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Abstract and Applied Analysis
Volume 2012, Article ID 236484, 19 pages
doi:10.1155/2012/236484
Research Article
Stability and Hopf Bifurcation in
a Modified Holling-Tanner Predator-Prey System
with Multiple Delays
Zizhen Zhang,1, 2 Huizhong Yang,1 and Juan Liu3
1
Key Laboratory of Advanced Process Control for Light Industry of Ministry of Education,
Jiangnan University, Wuxi 214122, China
2
School of Management Science and Engineering, Anhui University of Finance and Economics,
Bengbu 233030, China
3
Department of Science, Bengbu College, Bengbu 233030, China
Correspondence should be addressed to Huizhong Yang, [email protected]
Received 9 August 2012; Revised 17 September 2012; Accepted 4 October 2012
Academic Editor: Kunquan Lan
Copyright q 2012 Zizhen Zhang et al. 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 modified Holling-Tanner predator-prey system with multiple delays is investigated. By
analyzing the associated characteristic equation, the local stability and the existence of periodic
solutions via Hopf bifurcation with respect to both delays are established. Direction and stability
of the periodic solutions are obtained by using normal form and center manifold theory. Finally,
numerical simulations are carried out to substantiate the analytical results.
1. Introduction
Predator-prey dynamics has long been and will continue to be of interest to both applied
mathematicians and ecologists due to its universal existence and importance 1, 2. Many
population models investigating the dynamic relationship between predators and their
preys have been proposed and studied. For example, Lotka-Volterra model 3–5, LeslieGower model 6–10, and Holling-Tanner model 11–16. Among these widely used models,
Holling-Tanner model plays a special role in view of the interesting dynamics it possesses.
Holling-Tanner model for predator-prey interaction is governed by the following nonlinear
coupled ordinary differential equations:
X
mXY
dX
rX 1 −
−
,
dT
K
aX
dY
Y
Y s 1−h
,
dT
X
1.1
2
Abstract and Applied Analysis
where X and Y denote the population densities of prey species and predator species at time
T , respectively. The first equation in system 1.1 shows that the prey grows logistically with
the carrying capacity K and the intrinsic growth rate r in the absence of the predator. And
the growth of the prey is hampered by the predator at a rate proportional to the functional
response mX/a X in the presence of the predator. The second equation shows that the
predator consumes the prey according to the functional response mX/a X and grows
logistically with the intrinsic growth rate s and carrying capacity X/h proportional to the
number of the prey. The parameter m denotes the maximal predator per capita consumption
rate. a is a saturation value; it corresponds to the number of prey necessary to achieve one
half the maximum rate m. The parameter h denotes the number of prey required to support
one predator at equilibrium when y equals X/h.
Recently, there has been considerable interest in predator-prey systems with the
Beddington-DeAngelis functional response. And it has been shown that the predator-prey
systems with the Beddington-DeAngelis functional response have rich but biologically
reasonable dynamics. For more details about this functional response one can refer to 17–
21. Zhang 16, Lu and Liu 22 considered the following modified Holling-Tanner delayed
predator-prey system:
X
αXY
dX
rX 1 −
,
−
dT
K
a bX cY
dY
Y t − τ
Y s 1−h
,
dT
Xt − τ
1.2
where τ is incorporated in the negative feedback of the predator density. αXY/abX cY is
the Beddington-DeAngelis functional response. The parameters α, a, b, and c are assumed to
be positive. α is the maximum value at which per capita reduction rate of the prey can attain.
a measures the extent to which environment provides protection to the prey. b describes
the effect of handling time on the feeding rate. c describes the magnitude of interference
among predators. Zhang 16 investigated the local Hopf bifurcation of system 1.2. Lu and
Liu 22 proved that system 1.2 is permanent under some conditions and obtained the
sufficient conditions of local and global stability of system 1.2. Since both the species are
growing logistically, it is reasonable to assume delay in prey species as well. Based on this
consideration, we incorporate the negative feedback of the prey density into system 1.2 and
obtain the following system:
XT − T1 αXY
dX
rX 1 −
,
−
dT
K
a bX cY
dY
Y T − T2 Y s 1−h
,
dT
XT − T2 1.3
where T1 and T2 are the feedback time delays of the prey density and the predator density
respectively. Let X Kx, Y rK/αy, t rT , τ1 rT1 , τ2 rT2 , system 1.3 can be
Abstract and Applied Analysis
3
transformed into the following nondimensional form:
xy
dx
x1 − xt − τ1 −
,
dt
a1 bx c1 y
dy
yt − τ2 y δ−β
,
dt
xt − τ2 1.4
where a1 a/K, c1 cr/α, δ s/r, β sh/α are the non-dimensional parameters and they
are positive.
The main purpose of this paper is to consider the effect of multiple delays on system
1.4. The local stability of the positive equilibrium and the existence of Hopf bifurcation are
investigated. By employing normal form and center manifold theory, the direction of Hopf
bifurcation and the stability of the bifurcating periodic solutions are determined. Finally,
some numerical simulations are also included to illustrate the theoretical analysis.
2. Local Stability and the Existence of Hopf Bifurcation
In this section, we study the local stability of each of feasible equilibria and the existence of
Hopf bifurcation at the positive equilibrium. Obviously, system 1.4 has a unique boundary
equilibrium E1 1, 0 and a unique positive equilibrium E∗ x∗ , y∗ , where
2
− a1 − bβ 1 − c1 δ a1 − bβ 1 − c1 δ 4a1 β bβ c1 δ
,
x∗ 2 bβ c1 δ
y∗ δ
x∗ .
β
2.1
The Jacobian matrix of system 1.4 at E1 takes the form
⎞
1
−λτ1
−
−e
⎜
a1 b ⎟
JE1 ⎝
⎠.
0
δ
⎛
2.2
The characteristic equation of system 1.4 at E1 is of the form
λ − δ λ e−λτ1 0.
Clearly, the boundary equilibrium E1 1, 0 is unable.
2.3
4
Abstract and Applied Analysis
Next, we discuss the existence of Hopf bifurcation at the positive equilibrium Ex∗ , y∗ .
Let xt z1 t x∗ , yt z2 t y∗ , and still denote z1 t and z2 t by xt and yt,
respectively, then system 1.4 becomes
ijk
dx
a11 xt a12 yt b11 xt − τ1 f1 xi yj xk t − τ1 ,
dt
ijk≥2
ijk
dy
c21 xt − τ2 c22 yt − τ2 f2 yi xj t − τ2 yk t − τ2 ,
dt
ijk≥2
2.4
where
bx∗ y∗
a11 a1 bx∗ c1 y∗
2 ,
b11 −x∗ ,
f1 x1 − xt − τ1 −
a1 bx∗ c1 y∗
δ2
,
β
xy
,
a1 bx c1 y
ijk
f1 xi yj xk t − τ1 ijk
c21 f2 yi xj t − τ2 yk t − τ2 a1 bx∗ x∗
a12 − 2 ,
c22 −δ,
yt − τ2 f2 y δ − β
,
xt − τ2 2.5
∂ijk f1
1
|x ,y ,
i!j!k! ∂xi ∂yj ∂xk t − τ1 ∗ ∗
∂ijk f2
1
.
|
i!j!k! ∂yi ∂xj t − τ2 ∂yk t − τ2 x∗ ,y∗ Then we can obtain the linearized system of system 1.4
dx
a11 xt a12 yt b11 xt − τ1 ,
dt
2.6
dy
c21 xt − τ2 c22 yt − τ2 .
dt
The characteristic equation of system 2.6 is
λ2 − a11 λ − b11 λe−λτ1 a11 c22 − a12 c21 − c22 λe−λτ2 b11 c22 e−λτ1 τ2 0.
2.7
Case 1. τ1 τ2 τ 0.
The characteristic equation of system 1.4 becomes
2.8
λ2 A B Dλ C E 0,
where
A −a11 ,
B −b11 ,
C a11 c22 − a12 c21 ,
D −c22 ,
E b11 c22 .
2.9
Abstract and Applied Analysis
5
It is easy to verify that
a1 δ2 x∗
C E δx∗ 2 > 0.
β a1 bx∗ c1 y∗
2.10
Therefore, if H1 : A B D > 0, the roots of 2.8 must have negative real parts. Then, we
know that the positive equilibrium E∗ x∗ , y∗ of system 1.4 is locally stable in the absence
of delay, if H1 holds.
Case 2. τ1 τ2 τ > 0.
The associated characteristic equation of the system is
λ2 A1 λ B1 C1 λe−λτ D1 e−2λτ 0,
2.11
where
A1 −a11 ,
B1 a11 c22 − a12 c21 ,
C1 −b11 c22 ,
D1 b11 c22 .
2.12
Multiplying eλτ on both sides of 2.11, we can obtain
λ2 A1 λ eλτ B1 C1 λ D1 e−λτ 0.
2.13
Now, for τ > 0, if λ iωω > 0 be a root of 2.13. Then, we have
D1 − ω2 cos τω − A1 ω sin τω −B1 ,
D1 ω2 sin τω − A1 ω cos τω C1 ω.
2.14
It follows from 2.14 that
sin τω C1 ω2 A1 B1 − C1 D1 ω
,
ω4 A21 ω2 − D12
cos τω B1 − A1 C1 ω2 B1 D1
.
ω4 A21 ω2 − D12
2.15
Then we have
2.16
ω8 e3 ω6 e2 ω4 e1 ω2 e0 0,
where
e3 2A21 − C12 ,
e2 A41 2C12 D1 − A21 C12 − B12 − 2D12 ,
e1 4A1 B1 C1 D1 − A21 B12 − C12 D12 − 2A21 D12 − 2B12 D1 ,
e0 D14 − B12 D12 .
2.17
6
Abstract and Applied Analysis
Let v ω2 , then 2.16 becomes
2.18
v4 e3 v3 e2 v2 e1 v e0 0.
Next, we give the following assumption. H2 : 2.18 has at least one positive real root.
Suppose that H2 holds. Without loss of generality, we assume that 2.18 has four
real positive roots, which are defined by v1 , v2 , v3 , and v4 , respectively. Then 2.16 has four
√
positive roots ωk vk , k 1, 2, 3, 4. Therefore,
j
τk
1
arccos
ωk
B1 − A1 C1 ωk2 B1 D1
ωk4 A21 ωk2 − D12
2jπ ,
k 1, 2, 3, 4; j 0, 1, 2 . . .
2.19
j
Then we can know that ±iωk are a pair of purely imaginary roots of 2.11 with τ τk .
Define
0
0
τ0 τk min τk ,
ω0 ωk0 ,
k 1, 2, 3, 4.
2.20
Let λτ ατ iωτ be the root of 2.11 near τ τ0 which satisfies ατ0 0, ωτ0 ω0 .
Taking the derivative of λ with respect to τ in 2.13, we obtain
dλ
dλ
dλ
λτ dλ
2
λτ
−λτ
2λ A1 e
λ A1 λ e
− D1 e
0.
λτ
λτ
C1
dτ
dτ
dτ
dτ
2.21
it follows that
λ D1 e−λτ − λ2 A1 λ eλτ
dλ
.
dτ 2λ A1 eλτ C1 − τ D1 e−λτ − λ2 A1 λeλτ
2.22
Thus
dλ
dτ
−1
τ
2λ A1 eλτ C1
− .
−λτ
3
2
λτ
λ
D1 λe − λ A1 λ e
2.23
Let
Λ1 D1 ω0 − ω03 sin τ0 ω0 A1 ω02 cos τ0 ω0 ,
Λ3 A1 cos τ0 ω0 − 2ω0 sin τ0 ω0 C1 ,
Λ2 D1 ω0 ω03 cos τ0 ω0 A1 ω02 sin τ0 ω0 ,
Λ4 A1 sin τ0 ω0 − 2ω0 cos τ0 ω0 .
2.24
Substitute λ iω0 ω0 > 0 into 2.23, we can get
dλτ0 Re
dτ
−1
2λ A1 eλτ C1
Re
D1 λe−λτ − λ3 A1 λ2 eλτ
λiω0
Λ1 × Λ3 Λ2 × Λ4
.
Λ21 Λ22
2.25
Abstract and Applied Analysis
7
Noting that
d Re λτ0 −1
dλτ0 sign
.
signRe
dτ
dτ
2.26
Therefore, we make the following assumption in order to give the main results: H3 : Λ1 ×
0. Then, by Corollary 2.4 in 23, 24, we have the following theorem.
Λ3 Λ2 × Λ4 /
Theorem 2.1. For system 1.4, if the conditions H1 –H3 hold, then the equilibrium E∗ x∗ , y∗ of
system 1.4 is asymptotically stable for τ ∈ 0, τ0 and unstable when τ > τ0 . And system 1.4 has
a branch of periodic solution bifurcation from the zero solution near τ τ0 .
Case 3. τ1 / τ2 , τ1 > 0 and τ2 > 0.
The associated characteristic equation of the system is
λ2 A2 λ B2 λe−λτ1 C2 D2 λe−λτ2 E2 e−λτ1 τ2 ,
2.27
where
A2 −a11 ,
B2 −b11 ,
C2 a11 c22 − a12 c21 ,
D2 −c22 ,
E2 b11 c22 .
2.28
We consider 2.27 with τ2 in its stable interval, regarding τ1 as a parameter. Without loss of
generality, we consider system 1.4 under the case considered in 16, and τ2 ∈ 0, τ20 . τ20 is
defined as in 16 and can be obtained by
τ20
a11 c22 b11 c22 − a12 c21 ω2 − a11 b11 c22 ω2
1
,
arccos
ω
a11 c22 b11 c22 − a12 c21 2 c22 ω 2
2.29
with
ω 2
2
2
− a11 b11 2 − c2 b
−
c
4a11 c22 b11 c22 − a12 c21 2
a
11
11
22
22
2
.
2.30
Let λ iωω > 0 be a root of 2.27. Then we obtain
B2 ω − E2 sin τ2 ω sin τ1 ω E2 cos τ2 ω cos τ1 ω ω2 − C2 cos τ2 ω − D2 ω sin τ2 ω,
B2 ω − E2 sin τ2 ω cos τ1 ω − E2 cos τ2 ω sin τ1 ω C2 sin τ2 ω − D2 ω cos τ2 ω − A2 ω.
2.31
It follows from 2.31 that
sin τ1 ω M1 N1 − M2 N2
,
M12 M22
cos τ1 ω M1 N2 M2 N1
,
M12 M22
2.32
8
Abstract and Applied Analysis
With
M1 B2 ω − E2 sin τ2 ω,
N1 ω2 − C2 cos τ2 ω − D2 ω sin τ2 ω,
M2 E2 cos τ2 ω,
N2 −A2 ω C2 sin τ2 ω − D2 ω cos τ2 ω.
2.33
Then we have
P1 ω P2 ω sin τ2 ω P3 ω cos τ2 ω 0,
2.34
where
P1 ω ω4 A22 D22 − B22 ω2 C22 − E22 ,
P2 ω −2D2 ω − 2A2 C2 ω 2B2 E2 ω,
3
2.35
P3 ω 2A2 D2 − C2 ω .
2
Suppose that H4 : 2.34 has at least finite positive roots. If H4 holds, we define the roots
of 2.34 as ω1 , ω2 , . . . , ωk . Then, for every fixed ωi i 1, 2, . . . , k, there exists a sequence
j
{τ1i | j 1, 2, . . .} which satisfies 2.34. Let
j
τ1∗ min τ1i | i 1, 2, . . . , k, j 0, 1, . . . .
2.36
When τ1 τ1∗ , 2.27 has a pair of purely imaginary roots ±iω∗ for τ2 ∈ 0, τ20 .
To verify the transversality condition of Hopf bifurcation, we take the derivative of λ
with respect to τ1 in 2.27, we can obtain
λe−λτ1 B2 λ E2 e−λτ2
dλ
.
dτ1
2λ A2 B2 e−λτ1 D2 e−λτ2 − τ1 e−λτ1 B2 λ E2 e−λτ2 − τ2 e−λτ2 C2 D2 λ E2 e−λτ1
2.37
Thus
dλ
dτ1
−1
2λ A2 B2 e−λτ1 D2 e−λτ2 − τ2 e−λτ2 C2 D2 λ E2 e−λτ1
τ1
− .
λ
λe−λτ1 B2 λ E2 e−λτ2
2.38
Substitute λ iω∗ ω∗ > 0 into 2.38, we can get
Re
dλτ1∗ dτ1
−1
Δ1 × Δ3 Δ2 × Δ4
,
Δ21 Δ22
2.39
Abstract and Applied Analysis
9
where
Δ1 E2 ω∗ cos τ2 ω∗ sin τ1∗ ω∗ − ω∗ cos τ1∗ ω∗ B2 ω∗ − E2 ω∗ sin τ2 ω∗ ,
Δ2 E2 ω∗ cos τ2 ω∗ cos τ1∗ ω∗ ω∗ sin τ1∗ ω∗ B2 ω∗ − E2 ω∗ sin τ2 ω∗ ,
Δ3 A2 D2 − τ2 C2 cos τ2 ω∗ − τ2 D2 ω∗ sin τ2 ω∗
τ2 E2 sin τ2 ω∗ sin τ1∗ ω∗ B2 − τ2 E2 cos τ2 ω∗ cos τ1∗ ω∗ ,
2.40
Δ4 2ω∗ τ2 C2 τ2 D2 ω∗ − D2 sin τ2 ω∗
τ2 E2 sin τ2 ω∗ cos τ1∗ ω∗ − B2 − τ2 E2 cos τ2 ω∗ sin τ1∗ ω∗ .
Next, we make the following assumption: H5 : Δ1 × Δ3 Δ2 × Δ4 /
0.
Thus, by the discussion above and by the general Hopf bifurcation theorem for FDEs
in Hale 25, we have the following results.
Theorem 2.2. For τ2 ∈ 0, τ20 , τ20 is defined by 2.29. If the conditions H4 -H5 hold, then the
equilibrium E∗ x∗ , y∗ of system 1.4 is asymptotically stable for τ1 ∈ 0, τ1∗ and unstable when
τ > τ1∗ . System 1.4 has a branch of periodic solution bifurcation from the zero solution near τ τ1∗ .
3. Direction and Stability of Bifurcated Periodic Solutions
In this section, we shall investigate the direction of the Hopf bifurcation and the stability of
bifurcating periodic solution of system 1.4 w.r. to τ1 for τ2 ∈ 0, τ20 , and τ20 is defined by
2.29. The idea employed here is the normal form and center manifold theory described in
Hassard et al. 26. Throughout this section, it is considered that system 1.4 undergoes the
Hopf bifurcation at τ1 τ1∗ , τ2 ∈ 0, τ20 at E∗ x∗ , y∗ . Let τ1 τ1∗ μ, μ ∈ R so that the
Hopf bifurcation occurs at μ 0. Without loss of generality, we assume that τ2∗ < τ1∗ , where
τ2∗ ∈ 0, τ20 .
Let u1 t xt − x∗ , u2 t yt − y∗ , and rescaling the time delay t → t/τ1 , Then
system 1.4 can be transformed into an FDE in C C−1, 0, R2 as:
u̇t Lμ ut F μ, ut ,
3.1
where ut u1 t, u2 tT ∈ R2 and Lμ : C → R2 , F : R × C → R2 are given, respectively, by
τ2∗
B φ−1 ,
Lμ φ τ1∗ μ A φ0 C φ −
τ1∗
T
F μ, φ τ1∗ μ f1 , f2 ,
3.2
10
Abstract and Applied Analysis
with
A a11 a12
,
0 0
B b11 0
,
0 0
0 0
,
c21 c22
C f1 g1 φ12 0 g2 φ1 0φ2 0 g3 φ22 0 g4 φ1 0φ1 −1
h1 φ13 0 h2 φ12 0φ2 0 h3 φ1 0φ22 0 h4 φ23 0 · · · ,
τ2∗
τ2∗
τ2∗
τ2∗
g2 φ1 −
φ2 0 g3 φ1 −
φ2 −
,
f2 g1 φ12 −
τ1∗
τ1∗
τ1∗
τ1∗
τ2∗
τ2∗
τ2∗
g4 φ2 0φ2 −
h1 φ13 −
h2 φ12 −
φ2 0,
τ1∗
τ1∗
τ1∗
τ2∗
τ2∗
φ2 −
h4 φ23 0 · · · ,
h3 φ12 −
τ1∗
τ1∗
3.3
where
by∗ a1 c1 y∗
g1 3 ,
a1 bx∗ c1 y∗
g2 −
a21 a1 bx∗ a1 c1 y∗ 2bc1 x∗ y∗
,
3
a1 bx∗ c1 y∗
c1 x∗ a1 bx∗ g4 −1,
3 ,
a1 bx∗ c1 y∗
a21 b a1 b2 x∗ 2b2 c1 x∗ y∗ − bc12 y∗2
b2 y∗ a1 c1 y∗
h1 − ,
h
,
2
4
4
a1 bx∗ c1 y∗
a1 bx∗ c1 y∗
g3 h3 a21 c1
a1 c12 y∗
a1 bx∗ c1 y∗
g1 −
βy∗2
x∗3
,
h1 −b
4
2bc12 x∗ y∗
βy∗2
x∗4
2
c1 x∗2
g2 βy∗
,
h2 −
x∗2
,
g3 ,
βy∗
x∗3
,
3.4
bx∗ 4 ,
a1 bx∗ c1 y∗
h4 − βy∗
x∗2
,
h3 −
c12 x∗ a1
g4 −
βy∗
x∗3
β
,
x∗
.
Using Riesz representation theorem, there exists a 2 × 2 matrix function ηθ, μ, θ ∈ −1, 0
whose elements are of bounded variation, such that
Lμ φ 0
−1
dη θ, μ φθ,
φ ∈ C−1, 0, R2 .
3.5
Abstract and Applied Analysis
11
In fact, choosing
⎧
⎪
τ1∗μ A C B , θ 0,
⎪
⎪
⎪
⎪
τ2∗
⎪
⎪
θ ∈ − ,0 ,
⎨ τ1∗μ C B ,
τ1
η θ, μ τ2∗
⎪
⎪
,
θ
∈
−1,
−
,
B
τ
⎪ 1∗μ
⎪
⎪
τ1
⎪
⎪
⎩0,
θ −1.
3.6
For φ ∈ C−1, 0, we define
⎧
dφθ
⎪
⎪
,
⎪
⎨
dθ
A μ φ 0
⎪
⎪
⎪
⎩
dη θ, μ φθ,
−1
R μ φ
#
0,
F μ, φ ,
−1 ≤ θ < 0,
θ 0,
3.7
−1 ≤ θ < 0,
θ 0.
Then system 3.1 can be transformed into the following operator equation
u̇t A μ ut R μ ut ,
3.8
where ut ut θ u1 t θ, u2 t θ.
For ϕ ∈ C1 0, 1, R2 ∗ , where R2 ∗ is the 2-dimensional space of row vectors, we
further define the adjoint operator A∗ of A0:
⎧
dϕs
⎪
⎪
,
−
⎪
⎨
ds
∗
A ϕ 0
⎪
⎪
⎪
⎩
ϕ−ξdηξ, 0,
−1
0 < s ≤ 1,
3.9
s 0,
and a bilinear inner product:
$
%
T
ϕs, φθ ϕ 0φ0 −
0
θ−1
where ηθ ηθ, 0.
θ
ξ0
ϕT ξ − θdηθφξdξ,
3.10
12
Abstract and Applied Analysis
Since ±iω∗ τ1∗ are eigenvalues of A0, they are also eigenvalues of A∗ . Let
qθ 1, q2 T eiω∗ τ1∗ θ be the eigenvectors of A0 corresponding to iω∗ τ1∗ and q∗ s 1/ρ1, q2∗ T eiω∗ τ1∗ s be the eigenvectors of A∗ corresponding to −iω∗ τ1∗ . By a simple
computation, we can get
q2 ρ 1
iω∗ − a11 − b11 eiω∗ τ1∗
,
a12
q2 q∗2
b11 τ1∗ e
−iω∗ τ1∗
q2∗ −
iω∗ a11 b11 eiω∗ τ1∗
,
c21 eiω∗ τ2∗
c21 τ2∗ q∗2 e−iω∗ τ2∗
3.11
c22 τ2∗ q2 q∗2 e−iω∗ τ2∗ .
Then q∗ , q
1, q∗ , q
0.
In the remainder of this section, Following the algorithms given in 26 and using
similar computation process to that in 16, we can get that the coefficients which will be
used to determine the important qualities of the bifurcating periodic solutions,
g20
g11
g02
#
2
2τ1∗
g1 g2 q2 0 g3 q2 0 g4 q1 −1
ρ
2
τ2∗
τ2∗ 2
τ2∗ 2
τ2∗
∗
g2 q1 −
q2 g1 q1 −
q 0 g3 q1 −
q
−
τ1∗
τ1∗
τ1∗
τ1∗
&
τ2∗
,
g4 q2 0q2 −
τ1∗
'
τ1∗
2g1 g2 q2 0 q2 0 2g3 q2 0q2 0 g4 q1 −1 q1 −1
ρ
τ2∗ 1
τ2∗
τ2∗ 2
τ2∗ 2
q∗2 2g1 q1 −
q
−
g2 q1 −
q 0
q 0 q1 −
τ1∗
τ1∗
τ1∗
τ1∗
τ2∗ 2
τ2∗
τ2∗ 2
τ2∗
q
−
q1 −
q
−
g3 q1 −
τ1∗
τ1∗
τ1∗
τ1∗
(
τ2∗
τ2∗
q2 0q2 −
,
g4 q2 0q2 −
τ1∗
τ1∗
#
2
2τ1∗
g1 g2 q2 0 g3 q2 0 g4 q1 −1
ρ
2
τ2∗
τ2∗ 2
τ2∗ 2
τ2∗
∗
g2 q1 −
q 0 g3 q1 −
q
q2 g1 q1 −
−
τ1∗
τ1∗
τ1∗
τ1∗
(
τ2∗
g4 q2 0q2 −
,
τ1∗
Abstract and Applied Analysis
#
2τ1∗
1
1
g21 g1 W20 0 2W11 0
ρ
1 2
1 1
2
1
W20 0 W11 0 W20 0q2 0 W11 0q2 0
g2
2
2
2
2
g3 W20 0q2 0 2W11 0q2 0
13
1 1
1 1
1
1
1
1
W −1 W11 −1 W20 0q −1 W11 0q −1 3h1
g4
2 20
2
2 2
2
2
2
2
2
h2 2q 0 q 0 h3 2q 0q 0 q 0
3h4 q2 0 q2 0
τ2∗ 1
τ2∗
τ2∗ 1
τ2∗
1
1
q
g1 W20 −
−
2W11 −
q
−
τ1∗
τ1∗
τ1∗
τ1∗
τ2∗ 2
τ2∗ 2
τ2∗
1 2
1
1
1
1
W
g2
q 0 W11 −
−
q 0 W20 0q
−
2 20
τ1∗
τ1∗
2
τ1∗
τ2∗
2
W11 0q1 −
τ1∗
1 1
τ2∗ 2
τ2∗
τ2∗ 2
τ2∗
1
W20 −
q
−
W11 −
q
−
g3
2
τ1∗
τ1∗
τ1∗
τ1∗
1 2
τ2∗ 1
τ2∗
W20 −
q
−
2
τ1∗
τ1∗
1 2
τ2∗
τ2∗
1 2
τ2∗ 2
2
W20 0q2 −
g4
W11 0q2 −
W20 −
q 0
2
τ1∗
τ1∗
2
τ1∗
τ2∗ 2
2
q 0
W11 −
τ1∗
2
τ2∗
τ2∗
1
1
q
−
−
3h1 q
τ1∗
τ1∗
2
τ2∗
τ2∗ 1
τ2∗ 2
2
1
1
h2
q 0 2q
q
−
−
−
q 0
q
τ1∗
τ1∗
τ1∗
2
&
τ2∗
τ2∗
τ2∗ 1
τ2∗ 2
τ2∗
2
1
1
,
q
q
h3
−
−
2q
−
−
q
−
q
τ1∗
τ1∗
τ1∗
τ1∗
τ1∗
q∗2
3.12
with
W20 θ ig20 q0 iτ1∗ ω∗ θ ig 02 q0 −iτ1∗ ω∗ θ
e
e
E20 e2iτ1∗ ω∗ θ ,
τ1∗ ω∗
3τ1∗ ω∗
ig11 q0 iτ1∗ ω∗ θ ig 11 q0 −iτ1∗ ω∗ θ
e
e
E11 ,
W11 θ −
τ1∗ ω∗
τ1∗ ω∗
3.13
14
Abstract and Applied Analysis
where E20 and E11 can be computed as the following equations, respectively,
⎛
⎜
E20 2⎝
1
⎞
−1
−a12
2iω∗ − a11 − b11 e−2iτ1∗ ω∗
⎟
,
⎠×
−c21 e−2iτ2∗ ω∗
2iω∗ − c22 e−2iτ2∗ ω∗
2
E20
E20
⎞
⎛
E11
1
E
⎜ 11 ⎟
−1
a11 b11 a12
×
−⎝
,
⎠
c21
c22
2
E11
3.14
with
2
1
E20 g1 g2 q2 0 g3 q2 0 g4 q1 −1,
1
E11
2
E11
2
τ2∗
τ2∗ 2
2
g2 q1 −
E20 g1 q1 −
q 0
τ1∗
τ1∗
τ2∗ 2
τ2∗
τ2∗
q
−
g4 q2 0q2 −
,
g3 q1 −
τ1∗
τ1∗
τ1∗
2g1 g2 q2 0 q2 0 2g3 q2 0q2 0 g4 q1 −1 q1 −1 ,
3.15
τ2∗ 1
τ2∗
τ2∗ 2
τ2∗ 2
1
1
1
2g1 q
q
q 0 q
−
−
g2 q
−
−
q 0
τ1∗
τ1∗
τ1∗
τ1∗
τ2∗ 2
τ2∗
τ2∗ 2
τ2∗
1
1
q
−
−
q
−
q
−
g3 q
τ1∗
τ1∗
τ1∗
τ1∗
τ2∗
τ2∗
g4 q2 0q2 −
q−2 0q2 −
.
τ1∗
τ1∗
Therefore, we can calculate the following values:
C1 0 i
2τ1∗ ω∗
) ) ) )2 )g02 )2
g21
g11 g20 − 2)g11 ) −
,
3
2
μ2 −
Re{C1 0}
,
Re{λ τ1∗ }
3.16
β2 2 Re{C1 0},
T2 −
Im{C1 0} μ2 Im{λ τ1∗ }
.
τ1∗ ω∗
Based on the discussion above, we can obtain the following results.
Theorem 3.1. The direction of the Hopf bifurcation is determined by the sign of μ2 : if μ2 > 0 μ2 <
0, the Hopf bifurcation is supercritical (subcritical). The stability of bifurcating periodic solutions is
determined by the sign of β2 : if β2 < 0 β2 > 0, the bifurcating periodic solutions are stable (unstable).
Abstract and Applied Analysis
15
The period of the bifurcating periodic solutions is determined by the sign of T2 : if T2 > 0 T2 < 0, the
period of the bifurcating periodic solutions increases (decreases).
4. Numerical Example
In order to support the analytic results obtained above, we give some numerical simulations
in this section. We only study the most important steady state, namely, the positive steady
state. We consider the following system by taking the same coefficients as in 16:
xy
dx
x1 − xt − τ1 −
,
dt
0.01 3x y
dy
yt − τ2 y 3.5 − 2
,
dt
xt − τ2 4.1
where we get the positive equilibrium E∗ 0.6328, 1.1074. For system 4.1, we can get that
A B D 3.9017 > 0, namely, the condition H1 holds.
0. By a simple computation, we obtain that 2.18 has two positive
For τ1 τ2 τ /
roots: v1 12.0256, v2 0.3915. Thus, we know that the condition H2 holds. Further, we
get ω0 3.4678, τ0 0.4292. In addition, we have Λ1 × Λ3 Λ2 × Λ4 94.4826 > 0. Therefore,
the condition H3 is satisfied. Hence, from Theorem 2.1, we conclude that the positive
equilibrium E∗ 0.6328, 1.1074 is asymptotically stable when τ ∈ 0, τ0 . The corresponding
waveform and the phase plot are illustrated by Figure 1. When the delay τ passes through the
critical value τ0 the positive equilibrium E∗ 0.6328, 1.1074 will lose its stability and a Hopf
bifurcation occurs, and a family of periodic solution bifurcates from the positive equilibrium
E∗ 0.6328, 1.1074. This property is illustrated by the numerical simulation in Figure 2.
For τ1 /
τ2 , τ1 > 0 and τ2 > 0. Regard τ1 as a parameter and let τ2 0.36 ∈ 0, τ20 ,
and from 2.29 we can obtian τ20 0.4037. Then by a simple computation, we can obtain
2.34 has a positive root ω∗ 0.6792. Thus, the condition H4 holds. Further, we we can
get τ1∗ 2.3211 and Δ1 × Δ3 Δ2 × Δ4 3.7131 > 0. Namely, the condition H5 holds.
Thus, by Theorem 2.2, the positive equilibrium E∗ 0.6328, 1.1074 is asymptotically stable
when τ1 ∈ 0, τ1∗ and unstable when τ1 > τ1∗ , and a family of periodic solution bifurcates
from the positive equilibrium E∗ 0.6328, 1.1074. The corresponding waveform and the phase
plot are illustrated by Figures 3 and 4, respectively. In addition, from 3.16, we can get
C1 0 −26.2632 − 7.5707i, then μ2 12.1471, β2 −52.5264, T2 5.8586. From Theorem 3.1,
we know that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are
stable.
5. Conclusion
In this paper, a delayed predator-prey system with Beddington-DeAngelis functional
response has been investigated. The bifurcation of a predator-prey system with single delay
has been studied by many researchers 16, 27–30. However, there are few papers considering
the bifurcation of a predator-prey system with multiple delays see 31–33. Compared
with the system considered in 16, the system in this paper accounts for not only the
feedback delay of the prey density but also the feedback delay of the predator. The sufficient
conditions for the stability of the positive equilibrium and the existence of periodic solutions
16
Abstract and Applied Analysis
0.72
0.7
2.5
2.5
2
2
1.5
1.5
0.66
y(t)
y(t)
x(t)
0.68
1
1
0.5
0.5
0.64
0.62
0.6
0
20
40
0
60
0
20
40
0
0.6
60
Time t
Time t
0.65
0.7
x(t)
Predator
Prey
Figure 1: E∗ is stable for τ1 τ2 τ 0.4 < τ0 0.4292 with initial value 0.62,2.35.
0.74
2.5
2.5
2
2
1.5
1.5
0.72
0.7
0.66
y(t)
x(t)
y(t)
0.68
1
1
0.5
0.5
0.64
0.62
0.6
0
20
40
Time t
Prey
60
0
0
20
40
Time t
60
0
0.6
0.65
0.7
x(t)
Predator
Figure 2: E∗ is unstable for τ1 τ2 τ 0.44 > τ0 0.4292 with initial value 0.62,2.35.
via Hopf bifurcation at the positive equilibrium are obtained when τ1 τ2 and τ1 / τ2 with
τ2 ∈ 0, τ20 . Special attention is paid to the direction of the Hopf bifurcation and the stability
of the bifurcating periodic solutions. By computation, we find that the feedback delay of the
predator is marked because the critical value of τ2 is much smaller when we only consider it.
The feedback delay of the prey is unremarkable because the critical value of τ1 is much bigger
when we consider it with τ2 in its stable interval. Furthermore, Zhang 16 has obtained that
the two species in system 1.4 with only the feedback delay of the predator could coexist.
Abstract and Applied Analysis
17
0.71
2.5
2.5
2
2
1.5
1.5
0.7
0.69
y(t)
x(t)
0.67
y(t)
0.68
0.66
0.65
1
1
0.5
0.5
0.64
0.63
0.62
0.61
0
20
40
0
60
0
20
Time t
40
0
0.6
60
0.65
x(t)
Time t
0.7
Predator
Prey
Figure 3: E∗ is stable for τ1 0.8 < τ1 ∗ 2.3211, τ2 0.36 with initial value 0.62,2.35.
0.72
2.5
2.5
2
2
1.5
1.5
0.7
0.68
0.62
y(t)
0.64
y(t)
x(t)
0.66
1
1
0.5
0.5
0.6
0.58
0.56
0.54
0
20
40
Time t
Prey
60
0
0
20
40
Time t
60
0
0.55
0.6
0.65
0.7
x(t)
Predator
Figure 4: E∗ is unstable for τ1 2.4 > τ1 ∗ 2.3211, τ2 0.36 with initial value 0.62,2.35.
However, we get that the two species could also coexist with some available feedback delays
of the prey and the predator. This is valuable from the view of ecology.
Unfortunately, the existence of the periodic solutions remain valid only in a small
neighborhood of the critical value. It is definitely an interesting work to investigate whether
these nonconstant periodic solutions which are obtained through local Hopf bifurcation can
still exist for large values of the corresponding parameter time delay. The global continuation
of the local Hopf bifurcation is left as the future work.
18
Abstract and Applied Analysis
Acknowledgments
The authors are grateful to the two anonymous reviewers for their helpful comments and
valuable suggestions on improving the paper. This work was supported by Doctor Candidate
Foundation of Jiangnan University JUDCF12030 and Anhui Provincial Natural Science
Foundation under Grant no. 1208085QA11.
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