Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 527864, 14 pages
doi:10.1155/2010/527864
Research Article
Bifurcation Analysis for a Delayed Predator-Prey
System with Stage Structure
Zhichao Jiang and Guangtao Cheng
Fundamental Science Department, North China Institute of Astronautic Engineering,
Langfang Hebei 065000, China
Correspondence should be addressed to Zhichao Jiang, [email protected]
Received 9 August 2010; Revised 10 October 2010; Accepted 14 October 2010
Academic Editor: Massimo Furi
Copyright q 2010 Z. Jiang and G. Cheng. 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.
A delayed predator-prey system with stage structure is investigated. The existence and stability of
equilibria are obtained. An explicit algorithm for determining the direction of the Hopf bifurcation
and the stability of the bifurcating periodic solutions is derived by using the normal form and the
center manifold theory. Finally, a numerical example supporting the theoretical analysis is given.
1. Introduction
The age factors are important for the dynamics and evolution of many mammals. The rates of
survival, growth, and reproduction almost always depend heavily on age or developmental
stage, and it has been noticed that the life history of many species is composed of at
least two stages, immature and mature, with significantly different morphological and
behavioral characteristics. The study of stage-structured predator-prey systems has attracted
considerable attention in recent years see 1–6 and the reference therein.In4,Wang
considered the following predator-prey model with stage structure for predator, in which
the immature predators can neither hunt nor reproduce.
˙x
t
y
2
t
1 mx
t
−
D v
1
y
1
t
,
˙y
2
t
Dy
1
Moros¸anu 7 generalized the system 1.1 as
˙x
t
n
x
t
− f
x
t
y
2
t
,
˙y
1
t
kf
− v
2
y
2
t
,
1.2
satisfying the following hypotheses:
H1a fx is the predator functional response and satisfies that
f ∈ C
1
0, ∞
,
0, ∞
,f
0
0,f
x
> 0, lim
sufficient condition for the orbital stability of a periodic orbit and obtain the global stability
of the positive equilibrium for the general system. It is necessary to forsake some aspects of
realism, and one of the features of the real world which is commonly compromised in order
to achieve generality is the time delay. In general, delay differential equations exhibit much
more complicated dynamics than ordinary differential equations since a time delay could
cause a stable equilibrium to become unstable and cause the population to fluctuate. Time
delay due to gestation is a common example, because generally the consumption of prey by a
predator throughout its past history governs the present birth rate of the predator. Therefore,
more realistic models of population interactions should take into account the effect of time
delays. So, we introduce the delay τ due to gestation of mature predator into system 1.2
and consider the following system:
˙x
t
n
x
t
− f
x
t
y
2
˙y
2
t
Dy
1
t
− v
2
y
2
t
,
1.4
Fixed Point Theory and Applications 3
where all coefficients are positive constants and the detailed ecological meanings are the same
as in system 1.2. Some usual examples of fx and nx include fxm×
c
m>0, 0 <c
1,fxm1 −e
−cx
m, c > 0,fxαxe
−βx
α>0,β> 0,fxbx
p
∗
1
,y
∗
2
if
H2 v
2
D v
1
<kDfx
0
holds, where x
∗
,y
∗
1
and y
∗
2
satisfy
n
x
f
x
y
0
is
˙x
t
n
0
x
t
,
˙y
1
t
−
D v
1
y
1
t
λ
2
D v
1
v
2
λ v
2
D v
1
0. 2.3
From (H1), one knows that n
0 > 0. Hence, 2.3 has a positive real root and two negative real roots.
One has the following lemma.
Lemma 2.2. For system 1.4, E
0
is a saddle point.
4 Fixed Point Theory and Applications
The linear part of 1.4 at E
1
is
˙x
x
0
y
2
t − τ
−
D v
1
y
1
t
,
˙y
2
t
Dy
1
t
− v
D v
1
− kDf
x
0
e
−λτ
0. 2.5
From (H1), one has that n
x
0
< 0. Hence, the stability of E
1
is decided by the following equation:
λ
2
D v
1
v
2
λ v
2
/2,kDfx
0
} and
H5Δ
.
D v
1
v
2
2
4k
2
D
2
f
2
x
0
− 4v
2
D v
1
D v
1
v
2
> 0hold, then
2.6 has two pairs of purely imaginary roots noted by ±iω
11
v
2
−
√
Δ/2, τ
k
1j
1/ω
1k
{arccos −ω
2
1k
D v
1
v
2
/kDfx
0
2j 1π},j 0, 1, 2, ,k 1, 2.
Let λτατiωτ be the root of 2.6 satisfying ατ
k
1j
0,ωτ
k
1j
ω
1k
. Thus, the
following results hold.
− v
2
D v
1
D v
1
v
2
− τ
k
1j
ω
2
1k
− v
2
D v
1
iω
1k
is unstable for any τ 0; ii if (H2) holds, then E
1
is unstable and E
∗
exists;
iii if (H4) and (H5) hold, then E
1
is asymptotically stable for τ ∈ 0,τ
10
and unstable for τ>τ
10
,
where τ
10
min{τ
1
10
,τ
2
10
}.
Fixed Point Theory and Applications 5
The linear part of 1.4 at E
∗
is
˙x
t
,
˙y
1
t
kf
x
∗
y
∗
2
x
t − τ
−
D v
1
y
1
t
and the corresponding characteristic equation is
λ
3
f
x
∗
y
∗
2
− n
x
∗
D v
1
v
2
λ
2
v
2
D v
1
f
x
∗
y
∗
2
− n
x
∗
n
x
∗
∗
D v
1
v
2
λ
2
D v
1
v
2
f
x
∗
y
∗
2
− n
x
x
∗
D v
1
v
2
D v
1
v
2
f
x
∗
y
∗
2
−n
x
∗
>v
2
D v
1
f
x
∗
x
∗
y
∗
2
− n
x
∗
D v
1
v
2
, a
1
f
x
∗
y
∗
2
− n
x
∗
D v
1
v
0
v
2
D v
1
n
x
∗
.Froma
0
b
0
> 0,
we have that λ 0 is not the root of 2.11. Obviously, λ iω ω>0 is a root of 2.11 if and
only if
iω
3
a
2
ω
2
− ia
1
ω − a
0
−
b
1
ω
6
pω
4
qω
2
s 0,
2.14
6 Fixed Point Theory and Applications
where p a
2
2
−2a
1
,q a
2
1
−2a
0
a
2
−b
2
1
,s a
2
0
−b
2
0
The above Lemma can be seen in 8. Suppose that 2.15 has positive roots. Without
loss of generality, we assume that it has three positive roots z
1
,z
2
,z
3
. Then 2.14 has three
positive roots ω
1
√
z
1
,ω
2
√
z
2
,ω
3
√
z
3
.By2.13, we have
cos ωτ
b
1
.
2.16
Thus, if
τ
k
j
1
ω
k
arccos
b
1
ω
4
k
a
2
b
0
− a
1
b
1
ω
k
0
0
min
τ
k
0
,ω
0
ω
k
0
,k 1, 2, 3. 2.18
Thus, by Lemma2.2 and Corollary 2.4in9, we can easily get the following results.
Lemma 2.7. a If s 0 and Λp
2
− 3q 0, then for any τ 0, 2.9 and 2.10 have the same
number of roots with positive real parts.
b If either s<0 or s 0, Λp
2
−3q>0,z
∗
1
> 0 and Gz
∗
1
0 is satisfied, then 2.9 and
2.10 have the same number of roots with positive real parts when τ ∈ 0,τ
0. Furthermore, Sign{α
τ
k
j
} Sign{G
z
k
}.
Proof. By direct computation to 2.11,weobtain
dλ
dτ
−1
3λ
2
2a
2
λ a
1
e
λτ
b
1
λ
ib
0
ω
k
,
2.20
and
3λ
2
2a
2
λ a
1
e
λτ
|
ττ
k
j
a
1
− 3ω
2
k
− 3ω
2
k
sin ω
k
τ
k
j
.
2.21
From 2.19 to 2.21, we have
α
τ
k
j
−1
z
k
Ω
G
z
k
k
}
/
0.
By the above analyses, we can obtain the following theorem.
Theorem 2.9. If (H2) and (H6) are satisfied, then the following results hold.
a If s 0 and Λp
2
− 3q 0, then for any τ 0, all roots of 2.11 have negative real
parts. Furthermore, positive equilibrium E
∗
of 1.4 is absolutely stable for τ 0;
b If either s<0 or z
∗
1
> 0,Gz
∗
1
0,r 0 and Λp
2
−3q>0 hold, then G(z) has at least
one positive root z
k
, and when τ ∈ 0,τ
0
, all roots of 2.11 have negative real parts. So
the positive equilibrium E
∗
of 1.4 is asymptotically stable for τ ∈ 0,τ
0
by using the normal form and the center manifold theory
developed by Hassard et al. 10.
Throughout this section, we assume that b and c of Theorem 2.9 are satisfied.
Under the transformation u
1
txτt − x
∗
,u
2
ty
1
τt − y
∗
1
,u
3
ty
2
τt − y
∗
2
,τ τ
0
μ,
the system 1.2 is transformed into an FDE in C C−1, 0,R
3
as
˙u
t
τ
0
μ
B
1
ϕ
0
B
2
ϕ
−1
, 3.2
8 Fixed Point Theory and Applications
where B
1
and B
2
are defined as
B
1
⎛
⎜
⎞
⎟
⎟
⎠
,B
2
⎛
⎜
⎜
⎝
000
kf
x
∗
y
∗
2
0 kf
x
∗
000
⎞
⎟
⎟
n
x
∗
− f
x
∗
y
∗
2
ϕ
2
1
0
− 2f
x
∗
ϕ
1
3
1
0
− 3f
x
∗
ϕ
2
1
0
ϕ
3
0
O
4
k
2!
−1
ϕ
3
−1
k
3!
n
x
∗
− f
x
∗
y
∗
2
ϕ
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
3.4
By the Riesz representation theorem, there exists a matrix whose components are bounded
variation functions ηθ, ϕ in θ ∈ −1, 0 such that
L
μ
ϕ
0
−1
dη
θ, μ
ϕ
θ
1,θ 0,
0,θ
/
0.
3.7
For ϕ ∈ C
1
−1, 0,R
3
, define
A
μ
ϕ
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙ϕ
θ
,θ∈
−1, 0
,θ 0.
3.9
Fixed Point Theory and Applications 9
Leting u u
1
,u
2
,u
3
T
, then system 3.1 can be rewritten as
˙u
t
A
ϕ
u
t
R
ϕ
u
t
. 3.10
For ψ ∈ C
1
−1
dη
T
t, 0
α
−t
,s 0
3.11
and a bilinear form
ψ, ϕ
ψ
0
ϕ
0
−
0
−1
1,β,γ
T
e
iω
0
τ
0
θ
3.13
is the eigenvector of A0 corresponding to iω
0
τ
0
,and
q
∗
s
B
1,β
∗
,γ
∗
s
,
q
θ
0, 3.15
where
β
iω
0
τ
0
v
2
γ
D
,γ
n
x
∗
− f
− n
x
∗
− iω
0
τ
0
kf
x
∗
y
∗
2
e
iω
0
τ
0
,γ
∗
D v
1
∗
y
∗
2
γfx
∗
.
Using the same notations as in Hassard et al. 10,letu
t
be the solution of 3.1 when
τ τ
0
. Defining ztq
∗
,u
t
,u
t
x
t
,y
t
, then
˙z
t
q
,W
z, z
2Re
zq
,W
z, z
u
t
− 2Re
zq
,
W
z,
z
W
20
z
2
2
W
z
g
20
z
2
2
g
11
zz g
02
z
2
2
g
21
z
2
z
2
···.
3.20
Substituting 3.10 and 3.17 into
˙
W ˙u
t
− ˙zq −
˙
zq, we have
˙
θ
f, θ 0,
def
AW H
z, z, θ
, 3.21
where
H
z,
z, θ
H
20
θ
z
2
2
H
11
,AW
11
−H
11
θ
. 3.23
For u
t
ut θWz, z, θzqθzqθ, we have
z,
z
g
20
z
2
2
g
11
zz g
02
z
2
2
···
q
x
∗
y
∗
2
1 k
β
∗
e
−2iω
0
τ
0
− f
x
∗
γ −k
β
∗
γf
x
∗
y
∗
2
− f
x
∗
γ
γ
,
g
02
2τ
0
B
1
2
n
x
∗
γf
x
∗
e
2iω
0
τ
0
,
g
21
2τ
0
B
1
2
n
x
∗
− f
1
11
0
n
x
∗
− f
x
∗
y
∗
2
−
1
2
f
x
2
W
3
20
0
1
2
γW
1
20
0
γW
1
11
1
k
2
β
∗
n
−1
e
−iω
0
τ
0
− k
β
∗
f
x
∗
×
W
3
11
−1
e
−iω
0
τ
1
11
−1
e
−iω
0
τ
0
.
3.25
We still need to compute W
20
θ and W
11
θ. For θ ∈ −1, 0, we have
H
z,
z, θ
−2Re
q
∗
0
gq
θ
. 3.26
Comparing the coefficients with 3.22 gives that
H
20
θ
−g
20
q
θ
− g
02
q
θ
,H
11
θ
−g
11
θ
2iω
0
τ
0
W
20
θ
g
20
q
θ
g
20
q
θ
. 3.28
Solving for W
20
θ,weobtain
W
20
e
−iω
0
τ
0
θ
E
1
e
2iω
0
τ
0
θ
, 3.29
12 Fixed Point Theory and Applications
and similarly
W
11
θ
−ig
11
ω
0
τ
0
,
3.30
where E
1
and E
2
are both 3-dimensional vectors and can be determined by setting θ 0in
H. Hence combining the definition of A, we can get
0
−1
dη
θ
W
20
θ
AW
20
0
2iω
0
τ
0
W
0
τ
0
I −
0
−1
e
iω
0
τ
0
θ
dηθq00, −iω
0
τ
0
I −
0
−1
e
−iω
0
τ
0
θ
dηθq00, we have
2iω
−1
dη
θ
E
2
−
f
zz
. 3.33
Hence, we get
⎛
⎜
⎜
⎝
2iω
0
− n
x
∗
f
x
x
∗
e
−2iω
0
τ
0
0 −D 2iω
0
v
2
⎞
⎟
⎟
⎠
E
1
1
2
n
x
∗
− f
3.34
⎛
⎜
⎜
⎝
n
x
∗
− f
x
∗
y
∗
2
0 −f
x
∗
kf
x
∗
x
∗
f
x
∗
y
∗
2
⎛
⎝
1
k
0
⎞
⎠
.
3.35
Fixed Point Theory and Applications 13
0
100 200 300 400 500
0
1
2
6
7
a
0.5
1
1.5
0.2
0.4
0.6
0.8
1
1
2
3
4
5
6
b
Figure 2: When τ 10, the positive equilibrium E
∗
of system 4.1 is unstable, and small amplified periodic
solutions exist.
Then g
21
can be expressed by the parameters. Based on the above analysis, we can see that
each g
ij
can be determined by the parameters. Thus we can compute the following quantities:
C
1
2
g
21
2
,β
2
2Re
{
C
1
0
}
,
T
2
−
Im
{
C
1
0
}
μ
2
determines the directions of the Hopf bifurcation: if μ
2
> 0< 0, the Hopf
bifurcation is supercritical (subcritical); β
2
determines the stability of the bifurcation periodic
solutions: the bifurcation periodic solutions are orbitally stable (unstable) if β
2
< 0> 0; T
2
determines
the period of the bifurcating periodic solutions: the period increases (decreases) if T
2
> 0< 0.
14 Fixed Point Theory and Applications
4. Numerical Examples
In this section, we give a numerical example:
˙x
t
x
t
2 − x
t
t
0.8y
1
t
− 0.1y
2
t
.
4.1
Then we can conclude that the system 4.1 has a unique positive equilibrium
E
∗
0.4948, 0.6272, 5.0174. When τ 5, the dynamics behaviors of system 4.1 are shown
in Figure 1.FromSection 2, we can obtain ω
0
0.4867, τ
0
0
7.9367, τ
0
j
7.9367 2jπ/ω
0
j
predator,” Acta Mathematica Sinica. English Series, vol. 19, no. 2, pp. 1–11, 2003.
6 W. Wang and L. Chen, “A predator-prey s ystem with stage-structure for predator,” Computers &
Mathematics with Applications, vol. 33, no. 8, pp. 83–91, 1997.
7 P. Georgescu and G. Moros¸anu, “Global stability for a stage-structured predator-prey model,”
Mathematical Sciences Research Journal, vol. 10, no. 8, pp. 214–228, 2006.
8 Y. Song, M. Han, and J. Wei, “Stability and Hopf bifurcation analysis on a simplified BAM neural
network with delays,” Physica D, vol. 200, no. 3-4, pp. 185–204, 2005.
9 S. G. Ruan and J. J. Wei, “On the zeros of transcendental functions with applications to stability of
delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems.
Series A, vol. 10, no. 6, pp. 863–874, 2003.
10 B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of
London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1981.
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