
x
(
t
)
q
(
θ
)
+
z
¯
(
t
)
q
¯
(
θ
)
)
=
x
t
(
θ
)

2
Re
{
z
(
t
)
q
(
θ
)
}
.
On the center manifold
C
0
, we have
(37)
W
(
t
,
θ
)
=
W
(
z
(
t
)
,
z
¯
(
t
)
,
θ
)
,
where
(38)
W
(
z
,
z
¯
,
θ
)
=
W
20
(
θ
)
z
2
2
+
W
11
(
θ
)
z
z
¯
+
W
02
(
θ
)
z
¯
2
2
+
W
30
(
θ
)
z
3
6
+
⋯
,
where
z
and
z
¯
are local coordinates for center manifold
C
0
in the direction of
q
*
and
q
*
¯
. Note that
W
is real if
x
t
is real. Here we consider only real solutions. For the solution
x
t
∈
C
0
of (24), since
μ
=
0
, we have
(39)
z
˙
=
i
τ
(
j
)
ω
0
z
+
〈
q
*
(
θ
)
,
F
(
0
,
W
(
z
,
z
¯
,
θ
)
+
2
Re
{
z
q
(
θ
)
}
)
〉
=
i
τ
(
j
)
ω
0
z
+
q
*
¯
(
0
)
F
(
0
,
W
(
z
,
z
¯
,
0
)
+
2
Re
{
z
q
(
0
)
}
)
=
i
τ
(
j
)
ω
0
z
+
q
*
¯
(
0
)
F
0
(
z
,
z
¯
)
.
We rewrite this equation as
(40)
z
˙
(
t
)
=
i
τ
(
j
)
ω
0
z
(
t
)
+
g
(
z
,
z
¯
)
with
(41)
g
(
z
,
z
¯
)
=
q
*
¯
(
0
)
F
0
(
z
,
z
¯
)
=
g
20
z
2
2
+
g
11
z
z
¯
+
g
02
z
¯
2
2
+
g
21
z
2
z
¯
2
+
⋯
.
By (36), we have
x
t
(
θ
)
=
(
x
1
t
(
θ
)
,
x
2
t
(
θ
)
)
=
W
(
t
,
θ
)
+
z
q
(
θ
)
+
z
¯
q
¯
(
θ
)
and
q
(
θ
)
=
(
1
,
α
)
T
e
i
θ
ω
0
τ
(
j
)
, and then
(42)
x
1
t
(
0
)
=
z
+
z
¯
+
W
20
(
1
)
(
0
)
z
2
2
+
W
11
(
1
)
(
0
)
z
z
¯
+
W
02
(
1
)
(
0
)
z
¯
2
2
+
o
(

(
z
,
z
¯
)

3
)
,
x
2
t
(
0
)
=
z
α
e

i
ω
0
τ
(
j
)
+
z
¯
α
¯
e
i
ω
0
τ
(
j
)
+
W
20
(
2
)
(

1
)
z
2
2
+
W
11
(
2
)
(

1
)
z
z
¯
+
W
02
(
2
)
(

1
)
z
¯
2
2
+
o
(

(
z
,
z
¯
)

3
)
.
It follows, together with (23), that
(43)
x
2
t
(
0
)
=
α
z
+
α
¯
z
¯
+
W
20
(
2
)
(
0
)
z
2
2
+
W
02
(
2
)
(
0
)
z
¯
2
2
+
⋯
,
g
(
z
,
z
¯
)
=
q
*
¯
(
0
)
F
0
(
z
,
z
¯
)
=
D
¯
τ
(
j
)
(
1
,
α
*
¯
)
(

x
1
t
(
0
)
x
2
t
(

1
)

ɛ
x
1
t
2
(
0
)
+
h
.
o
.
t
Σ
1
i
!
j
!
c
i
j
x
1
t
2
(
0
)
x
2
t
j
(
0
)
+
h
.
o
.
t
)
=
D
¯
τ
(
j
)
(

x
1
t
(
0
)
x
2
t
(

1
)

ɛ
x
1
t
2
(
0
)
∑
1
i
!
j
!
00000000000
+
α
*
¯
∑
1
i
!
j
!
c
i
j
x
1
t
2
(
0
)
x
2
t
j
(
0
)
+
h
.
o
.
t
)
=
D
¯
τ
(
j
)
{
(

α
e

i
ω
0
τ
(
j
)

ɛ
+
α
*
¯
2
!
(
c
20
+
c
02
α
2
)
)
z
2
0000000000
+
(
(
c
20
+
c
02
α
α
¯
2
)

α
e

i
ω
0
τ
(
j
)

α
¯
e
i
ω
0
τ
(
j
)

2
ɛ
000000000000000
+
α
*
¯
(
c
20
+
c
02
α
α
¯
2
)
)
z
z
¯
0000000000
+
(

α
¯
e
i
ω
0
τ
(
j
)

ɛ
+
α
*
¯
c
20
2
!
+
α
¯
2
α
*
¯
c
02
2
!
)
z
¯
2
0000000000
+
(

α
¯
2
e
i
ω
0
τ
(
j
)
W
20
(
1
)
(
0
)

1
2
W
20
(
2
)
(

1
)
00000000000000

ɛ
W
20
(
1
)
(
0
)
+
α
*
¯
(
1
2
c
20
W
20
(
1
)
(
0
)
000000000000000000000000000000
+
1
2
α
¯
c
02
W
20
(
2
)
)
0000000000000000
×
c
20
W
11
(
1
)
(
0
)
00000000000000000
+
α
c
02
W
(
2
)
11
(
0
)

α
e

i
ω
0
τ
(
j
)

ɛ
+
2
!
(
c
20
+
c
02
α
2
)
)
z
2
z
¯
+
⋯
}
.
Comparing the coefficients with (41), we have
(44)
g
20
=
D
¯
τ
(
j
)
(

2
α
e

i
ω
0
τ
(
j
)

2
ɛ
+
α
*
¯
(
c
20
+
c
02
α
2
)
)
,
g
11
=
D
¯
τ
(
j
)
(

α
e

i
ω
0
τ
(
j
)

α
¯
e
i
ω
0
τ
(
j
)

2
ɛ
(
c
20
+
c
02
α
α
¯
2
)
000000000
+
α
*
¯
(
c
20
+
c
02
α
α
¯
2
)
)
,
g
02
=
D
¯
τ
(
j
)
(

2
α
¯
e
i
ω
0
τ
(
j
)

2
ɛ
+
α
¯
(
c
20
+
α
¯
c
02
)
)
,
g
21
=

α
¯
e
i
ω
0
τ
(
j
)
W
20
(
1
)
(
0
)

W
20
(
2
)
(

1
)

2
ɛ
W
20
(
1
)
(
0
)
+
α
*
¯
(
c
20
W
20
(
1
)
(
0
)
+
c
02
α
¯
W
20
(
2
)
(
0
)
)
+
2
c
20
W
11
(
1
)
(
0
)
+
2
c
02
α
W
11
(
2
)
(
0
)
.
In order to determine
g
21
, we need to compute
W
20
(
θ
)
and
W
11
(
θ
)
. From (28) and (36), we have
(45)
W
˙
=
x
˙
t

z
˙
q

z
¯
˙
q
¯
=
{
A
W

2
R
{
q
*
(
0
)
¯
F
0
q
(
θ
)
}
,
θ
∈
[

1,0
]
,
A
W

2
R
{
q
*
(
0
)
¯
F
0
q
(
θ
)
}
+
F
0
,
θ
=
0
≡
A
W
+
H
(
z
,
z
¯
,
θ
)
,
where
(46)
H
(
z
,
z
¯
,
θ
)
=
H
20
(
θ
)
z
2
2
+
H
11
(
θ
)
z
z
¯
+
H
02
(
θ
)
z
¯
2
2
+
⋯
.
Expanding the above series and comparing the corresponding coefficients, we obtain
(47)
(
A

2
i
τ
(
j
)
ω
0
)
W
20
(
θ
)
=

H
20
(
θ
)
,
A
W
11
(
θ
)
=

H
11
(
θ
)
,
…
.
Following (45), we know that for
θ
∈
[

1,0
]
,
(48)
H
(
z
,
z
¯
,
θ
)
=

q
*
(
0
)
¯
F
0
q
(
θ
)

q
*
(
0
)
F
0
¯
q
(
θ
)
¯
=

g
q
(
θ
)

g
¯
q
(
θ
)
¯
.
Comparing the coefficients with (46), we get
(49)
H
20
(
θ
)
=

g
20
q
(
θ
)

g
02
¯
q
(
θ
)
¯
,
H
11
(
θ
)
=

g
11
q
(
θ
)

g
11
¯
q
(
θ
)
¯
.
Substituting these relations into (47), we obtain
(50)
W
˙
20
(
θ
)
=
2
i
τ
(
j
)
ω
0
W
20
(
θ
)
+
g
20
q
(
θ
)
+
g
02
¯
q
(
θ
)
¯
.
Solving
W
20
(
θ
)
, we obtain
(51)
W
20
(
θ
)
=
i
g
20
q
(
0
)
τ
(
j
)
ω
0
e
i
τ
(
j
)
ω
0
θ
+
i
g
0
2
¯
q
(
0
)
¯
3
τ
(
j
)
ω
0
e

i
τ
(
j
)
ω
0
θ
+
E
1
e
2
i
τ
(
j
)
ω
0
θ
,
where
E
1
=
(
E
1
(
1
)
,
E
1
(
2
)
)
∈
R
2
is a constant vector.
Similarly, we can obtain
(52)
W
11
(
θ
)
=

i
g
11
q
(
0
)
τ
(
j
)
ω
0
e
i
τ
(
j
)
ω
0
θ
+
i
g
11
¯
q
(
0
)
¯
τ
(
j
)
ω
0
e

i
τ
(
j
)
ω
0
θ
+
E
2
,
where
E
2
=
(
E
2
(
1
)
,
E
2
(
2
)
)
∈
R
2
is also a constant vector.
In what follows, we determine the constant vectors
E
1
and
E
2
. From (47) and the definition of
A
, we obtain
(53)
∫

1
0
d
η
(
θ
)
W
20
(
θ
)
=
2
i
τ
(
j
)
ω
0
W
20
(
0
)

H
20
(
0
)
,
(54)
∫

1
0
d
η
(
θ
)
W
11
(
θ
)
=

H
11
(
θ
)
,
where
η
(
θ
)
=
η
(
0
,
θ
)
. From (45) and (46), we have
(55)
H
20
(
0
)
=

g
20
q
(
0
)

g
02
¯
q
(
0
)
¯
+
2
τ
(
j
)
(

2
α
e

i
ω
0
τ
(
j
)

2
ɛ
c
20
+
c
02
α
2
)
,
(56)
H
11
(
0
)
=

g
11
q
(
0
)

g
11
¯
q
(
0
)
¯
+
2
τ
(
j
)
(

2
α
e

i
ω
0
τ
(
j
)

α
¯
e
i
ω
0
τ
(
j
)

2
ɛ
c
20
+
c
02
α
α
¯
)
.
Substituting (51) and (55) into (53) and noticing that
(57)
(
i
τ
(
j
)
ω
0
I

∫

1
0
e
i
θ
ω
0
τ
(
j
)
d
η
(
θ
)
)
q
(
0
)
=
0
,
(

i
τ
(
j
)
ω
0
I

∫

1
0
e

i
θ
ω
0
τ
(
j
)
d
η
(
θ
)
)
q
¯
(
0
)
=
0
,
we get
(58)
(
2
i
τ
(
j
)
ω
0
I

∫

1
0
e
2
i
θ
ω
0
τ
(
j
)
d
η
(
θ
)
)
E
1
=
2
τ
(
j
)
(

2
α
e

i
ω
0
τ
(
j
)

2
ɛ
c
20
+
c
02
α
2
)
;
that is,
(59)
(
2
i
ω
0
+
α
1

α
2
e

2
i
ω
0
τ
(
j
)

r
1
2
i
ω
0
)
E
1
=
2
(

2
α
e

i
ω
0
τ
(
j
)

2
ɛ
c
20
+
c
02
α
2
)
.
It follows that
(60)
E
1
(
1
)
=
2
A


2
α
e

i
ω
0
τ
(
j
)

2
ɛ

α
2
e

2
i
ω
0
τ
(
j
)
c
20
+
c
02
α
2
2
i
ω
0

,
E
1
(
2
)
=
2
A

2
i
ω
0
+
α
1

2
α
e

i
ω
0
τ
(
j
)

2
ɛ

r
1
c
20
+
c
02
α
2

,
where
A
=

2
i
ω
0
+
α
1

α
2
e

2
i
ω
0
τ
(
j
)

r
1
2
i
ω
0

.
Similarly, substituting (52) and (56) into (54), we have
(61)
(
α
1
α
2

r
1

r
2
)
E
2
=
2
(

2
α
e

i
ω
0
τ
(
j
)

α
¯
e
i
ω
0
τ
(
j
)

2
ɛ
c
20
+
c
02
α
α
¯
)
.
Then we obtain
(62)
E
2
(
1
)
=
2
B


2
α
e

i
ω
0
τ
(
j
)

α
¯
e
i
ω
0
τ
(
j
)

2
ɛ

α
2
c
20
+
c
02
α
α
¯

r
2

,
E
2
(
2
)
=
2
B

α
1

2
α
e

i
ω
0
τ
(
j
)

α
¯
e
i
ω
0
τ
(
j
)

2
ɛ

r
1
c
20
+
c
02
α
α
¯

,
where
B
=

α
1
α
2

r
1

r
2

.
Therefore, all
g
i
j
in (41) have been expressed in terms of the parameters and the delay given in (2). Substituting expressions of
g
02
,
g
11
,
g
20
, and
g
21
into the following relations,
(63)
C
1
(
0
)
=
i
2
ω
0
τ
0
(
g
20
g
11

2

g
11

2

1
3

g
02

2
+
g
21
2
)
,
we obtain
(64)
K
2
=

Re
{
c
1
(
0
)
}
Re
{
λ
′
(
τ
)
}
,
β
2
=
2
Re
{
C
1
(
0
)
}
,
T
2
=

Im
{
C
1
(
0
)
}
+
K
2
Im
{
λ
′
(
τ
(
j
)
)
}
τ
(
j
)
ω
0
.
We follow the idea in Hassard et al. [18] and Song et al. [19], which implies that the direction of the Hopf bifurcation is determined by the sign of
β
2
, and the stability of the bifurcating periodic solutions is determined by the sign of
K
2
and
T
2
determines the period of the bifurcating periodic solution. Thus we have the following.
Theorem 4.
(1) If
K
2
>
0
(
K
2
<
0
)
, then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for
τ
>
τ
(
j
)
(
τ
<
τ
(
j
)
)
.
(2) If
β
2
<
0
(
β
2
>
0
)
, then the bifurcating periodic solutions are stable (unstable).
(3) If
T
2
>
0
(
T
2
<
0
)
, then the periodic of the bifurcating periodic solutions increase (decrease).