1. Introduction
The classical predatorprey systems have been extensively investigated in recent years, and they will continue to be one of the dominant themes in the future due to their universal existence and importance. Many biological phenomena are always described by differential equations, difference equations, and other type equations. In general, delay differential equations exhibit more complicated dynamical behaviors than ordinary ones; for example, the delay can induce the loss of stability, various oscillations, and periodic solutions. The dynamical behaviors of delay differential equations, stability, bifurcation and chaos, and so forth have been paid much attention by many researchers. Especially, the direction and stability of Hopf bifurcation to delay differential equations have been investigated extensively in recent work (see [1–7] and references therein).
After the classical predatorprey model was first proposed and discussed by May in [8], there were some similar topics, regarding persistence, local and global stabilities of equilibria, and other dynamical behaviors (see [5, 9, 10] and references therein). Recently, Song and Wei in [7] had considered a delayed predatorprey system as follows:
(1)
x
˙
(
t
)
=
x
(
t
)
[
r
1

a
11
x
(
t

τ
)

a
12
y
(
t
)
]
,
y
˙
(
t
)
=
y
(
t
)
[

r
2
+
a
21
x
(
t
)

a
22
y
(
t
)
]
,
where
x
(
t
)
and
y
(
t
)
were the densities of prey species and predator species at time
t
, respectively. The local Hopf bifurcation and the existence of the periodic solution bifurcating of system (1) was investigated in [7]. When selective harvesting was put into the predatorprey model similar to (1), Kar [11] studied two predatorprey models with selective harvesting; that is, in the first model, selective harvesting of predator species:
(2)
x
˙
(
t
)
=
x
(
t
)
[
g
(
x
)

y
p
(
x
)
]
,
y
˙
(
t
)
=
y
(
t
)
[

d
+
α
x
p
(
x
)
]

q
E
y
(
t

τ
)
,
and, in the second model, selective harvesting of prey species:
(3)
x
˙
(
t
)
=
x
(
t
)
[
g
(
x
)

y
p
(
x
)
]

q
E
x
(
t

τ
)
,
y
˙
(
t
)
=
y
(
t
)
[

d
+
α
x
p
(
x
)
]
had been considered by incorporating time delay on the harvesting term. They found that the delay for selective harvesting could induce the switching of stability and Hopf bifurcation occurred at
τ
=
τ
0
.
Recently, Kar and Ghorai [9] had investigated a predatorprey model with harvesting:
(4)
x
˙
(
t
)
=
r
1
x
(
t
)

b
1
x
2
(
t
)

a
1
x
(
t
)
y
(
t
)
x
(
t
)
+
k
1

c
1
x
(
t
)
,
y
˙
(
t
)
=
y
(
t
)
[
r
2

a
2
y
(
t

τ
)
x
(
t

τ
)
+
k
2
]

c
2
y
(
t
)
.
They obtained the local stability, global stability, influence of the harvesting, direction of Hopf bifurcation and the stability to system (4). Motivated by models (1)–(4), we will consider a predatorprey system with delay incorporating harvests to predator and prey:
(5)
x
˙
(
t
)
=
x
(
t
)
[
1

x
(
t
)
k
1

a
y
(
t
)
]

h
1
x
(
t
)
,
y
˙
(
t
)
=
y
(
t
)
[
1

y
(
t

τ
)
k
2
+
b
x
(
t

τ
)
]

h
2
y
(
t
)
,
where
x
(
t
)
and
y
(
t
)
represent the population densities of prey species and predator species, respectively, at time
t
;
a
,
b
,
h
1
,
h
2
,
k
1
, and
k
2
are model parameters assuming only positive values;
k
1
measures the scale whose environment provides protection to prey
x
;
k
2
denotes the scale whose environment provides protection to predator
y
;
τ
means the period of pregnancy;
x
(
t

τ
)
represents the number of prey species which was born at time
t

τ
and still survived at time
t
;
h
1
and
h
2
represent the coefficients of prey species and predator species, respectively. We always assume that
0
≤
h
1
≤
h
2
<
1
in this paper.
The organization of the paper is as follows. The stability of the positive equilibrium and the existence of the Hopf bifurcation are discussed in Section 2. The effect of harvesting to prey species and predator species is investigated in Section 3. The direction of Hopf bifurcation and stability of the corresponding periodic solution are obtained in Section 4. Numerical simulations are carried out to illustrate our results in Section 5.
4. Direction and Stability of Hopf Bifurcation
Motivated by the ideas of Hassard et al. [12], by applying the normal form theory and the center manifold theorem, the properties of the Hopf bifurcation at the critical value
τ
=
τ
j
are derived in this section.
Let
t
=
s
τ
,
x
i
(
s
τ
)
=
x
^
i
(
s
)
,
i
=
1,2
,
τ
=
τ
0
+
μ
,
μ
∈
R
;
τ
0
is defined by (14), we still denote
x
^
i
(
s
)
=
u
i
(
s
)
and
s
=
t
, then system (5) is transformed into functional differential equations in
C
(
[

1,0
]
,
R
2
)
as
(25)
u
˙
(
t
)
=
L
μ
(
u
t
)
+
f
(
μ
,
u
t
)
,
where
u
(
t
)
=
(
u
1
(
t
)
,
u
2
(
t
)
)
T
∈
R
2
,
u
t
(
θ
)
=
u
(
t
+
θ
)
,
θ
∈
[

1,0
]
, and
L
μ
:
C
(
[

1,0
]
;
R
2
)
→
R
,
f
:
R
×
C
(
[

1,0
]
;
R
2
)
→
R
are given by
(26)
L
μ
(
ϕ
)
=
(
τ
0
+
μ
)
(

a
11

a
12
0
0
)
(
ϕ
1
(
0
)
ϕ
2
(
0
)
)
+
(
τ
0
+
μ
)
(
0
0
a
21

a
22
)
(
ϕ
1
(

1
)
ϕ
2
(

1
)
)
,
(27)
f
(
μ
,
ϕ
)
=
(
τ
0
+
μ
)
(
c
1
ϕ
2
2
(
0
)
+
c
2
ϕ
1
(
0
)
ϕ
2
(
0
)
+
c
3
ϕ
1
2
(
0
)
e
1

e
2
ϕ
2
(
0
)
c
4
ϕ
1
(

1
)
ϕ
2
(
0
)

c
5
ϕ
1
2
(

1
)
+
c
6
ϕ
1
(

1
)
ϕ
2
(

1
)

c
7
ϕ
2
(
0
)
ϕ
2
(

1
)
e
3
+
e
4
ϕ
1
(

1
)
)
,
where
(28)
c
1
=
a
2
(
x
*
)
2
,
c
2
=
2
a
2
x
*
y
*

2
a
k
1
x
*
,
c
3
=
2
a
k
1
y
*

a
2
c
k
1
2
y
*
,
c
4
=
b
k
2
y
*
+
b
2
x
*
y
*
,
c
5
=
b
2
(
y
*
)
2
,
c
6
=
k
2
b
y
*
+
b
2
x
*
y
*
,
c
7
=
2
k
2
b
x
*
+
k
2
2
+
b
2
(
x
*
)
2
,
e
1
=
(
k
1

a
y
*
)
3
,
e
2
=
a
(
k
1

a
y
*
)
2
,
e
3
=
(
k
2
+
b
x
*
)
3
,
e
4
=
b
(
k
2
+
b
x
*
)
2
.
By Riesz representation theorem, there exists a function
η
(
θ
,
μ
)
of bounded variation for
θ
∈
[

1,0
]
, such that
(29)
L
μ
ϕ
=
∫

1
0
d
η
(
θ
,
μ
)
ϕ
(
θ
)
.
We choose
(30)
η
(
θ
,
μ
)
=
(
τ
0
+
μ
)
(

a
11

a
12
0
0
)
δ
(
θ
)
+
(
τ
0
+
μ
)
(
0
0
a
21

a
22
)
δ
(
θ
+
1
)
,
where
δ
is the Dirac delta function. For
ϕ
∈
C
1
(
[

1,0
]
,
R
2
)
, we define
(31)
A
(
μ
)
ϕ
(
θ
)
=
{
d
ϕ
(
θ
)
d
θ
,

1
≤
θ
<
0
∫

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

1
≤
θ
<
0
f
(
μ
,
ϕ
)
,
θ
=
0
.
Then, system (25) can be transformed into an operator differential equation of the form
(32)
u
˙
t
=
A
(
μ
)
u
t
+
R
(
μ
)
u
t
,
where
u
t
(
θ
)
=
u
(
t
+
θ
)
, for
θ
∈
[

1,0
]
. For
ψ
∈
C
1
(
[
0,1
]
,
(
R
2
)
*
)
, we define
(33)
A
*
(
μ
)
ψ
(
s
)
=
{

d
ψ
(
s
)
d
s
,
0
<
s
≤
1
∫

1
0
d
η
T
(
t
,
0
)
ψ
(

t
)
,
s
=
0
and a bilinear inner product
(34)
〈
ψ
(
θ
)
,
ϕ
(
θ
)
〉
=
ψ
¯
T
(
0
)
ϕ
(
0
)

∫

1
0
∫
ξ
=
0
θ
ψ
¯
T
(
ξ

θ
)
d
η
(
θ
)
ϕ
(
ξ
)
d
ξ
,
where
η
(
θ
)
=
η
(
θ
,
0
)
; then,
A
(
0
)
and
A
*
are adjoint operators. Noting that
±
i
ω
τ
0
are eigenvalues of
A
(
0
)
, thus, they are also eigenvalues of
A
*
. In order to calculate the eigenvector
q
(
θ
)
of
A
(
0
)
corresponding to the eigenvalue
i
ω
τ
0
and
p
(
s
)
of
A
*
corresponding to the eigenvalue

i
ω
τ
0
, let
q
(
θ
)
=
(
1
,
α
)
T
e
i
ω
τ
0
θ
be the eigenvector of
A
(
0
)
corresponding to
i
ω
τ
0
; then,
A
(
0
)
q
(
θ
)
=
i
ω
τ
0
q
(
θ
)
.
By the definition of
A
(
0
)
and (26), (30), then,
(35)
τ
0
(

i
ω

a
11

a
12
a
21
e

i
ω
τ
0

i
ω

a
22
e

i
ω
τ
0
)
q
(
0
)
=
(
0
0
)
.
Thus, we can get
(36)
q
(
0
)
=
(
1
,
α
)
T
=
(
1
,
a
11
+
i
ω

a
12
)
T
.
Similarly, let
p
(
s
)
=
D
(
1
,
β
)
T
e
i
ω
τ
0
s
be the eigenvector of
A
*
corresponding to

i
ω
τ
0
; by similar discussion, we get
β
=
(
a
11

i
ω
)
/
a
21
e
i
ω
τ
0
.
In view of standardization of
p
(
s
)
and
q
(
θ
)
; that is,
〈
p
(
s
)
,
q
(
θ
)
〉
=
1
, we have
(37)
〈
p
(
s
)
,
q
(
θ
)
〉
=
D
¯
(
1
,
β
¯
)
(
1
,
α
)
T
l
m
m

∫

1
0
∫
ξ
=
0
θ
D
¯
(
1
,
β
¯
)
e

i
ω
τ
0
(
ξ

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

∫

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

i
ω
τ
0
(
a
21

α
a
22
)
}
.
Thus, choose
D
=
[
1
+
β
α
¯
+
τ
0
β
e
i
ω
τ
0
(
a
21

α
¯
a
22
)
]

1
. Next, we will quote the same notation (see [13]), we first compute the coordinates to describe the center manifold
C
0
at
μ
=
0
. Define
(38)
z
(
t
)
=
〈
p
,
u
t
〉
,
W
(
t
,
θ
)
=
u
t
(
θ
)

2
Re
{
z
(
t
)
q
(
θ
)
}
.
On the center manifold
C
0
, we have
(39)
W
(
t
,
θ
)
=
W
(
z
(
t
)
,
z
¯
(
t
)
,
θ
)
=
W
20
(
θ
)
z
2
2
+
W
11
(
θ
)
z
z
¯
+
W
02
(
θ
)
z
¯
2
2
+
⋯
z
and
z
¯
are local coordinates for center manifold
C
0
in the direction
p
and
p
¯
; noting that
W
is real if
u
t
is real, we only consider real solution
u
t
∈
C
0
of (25). Since
μ
=
0
, then we have
(40)
z
˙
(
t
)
=
i
ω
τ
0
z
+
p
¯
(
0
)
f
(
0
,
W
(
z
,
z
¯
,
θ
)
+
2
Re
{
z
(
t
)
q
(
0
)
}
)
=
def
i
ω
τ
0
z
+
p
¯
(
0
)
f
0
(
z
,
z
¯
)
.
We rewrite this equation as
(41)
z
˙
(
t
)
=
i
ω
τ
0
z
+
g
(
z
,
z
¯
)
,
where
(42)
g
(
z
,
z
¯
)
=
p
¯
(
0
)
f
0
(
z
,
z
¯
)
=
g
20
(
θ
)
z
2
2
+
g
11
(
θ
)
z
z
¯
+
g
02
(
θ
)
z
¯
2
2
+
⋯
.
Noting
u
t
(
θ
)
=
(
ϕ
1
(
θ
)
,
ϕ
2
(
θ
)
)
T
=
W
(
t
,
θ
)
+
z
q
(
θ
)
+
z
¯
q
¯
(
θ
)
and
q
(
θ
)
=
(
1
,
α
)
T
e
i
ω
τ
0
θ
, we have
(43)
ϕ
1
(
0
)
=
z
+
z
¯
+
W
20
(
1
)
(
0
)
z
2
2
+
W
11
(
1
)
(
0
)
z
z
¯
+
W
02
(
1
)
(
0
)
z
¯
2
2
+
⋯
,
ϕ
2
(
0
)
=
z
α
+
z
¯
α
¯
+
W
20
(
2
)
(
0
)
z
2
2
+
W
11
(
2
)
(
0
)
z
z
¯
+
W
02
(
2
)
(
0
)
z
¯
2
2
+
⋯
,
ϕ
1
(

1
)
=
z
e

i
ω
τ
0
+
z
¯
e
i
ω
τ
0
+
W
20
(
1
)
(

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

1
)
z
z
¯
+
W
02
(
1
)
(

1
)
z
¯
2
2
+
⋯
,
ϕ
2
(

1
)
=
z
α
e

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

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

1
)
z
z
¯
+
W
02
(
2
)
(

1
)
z
¯
2
2
+
⋯
.
From (27), (42), we obtain that
(44)
g
20
=
2
D

τ
0
[
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
+
β

e
3
m
m
m
m
m
×
(

c
5
e

2
i
ω
τ
0
+
c
6
α
e

2
i
ω
τ
0
m
m
m
m
m
m
l
l
+
c
4
α
e

i
ω
τ
0

c
7
α
2
e

i
ω
τ
0
)
1
e
1
β

e
3
]
,
g
11
=
D

τ
0
{
1
e
1
[
2
c
3
+
2
c
1
α
α

+
c
2
(
α
+
α

)
]
+
β

e
3
m
m
m
m
l
l
×
[

2
c
5
+
c
6
(
α
+
α

)
+
c
4
(
α

e

i
ω
τ
0
+
α
e
i
ω
τ
0
)
m
m
m
m
m
m

c
7
(
α
α

e

i
ω
τ
0
+
α
α

e
i
ω
τ
0
)
]
β

e
3
}
,
g
02
=
2
D

τ
0
[
1
e
1
(
c
3
+
c
1
α

2
+
c
2
α

)
+
β

e
3
m
m
m
m
m
×
(

c
5
e
2
i
ω
τ
0
+
c
6
α

e
2
i
ω
τ
0
m
m
m
m
m
m
l
l
+
c
4
α

e
i
ω
τ
0

c
7
α

2
e
i
ω
τ
0
)
β

e
3
]
,
g
21
=
D

τ
0
{
1
e
1
[
e
2
c
2
e
1
c
3
(
2
W
20
(
1
)
(
0
)
+
4
W
11
(
1
)
(
0
)
)
m
m
m
m
m
m
l
l
+
c
1
(
2
α

W
20
(
2
)
(
0
)
+
4
α
W
11
(
2
)
(
0
)
)
m
m
m
m
m
m
l
l
+
c
2
(
α

W
20
(
1
)
(
0
)
+
W
20
(
2
)
(
0
)
+
2
α
W
11
(
1
)
(
0
)
m
m
m
m
m
m
m
m
m
m
+
2
W
11
(
2
)
(
0
)
)
+
6
c
1
e
2
e
1
+
2
(
α

+
2
α
)
m
m
m
m
m
m
l
l
×
e
2
c
3
e
1
+
2
(
α
2
+
2
α
α

)
e
2
c
2
e
1
]
+
β

e
3
m
m
m
m
l
l
l
×
[

c
5
(
2
e
i
ω
τ
0
W
20
(
1
)
(

1
)
+
4
e

i
ω
τ
0
W
11
(
1
)
(

1
)
)
m
m
m
m
m
m
l
+
c
6
(
α

e
i
ω
τ
0
W
20
(
1
)
(

1
)
+
e
i
ω
τ
0
W
20
(
2
)
(

1
)
m
m
m
m
m
m
m
m
m
l
+
2
e

i
ω
τ
0
W
11
(
2
)
(

1
)
+
2
α
e

i
ω
τ
0
W
11
(
1
)
(

1
)
)
m
m
m
m
m
m
l
+
c
4
(
α

W
20
(
1
)
(

1
)
+
e
i
ω
τ
0
W
20
(
2
)
(
0
)
m
m
m
m
m
m
m
m
m
l
+
2
e

i
ω
τ
0
W
11
(
2
)
(
0
)
+
2
α
W
11
(
1
)
(

1
)
)
m
m
m
m
m
m
l

c
7
(
α

W
20
(
2
)
(

1
)
+
α

e
i
ω
τ
0
W
20
(
2
)
(
0
)
m
m
m
m
m
m
m
m
m
l
+
2
α
e

i
ω
τ
0
W
11
(
2
)
(
0
)
+
2
α
W
11
(
2
)
(

1
)
)
m
m
m
m
m
m
l
+
6
e
4
c
5
e
3
e

i
ω
τ
0

2
e
4
c
6
e
3
m
m
m
m
m
m
l
×
(
α

e

i
ω
τ
0
+
2
α
e

i
ω
τ
0
)

2
e
4
c
4
e
3
m
m
m
m
m
m
l
×
(
α

e

2
i
ω
τ
0
+
2
α
)
+
2
e
4
c
7
e
3
m
m
m
m
m
m
l
×
(
α

α
e

2
i
ω
τ
0
+
α
α

+
α
2
)
2
e
i
ω
τ
0
W
20
(
1
)
(

1
)
]
1
e
1
}
.
Because
g
21
contains
W
20
and
W
11
, from (32) and (38), we have
(45)
W
˙
=
u
˙
t

z
˙
q

z
¯
˙
q

=
{
A
W

2
Re
{
p
¯
(
0
)
f
0
q
(
θ
)
}
,

1
≤
θ
<
0
A
W

2
Re
{
p
¯
(
0
)
f
0
q
(
0
)
}
+
f
0
,
θ
=
0
=
def
A
W
+
H
(
z
,
z
¯
,
θ
)
,
where
(46)
H
(
z
(
t
)
,
z
¯
(
t
)
,
θ
)
=
H
20
(
θ
)
z
2
2
+
H
11
(
θ
)
z
z
¯
+
H
02
(
θ
)
z
¯
2
2
+
⋯
.
Substituting the corresponding series into (45) and comparing the coefficients, we have
(47)
(
A

2
i
ω
τ
0
)
W
20
(
θ
)
=

H
20
(
θ
)
,
A
W
11
(
θ
)
=

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

1,0
)
, we have
(48)
H
(
z
(
t
)
,
z
¯
(
t
)
,
θ
)
=

p
¯
(
0
)
f
0
q
(
θ
)

p
(
0
)
f
¯
0
q
¯
(
θ
)
=

g
(
z
,
z
¯
)
q
(
θ
)

g
¯
(
z
,
z
¯
)
q
¯
(
θ
)
.
Comparing the coefficient with (46) yields that for
θ
∈
[

1,0
)
(49)
H
20
(
θ
)
=

g
20
(
θ
)

g
¯
02
q
¯
(
θ
)
,
(50)
H
11
(
θ
)
=

g
11
q
(
θ
)

g
¯
11
q
¯
(
θ
)
.
From (47), (49) and the definition of
A
, it follows that
(51)
W
˙
20
(
θ
)
=
2
i
ω
τ
0
W
20
(
θ
)
+
g
20
q
(
θ
)
+
g
¯
02
q
¯
(
θ
)
,
taking notice of
q
(
θ
)
=
(
1
,
α
)
T
e
i
ω
τ
0
θ
; hence,
(52)
W
20
(
θ
)
=
i
g
20
ω
τ
0
q
(
0
)
e
i
ω
τ
0
θ

g
02
¯
3
i
ω
τ
0
q
¯
(
0
)
e

i
ω
τ
0
θ
+
E
1
e
2
i
ω
τ
0
θ
,
where
E
1
=
(
E
1
(
1
)
,
E
1
(
2
)
)
∈
R
2
is a constant vector. By the similar way, we have
(53)
W
11
(
θ
)
=

i
g
11
ω
τ
0
q
(
0
)
e
i
ω
τ
0
θ

g
¯
11
i
ω
τ
0
q
¯
(
0
)
e

i
ω
τ
0
θ
+
E
2
,
where
E
2
=
(
E
2
(
1
)
,
E
2
(
2
)
)
∈
R
2
is a constant vector.
Next, computing
E
1
and
E
2
, from the definition of
A
and (47), one then obtains
(54)
∫

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

H
20
(
0
)
,
(55)
∫

1
0
d
η
(
θ
)
W
11
(
θ
)
=

H
11
(
0
)
,
where
η
(
θ
)
=
η
(
0
,
θ
)
. Furthermore, we have
(56)
H
20
(
0
)
=

g
20
q
(
0
)

g
¯
02
q
¯
(
0
)
+
2
τ
0
(
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
1
e
3
(

c
5
e

2
i
ω
τ
0
+
c
6
α
e

2
i
ω
τ
0
+
c
4
α
e

i
ω
τ
0

c
7
α
2
e

i
ω
τ
0
)
)
,
(57)
H
11
(
0
)
=

g
11
q
(
0
)

g
¯
11
q
¯
(
0
)
+
2
τ
0
(
1
e
1
(
c
3
+
c
1
α
α
¯
+
c
2
Re
{
α
}
)
1
e
3
(

c
5
+
c
6
Re
{
α
}
+
c
4
Re
{
α
e
i
ω
τ
0
}

c
7
Re
{
α
α
¯
e
i
ω
τ
0
}
)
)
.
Substituting (52) and (56) into (54) and noting that
(58)
(
i
ω
τ
0
I

∫

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

i
ω
τ
0
I

∫

1
0
e

i
ω
τ
0
θ
d
η
(
θ
)
)
q
¯
(
0
)
=
0
,
it implies that
(59)
(
2
i
ω
τ
0
I

∫

1
0
e
2
i
ω
τ
0
θ
d
η
(
θ
)
)
E
1
=
2
(
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
1
e
3
(

c
5
e

2
i
ω
τ
0
+
c
6
α
e

2
i
ω
τ
0
+
c
4
α
e

i
ω
τ
0

c
7
α
2
e

i
ω
τ
0
)
)
,
Namely,
(60)
(
2
i
ω
+
a
11
a
12

a
21
e

2
i
ω
τ
0
2
i
ω
+
a
22
e

2
i
ω
τ
0
)
E
1
=
2
(
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
1
e
3
(

c
5
e

2
i
ω
τ
0
+
c
6
α
e

2
i
ω
τ
0
+
c
4
α
e

i
ω
τ
0

c
7
α
2
e

i
ω
τ
0
)
)
.
Then it yields that
(61)
E
1
(
1
)
=
2
A
1

1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
a
12
1
e
3
[
(

c
5
+
c
6
α
)
e

2
i
ω
τ
0
+
(
c
4
α

c
7
α
2
)
e

i
ω
τ
0
]
2
i
ω
+
a
22
e

2
i
ω
τ
0

,
E
1
(
2
)
=
2
A
1

2
i
ω
+
a
11
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)

a
21
e

2
i
ω
τ
0
1
e
3
[
(

c
5
+
c
6
α
)
e

2
i
ω
τ
0
+
(
c
4
α

c
7
α
2
)
e

i
ω
τ
0
]

,
where
(62)
A
1
=

2
i
ω
+
a
11
a
12

a
21
e

2
i
ω
τ
0
2
i
ω
+
a
22
e

2
i
ω
τ
0

.
Similarly, we get
(63)
E
2
(
1
)
=
2
A
2

1
e
1
(
c
3
+
c
1
α
α
¯
+
c
2
Re
{
α
}
)
a
12
1
e
3
(

c
5
+
c
6
Re
{
α
}
+
c
4
Re
{
α
e
i
ω
τ
0
}

c
7
Re
{
α
α
¯
e
i
ω
τ
0
}
)
a
22

,
E
2
(
2
)
=
2
A
2

a
11
1
e
1
(
c
3
+
c
1
α
α
¯
+
c
2
Re
{
α
}
)

a
21
1
e
3
(

c
5
+
c
6
Re
{
α
}
+
c
4
Re
{
α
e
i
ω
τ
0
}

c
7
Re
{
α
α
¯
e
i
ω
τ
0
}
)

,
where
(64)
A
2
=

a
11
a
12

a
21
a
22

.
Through simple computation, we determine
W
20
,
W
11
from (52) and (53); further, we can determine
g
21
. Therefore,
g
i
j
in (44) can be expressed by the parameter and delay; hence,
(65)
C
1
(
0
)
=
i
2
ω
τ
0
(
g
20
g
11

2

g
11

2


g
02

2
3
)
+
g
21
2
,
μ
2
=

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

Im
{
C
1
(
0
)
}
+
μ
Im
{
λ
′
(
τ
0
)
}
ω
τ
0
,
which determine the qualities of bifurcation periodic solution of the critical value
τ
0
.
Theorem 5.
(i)
μ
2
determines the direction of Hopf bifurcation: if
μ
2
>
0
(
<
0
), then Hopf bifurcation is supercritical (subcritical), and the bifurcating periodic solutions exist for
τ
>
τ
0
(
τ
<
τ
0
).
(ii)
ζ
determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if
ζ
<
0
(
ζ
>
0
).
T
determines the period of the bifurcating periodic solution: the period increases (decrease) if
T
>
0
(
T
<
0
).