# Dynamical Analysis in a Delayed Predator-Prey System with Stage-Structure for Both the Predator and the Prey

Authors: Zizhen Zhang; Huizhong Yang
Journal: Discrete Dynamics in Nature and Society
 (2014)

---

## Abstract

A predator-prey system with two delays and stage-structure for both the predator and the prey
is considered. Sufficient conditions for the local stability and the existence of periodic solutions
via Hopf bifurcation with respect to both delays are obtained by analyzing the distribution
of the roots of the associated characteristic equation. Specially, the direction of the Hopf
bifurcation and the stability of the periodic solutions bifurcating from the Hopf bifurcation are
determined by applying the normal form theory and center manifold argument. Some numerical
simulations for justifying the theoretical analysis are also provided.


---

## Body

## 1. Introduction

The predator-prey systems with stage-structure have been studied by many authors [1–5]. In [2], Aiello and Freedman built a stage-structured model of single species:
(1)
d
x
i
(
t
)
d
t
=
α
x
m
(
t
)
-
γ
x
i
(
t
)
-
α
e
-
γ
τ
x
m
(
t
-
τ
)
,
d
x
m
(
t
)
d
t
=
α
e
-
γ
τ
x
m
(
t
-
τ
)
-
β
x
m
2
(
t
)
,
where x
i
(
t
) and x
m
(
t
) denote the densities of the immature population and the mature population at time t, respectively. However, it is well known that the predator-prey relationship often appears in population ecology. Multispecies predator-prey systems with stage structure are very important and have received much attention in recent years [6–16].In [13], Xu investigated a predator-prey system with stage structure for the predator:
(2)
d
x
(
t
)
d
t
=
x
(
t
)
(
r
-
a
x
(
t
)
-
a
1
y
2
(
t
)
1
+
m
x
(
t
)
)
,
d
y
1
(
t
)
d
t
=
a
2
x
(
t
-
τ
)
y
2
(
t
-
τ
)
1
+
m
x
(
t
-
τ
)
-
r
1
y
1
(
t
)
-
D
y
1
(
t
)
,
d
y
2
(
t
)
d
t
=
D
y
1
(
t
)
-
r
2
y
2
(
t
)
,
where x
(
t
) denotes the density of the prey at time t. y
1
(
t
) and y
2
(
t
) denote the densities of the immature predator and the mature predator at time t, respectively. Xu [13] proved that system (2) is permanent under certain conditions by means of the persistence theory on infinite-dimensional systems. And sufficient conditions were derived for the local and global stability of the coexistence equilibrium of the system. In [15], F. Li and H. Li investigated a predator-prey system with stage structure for the prey:
(3)
d
x
1
(
t
)
d
t
=
a
x
2
(
t
)
-
r
1
x
1
(
t
)
-
b
x
1
(
t
)
,
d
x
2
(
t
)
d
t
=
b
x
1
(
t
)
-
r
2
x
2
(
t
)
-
b
1
x
2
2
(
t
)
-
a
1
x
2
2
(
t
)
y
(
t
)
1
+
m
x
2
2
(
t
)
,
d
y
(
t
)
d
t
=
a
2
x
2
2
(
t
-
τ
)
y
(
t
-
τ
)
1
+
m
x
2
2
(
t
-
τ
)
-
r
y
(
t
)
,
where x
1
(
t
) and x
2
(
t
) denote the densities of the immature prey and the mature prey at time t, respectively. y
(
t
) denotes the density of the predator at time t. F. Li and H. Li [15] studied the effect of the gestation time of the predator on the dynamics of system (3).Since both the predator and the prey have a life history that takes them through immature stage and mature stage, it is reasonable to consider the predator-prey system with stage-structure for both the predator and the prey. Starting from this point, Wang [17] proposed the following delayed system:
(4)
d
x
1
(
t
)
d
t
=
r
x
2
(
t
)
-
a
x
1
2
(
t
)
-
(
d
1
+
r
1
)
x
1
(
t
)
-
a
1
x
1
(
t
)
y
2
(
t
)
,
d
x
2
(
t
)
d
t
=
r
1
x
1
(
t
)
-
d
2
x
2
(
t
)
,
d
y
1
(
t
)
d
t
=
a
2
x
1
(
t
-
τ
)
y
2
(
t
-
τ
)
-
(
d
3
+
r
2
)
y
1
(
t
)
,
d
y
2
(
t
)
d
t
=
r
2
y
1
(
t
)
-
d
4
y
2
(
t
)
,
where x
1
(
t
) and x
2
(
t
) denote the densities of the immature prey and the mature prey at time t, respectively. y
1
(
t
) and y
2
(
t
) denote the densities of the immature predator and the mature predator at time t, respectively. a is the intraspecific competition coefficient of the immature prey. a
1 and a
2 are the interspecific interaction coefficients between the immature prey and the mature predator. d
1 and d
2 are the death rates of the immature prey and the mature prey, respectively. d
3 and d
4 are the death rates of the immature predator and the mature predator, respectively. r is the birth rate of the immature prey. r
1 and r
2 are the transformation rates from immature individuals to mature individuals for the prey and the predator, respectively. All the parameters in system (4) are assumed to be positive constants. And τ
≥
0 is a constant delay due to the gestation of the mature predator. Wang [17] considered the bifurcation phenomenon and the properties of periodic solutions of system (4).The predator-prey systems with single delay have been investigated by many researchers. There are also many papers on the bifurcations of predator-prey systems with two or multiple delays [18–24]. Cui and Yan [20] investigated the stability and Hopf bifurcations of a three-species Lotka-Volterra food chain system by taking the sum of the two delays in the system as the bifurcation parameter. Gakkhar and Singh [23] investigated a modified Leslie-Gower predator-prey system with two delays and obtained the sufficient conditions for existence of Hopf bifurcation for possible combinations of the two delays. Motivated by the work above, we consider the following predator-prey system with two delays:
(5)
d
x
1
(
t
)
d
t
=
r
x
2
(
t
)
-
a
x
1
(
t
)
x
1
(
t
-
τ
1
)
-
(
d
1
+
r
1
)
x
1
(
t
)
-
a
1
x
1
(
t
)
y
2
(
t
)
,
d
x
2
(
t
)
d
t
=
r
1
x
1
(
t
)
-
d
2
x
2
(
t
)
,
d
y
1
(
t
)
d
t
=
a
2
x
1
(
t
-
τ
2
)
y
2
(
t
-
τ
2
)
-
(
d
3
+
r
2
)
y
1
(
t
)
,
d
y
2
(
t
)
d
t
=
r
2
y
1
(
t
)
-
d
4
y
2
(
t
)
,
where τ
1
≥
0 is feedback delay of the immature prey to the growth of the species itself and τ
2
≥
0 is the time delay due to the gestation of the mature predator.The rest of this paper is organized as follows. In Section2, sufficient conditions are established for the local stability of the positive equilibrium and the existence of Hopf bifurcation for possible combinations of the two delays in system (5). Section 3 is devoted to the properties of the Hopf bifurcation on the normal form theory and center manifold argument. Numerical simulations supporting the theoretical analysis are presented in Section 4. Finally, conclusions are given in Section 5.


## 2. Local Stability and Existence of Hopf Bifurcation

Considering the significance of ecology, we are interested only in positive equilibrium of system (5). It is not difficult to verify that if the condition (
H
):
(6)
a
2
r
r
1
r
2
>
a
d
2
d
4
(
d
3
+
r
2
)
+
a
2
d
2
r
2
(
d
1
+
r
1
)
holds, then system (5) has a unique positive equilibrium E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
), where
(7)
x
1
*
=
d
4
(
d
3
+
r
2
)
a
2
r
2
,
x
2
*
=
d
4
r
1
(
d
3
+
r
2
)
a
2
d
2
r
2
,
y
1
*
=
d
4
r
2
y
2
*
,
y
2
*
=
r
r
1
-
d
2
(
d
1
+
r
1
)
a
1
d
2
-
a
d
4
(
d
3
+
r
2
)
a
1
a
2
r
2
.
Let x
-
1
=
x
1
-
x
1
*, x
-
2
=
x
2
-
x
2
*, y
-
1
=
y
1
-
y
1
*, and y
-
2
=
y
2
-
y
2
*. Dropping the bars, system (5) can be transformed to the following form:
(8)
d
x
1
(
t
)
d
t
=
a
11
x
1
(
t
)
+
a
12
x
2
(
t
)
+
a
14
y
2
(
t
)
+
b
11
x
1
(
t
-
τ
1
)
+
g
1
x
1
(
t
)
y
2
(
t
)
+
g
2
x
1
(
t
)
x
1
(
t
-
τ
1
)
,
d
x
2
(
t
)
d
t
=
a
21
x
1
(
t
)
+
a
22
x
2
(
t
)
,
d
y
1
(
t
)
d
t
=
a
33
y
1
(
t
)
+
c
31
x
1
(
t
-
τ
2
)
+
c
34
y
2
(
t
-
τ
2
)
+
h
1
x
1
(
t
-
τ
2
)
y
2
(
t
-
τ
2
)
,
d
y
2
(
t
)
d
t
=
a
43
y
1
(
t
)
+
a
44
y
2
(
t
)
,
where
(9)
a
11
=
-
a
x
1
*
-
a
1
y
2
*
-
(
d
1
+
r
1
)
,
a
12
=
r
,
a
14
=
-
a
1
x
1
*
,
a
21
=
r
1
,
a
22
=
-
d
2
,
a
33
=
-
(
d
3
+
r
2
)
,
a
43
=
r
2
,
a
44
=
-
d
4
,
b
11
=
-
a
x
1
*
,
c
31
=
a
2
y
2
*
,
c
34
=
a
2
x
1
*
,
g
1
=
-
a
1
,
g
2
=
-
a
,
h
1
=
a
2
.
The linear system of system (8) is
(10)
d
x
1
(
t
)
d
t
=
a
11
x
1
(
t
)
+
a
12
x
2
(
t
)
+
a
14
y
2
(
t
)
+
b
11
x
1
(
t
-
τ
1
)
,
d
x
2
(
t
)
d
t
=
a
21
x
1
(
t
)
+
a
22
x
2
(
t
)
,
d
y
1
(
t
)
d
t
=
a
33
y
1
(
t
)
+
c
31
x
1
(
t
-
τ
2
)
+
c
34
y
2
(
t
-
τ
2
)
,
d
y
2
(
t
)
d
t
=
a
43
y
1
(
t
)
+
a
44
y
2
(
t
)
.
The characteristic equation of system (10) is as follows:
(11)
λ
4
+
m
3
λ
3
+
m
2
λ
2
+
m
1
λ
+
m
0
+
(
n
3
λ
3
+
n
2
λ
2
+
n
1
λ
+
n
0
)
e
-
λ
τ
1
+
(
p
2
λ
2
+
p
1
λ
+
p
0
)
e
-
λ
τ
2
+
(
q
1
λ
+
q
0
)
e
-
λ
(
τ
1
+
τ
2
)
=
0
,
where
(12)
m
3
=
-
(
a
11
+
a
22
+
a
33
+
a
44
)
,
m
2
=
a
11
a
22
+
a
33
a
44
-
a
12
a
21
+
(
a
11
+
a
22
)
(
a
33
+
a
44
)
,
m
1
=
(
a
12
a
21
-
a
11
a
22
)
(
a
33
+
a
44
)
-
(
a
11
+
a
22
)
a
33
a
44
,
m
0
=
(
a
11
a
22
-
a
12
a
21
)
a
33
a
44
,
n
3
=
-
b
11
,
n
2
=
(
a
22
+
a
33
+
a
44
)
b
11
,
n
1
=
-
a
22
b
11
(
a
33
+
a
44
)
-
a
33
a
44
b
11
,
n
0
=
a
22
a
33
a
44
b
11
,
p
2
=
-
a
43
c
34
,
p
1
=
(
a
11
+
a
22
)
a
43
c
34
-
a
14
a
43
c
31
,
p
0
=
(
a
12
a
21
-
a
11
a
22
)
a
43
c
34
+
a
14
a
22
a
43
c
31
,
q
1
=
a
43
b
11
c
34
,
q
0
=
-
a
22
a
43
b
11
c
34
.Case 1 (τ
1
=
τ
2
=
0).
Forτ
1
=
τ
2
=
0, (11) can be rewritten in the following form:
(13)
λ
4
+
A
13
λ
3
+
A
12
λ
2
+
A
11
λ
+
A
10
=
0
,
where
(14)
A
13
=
m
3
+
n
3
=
2
a
x
1
*
+
a
1
y
2
*
+
d
1
+
d
2
+
d
3
+
d
4
+
r
1
+
r
2
,
A
12
=
m
2
+
n
2
+
p
2
,
A
11
=
m
1
+
n
1
+
p
1
+
q
1
,
A
10
=
m
0
+
n
0
+
p
0
+
q
0
.
Clearly,A
13
>
0. Therefore, by the Routh-Hurwitz theorem, if the conditions (
H
11
)
A
10
>
0, A
13
A
12
>
A
11 and (
H
12
)
A
11
A
12
A
13
>
A
11
2
+
A
10
A
13
2 hold, then the positive equilibrium E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) of system (5) without time delay is locally asymptotically stable.Case 2 (τ
1
>
0
,
τ
2
=
0).
On substitutingτ
2
=
0, (11) becomes
(15)
λ
4
+
A
23
λ
3
+
A
22
λ
2
+
A
21
λ
+
A
20
+
(
B
23
λ
3
+
B
22
λ
2
+
B
21
λ
+
B
20
)
e
-
λ
τ
1
=
0
,
where
(16)
A
23
=
m
3
,
A
22
=
m
2
+
p
2
,
A
21
=
m
1
+
p
1
,
A
20
=
m
0
+
p
0
,
B
23
=
n
3
,
B
22
=
n
2
,
B
21
=
n
1
+
q
1
,
B
20
=
n
0
+
q
0
.Letλ
=
i
ω
1
(
ω
1
>
0
) be the root of (15). Then, we have
(17)
(
B
21
ω
1
-
B
23
ω
1
3
)
sin
τ
1
ω
1
+
(
B
20
-
B
22
ω
1
2
)
cos
⁡
τ
1
ω
1
=
A
22
ω
1
2
-
ω
1
4
-
A
20
,
(
B
21
ω
1
-
B
23
ω
1
3
)
cos
⁡
τ
1
ω
1
-
(
B
20
-
B
22
ω
1
2
)
sin
τ
1
ω
1
=
A
23
ω
1
3
-
A
21
ω
1
,which follows that(18)
ω
1
8
+
e
23
ω
1
6
+
e
22
ω
1
4
+
e
21
ω
1
+
e
20
=
0with(19)
e
23
=
A
23
2
-
B
23
2
-
2
A
22
,
e
22
=
A
22
2
-
B
22
2
+
2
B
21
B
23
-
2
A
21
A
23
+
2
A
20
,
e
21
=
A
21
2
-
B
21
2
-
2
A
20
A
22
+
2
B
20
B
22
,
e
20
=
A
20
2
-
B
20
2
.Letω
1
2
=
v
1, then (18) becomes
(20)
v
1
4
+
e
23
v
1
3
+
e
22
v
1
2
+
e
21
v
1
+
e
20
=
0
.Define(21)
f
1
(
v
1
)
=
v
1
4
+
e
23
v
1
3
+
e
22
v
1
2
+
e
21
v
1
+
e
20
,
p
=
1
2
e
22
-
3
16
e
23
2
,
q
=
1
32
e
23
3
-
1
8
e
22
e
23
+
e
21
,
α
1
=
(
q
2
)
3
+
(
p
3
)
3
,
β
1
=
-
1
2
+
3
2
i
,
y
1
=
-
q
2
+
α
1
3
+
-
q
2
-
α
1
3
,
y
2
=
β
1
-
q
2
+
α
1
3
+
β
1
2
-
q
2
-
α
1
3
,
y
3
=
β
1
2
-
q
2
+
α
1
3
+
β
1
-
q
2
-
α
1
3
,
v
1
i
=
y
i
-
3
e
23
4
,
i
=
1,2
,
3
.Discussion about the roots of (20) is similar to that in [25]. Therefore, we have the following lemma.Lemma 1.
For (20), one has the following.(i)
Ife
20
<
0, (20) has at least one positive root.
(ii)
Ife
20
≥
0 and α
1
≥
0, (20) has positive roots if and only if v
11
>
0 and f
1
(
v
11
)
<
0.
(iii)
Ife
20
≥
0 and α
1
<
0, (20) has positive roots if and only if there exists at least one v
1
*
∈
{
v
11
,
v
12
,
v
13
} such that v
1
*
>
0 and f
1
(
v
1
*
)
≤
0.
In what follows, one assumes that(
H
21
): the coefficients in f
1
(
v
1
) satisfy one of the following conditions in (
a
)
–
(
c
):
(a)e
20
<
0;
(b)e
20
≥
0, α
1
≥
0, v
11
>
0 and f
1
(
v
11
)
<
0;
(c)e
20
≥
0, α
1
<
0, and there exists at least one v
1
*
∈
{
v
11
,
v
12
,
v
13
} such that v
1
*
>
0 and f
1
(
v
1
*
)
≤
0.
If(
H
21
) holds, one knows that (18) has at least a positive root ω
10 such that (15) has a pair of purely imaginary roots ±
i
ω
10. The corresponding critical value of the delay is
(22)
τ
1
k
=
1
ω
10
×
arccos
{
{
+
B
20
2
)
-
1
}
(
(
B
22
-
A
23
B
23
)
ω
10
6
+
(
A
21
B
23
+
A
23
B
21
-
A
22
B
22
-
B
20
)
ω
10
4
(
(
B
22
-
A
23
B
23
)
ω
10
6
)
×
(
B
23
2
ω
10
6
+
(
B
22
2
-
2
B
21
B
23
)
ω
10
4
+
(
B
21
2
-
2
B
20
B
22
)
ω
10
2
+
B
20
2
)
-
1
+
(
(
A
20
B
22
+
A
22
B
20
-
A
21
B
21
)
ω
10
2
-
A
20
B
20
(
A
20
B
22
+
A
22
B
20
-
A
21
B
21
)
)
×
(
B
23
2
ω
10
6
+
(
B
22
2
-
2
B
21
B
23
)
ω
10
4
+
(
B
21
2
-
2
B
20
B
22
)
ω
10
2
+
B
20
2
)
-
1
(
(
B
22
-
A
23
B
23
)
ω
10
6
}
+
2
k
π
ω
10
,
k
=
0,1
,
2
,
…
.
Differentiating both sides of (15) regarding τ
1, one gets
(23)
[
d
λ
d
τ
1
]
-
1
=
-
4
λ
3
+
3
A
23
λ
2
+
2
A
22
λ
+
A
21
λ
(
λ
4
+
A
23
λ
3
+
A
22
λ
2
+
A
21
λ
+
A
20
)
+
3
B
23
λ
2
+
2
B
22
λ
+
B
21
λ
(
B
23
λ
3
+
B
22
λ
2
+
B
21
λ
+
B
20
)
-
τ
1
λ
.
Thus,
(24)
Re
[
d
λ
d
τ
1
]
τ
1
=
τ
10
-
1
=
f
1
′
(
v
1
*
)
×
(
B
23
2
ω
10
6
+
(
B
22
2
-
2
B
21
B
23
)
ω
10
4
+
(
B
21
2
-
2
B
20
B
22
)
ω
10
2
+
B
20
2
)
-
1
,
where v
1
*
=
ω
10
2. Therefore, if the condition (
H
22
)
:
f
1
′
(
v
1
*
)
≠
0 holds, then Re
[
d
λ
/
d
τ
1
]
τ
1
=
τ
10
-
1
≠
0. Namely, if the condition (
H
22
)
:
f
1
′
(
v
1
*
)
≠
0 holds, the transversality condition is satisfied. By the Hopf bifurcation theorem in [26], one has the following results.Theorem 2.
If the conditions(
H
21
)
-
(
H
22
) hold, then
(i) the positive equilibriumE
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) of system (5) is asymptotically stable for τ
1
∈
[
0
,
τ
10
);
(ii) system (5) undergoes a Hopf bifurcation at the equilibrium E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) when τ
1
=
τ
10, and a family of periodic solutions bifurcate from E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) near τ
1
=
τ
10.Case 3 (τ
1
=
0
,
τ
2
>
0).
On substitutingτ
1
=
0, (11) becomes
(25)
λ
4
+
A
33
λ
3
+
A
32
λ
2
+
A
31
λ
+
A
30
+
(
B
32
λ
2
+
B
31
λ
+
B
30
)
e
-
λ
τ
2
=
0
,
where
(26)
A
33
=
m
3
+
n
3
,
A
32
=
m
2
+
n
2
,
A
31
=
m
1
+
n
1
,
A
30
=
m
0
+
n
0
,
B
32
=
p
2
,
B
31
=
p
1
+
q
1
,
B
30
=
p
0
+
q
0
.
Let λ
=
i
ω
2
(
ω
2
>
0
) be the root of (25). Then, we have
(27)
B
31
sin
τ
2
ω
2
+
(
B
30
-
B
32
ω
2
2
)
cos
⁡
τ
2
ω
2
=
A
32
ω
2
2
-
ω
2
4
-
A
30
,
B
31
cos
⁡
τ
2
ω
2
-
(
B
30
-
B
32
ω
2
2
)
sin
τ
2
ω
2
=
A
33
ω
2
3
-
A
31
ω
2
.
It follows that
(28)
ω
2
8
+
e
33
ω
2
6
+
e
32
ω
2
4
+
e
31
ω
2
+
e
30
=
0
,
where
(29)
e
33
=
A
33
2
-
2
A
32
,
e
32
=
A
32
2
-
B
32
2
+
2
A
30
-
2
A
31
A
33
,
e
31
=
A
31
2
-
B
31
2
-
2
A
30
A
32
+
2
B
30
B
32
,
e
30
=
A
30
2
-
B
30
2
.
Let ω
2
2
=
v
2, then (28) becomes
(30)
v
2
4
+
e
33
v
2
3
+
e
32
v
2
2
+
e
31
v
2
+
e
30
=
0
.
Define
(31)
f
2
(
v
2
)
=
v
2
4
+
e
33
v
2
3
+
e
32
v
2
2
+
e
31
v
2
+
e
30
,
m
=
1
2
e
32
-
3
16
e
33
2
,
n
=
1
32
e
33
3
-
1
8
e
32
e
33
+
e
31
,
α
2
=
(
n
2
)
3
+
(
m
3
)
3
,
β
2
=
-
1
2
+
3
2
i
,
z
1
=
-
n
2
+
α
2
3
+
-
n
2
-
α
2
3
,
z
2
=
β
2
-
n
2
+
α
2
3
+
β
2
2
-
n
2
-
α
2
3
,
z
3
=
β
2
2
-
n
2
+
α
2
3
+
β
2
-
n
2
-
α
2
3
,
v
2
i
=
z
i
-
3
e
33
4
,
i
=
1,2
,
3
.According to Lemma1, we know that if (
H
31
) the coefficients in f
2
(
v
2
) satisfy one of the following conditions in (
a
′
)
–
(
c
′
):(a′)
e
30
<
0;
(b′)
e
30
≥
0, α
2
≥
0, v
21
>
0, and f
2
(
v
21
)
<
0;
(c′)
e
30
≥
0, α
2
<
0, and there exists at least one v
2
*
∈
{
v
21
,
v
22
,
v
23
} such that v
2
*
>
0 and f
2
(
v
2
*
)
≤
0.Then, (28) has at least a positive roots ω
20 such that (25) has a pair of purely imaginary roots ±
i
ω
20. The corresponding critical value of the delay is
(32)
τ
2
k
=
1
ω
20
<
arccos
⁡
(
ω
20
2
+
B
30
2
)
-
1
(
B
32
ω
20
6
+
(
A
33
B
31
-
A
32
B
32
-
B
30
)
ω
20
4
+
(
A
30
B
32
+
A
32
B
30
-
A
31
B
31
)
ω
20
2
)
×
(
B
32
2
ω
20
4
+
(
B
31
2
-
2
B
30
B
32
)
×
ω
20
2
+
B
30
2
)
-
1
)
+
2
k
π
ω
20
,
k
=
0,1
,
2
,
…
.
Similarly as in Case 2, we can conclude that if the condition (
H
32
)
f
2
′
(
v
2
*
)
≠
0
,
(
v
2
*
=
ω
20
2
), then Re
[
d
λ
/
d
τ
2
]
τ
2
=
τ
20
-
1
≠
0. That is, if the condition (
H
32
)
f
2
′
(
v
2
*
)
≠
0 holds, the transversality condition is satisfied. According to the Hopf bifurcation theorem in [26], we obtain the following results.Theorem 3.
If the conditions(
H
31
)
-
(
H
32
) hold, then(i)
the positive equilibriumE
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) of system (5) is asymptotically stable for τ
2
∈
[
0
,
τ
20
);
(ii)
system (5) undergoes a Hopf bifurcation at the positive equilibrium E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) when τ
2
=
τ
20, and a family of periodic solutions bifurcate from E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) near τ
2
=
τ
20.Case 4 (τ
1
=
τ
2
=
τ
>
0).
Forτ
1
=
τ
2
=
τ
>
0, (11) can be transformed into the following form:
(33)
λ
4
+
A
43
λ
3
+
A
42
λ
2
+
A
41
λ
+
A
40
+
(
B
43
λ
3
+
B
42
λ
2
+
B
41
λ
+
B
40
)
e
-
λ
τ
+
(
C
41
λ
+
C
40
)
e
-
2
λ
τ
=
0
,
where
(34)
A
43
=
m
3
,
A
42
=
m
2
,
A
41
=
m
1
,
A
40
=
m
0
,
B
43
=
n
3
,
B
42
=
n
2
+
p
2
,
B
41
=
n
1
+
p
1
,
B
40
=
n
0
+
p
0
,
C
41
=
q
1
,
C
40
=
q
0
.
Multiplying by e
λ
τ, (33) becomes
(35)
(
λ
4
+
A
43
λ
3
+
A
42
λ
2
+
A
41
λ
+
A
40
)
e
λ
τ
+
(
C
41
λ
+
C
40
)
e
-
λ
τ
+
B
43
λ
3
+
B
42
λ
2
+
B
41
λ
+
B
40
=
0
.
Let λ
=
i
ω
(
ω
>
0
) be the root of (35). Then, we can get
(36)
(
ω
4
-
A
42
ω
2
+
A
40
+
C
40
)
cos
⁡
τ
ω
+
(
A
43
ω
3
-
A
41
ω
+
C
41
ω
)
sin
τ
ω
=
B
42
ω
2
-
B
40
,
(
ω
4
-
A
42
ω
2
+
A
40
-
C
40
)
sin
τ
ω
-
(
A
43
ω
3
-
A
41
ω
-
C
41
ω
)
cos
⁡
τ
ω
=
B
43
ω
3
-
B
41
ω
.
Then, we obtain
(37)
sin
τ
ω
=
A
7
ω
7
+
A
5
ω
5
+
A
3
ω
3
+
A
1
ω
ω
8
+
B
6
ω
6
+
B
4
ω
4
+
B
2
ω
2
+
B
0
,
cos
⁡
τ
ω
=
A
6
ω
6
+
A
4
ω
4
+
A
2
ω
2
+
A
0
ω
8
+
B
6
ω
6
+
B
4
ω
4
+
B
2
ω
2
+
B
0
,
where
(38)
A
0
=
(
C
40
-
A
40
)
B
40
,
A
1
=
(
A
41
+
C
41
)
B
40
-
(
A
40
+
C
40
)
B
41
,
A
2
=
A
40
B
42
+
A
42
B
40
+
B
41
C
41
-
A
41
B
41
-
B
42
C
40
,
A
3
=
A
40
B
43
+
A
42
B
41
+
B
43
C
40
-
A
41
B
42
-
A
43
B
40
-
B
42
C
41
,
A
4
=
A
41
B
43
+
A
43
B
41
-
A
42
B
42
-
B
43
C
41
-
B
40
,
A
5
=
A
43
B
42
-
A
42
B
43
-
B
41
,
A
6
=
B
42
-
A
43
B
43
,
A
7
=
B
43
,
B
0
=
A
40
2
-
C
40
2
,
B
2
=
A
41
2
-
C
41
2
-
2
A
40
A
42
,
B
4
=
A
42
2
+
2
A
40
-
2
A
41
A
43
,
B
6
=
A
43
2
-
2
A
42
.
It is well known that sin
2
τ
ω
+
cos
⁡
2
τ
ω
=
1. Therefore, we have
(39)
ω
16
+
e
47
ω
14
+
e
46
ω
12
+
e
45
ω
10
+
e
44
ω
8
+
e
43
ω
6
+
e
42
ω
4
+
e
41
ω
2
+
e
40
=
0
,
where
(40)
e
47
=
2
B
6
-
A
7
2
,
e
46
=
B
6
2
-
A
6
2
+
2
B
4
-
2
A
5
A
7
,
e
45
=
2
B
2
+
2
B
4
B
6
-
A
5
2
-
2
A
3
A
7
-
2
A
4
A
6
,
e
44
=
B
4
2
+
2
B
0
+
2
B
2
B
6
-
A
4
2
-
2
A
1
A
7
-
2
A
3
A
5
-
2
A
2
A
6
,
e
43
=
2
B
0
B
6
+
2
B
2
B
4
-
2
A
1
A
5
-
2
A
0
A
6
-
2
A
2
A
4
-
A
3
2
,
e
42
=
B
2
2
-
A
2
2
+
2
B
0
B
4
-
2
A
1
A
3
-
2
A
0
A
4
,
e
42
=
2
B
0
B
2
-
2
A
0
A
2
-
A
1
2
,
e
40
=
B
0
2
-
A
0
2
.
Let ω
2
=
v, then (39) becomes
(41)
v
8
+
e
47
v
7
+
e
46
v
6
+
e
45
v
5
+
e
44
v
4
+
e
43
v
3
+
e
42
v
2
+
e
41
v
+
e
40
=
0
.If all the parameters of system (5) are given, it is easy to calculate the roots of (41). Thus, we suppose the following:(
H
41
) Equation (41) has at least one positive real root.Without loss of generality, we assume that (41) has eight positive real roots, which are denoted as v
1, v
2, v
3, …, v
8. Then (39) has eight positive roots ω
k
=
v
k
,
k
=
1,2
,
3
,
…
,
8. And for every fixed ω
k, k
=
1,2
,
3
,
…
,
8, the corresponding critical value of time delay τ
k
(
j
) is
(42)
τ
k
(
j
)
=
1
ω
k
arccos
A
6
ω
k
6
+
A
4
ω
k
4
+
A
2
ω
k
2
+
A
0
ω
k
8
+
B
6
ω
k
6
+
B
4
ω
k
4
+
B
2
ω
k
2
+
B
0
+
2
j
π
ω
k
,
k
=
1,2
,
3
,
…
,
8
;
j
=
0,1
,
2
,
…
Let
(43)
τ
0
=
min
⁡
{
τ
k
(
0
)
}
,
ω
0
=
ω
k
∣
τ
=
τ
0
,
k
=
1,2
,
3
,
…
,
8
.Next, we verify the transversality condition. Differentiating both sides of (35) with respect to τ, we get
(44)
[
d
λ
d
τ
]
-
1
=
-
(
-
(
C
41
λ
+
C
40
)
e
-
λ
τ
)
)
-
1
(
(
4
λ
3
+
3
A
43
λ
2
+
2
A
42
λ
+
A
41
)
e
λ
τ
+
C
41
e
-
λ
τ
+
3
B
43
λ
2
+
2
B
42
λ
+
B
41
)
×
(
λ
(
(
λ
4
+
A
43
λ
3
+
A
42
λ
2
+
A
41
λ
+
A
40
)
×
e
λ
τ
-
(
C
41
λ
+
C
40
)
e
-
λ
τ
)
)
-
1
)
-
τ
λ
.
Thus,
(45)
Re
[
d
λ
d
τ
]
τ
=
τ
0
-
1
=
-
P
41
Q
41
+
P
42
Q
42
Q
41
2
+
Q
42
2
,
where
(46)
P
41
=
(
A
41
+
C
41
-
3
A
43
ω
0
2
)
cos
⁡
τ
0
ω
0
-
(
2
A
42
ω
0
-
4
ω
0
3
)
sin
τ
0
ω
0
-
3
B
43
ω
0
2
+
B
41
,
P
42
=
(
A
41
-
C
41
-
3
A
43
ω
0
2
)
sin
τ
0
ω
0
+
(
2
A
42
ω
0
-
4
ω
0
3
)
cos
⁡
τ
0
ω
0
+
2
B
42
ω
0
,
Q
41
=
(
A
43
ω
0
4
-
A
41
ω
0
2
-
C
41
ω
0
2
)
cos
⁡
τ
0
ω
0
-
(
ω
0
5
-
A
42
ω
0
3
+
A
40
ω
0
+
C
40
ω
0
)
sin
τ
0
ω
0
,
Q
42
=
(
A
43
ω
0
4
-
A
41
ω
0
2
+
C
41
ω
0
2
)
sin
τ
0
ω
0
+
(
ω
0
5
-
A
42
ω
0
3
+
A
40
ω
0
-
C
40
ω
0
)
cos
⁡
τ
0
ω
0
.Obviously, if the condition(
H
42
)
:
P
41
Q
41
+
P
42
Q
42
≠
0 holds, then Re
[
d
λ
/
d
τ
]
τ
=
τ
0
-
1
≠
0. Namely, if the condition (
H
42
)
:
P
41
Q
41
+
P
42
Q
42
≠
0 holds, the transversality condition is satisfied. In conclusion, we have the following results.Theorem 4.
If the conditions(
H
41
)
-
(
H
42
) hold, then(i)
the positive equilibriumE
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) of system (5) is asymptotically stable for τ
∈
[
0
,
τ
0
);
(ii)
system (5) undergoes a Hopf bifurcation at the equilibrium E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) when τ
=
τ
0, and a family of periodic solutions bifurcate from E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) near τ
=
τ
0.Case 55 (τ
1
>
0
,
τ
2
>
0 and τ
1
∈
(
0
,
τ
10
)).
We consider system (5) under Case 2. That is, τ
1 is in its stable interval and τ
2 is considered as a parameter. Let λ
=
i
ω
2
′
(
ω
2
′
>
0
) be the root of (11). Then, we can get
(47)
Δ
51
sin
τ
2
ω
2
′
+
Δ
52
cos
⁡
τ
2
ω
2
′
=
Δ
53
,
Δ
51
cos
⁡
τ
2
ω
2
′
-
Δ
52
sin
τ
2
ω
2
′
=
Δ
54
,
where
(48)
Δ
51
=
q
1
ω
2
′
cos
⁡
τ
1
ω
2
′
-
q
0
sin
τ
1
ω
2
′
+
p
1
ω
2
′
,
Δ
52
=
q
1
ω
2
′
sin
τ
1
ω
2
′
+
q
0
cos
⁡
τ
1
ω
2
′
-
p
2
(
ω
2
′
)
2
+
p
0
,
Δ
53
=
(
n
2
(
ω
2
′
)
2
-
n
0
)
cos
⁡
τ
1
ω
2
′
+
(
n
3
(
ω
2
′
)
3
-
n
1
ω
2
′
)
×
sin
τ
1
ω
2
′
-
(
ω
2
′
)
4
+
m
2
(
ω
2
′
)
2
-
m
0
,
Δ
54
=
(
n
0
-
n
2
(
ω
2
′
)
2
)
sin
τ
1
ω
2
′
+
(
n
3
(
ω
2
′
)
3
-
n
1
ω
2
′
)
×
cos
⁡
τ
1
ω
2
′
+
m
3
(
ω
2
′
)
3
-
m
1
ω
2
′
.
It follows that(49)
c
0
(
ω
2
′
)
+
c
1
(
ω
2
′
)
cos
⁡
τ
1
ω
2
′
+
c
2
(
ω
2
′
)
sin
τ
1
ω
2
′
=
0
with
(50)
c
0
(
ω
2
′
)
=
(
ω
2
′
)
8
+
(
m
3
2
+
n
3
2
-
2
m
2
)
(
ω
2
′
)
6
+
(
m
2
2
+
n
2
2
-
p
2
2
+
2
m
0
-
2
m
1
m
3
-
2
n
1
n
3
)
(
ω
2
′
)
4
+
(
m
1
2
+
n
1
2
-
p
1
2
-
q
1
2
-
2
m
0
m
2
-
2
n
0
n
2
+
2
p
0
p
2
m
1
2
+
n
1
2
-
p
1
2
-
q
1
2
-
2
m
0
m
2
)
(
ω
2
′
)
2
+
m
0
2
+
n
0
2
-
p
0
2
-
q
0
2
,
c
1
(
ω
2
′
)
=
2
(
m
3
n
3
-
n
2
)
(
ω
2
′
)
6
+
2
(
m
2
n
2
+
n
0
-
m
3
n
1
-
m
1
n
3
)
(
ω
2
′
)
4
+
2
(
m
1
n
1
-
m
0
n
2
-
m
2
n
0
-
p
1
q
1
+
p
2
q
0
)
(
ω
2
′
)
2
+
2
(
m
0
n
0
-
p
0
q
0
)
,
c
2
(
ω
2
′
)
=
-
2
n
3
(
ω
2
′
)
7
+
2
(
m
2
n
3
-
m
3
n
2
+
n
1
)
(
ω
2
′
)
5
+
2
(
m
3
n
0
+
m
1
n
2
-
m
0
n
3
-
m
2
n
1
+
p
2
q
1
)
(
ω
2
′
)
3
+
2
(
m
0
n
1
-
m
1
n
0
+
p
1
q
0
-
p
0
q
1
)
ω
2
′
.Suppose that(
H
51
): Equation (49) has at least finite positive real roots. If the condition (
H
51
) holds, we denote the positive roots of (49) as ω
21
′, ω
22
′, …, ω
2
k
′. Then, for every fixed ω
2
i
′
(
i
=
1,2
,
3
,
…
,
k
), the corresponding critical value of time delay {
τ
2
i
′
(
j
)
∣
j
=
0,1
,
2
,
…
} is
(51)
τ
2
i
′
(
j
)
=
1
ω
2
i
′
arccos
Δ
51
Δ
54
+
Δ
52
Δ
53
Δ
51
2
+
Δ
52
2
+
2
j
π
ω
2
i
′
,
i
=
1,2
,
3
,
…
,
k
,
j
=
0,1
,
2
,
…
.
Let
(52)
τ
2
*
=
min
⁡
{
τ
2
i
′
(
0
)
∣
i
=
1,2
,
3
,
…
,
k
,
}
,
ω
2
*
=
ω
2
i
′
∣
τ
2
=
τ
2
*
.
Differentiating (11) regarding τ
2, we obtain
(53)
[
d
λ
d
τ
2
]
-
1
=
g
0
(
λ
)
+
g
1
(
λ
)
e
-
λ
τ
1
+
g
2
(
λ
)
e
-
λ
τ
2
+
g
3
(
λ
)
e
-
λ
(
τ
1
+
τ
2
)
h
2
(
λ
)
e
-
λ
τ
2
+
h
3
(
λ
)
e
-
λ
(
τ
1
+
τ
2
)
-
τ
2
λ
with
(54)
g
0
(
λ
)
=
4
λ
3
+
3
m
3
λ
2
+
2
m
2
λ
+
m
1
,
g
1
(
λ
)
=
-
τ
1
n
3
λ
3
+
(
3
n
3
-
τ
1
n
2
)
λ
2
+
(
2
n
2
-
τ
1
n
1
)
λ
+
n
1
-
τ
1
n
0
,
g
2
(
λ
)
=
2
p
2
λ
+
p
1
,
g
3
(
λ
)
=
q
1
,
h
2
(
λ
)
=
p
2
λ
3
+
p
1
λ
2
+
p
0
λ
,
h
3
(
λ
)
=
q
1
λ
2
+
q
0
λ
.
Hence,
(55)
Re
[
d
λ
d
τ
2
]
τ
2
=
τ
2
*
-
1
=
-
P
51
Q
51
-
P
52
Q
52
Q
51
2
+
Q
52
2
,
where
(56)
P
51
=
(
2
p
2
ω
2
*
-
q
1
sin
τ
1
ω
2
*
)
sin
τ
2
*
ω
2
*
+
(
p
1
+
q
1
cos
⁡
τ
1
ω
2
*
)
cos
⁡
τ
2
*
ω
2
*
+
(
τ
1
n
3
(
ω
2
*
)
3
+
(
2
n
2
-
τ
1
n
1
)
ω
2
*
)
sin
τ
1
ω
2
*
+
(
(
τ
1
n
2
-
3
n
3
)
(
ω
2
*
)
2
+
n
1
-
τ
1
n
0
)
cos
⁡
τ
1
ω
2
*
-
3
m
3
(
ω
2
*
)
2
+
m
1
,
P
52
=
(
2
p
2
ω
2
*
-
q
1
sin
τ
1
ω
2
*
)
cos
⁡
τ
2
*
ω
2
*
-
(
p
1
+
q
1
cos
⁡
τ
1
ω
2
*
)
sin
τ
2
*
ω
2
*
+
(
τ
1
n
3
(
ω
2
*
)
3
+
(
2
n
2
-
τ
1
n
1
)
ω
2
*
)
cos
⁡
τ
1
ω
2
*
-
(
(
τ
1
n
2
-
3
n
3
)
(
ω
2
*
)
2
+
n
1
-
τ
1
n
0
)
sin
τ
1
ω
2
*
-
4
(
ω
2
*
)
3
+
2
m
2
ω
2
*
,
Q
51
=
(
p
2
(
ω
2
*
)
3
-
p
0
ω
2
*
-
q
1
(
ω
2
*
)
2
sin
τ
1
ω
2
*
-
q
0
ω
2
*
cos
⁡
τ
1
ω
2
*
)
sin
τ
2
*
ω
2
*
+
(
p
1
(
ω
2
*
)
2
+
q
1
(
ω
2
*
)
2
cos
⁡
τ
1
ω
2
*
-
q
0
ω
2
*
sin
τ
1
ω
2
*
)
cos
⁡
τ
2
*
ω
2
*
,
Q
52
=
(
p
2
(
ω
2
*
)
3
-
p
0
ω
2
*
-
q
1
(
ω
2
*
)
2
sin
τ
1
ω
2
*
-
q
0
ω
2
*
cos
⁡
τ
1
ω
2
*
)
cos
⁡
τ
2
*
ω
2
*
-
(
p
1
(
ω
2
*
)
2
+
q
1
(
ω
2
*
)
2
cos
⁡
τ
1
ω
2
*
-
q
0
ω
2
*
sin
τ
1
ω
2
*
)
sin
τ
2
*
ω
2
*
.Clearly, if the condition(
H
52
)
:
P
51
Q
51
-
P
52
Q
52
≠
0 holds, then Re
[
d
λ
/
d
τ
2
]
τ
2
=
τ
2
*
-
1
≠
0. Thus, we have the following results.Theorem 5.
If the conditions(
H
51
)
-
(
H
52
) hold and τ
1
∈
(
0
,
τ
10
), then(i)
the positive equilibriumE
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) of system (5) is asymptotically stable for τ
2
∈
[
0
,
τ
2
*
);
(ii)
system (5) undergoes a Hopf bifurcation at the equilibrium E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) when τ
2
=
τ
2
*, and a family of periodic solutions bifurcate from E
*
(
x
1
*
,
x
2
*
,
y
1
*
,
y
2
*
) near τ
2
=
τ
2
*.


## 3. Stability of the Bifurcating Periodic Solutions

In this section, we will determine the direction of Hopf bifurcation and stability of the bifurcated periodic solutions of system (5) with respect to τ
2 for τ
1
∈
(
0
,
τ
10
) by using the normal form method and center manifold theorem introduced by Hassard et al. in [25]. Without loss of generality, we assume that τ
1
*
<
τ
2
*, where τ
1
*
∈
(
0
,
τ
10
).Letτ
2
=
τ
2
*
+
μ, μ
∈
R. Then, μ is the Hopf bifurcation value of system (5). Rescaling the time t
→
t
/
τ
2, then system (5) can be written as
(57)
u
˙
(
t
)
=
L
μ
u
t
+
F
(
μ
,
u
t
)
,
where
(58)
L
μ
ϕ
=
(
τ
2
*
+
μ
)
(
A
′
ϕ
(
0
)
+
B
′
ϕ
(
-
τ
1
*
τ
2
*
)
+
C
′
ϕ
(
-
1
)
)
,
F
(
μ
,
ϕ
)
=
(
τ
2
*
+
μ
)
×
(
g
1
x
1
(
t
)
y
2
(
t
)
+
g
2
x
1
(
t
)
x
1
(
t
-
τ
1
)
0
h
1
x
1
(
t
-
τ
2
)
y
2
(
t
-
τ
2
)
0
)
with
(59)
ϕ
(
θ
)
=
(
ϕ
1
(
θ
)
,
ϕ
2
(
θ
)
,
ϕ
3
(
θ
)
,
ϕ
4
(
θ
)
)
T
∈
C
(
[
-
1,0
]
,
R
4
)
,
A
′
=
(
a
11
a
12
0
a
14
a
21
a
22
0
0
0
0
a
33
0
0
0
a
43
a
44
)
,
B
′
=
(
b
11
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
)
,
C
′
=
(
0
0
0
0
0
0
0
0
c
31
0
0
c
34
0
0
0
0
)
.Using the Riesz representation theorem, there exists a4
×
4 matrix function η
(
θ
,
μ
)
:
[
-
1,0
]
→
R
4 whose elements are of bounded variation, such that
(60)
L
μ
ϕ
=
∫
-
1
0
d
η
(
θ
,
μ
)
ϕ
(
θ
)
,
ϕ
∈
C
(
[
-
1,0
]
,
R
4
)
.
In fact, we can choose
(61)
η
(
θ
,
μ
)
=
{
(
τ
2
*
+
μ
)
(
A
′
+
B
′
+
C
′
)
,
θ
=
0
,
(
τ
2
*
+
μ
)
(
B
′
+
C
′
)
,
θ
∈
[
-
τ
1
*
τ
2
*
,
0
)
,
(
τ
2
*
+
μ
)
C
′
,
θ
∈
(
-
1
,
-
τ
1
*
τ
2
*
)
,
0
,
θ
=
-
1
.
For ϕ
∈
C
(
[
-
1,0
]
,
R
4
), we define
(62)
A
(
μ
)
ϕ
=
{
d
ϕ
(
θ
)
d
θ
,
-
1
≤
θ
<
0
,
∫
-
1
0
d
η
(
θ
,
μ
)
ϕ
(
θ
)
,
θ
=
0
,
R
(
μ
)
ϕ
=
{
0
,
-
1
≤
θ
<
0
,
F
(
μ
,
ϕ
)
,
θ
=
0
.
Then system (57) can be transformed into the following operator equation:
(63)
u
˙
(
t
)
=
A
(
μ
)
u
t
+
R
(
μ
)
u
t
,
where u
t
=
u
(
t
+
θ
)
=
(
u
1
(
t
+
θ
)
,
u
2
(
t
+
θ
)
,
u
3
(
t
+
θ
)
,
u
4
(
t
+
θ
)
).The adjoint operatorA
* of A is defined by
(64)
A
*
(
φ
)
=
{
-
d
φ
(
s
)
d
s
,
0
<
s
≤
1
,
∫
-
1
0
d
η
T
(
s
,
μ
)
φ
(
-
s
)
,
s
=
0
,
associated with a bilinear inner product:
(65)
〈
φ
(
s
)
,
ϕ
(
θ
)
〉
=
φ
-
(
0
)
ϕ
(
0
)
-
∫
θ
=
-
1
0
‍
∫
ξ
=
0
θ
φ
-
(
ξ
-
θ
)
d
η
(
θ
)
ϕ
(
ξ
)
d
ξ
,
where η
(
θ
)
=
η
(
θ
,
0
).By the discussion above, we know that±
i
τ
2
*
ω
2
* are eigenvalues of A
(
0
) and they are also eigenvalues of A
*
(
0
). We assume that q
(
θ
)
=
(
1
,
q
2
,
q
3
,
q
4
)
T
e
i
τ
2
*
ω
2
*
θ are the eigenvectors of A
(
0
) belonging to the eigenvalue +
i
τ
2
*
ω
2
* and q
*
(
s
)
=
D
(
1
,
q
2
*
,
q
3
*
,
q
4
*
)
e
i
τ
2
*
ω
2
*
s are the eigenvectors of A
*
(
0
) belonging to the eigenvalue -
i
τ
2
*
ω
2
*.By a simple computation, we can get(66)
q
2
=
a
21
i
ω
2
*
-
a
22
,
q
3
=
c
34
(
i
ω
2
*
-
a
11
)
+
a
14
c
31
-
b
11
c
34
e
-
i
τ
1
*
ω
2
*
a
14
(
i
ω
2
*
-
a
33
)
e
i
τ
2
*
ω
2
*
-
a
12
a
21
c
34
a
14
(
i
ω
2
*
-
a
22
)
(
i
ω
2
*
-
a
33
)
e
i
τ
2
*
ω
2
*
,
q
4
=
(
e
-
i
τ
1
*
ω
2
*
(
i
ω
2
*
-
a
11
)
(
i
ω
2
*
-
a
22
)
-
a
12
a
21
-
b
11
(
i
ω
2
*
-
a
22
)
e
-
i
τ
1
*
ω
2
*
)
×
(
a
14
(
i
ω
2
*
-
a
22
)
)
-
1
,
q
2
*
=
-
a
12
i
ω
2
*
+
a
22
,
q
3
*
=
-
(
(
i
ω
2
*
+
a
22
)
e
i
τ
1
*
ω
2
*
(
i
ω
2
*
+
a
11
)
(
i
ω
2
*
+
a
22
)
-
a
12
a
21
+
b
11
×
(
i
ω
2
*
+
a
22
)
e
i
τ
1
*
ω
2
*
)
×
(
c
31
(
i
ω
2
*
+
a
33
)
e
i
τ
2
*
ω
2
*
)
-
1
,
q
4
*
=
c
34
(
i
ω
2
*
+
a
11
)
-
a
14
c
31
-
b
11
c
34
e
i
τ
1
*
ω
2
*
c
31
(
i
ω
2
*
+
a
44
)
-
a
12
a
21
c
34
c
31
(
i
ω
2
*
+
a
22
)
(
i
ω
2
*
+
a
44
)
.
From (65), we can get
(67)
〈
q
*
(
s
)
,
q
(
θ
)
〉
=
q
-
*
(
0
)
q
(
0
)
-
∫
θ
=
-
1
0
‍
∫
ξ
=
0
θ
‍
q
-
*
(
ξ
-
θ
)
d
η
(
θ
)
q
(
ξ
)
d
ξ
=
D
-
(
1
,
q
-
2
*
,
q
-
3
*
,
q
-
4
*
)
(
1
,
q
2
,
q
3
,
q
4
)
T
-
D
-
∫
θ
=
-
1
0
‍
∫
ξ
=
0
θ
‍
(
1
,
q
-
2
*
,
q
-
3
*
,
q
-
4
*
)
×
e
i
τ
2
*
ω
2
*
(
θ
-
ξ
)
d
η
(
θ
)
×
(
1
,
q
2
,
q
3
,
q
4
)
T
e
i
τ
2
*
ω
2
*
ξ
d
ξ
=
D
-
[
×
∫
-
1
0
‍
θ
e
i
τ
2
*
ω
2
*
θ
d
η
(
θ
)
(
1
,
q
2
,
q
3
,
q
4
)
T
]
1
+
q
2
q
-
2
*
+
q
3
q
-
3
*
+
q
4
q
-
4
*
-
(
1
,
q
-
2
*
,
q
-
3
*
,
q
-
4
*
)
×
∫
-
1
0
‍
θ
e
i
τ
2
*
ω
2
*
θ
d
η
(
θ
)
(
1
,
q
2
,
q
3
,
q
4
)
T
]
=
D
-
[
1
+
q
2
q
-
2
*
+
q
3
q
-
3
*
+
q
4
q
-
4
*
+
b
11
τ
1
*
e
-
i
τ
1
*
ω
2
*
+
q
-
3
*
τ
2
*
e
-
i
τ
2
*
ω
2
*
(
c
31
+
c
34
q
4
)
]
.
Normalizing q and q
* by the condition 〈
q
*
,
q
〉
=
1 and 〈
q
*
,
q
-
〉, one can get
(68)
D
-
=
[
1
+
q
2
q
-
2
*
+
q
3
q
-
3
*
+
q
4
q
-
4
*
+
b
11
τ
1
*
e
-
i
τ
1
*
ω
2
*
+
q
-
3
*
τ
2
*
e
-
i
τ
2
*
ω
2
*
(
c
31
+
c
34
q
4
)
]
-
1
.Next, we can get the coefficients determining the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions by the algorithms given in [26]:
(69)
g
20
=
2
τ
2
*
D
-
[
g
1
q
(
4
)
(
0
)
+
g
2
q
(
1
)
(
-
τ
1
*
τ
2
*
)
+
h
1
q
-
3
*
q
(
1
)
(
-
1
)
q
(
4
)
(
-
1
)
(
-
τ
1
*
τ
2
*
)
]
,
g
11
=
τ
2
*
D
-
[
{
(
-
τ
1
*
τ
2
*
)
}
g
1
(
q
-
(
4
)
(
0
)
+
q
(
4
)
(
0
)
)
+
g
2
(
q
-
(
1
)
(
-
τ
1
*
τ
2
*
)
+
q
(
1
)
(
-
τ
1
*
τ
2
*
)
)
+
h
1
q
-
3
*
(
q
(
1
)
(
-
1
)
q
-
(
4
)
(
-
1
)
+
q
-
(
1
)
(
-
1
)
q
(
4
)
(
-
1
)
)
{
(
-
τ
1
*
τ
2
*
)
}
]
,
g
02
=
2
τ
2
*
D
-
[
g
1
q
-
(
4
)
(
0
)
+
g
2
q
-
(
1
)
(
-
τ
1
*
τ
2
*
)
+
h
1
q
-
3
*
q
-
(
1
)
(
-
1
)
q
-
(
4
)
(
-
1
)
g
1
q
-
(
4
)
(
0
)
+
g
2
q
-
(
1
)
(
-
τ
1
*
τ
2
*
)
]
,
g
21
=
2
τ
2
*
D
-
[
{
+
1
2
W
20
(
1
)
(
-
τ
1
*
τ
2
*
)
)
}
g
1
(
W
11
(
1
)
(
0
)
q
(
4
)
(
0
)
+
1
2
W
20
(
1
)
(
0
)
q
-
(
4
)
(
0
)
+
W
11
(
4
)
(
0
)
+
1
2
W
20
(
4
)
(
0
)
)
+
g
2
(
W
11
(
1
)
(
0
)
q
(
1
)
(
-
τ
1
*
τ
2
*
)
+
1
2
W
20
(
1
)
(
0
)
q
-
(
1
)
×
(
-
τ
1
*
τ
2
*
)
+
W
11
(
1
)
(
-
τ
1
*
τ
2
*
)
+
1
2
W
20
(
1
)
(
-
τ
1
*
τ
2
*
)
)
+
h
1
q
-
3
*
(
+
1
2
W
20
(
4
)
(
-
1
)
q
-
(
1
)
(
-
1
)
W
11
(
1
)
(
-
1
)
q
(
4
)
(
-
1
)
+
1
2
W
20
(
1
)
(
-
1
)
q
-
(
4
)
(
-
1
)
+
W
11
(
4
)
(
-
1
)
q
(
1
)
(
-
1
)
+
1
2
W
20
(
4
)
(
-
1
)
q
-
(
1
)
(
-
1
)
)
{
+
1
2
W
20
(
1
)
(
-
τ
1
*
τ
2
*
)
)
}
]
with
(70)
W
20
(
θ
)
=
i
g
20
q
(
0
)
τ
2
*
ω
2
*
e
i
τ
2
*
ω
2
*
θ
+
i
g
-
02
q
-
(
0
)
3
τ
2
*
ω
2
*
e
-
i
τ
2
*
ω
2
*
θ
+
E
1
e
2
i
τ
2
*
ω
2
*
θ
,
W
11
(
θ
)
=
-
i
g
11
q
(
0
)
τ
2
*
ω
2
*
e
i
τ
2
*
ω
2
*
θ
+
i
g
-
11
q
-
(
0
)
τ
2
*
ω
2
*
e
-
i
τ
2
*
ω
2
*
θ
+
E
2
,
where E
1 and E
2 can be determined by the following equations, respectively:(71)
E
1
=
2
(
2
i
ω
2
*
-
a
11
-
b
11
e
-
2
i
τ
1
*
ω
2
*
-
a
12
0
-
a
14
-
a
21
2
i
ω
2
*
-
a
22
0
0
-
c
31
e
-
2
i
τ
2
*
ω
2
*
0
2
i
ω
2
*
-
a
33
-
c
34
e
-
2
i
τ
2
*
ω
2
*
0
0
-
a
43
2
i
ω
2
*
-
a
44
)
-
1
×
(
g
1
q
(
4
)
(
0
)
+
g
2
q
(
1
)
(
-
τ
1
*
τ
2
*
)
0
h
1
q
(
1
)
(
-
1
)
q
(
4
)
(
-
1
)
0
)
,
E
2
=
-
(
a
11
+
b
11
a
12
0
a
14
a
21
a
22
0
0
c
31
0
a
33
c
34
0
0
a
43
a
44
)
-
1
×
(
g
1
(
q
-
(
4
)
(
0
)
+
q
(
4
)
(
0
)
)
+
g
2
(
q
-
(
1
)
(
-
τ
1
*
τ
2
*
)
+
q
(
1
)
(
-
τ
1
*
τ
2
*
)
)
0
h
1
(
q
(
1
)
(
-
1
)
q
-
(
4
)
(
-
1
)
+
q
-
(
1
)
(
-
1
)
q
(
4
)
(
-
1
)
)
0
)
.Therefore, we can calculate the following values:
(72)
C
1
(
0
)
=
i
2
τ
2
*
ω
2
*
(
g
11
g
20
-
2
|
g
11
|
2
-
|
g
02
|
2
3
)
+
g
21
2
,
μ
2
=
-
Re
{
C
1
(
0
)
}
Re
{
λ
′
(
τ
2
*
)
}
,
β
2
=
2
Re
{
C
1
(
0
)
}
,
T
2
=
-
Im
⁡
{
C
1
(
0
)
}
+
μ
2
Im
⁡
{
λ
′
(
τ
2
*
)
}
τ
2
*
ω
2
*
.
By the results of Hassard et al. [26], the following results are obtained.Theorem 6.
For system (5),(i)
the direction of the Hopf bifurcation is determined byμ
2: if μ
2
>
0, the Hopf bifurcation is supercritical; if μ
2
<
0, the Hopf bifurcation is subcritical;
(ii)
the stability of the bifurcating periodic solutions is determined byβ
2: if β
2
<
0, the bifurcating periodic solutions are stable; if β
2
>
0, the bifurcating periodic solutions are unstable;
(iii)
the period of the bifurcating periodic solution is determined byT
2: if T
2
>
0, the period of the bifurcating periodic solutions increases; if T
2
<
0, the period of the bifurcating periodic solutions decreases.


## 4. Numerical Simulation

In this section, we present some numerical simulations to support the theoretical analysis in Sections2 and 3. We consider the following system by taking the same coefficients as in [17]:
(73)
d
x
1
(
t
)
d
t
=
5
x
2
(
t
)
-
x
1
(
t
)
x
1
(
t
-
τ
1
)
-
2
x
1
(
t
)
-
x
1
(
t
)
y
2
(
t
)
,
d
x
2
(
t
)
d
t
=
x
1
(
t
)
-
x
2
(
t
)
,
d
y
1
(
t
)
d
t
=
2
x
1
(
t
-
τ
2
)
y
2
(
t
-
τ
2
)
-
2.2
y
1
(
t
)
,
d
y
2
(
t
)
d
t
=
1.2
y
1
(
t
)
-
y
2
(
t
)
,
from which we can get a
2
r
r
1
r
2
=
12, a
d
2
d
4
(
d
3
+
r
2
)
+
a
2
d
2
r
2
(
d
1
+
r
1
)
=
7. Obviously, a
2
r
r
1
r
2
>
a
d
2
d
4
(
d
3
+
r
2
)
+
a
2
d
2
r
2
(
d
1
+
r
1
). Therefore, the condition (
H
) is satisfied. Further, we get the unique positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) of system (73) by MATLAB software package. Then, we have A
10
=
4.5833, A
11
=
7.5163, A
13
A
12
=
233.1899, A
11
A
12
A
13
=
1752.7, and A
11
2
+
A
10
A
13
2
=
525.5744. Obviously, A
10
=
4.5833
>
0, A
13
A
12
>
A
11, and A
11
A
12
A
13
>
A
11
2
+
A
10
A
13
2. Namely, the conditions (
H
11
) and (
H
12
) hold.Forτ
1
>
0
,
τ
2
=
0. From (18), we have
(74)
ω
1
8
+
45.3999
ω
1
6
+
284.0324
ω
1
4
-
163.6041
ω
1
2
+
21.0076
=
0
.
Then, (20) becomes
(75)
v
1
4
+
45.3999
v
1
3
+
284.0324
v
1
2
-
163.6041
v
1
+
21.0076
=
0
and it has two positive real roots: v
11
=
0.3397, v
12
=
0.2003. Thus, the condition (
H
21
) is satisfied. Further, we have ω
10
=
0.4475, τ
10
=
4.5061. Then, we get f
1
′
(
v
1
*
)
=
-
44.3242
≠
0. Therefore, the condition (
H
22
) holds. From Theorem 2, we know that the positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) is asymptotically stable for τ
1
∈
[
0,4.5061
), which can be shown in Figures 1, 2, and 3. The positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) is unstable when τ
1
>
4.5061 and system (73) undergoes a Hopf bifurcation at E
*
(
0.9167,0.9167,1.7361,2.0833
), and a family of periodic solutions bifurcate from E
*
(
0.9167,0.9167,1.7361,2.0833
). This property can be illustrated by Figures 4–6. Similarly, we have ω
20
=
0.1517, τ
20
=
9.9183 for τ
1
=
0
,
τ
2
>
0. For τ
2
=
7.7500
∈
[
0
,
τ
20
), the positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) is asymptotically stable from Theorem 3 and this property can be shown in Figures 7, 8, and 9. If τ
2
=
10.7500
>
τ
20
=
9.9183, the positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) is unstable and a Hopf bifurcation occurs, and the corresponding waveforms and phase plots are shown in Figures 10, 11, and 12.Figure 1
The track of the statesx
1, x
2, y
1, y
2 for τ
1
=
4.2500
<
τ
10
=
4.5061.Figure 2
The phase plot of the statesx
1, x
2, y
2 for τ
1
=
4.2500
<
τ
10
=
4.5061.Figure 3
The phase plot of the statesx
2, y
1, y
2 for τ
1
=
4.2500
<
τ
10
=
4.5061.Figure 4
The track of the statesx
1, x
2, y
1, y
2 for τ
1
=
4.71500
>
τ
10
=
4.5061.Figure 5
The phase plot of the statesx
1, x
2, y
2 for τ
1
=
4.71500
>
τ
10
=
4.5061.Figure 6
The phase plot of the statesx
2, y
1, y
2 for τ
1
=
4.71500
>
τ
10
=
4.5061.Figure 7
The track of the statesx
1, x
2, y
1, y
2 for τ
2
=
7.7500
<
τ
20
=
9.9183.Figure 8
The phase plot of the statesx
1, x
2, y
2 for τ
2
=
7.7500
<
τ
20
=
9.9183.Figure 9
The phase plot of the statesx
2, y
1, y
2 for τ
2
=
7.7500
<
τ
20
=
9.9183.Figure 10
The track of the statesx
1, x
2, y
1, y
2 for τ
2
=
10.7500
>
τ
20
=
9.9183.Figure 11
The phase plot of the statesx
1, x
2, y
2 for τ
2
=
10.7500
>
τ
20
=
9.9183.Figure 12
The phase plot of the statesx
2, y
1, y
2 for τ
2
=
10.7500
>
τ
20
=
9.9183.Forτ
1
=
τ
2
=
τ
>
0, we can get ω
0
=
0.3412
,
τ
0
=
3.1043. From Theorem 4, we know that the positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) is asymptotically stable for τ
∈
[
0
,
τ
0
), which can be illustrated by Figures 13, 14, and 15. As can be seen from Figure 5 that when τ
=
2.7500
∈
[
0
,
τ
0
), the positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) is asymptotically stable. However, if τ
=
3.2500
>
τ
0
=
3.1043, then the positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) becomes unstable and a family of bifurcating periodic solutions occur, which is illustrated by Figures 16, 17, and 18.Figure 13
The track of the statesx
1, x
2, y
1, y
2 for τ
=
2.7500
<
τ
0
=
3.1043.Figure 14
The phase plot of the statesx
1, x
2, y
2 for τ
=
2.7500
<
τ
0
=
3.1043.Figure 15
The phase plot of the statesx
2, y
1, y
2 for τ
=
2.7500
<
τ
0
=
3.1043.Figure 16
The track of the statesx
1, x
2, y
1, y
2 for τ
=
3.2500
>
τ
0
=
3.1043.Figure 17
The phase plot of the statesx
1, x
2, y
2 for τ
=
3.2500
>
τ
0
=
3.1043.Figure 18
The phase plot of the statesx
2, y
1, y
2 for τ
=
3.2500
>
τ
0
=
3.1043.Lastly, regardingτ
2 as a parameter and letting τ
1
=
2.5
∈
(
0
,
τ
10
), we can obtain that ω
2
*
=
0.2253. Further, we have τ
2
*
=
6.6494 and λ
′
(
τ
2
*
)
=
0.0466
+
0.0314
i. Letting τ
2
=
5.5500
∈
[
0
,
τ
1
*
), we can know that the positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) is asymptotically stable from Theorem 5. When τ
2
=
7.5000
>
τ
2
*
=
6.6494, then the positive equilibrium E
*
(
0.9167,0.9167,1.7361,2.0833
) becomes unstable and a Hopf bifurcation occurs. The corresponding waveforms and phase plots are illustrated in Figures 19, 20, 21, 22, 23, and 24.Figure 19
The track of the statesx
1, x
2, y
1, y
2 for τ
2
=
5.5500
<
τ
2
*
=
6.6494 and τ
1
*
=
2.5000.Figure 20
The phase plot of the statesx
1, x
2, y
2 for τ
2
=
5.5500
<
τ
2
*
=
6.6494 and τ
1
*
=
2.5000.Figure 21
The phase plot of the statesx
2, y
1, y
2 for τ
2
=
5.5500
<
τ
2
*
=
6.6494 and τ
1
*
=
2.5000.Figure 22
The track of the statesx
1, x
2, y
1, y
2 for τ
2
=
7.5000
>
τ
2
*
=
6.6494 and τ
1
*
=
2.5000.Figure 23
The phase plot of the statesx
1, x
2, y
2 for τ
2
=
7.5000
>
τ
2
*
=
6.6494 and τ
1
*
=
2.5000.Figure 24
The phase plot of the statesx
2, y
1, y
2 for τ
2
=
7.5000
>
τ
2
*
=
6.6494 and τ
1
*
=
2.5000.In addition, for system (73), we have C
1
(
0
)
=
-
31.3276
+
4.4079
i by a simple computation. Further, from (72), we have μ
2
=
672.2661
>
0, β
2
=
-
62.6552
<
0, and T
2
=
-
17.0330
<
0. Therefore, by Theorem 6, we can know that the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable, and the period of the bifurcated periodic solutions decrease. Since the bifurcating periodic solutions of system (73) are stable, the predator and the prey species in system (73) can coexist in an oscillatory mode from the view of biology.


## 5. Conclusions

In this paper, a delayed predator-prey system with stage-structure for both the predator and the prey population is investigated. The sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation for the possible combinations of two delays are obtained. The main results are given in Theorems2–5. The results show that the two time delays can play a complicated role in system (5). Furthermore, the explicit formulae determining the direction of the bifurcation and the stability of the bifurcating periodic solutions are established when τ
2
>
0 and τ
1
∈
(
0
,
τ
10
) by using the normal form theory and center manifold theorem. The main results are given in Theorem 6. If the bifurcating periodic solutions are stable, the predator and the prey species can coexist in an oscillatory mode from the view of biology. Compared with literature [17], we consider not only the time delay due to the gestation of the mature predator but also the negative feedback delay of the immature prey. Wang [17] obtained that the species in system (5) with only the time delay due to the gestation of the mature predator could coexist under some conditions. However, we get that the species could also coexist with some available time delays of the immature prey and the mature predator under certain conditions. This is valuable from the view of ecology. Some numerical simulations are also included to support the obtained theoretical results.



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*Source: 909238-2014-02-19.xml*