Volume 2010, Article ID 527864,14pages 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
Copyrightq2010 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.
˙
xt xt
r−axt− by2t
1mxt
,
˙
y1t
kbxty2t
1mxt −Dv1y1t,
˙
y2t Dy1t−v2y2t,
where xt denotes the density of prey at time t, y1t denotes the density of immature predator at time t,y2t denotes the density of mature predator at timet, b is the search rate,mis the search rate multiplied by the handling time, andr is the intrinsic growth rate. It is assumed that the reproduction rate of the mature predator depends on the quality of prey considered, the efficiency of conversion of prey into newborn immature predators being denoted byk.D denotes the rate at which immature predators become mature predators.
v1 and v2 denote the mortality rates of immature and mature predators, respectively. All coefficients are positive constants. In4, he concluded that the system under some conditions has a unique positive equilibrium, which is globally asymptotically stable. Georgescu and Moros¸anu7generalized the system1.1as
˙
xt nxt−fxty2t,
˙
y1t kfxty2t−Dv1y1t,
˙
y2t Dy1t−v2y2t,
1.2
satisfying the following hypotheses:
H1 afxis the predator functional response and satisfies that
f∈C10,∞,0,∞, f0 0, fx>0, lim
x→ ∞
fx
x <∞. 1.3
bnxis the growth function and satisfies thatn∈C10,∞, R, nx 0 if and only if x ∈ {0, x0},withx0 > 0 andnx > 0 forx ∈ 0, x0, andnxis strictly decreasing on
xp,∞,0< xp< x0.
c The prey isocline is given by hx : nx/fx and is assumed to be concave down, that is,hx<0 forx0.
In7, they employ the theory of competitive systems and Muldowney’s necessary and 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 delayτ due to gestation of mature predator into system1.2
and consider the following system:
˙
xt nxt−fxty2t,
˙
y1t kfxt−τy2t−τ−Dv1y1t,
˙
y2t Dy1t−v2y2t,
where all coefficients are positive constants and the detailed ecological meanings are the same as in system1.2. Some usual examples offxandnxincludefx m×cm >0,0< c
1, fx m1−e−cx m, c >0, fx αxe−βxα >0, β >0, fx bxp/1mxpp >0and
nx xr−ax/1εxε >0, nx rx1−x/r/ac 0< c1,ornx xre1−x/k−d,
and so forth.
Our main purpose of this paper is to investigate the dynamic behaviors of system1.4
and the frame of this paper is organized as follows. In the next section, we will investigate the stability of equilibria and the existence of local Hopf bifurcation. InSection 3, the direction and stability of the bifurcating periodic solutions are determined by applying the center manifold theorem and normal form theory. In Section 4, a numerical example supporting the theoretical analysis is given.
2. Stability of the Equilibrium and Local Hopf Bifurcations
It is known that time delay does not change the location and number of positive equilibrium. We have the following lemma.
Lemma 2.1. The system1.4has two nonnegative equilibria,E00,0,0,E1x0,0,0, and a positive
equilibriumE∗x∗, y∗1, y∗2if
H2v2Dv1< kDfx0holds, wherex∗, y∗1andy∗2satisfy
nx fxy2,
kfxty2t Dv1y1,
Dy1v2y2.
2.1
The linear part of 1.4atE0is
˙
xt n0xt,
˙
y1t −Dv1y1t,
˙
y2t Dy1t−v2y2t,
2.2
and the corresponding characteristic equation is
λ−n0λ2 Dv1v2λv2Dv1
0. 2.3
From (H1), one knows thatn0>0.Hence,2.3has a positive real root and two negative real roots. One has the following lemma.
The linear part of 1.4atE1is
˙
xt nx0xt−fx0y2t,
˙
y1t kfx0y2t−τ−Dv1y1t,
˙
y2t Dy1t−v2y2t,
2.4
and the corresponding characteristic equation is
λ−nx0
λ2 Dv1v2λv2Dv1−kDfx0e−λτ
0. 2.5
From (H1), one has thatnx0<0. Hence, the stability ofE1is decided by the following equation:
λ2 Dv1v2λv2Dv1−kDfx0e−λτ 0. 2.6
IfH3v2Dv1 > kDfx0holds, thenλ 0is not the root of 2.6, and all the roots of 2.6
have strictly negative real parts whenτ 0. Furthermore, one has the following conclusion.
Lemma 2.3. If
H4v2Dv1>max{Dv1v2/2, kDfx0}and
H5 Δ . D v1 v22 4k2D2f2x0−4v2D v1D v1 v2 > 0hold, then
2.6 has two pairs of purely imaginary roots noted by±iω11 and ±iω12 when τ τ1kj, and the
other roots have negative real parts, whereω11
2v2Dv1−Dv1v2
√
Δ/2,ω12
2v2Dv1−Dv1v2−
√
Δ/2,τk
1j 1/ω1k{arccos−ω21k Dv1v2/kDfx0
2j1π}, j 0,1,2, . . . , k1,2.
Letλτ ατ iωτbe the root of 2.6satisfyingατk
1j 0, ωτ1kj ω1k. Thus, the
following results hold.
Lemma 2.4. ατk
1j>0.
Proof. By2.6, we have
λ τ1kj ω
2
1kDv1v2 iω1k
ω2
1k−v2Dv1
Dv1v2−τ1kj
ω2
1k−v2Dv1
iω1k
2τk
1jDv1v2
2.7
andατ1kj ω2
1kDv1
22ω2 1k>0.
From the above discussion, we have the following.
Theorem 2.5. iE0is unstable for anyτ 0;iiif (H2) holds, thenE1is unstable andE∗exists;
iiiif (H4) and (H5) hold, thenE1is asymptotically stable forτ ∈0, τ10and unstable forτ > τ10,
The linear part of1.4atE∗is
˙
xt nx∗−fx∗y2∗xt−fx∗y2t,
˙
y1t kfx∗y∗2xt−τ−Dv1y1t kfx∗y2t−τ,
˙
y2t Dy1t−v2y2t,
2.8
and the corresponding characteristic equation is
λ3fx∗y2∗−nx∗ Dv1v2
λ2
v2Dv1 Dv1v2
fx∗y2∗−nx∗
×λv2Dv1
fx∗y2∗−nx∗nx∗−λkDfx∗e−λτ0.
2.9
Next, we will investigate the distribution of roots of2.9. Whenτ 0,2.9can be reduced to
λ3fx∗y∗2−nx∗ Dv1v2
λ2 Dv1v2
fx∗y∗2−nx∗λ
v2Dv1fx∗y2∗0.
2.10
By Routh-Hurwitz criteria, if
H6 fx∗y∗2−nx∗Dv1v2Dv1v2fx∗y∗2−nx∗> v2Dv1fx∗y2∗ holds, then all roots of2.10have strictly negative real parts andλ0 is not the root of2.9. If the reverse of2.10is satisfied, then two characteristic roots have positive real parts. For convenience, we denote2.9as follows
λ3a2λ2a1λa0 b1λb0e−λτ0, 2.11
wherea2 fx∗y2∗−nx∗ Dv1v2,a1 fx∗y∗2−nx∗Dv1v2 v2Dv1,
a0v2Dv1fx∗y∗2−nx∗,b1 −v2Dv1, b0 v2Dv1nx∗. Froma0b0 >0, we have thatλ0 is not the root of2.11. Obviously,λiωω >0is a root of2.11if and only if
iω3a2ω2−ia1ω−a0−b1ωib0cosωτ−isinωτ 0. 2.12
Separating the real part and imaginary part, we can obtain
a2ω2−a0b0cosωτb1ωsinωτ,
a1ω−ω3b0sinωτ−b1ωcosωτ,
2.13
which yields
wherepa22−2a1, qa21−2a0a2−b21, sa20−b02. Setzω2. Then2.14takes the following form:
Gzdef z3pz2qzs0. 2.15
Lemma 2.6see8. aIfs <0, then2.15has at least one positive root.
bIfs0andΛ p2−3q0, then2.15has no positive roots.
cIfs0andΛ p2−3q >0, then2.15has positive roots if and only ifz∗
1 1/3−p
√
ΛandGz∗10.
The above Lemma can be seen in8. Suppose that2.15has positive roots. Without loss of generality, we assume that it has three positive rootsz1, z2, z3. Then2.14has three positive rootsω1√z1, ω2√z2, ω3√z3. By2.13, we have
cosωτ b1ω
4
k a2b0−a1b1ωk2−a0b0
b2 0b21ωk2
. 2.16
Thus, if
τjk 1 ωk
arccos
b1ω4k a2b0−a1b1ωk2−a0b0
b20b21ω2k
2jπ
, 2.17
wherek 1,2,3, j 0,1,2, . . . ,then±iωkare a pair of purely imaginary roots of2.11with
τ τjk. Suppose that
τ0 τ0k0min
τ0k, ω0ωk0, k1,2,3. 2.18
Thus, by Lemma2.2 and Corollary 2.4 in9, we can easily get the following results.
Lemma 2.7. aIfs0andΛ p2−3q0, then for anyτ 0,2.9and2.10have the same
number of roots with positive real parts.
bIf eithers <0ors0,Λ p2−3q >0, z∗
1>0andGz1∗0is satisfied, then2.9and
2.10have the same number of roots with positive real parts whenτ∈0, τ0.
Letλτ ατ iωτbe the root of2.9satisfyingατk
j 0, ωτjk ωk. Thus, the
following transversality condition holds.
Lemma 2.8. Ifzkω2kandGzk/0, thenατjk/0. Furthermore,Sign{ατjk}Sign{Gzk}.
Proof. By direct computation to2.11, we obtain
dλ dτ
−1
3λ22a 2λa1
eλτb
1
λb1λb0
τ
By2.13, we have
λb1λb0|ττk
j −b1ω 2
kib0ωk, 2.20
and
3λ22a2λa1
eλτ|ττk
j a1−3ω 2
k
cosωkτjk−2a2ωksinωkτjk
i2a2ωkcosωkτjk a1−3ωk2
sinωkτjk
.
2.21
From2.19to2.21, we have
α τjk−1 zk
ΩGzk, 2.22
whereΩ b2
1ωk4b02ω2k. Thus Sign{ατjk}Sign{ατjk}
−1
Sign{Gzk}/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.
aIfs0andΛ p2−3q0, then for anyτ 0, all roots of 2.11have negative real
parts. Furthermore, positive equilibriumE∗of1.4is absolutely stable forτ 0;
bIf eithers <0orz∗1>0, Gz1∗0, r 0andΛ p2−3q >0hold, then G(z) has at least
one positive rootzk, and whenτ ∈0, τ0, all roots of 2.11have negative real parts. So
the positive equilibriumE∗of 1.4is asymptotically stable forτ ∈0, τ0.
cIf the conditions in (b) andGzk/0, then Hopf bifurcation for1.4occurs at positive
equilibrium E∗ when τ τjk, which means that small amplified periodic solutions will bifurcate fromE∗.
3. Properties of the Hopf Bifurcation
InSection 2, we obtain the conditions which guarantee that system1.4undergoes the Hopf bifurcation at the positive equilibriumE∗ whenτ τk
j.In this section, we will investigate
the direction of the Hopf bifurcation whenτ τ0and the stability of the bifurcating periodic solutions from the equilibriumE∗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 transformationu1t xτt−x∗, u2t y1τt−y∗1, u3t y2τt−y∗2, τ τ0μ, the system1.2is transformed into an FDE inCC−1,0, R3as
˙
ut Lμut f
μ, ut
, 3.1
whereut u1t, u2t, u3tT∈R3and
Lμ
ϕτ0μ
B1ϕ0 B2ϕ−1
whereB1andB2are defined as B1 ⎛ ⎜ ⎜ ⎝
nx∗−fx∗y2∗ 0 −fx∗
0 −Dv1 0
0 D −v2
⎞ ⎟ ⎟
⎠, B2
⎛ ⎜ ⎜ ⎝
0 0 0
kfx∗y∗2 0 kfx∗
0 0 0
⎞ ⎟ ⎟
⎠, 3.3
fμ, ϕτ0μ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2!
nx∗−fx∗y2∗ϕ2
10−2fx∗ϕ10ϕ30
1
3!
nx∗−fx∗y2∗ϕ310−3fx∗ϕ2
10ϕ30
O4
k
2!
nx∗−fx∗y∗2ϕ21−1−2fx∗ϕ1−1ϕ3−1
k
3!
nx∗−fx∗y∗2ϕ3
1−1−3fx∗ϕ21−1ϕ3−1
O4 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . 3.4
By the Riesz representation theorem, there exists a matrix whose components are bounded variation functionsηθ, ϕinθ∈−1,0such that
Lμϕ
0
−1
dηθ, μϕθ, 3.5
whereϕ∈C.In fact, we can choose
ηθ, μτB1δθ−τB2δθ1, 3.6
where
δθ
⎧ ⎨ ⎩
1, θ0,
0, θ /0. 3.7
Forϕ∈C1−1,0, R3, define
Aμϕ
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙
ϕθ, θ∈−1,0,
0
−1
dηs, μϕs, θ0,
3.8
Rμϕ
⎧ ⎨ ⎩
0, θ∈−1,0,
fϕ, μ, θ0.
Letingu u1, u2, u3T,then system3.1can be rewritten as
˙
utA
ϕutR
ϕut. 3.10
Forψ∈C10,1,R3∗, define
A∗αs
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
−α˙s, s∈0,1,
0
−1
dηTt,0α−t, s0 3.11
and a bilinear form
"
ψ, ϕ#ψ0ϕ0− 0
−1 θ
0
ψξ−θdηθρξdξ, 3.12
whereηθ ηθ,0.ThenA∗ andAare adjoint operators. In addition, fromSection 2we know that±iω0τ0 are eigenvalues of A0. Thus they are also eigenvalues of A∗. By direct computation, we conclude that
qθ 1, β, γTeiω0τ0θ 3.13
is the eigenvector ofA0corresponding toiω0τ0, and
q∗s B1, β∗, γ∗eiω0τ0s 3.14
is the eigenvectorA∗corresponding to−iω0τ0.Moreover, "
q∗s, qθ#1, "q∗s, qθ#0, 3.15
where
β iω0τ0v2γ D , γ
nx∗−fx∗y∗2−iω0τ0
fx∗ ,
β∗ f
x∗y∗
2−nx∗−iω0τ0
kfx∗y∗2eiω0τ0 , γ
∗ Dv1−iω0τ0β∗
D , B
1
Γ,
3.16
whereΓ 1ββ∗γγ∗kτ0β∗e−iω0τ0fx∗y2∗γfx∗.
Using the same notations as in Hassard et al.10, letutbe the solution of3.1when
τ τ0. Definingzt q∗, ut , ut xt, yt, then
˙
zt "q∗,u˙t
#
where
$
ffτ0, Wz, z 2 Re
zq, Wz, z ut−2 Re
zq,
Wz, z W20z 2
2 W11zzW02
z2
2 · · ·.
3.18
Notice thatWis real ifutis real. We consider only real solutions. Rewrite3.19as
˙
utiω0τ0zt gz, z, 3.19
where
gz, z g20
z2
2 g11zzg02
z2
2 g21
z2z
2 · · ·. 3.20
Substituting3.10and3.17into ˙Wu˙t−zq˙ −zq˙ , we have
˙
W
⎧ ⎨ ⎩
AW−2Req∗0fq$ θ, θ∈−τ,0
AW−2Req∗0fq$ θf,$ θ0,
def
AWHz, z, θ, 3.21
where
Hz, z, θ H20θ
z2
2 H11θzzH02θ
z2
2 · · ·. 3.22
Expanding the above series and comparing the coefficients, we obtain
A−2iω0τ0IW20θ −H20θ, AW11−H11θ. 3.23
Forututθ Wz, z, θ zqθ zqθ,we have
z, z g20
z2
2 g11zzg02
z2
2 · · ·q
Comparing the coefficients, we obtain
g20 2τ0B %
1 2
nx∗−fx∗y∗2 1kβ∗e−2iω0τ0−fx∗γ−kβ∗γfx∗e−2iω0τ0
&
,
g11 τ0B 1kβ∗nx∗−fx∗y∗2−fx∗
γγ,
g02 2τ0B %
1 2
nx∗−fx∗y∗2 1kβ∗e2iω0τ0−fx∗γ−kβ∗γfx∗e2iω0τ0
&
,
g21 2τ0B %
1 2
nx∗−fx∗y∗21kβ∗e−iω0τ0
1
2
W2010 2W1110nx∗−fx∗y2∗
−1
2f
x∗γ2γ 1kβ∗e−iω0τ0
−fx∗
W1130 1 2W
3 200
1 2γW
1
20 0 γW
1 11 1
k
2β∗
nx∗−fx∗y2∗W201−1eiω0τ02W1
11−1e−iω0τ0
−kβ∗fx∗
×
W113−1e−iω0τ01
2W
3
20 −1eiω0τ0 1 2γW
1
20 −1eiω0τ0γW 1
11 −1e−iω0τ0 &
.
3.25
We still need to computeW20θandW11θ.Forθ∈−1,0, we have
Hz, z, θ −2 Req∗0fq$ θ−q∗fq$ θ−q∗0fq$ θ −gqθ−gqθ. 3.26
Comparing the coefficients with3.22gives that
H20θ −g20qθ−g02qθ, H11θ −g11qθ−g11qθ. 3.27
It follows from the definition ofWthat
˙
W20θ 2iω0τ0W20θ−H20θ 2iω0τ0W20θ g20qθ g20qθ. 3.28
Solving forW20θ, we obtain
W20θ
ig20
ω0τ0
q0eiω0τ0θ ig20
3ω0τ0
and similarly
W11θ
−ig11
ω0τ0
q0eiω0τ0θ ig11 ω0τ0
q0e−iω0τ0θE2, 3.30
whereE1 andE2are both 3-dimensional vectors and can be determined by settingθ 0 in
H. Hence combining the definition ofA, we can get
0
−1
dηθW20θ AW200 2iω0τ0W200−H200,
0
−1
dηθW11θ −H110.
3.31
Noticing thatiω0τ0I− '0
−1eiω0τ0θdηθq0 0,−iω0τ0I− '0
−1e−iω0τ0θdηθq0 0,we have
2iω0τ0I− 0
−1
e2iω0τ0θdηθ
E1f$z2. 3.32
Similarly, we have
0
−1
dηθ
E2−f$zz. 3.33
Hence, we get
⎛ ⎜ ⎜ ⎝
2iω0−nx∗ fx∗y∗2 0 fx∗
−kfx∗y∗
2e−2iω0τ0 2iω0Dv1 −kfx∗e−2iω0τ0
0 −D 2iω0v2
⎞ ⎟ ⎟ ⎠E1
1 2
nx∗−fx∗y2∗−γfx∗
⎛
⎜ ⎝
1
ke−2iω0τ0
0 ⎞ ⎟ ⎠,
3.34
⎛ ⎜ ⎜ ⎝
nx∗−fx∗y∗2 0 −fx∗ kfx∗y∗2 Dv1 kfx∗
0 D −v2
⎞ ⎟ ⎟ ⎠E2
γγfx∗−nx∗ fx∗y2∗
⎛ ⎝k1
0 ⎞ ⎠.
0 100 200 300 400 500 0
1 2 3 4 5 6
a
0.5
1
1.5
0.2 0.4 0.6 0.8 1 1 2 3 4 5 6
b
Figure 1:Whenτ5, the positive equilibriumE∗of system4.1is asymptotically stable.
0 100 200 300 400 500
0 1 2 3 4 5 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:Whenτ10, the positive equilibriumE∗of system4.1is unstable, and small amplified periodic solutions exist.
Theng21can be expressed by the parameters. Based on the above analysis, we can see that eachgijcan be determined by the parameters. Thus we can compute the following quantities:
C10
i
2ω0τ0
g20g11−2((g11((2− 1 3((g02((
2g21
2 , β22 Re{C10},
T2−
Im{C10}μ2Imλτ0
ω0
, μ2−
Re{C10} Reλτ0
.
3.36
Hence, we have following theorem.
Theorem 3.1. μ2 determines the directions of the Hopf bifurcation: if μ2 > 0< 0, the Hopf
bifurcation is supercritical (subcritical); β2 determines the stability of the bifurcation periodic
solutions: the bifurcation periodic solutions are orbitally stable (unstable) ifβ2<0>0;T2determines
4. Numerical Examples
In this section, we give a numerical example:
˙
xt xt2−xt−0.3y2t
,
˙
y1t 0.24xt−τy2t−τ−0.95y1t,
˙
y2t 0.8y1t−0.1y2t.
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 system4.1 are shown inFigure 1. FromSection 2, we can obtainω00.4867,τ007.9367,τj0 7.93672jπ/ω0j
1,2,3, . . .. From the formulas inSection 3, whenτ0 10, it follows thatGω02 0.9401>0,
Ω 0.0027, ReC0 −34.9505 < 0, μ2 0.4220 > 0, and β2 −69.9010 < 0. Therefore, the Hopf bifurcation is supercritical, and the bifurcating periodic solution is orbitally asymptotically stable. The plots are shown inFigure 2.
Acknowledgments
The authors wish to thank the reviewers for their valuable comments and suggestions that led to a truly significant improvement of the paper. This research is supported by the Natural Science Research Project from Institutions of Higher Education of Hebei Province no. Z 2010105.
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