Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 652947, 22 pages
doi:10.1155/2012/652947
Research Article
Dynamical Analysis in a Delayed Predator-Prey
Model with Two Delays
Changjin Xu1 and Peiluan Li2
1
Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics,
Guiyang 550004, China
2
Department of Mathematics and Statistics, Henan University of Science and Technology,
Luoyang 471003, China
Correspondence should be addressed to Changjin Xu, xcj403@126.com
Received 7 December 2011; Accepted 19 February 2012
Academic Editor: Xue He
Copyright q 2012 C. Xu and P. Li. 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 class of Beddington-DeAngelis functional response predator-prey model is considered. The conditions for the local stability and the existence of Hopf bifurcation at the positive equilibrium of the
system are derived. Some explicit formulae for determining the stability and the direction of the
Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the
normal form theory and center manifold theory. Some numerical simulations for justifying the
theoretical analysis are also provided. Finally, main conclusions are given.
1. Introduction
In recent years, population dynamics including stable, unstable, persistent, and oscillatory
behavior has become very popular since Vito Volterra and James Lotka proposed the seminal models of predator-prey models in the mid-1920s. Great attention has been paid to the dynamics properties of the predator-prey models which have significant biological background.
Many excellent and interesting results have been obtained 1–21. In 2009, Gakkhar et al. 4
investigated the local stability and Hopf bifurcation of the autonomous delayed predatorprey system with Beddington-DeAngelis functional response:
a1 u1 tu2 t
,
a u1 t bu2 t
a1 u1 t − τ2 u̇2 t d1 u2 t −d ,
a u1 t − τ2 bu2 t − τ2 u̇1 t u1 t1 − u1 t − τ1 −
1.1
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where u1 t, u2 t represent the prey density and the predator density, respectively. The delay
terms occur in growth as well as interaction terms. For this, it means that the prey takes time
τ1 to convert the food into its growth 11, whereas the predator takes time τ2 for the same
22. All the parameters in the model take positive values, that is, a1 > 0, d > 0, d1 > 0, a > 0,
b > 0. The more detail biological meaning of the coefficients of system 1.1, one can see 11
or 22.
We would like to point out that Gakkhar et al. 4 studied the local stability and Hopf
bifurcation of system 1.1 under the assumption: τ1 τ2 τ and obtained some excellent
τ2 . Considering the factor, we further investigate the model
results. While in most cases, τ1 /
τ2 as a complementarity.
1.1 with τ1 /
In this paper, we go on to study the stability, the local Hopf bifurcation for system 1.1.
To the best of our knowledge, it is the first time to deal with the research of Hopf bifurcation
for model 1.1 under the assumption τ1 / τ2 .
The remainder of the paper is organized as follows. In Section 2, we investigate the stability of the positive equilibrium and the occurrence of local Hopf bifurcations. In Section 3,
the direction and stability of the local Hopf bifurcation are established. In Section 4, numerical simulations are carried out to illustrate the validity of the main results. Some main conclusions are drawn in Section 5.
2. Stability of the Positive Equilibrium and Local Hopf Bifurcations
In this section, we shall study the stability of the positive equilibrium and the existence of
local Hopf bifurcations.
Since time delay does not change the equilibrium of system and according to 4, we
know that the delayed prey predator model 1.1 has three equilibrium points: two boundary
equilibrium E1 0, 0 and E2 1, 0, and a nontrivial equilibrium point E0 u∗1 , u∗2 , where u∗1 , u∗2
are the positive solutions of the following quadratic equations:
u21 α1 u1 β1 0,
2.1
u22 α2 u2 β2 0,
where
α1 a1 − da1 − b
,
b
β1 −
a1 d − 12 b2ad d − 1
,
α2 b2 d
ada1
,
d
2.2
aad d − 1
β2 −
.
b2 d
Since β1 < 0, 2.1 in u1 admits a unique positive solution. If one of the following conditions:
a α2 > 0,
β2 < 0,
b α2 < 0,
β2 α22
,
4
c α2 < 0,
0 < β2 <
holds, then system 2.1 has at least one positive equilibrium point E0 u∗1 , u∗2 .
α22
,
4
2.3
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Let u1 t u1 t−u∗1 , u2 t u2 t−u∗2 and still denote ui t i 1, 2 by ui t i 1, 2,
respectively, then 1.1 becomes
u̇1 t m1 u1 t m2 u2 t m3 u1 t − τ1 F1 ,
u̇2 t n1 u2 t n2 u1 t − τ2 n3 u2 t − τ2 F2 ,
2.4
where mi , ni i 1, 2, 3 and Fj j 1, 2 are defined by Appendix A.
The linearization of 2.4 at 0, 0 is
u̇1 t m1 u1 t m2 u2 t m3 u1 t − τ1 ,
u̇2 t n1 u2 t n2 u1 t − τ2 n3 u2 t − τ2 ,
2.5
whose characteristic equation is
λ2 − m1 n1 λ m1 n1 − n3 λ − m1 n3 m2 n2 e−λτ2
− m3 λ m3 n1 e−λτ1 m3 n3 e−λτ1 τ2 0.
2.6
In order to investigate the distribution of roots of the transcendental equation 2.6, the following Lemma is useful.
Lemma 2.1 see 23. For the transcendental equation
0
0
0
P λ, e−λτ1 , . . . , e−λτm λn p1 λn−1 · · · pn−1 λ pn
1
1
1
p1 λn−1 · · · pn−1 λ pn e−λτ1 · · ·
2.7
m
m
m
p1 λn−1 · · · pn−1 λ pn e−λτm 0,
as τ1 , τ2 , τ3 , . . . , τm vary, the sum of orders of the zeros of P λ, e−λτ1 , . . . , e−λτm in the open right
half plane can change, and only a zero appears on or crosses the imaginary axis.
In the sequel, we consider three cases.
Case a. τ1 τ2 0, 2.6 becomes
λ2 − m1 m3 n1 n3 λ m1 n1 m1 n3 m3 n3 − m2 n2 − m3 n1 0.
2.8
A set of necessary and sufficient conditions for all roots of 2.8 to have a negative real part are
given in the following form:
H1 m1 m3 n1 n3 < 0,
m1 n1 m1 n3 m3 n3 − m2 n2 − m3 n1 > 0.
2.9
Then, the equilibrium point E0 u∗1 , u∗2 is locally asymptotically stable when the condition
H1 holds.
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Case b. τ1 0, τ2 > 0, 2.6 becomes
λ2 pλ r sλ q e−λτ2 0,
2.10
where
p −m1 m3 n1 ,
r m1 n1 − m3 n1 ,
s −n3 ,
q m1 n3 m3 n3 − m2 n2 .
2.11
For ω > 0, iω be a root of 2.10, then it follows that
q cos ωτ2 sω sin ωτ2 ω2 − r,
2.12
sω cos ωτ2 − q sin ωτ2 −pω,
which leads to
ω4 p2 − s2 − 2r ω2 r 2 − q2 0.
2.13
It is easy to see that if the condition
H2 p2 − s2 − 2r > 0,
2.14
r 2 − q2 > 0
holds, then 2.13 has no positive roots. Hence, all roots of 2.10 have negative real parts
when τ2 ∈ 0, ∞ under the conditions H1 and H2.
If H1 and
2.15
H3 r 2 − q2 < 0
hold, then 2.13 has a unique positive root ω02 . Substituting ω02 into 2.12, we obtain
τ2n
q ω02 − r − psω02
1
arccos
2nπ ,
ω0
s2 ω02 q2
n 0, 1, 2, . . . .
2.16
If H1 and
H4 p2 − s2 − 2r < 0,
r 2 − q2 > 0,
p2 − s2 − 2r
2
> 4 r 2 − q2
2.17
hold, then 2.10 has two positive roots ω2 and ω−2 . Substituting ω±2 into 2.12, we obtain
τ2±k
q ω±2 − r − psω±2
1
arccos
2kπ ,
ω±
s2 ω±2 q2
k 0, 1, 2, . . . .
2.18
Let λτ2 ατ2 iωτ2 be a root of 2.10 near τ2 τ2n and ατ2n 0, ωτ2n ω0 . Due to
functional differential equation theory, for every τ2n , n 0, 1, 2, . . ., there exists ε > 0 such that
Discrete Dynamics in Nature and Society
5
λτ2 is continuously differentiable in τ2 for |τ2 − τ2n | < ε. Substituting λτ2 into the left-hand
side of 2.10 and taking derivative with respect to τ2 , we have
dλ
dτ2
−1
2λ p eλτ2
s
τ2
− ,
λ
λ sλ q
λ sλ q
2.19
which leads to
dRe λτ
dτ2
−1
τ2 τ2n
2λ p eλτ2 s
Re
λ sλ q
λ
sλ
q
τ τ
τ
Re
Re
2
s
Re
−sω02 iqω0
2 τ2n
p cos ω0 τ2n − 2ω0 sin ω0 τ2n i 2ω0 cos ω0 τ2n p sin ω0 τ2n
2n
−sω02 iqω0
1
−sω02 p cos ω0 τ2n − 2ω0 sin ω0 τ2n
Λ
qω0 2ω0 cos ω0 τ2n p sin ω0 τ2n − s2 ω02
1
Λ
2.20
pω0 q sin ω0 τ2n − sω0 cos ω0 τ2n
2ω02 q cos ω0 τ2n sω0 sin ω0 τ2n − s2 ω02
1 2 2
p ω0 2ω04 − 2rω02 − s2 ω02
Λ
ω2 0 2ω02 p2 − s2 − 2r
Λ
ω2 1 2
−p s2 2r Δ∗ p2 − s2 − 2r 0 Δ∗ > 0,
Λ
Λ
where
Λ s2 ω04 q2 ω02 > 0,
2
Δ∗ s2 − p2 2r − 4 r 2 − q2 .
2.21
Noting that
sign
dRe λ dλ
sign
Re
1,
dτ2
dτ2 τ2 τ2n
τ2 τ2n
2.22
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Discrete Dynamics in Nature and Society
we have
dRe λ > 0.
dτ2 τ2 τ2n
2.23
Similarly, we can obtain
dRe λ > 0,
dτ2 τ2 τ2
k
dRe λ < 0.
dτ2 τ2 τ2−
2.24
k
According to above analysis the Corollary 2.4 in Ruan and Wei 23, we have the following
results.
Lemma 2.2. For τ1 0, assume that one of the conditions (a), (b), (c), and (d) holds and H1 is satisfied. Then, the following conclusions hold.
i If H2 holds, then the positive equilibrium E0 u∗1 , u∗2 of system 1.1 is asymptotically
stable for all τ2 ≥ 0.
ii If H3 holds, then the positive equilibrium E0 u∗1 , u∗2 of system 1.1 is asymptotically
stable for τ2 < τ20 , and unstable for τ2 < τ20 . Furthermore, system 1.1 undergoes a Hopf
bifurcation at the positive equilibrium E0 u∗1 , u∗2 when τ2 τ20 .
iii If H4 holds, then there is a positive integer m such that the positive equilibrium E0 u∗1 , u∗2 is stable when τ2 ∈ 0, τ20 ∪ τ2−0 , τ21 ∪ · · · ∪ τ2−m−1 , τ2m , and unstable when τ2 ∈
τ20 , τ2−0 ∪ τ21 , τ2−1 ∪ · · · ∪ τ2m , τ2−m ∪ τ2m , ∞. Furthermore, system 1.1 undergoes
a Hopf bifurcation at the positive equilibrium E0 u∗1 , u∗2 when τ2 τ2±k , k 0, 1, 2, . . ..
Case c (τ1 > 0, τ2 > 0)
We consider 2.6 with τ2 in its stable interval. Regarding τ1 as a parameter, without loss of
generality, we consider system 1.1 under the assumptions H1 and H3. Let iωω > 0 be
a root of 2.6, then we can obtain
ω4 k1 ω3 k2 ω2 k3 ω k4 0,
2.25
where
k1 2n3 sin ωτ2 ,
k2 2m2 n2 − m1 n3 cos ωτ2 − m1 n1 m1 n1 n3 cos ωτ2 2 − m23 ,
k3 2m2 n2 − m1 n3 cos ωτ2 n3 sin ωτ2 − 2m23 n3 sin ωτ2
2m1 n1 n3 cos ωτ2 m2 n2 − m1 n3 sin ωτ2 ,
k4 m2 n2 − m1 n3 cos ωτ2 − m1 n1 2 m2 n2 − m1 n3 sin ωτ2 2
− m3 n1 − m3 n3 cos ωτ2 2 − m3 n3 sin ωτ2 2 .
2.26
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Denote
Hω ω4 k1 ω3 k2 ω2 k3 ω k4 .
2.27
H5 m2 n2 − m1 n3 − m1 n1 2 < m3 n1 − m3 n3 2 .
2.28
Assume that
It is easy to check that H0 < 0 if H5 holds and limω → ∞ Hω ∞. We can obtain that
2.25 has finite positive roots ω1 , ω2 , . . . , ωn . For every fixed ωi , i 1, 2, 3, . . . , k, there exists
j
a sequence {τ1i | j 1, 2, 3, . . .}, such that 2.25 holds. Let
j
τ10 min τ1i | i 1, 2, . . . , k; j 1, 2, . . . .
2.29
When τ1 τ10 , 2.6 has a pair of purely imaginary roots ±iω∗ for τ2 ∈ 0, τ20 .
In the following, we assume that
H6
dRe λ
dτ1
λiω∗
0.
/
2.30
Thus, by the general Hopf bifurcation theorem for FDEs in Hale 24, we have the following
result on the stability and Hopf bifurcation in system 1.1.
Theorem 2.3. For system 1.1, assume that one of the conditions (a), (b), (c), and (d) holds and suppose H1, H3, and H5 are satisfied, and τ2 ∈ 0, τ20 , then the positive equilibrium E0 u∗1 , u∗2 is
asymptotically stable when τ1 ∈ 0, τ10 , and system 1.1 undergoes a Hopf bifurcation at the positive
equilibrium E0 u∗1 , u∗2 when τ1 τ10 .
3. Direction and Stability of the Hopf Bifurcation
In the previous section, we obtained conditions for Hopf bifurcation to occur when τ1 τ10 . In
this section, we shall derive the explicit formulae determining the direction, stability, and period of these periodic solutions bifurcating from the positive equilibrium E0 u∗1 , u∗2 at this critical value of τ1 , by using techniques from normal form and center manifold theory 7.
Throughout this section, we always assume that system 1.1 undergoes Hopf bifurcation at
the positive equilibrium E0 u∗1 , u∗2 for τ1 τ10 , and then ±iω∗ is corresponding purely imaginary roots of the characteristic equation at the positive equilibrium E0 u∗1 , u∗2 .
Without loss of generality, we assume that τ2∗ < τ10 , where τ2∗ ∈ 0, τ20 . For convenience, let ui t ui τt i 1, 2 and τ1 τ10 μ, where τ10 is defined by 2.28 and μ ∈ R,
drop the bar for the simplification of notations, then system 1.1 can be written as an FDE in
C C−1, 0, R2 as
u̇t Lμ ut F μ, ut ,
3.1
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where ut u1 t, u2 tT ∈ C and ut θ ut θ u1 t θ, u2 t θT ∈ C, and Lμ :
C → R, F : R × C → R are given by
∗ ⎞
τ2
−
φ
1
φ
0
⎜
1
τ10 ⎟
⎟
Lμ φ τ10 μ B
τ10 μ C⎜
⎝ τ ∗ ⎠
2
φ2 0
φ1 −
τ10
φ1 −1
,
τ10 μ D
φ2 −1
⎛
T
F μ, φ τ10 μ f1 , f2 ,
3.2
3.3
respectively, where φθ φ1 θ, φ2 θT ∈ C,
B
m1 m2
0
n1
,
C
0
0
n2 n3
,
D
m3 0
0 0
,
f1 −φ1 0φ1 −1 l1 φ12 0 l2 φ22 0 l3 φ1 0φ2 0 l4 φ13 0
l5 φ23 0 l6 φ12 0φ2 0 l7 φ1 0φ22 0 h.o.t.,
∗
∗
∗
τ
τ
τ
f2 k1 φ1 − 2 φ2 0 k2 φ2 0φ2 − 2 k3 φ12 − 2
τ10
τ10
τ10
∗ ∗
∗
∗ ∗
τ
τ
τ
τ
τ
k4 φ1 − 2 φ2 − 2 e1 φ12 − 2 φ2 0 e2 φ1 − 2 ϕ2 − 2 φ2 0
τ10
τ10
τ10
τ10
τ10
∗
∗ ∗
∗
τ
τ
τ
τ
e3 φ13 − 2 e4 φ1 − 2 φ22 − 2 e5 φ23 − 2 h.o.t.
τ10
τ10
τ10
τ10
3.4
From the discussion in Section 2, we know that, if μ 0, then system 3.1 undergoes a Hopf
bifurcation at the positive equilibrium E0 u∗1 , u∗2 and the associated characteristic equation of
system 3.1 has a pair of simple imaginary roots ±ω∗ τ10 .
By the representation theorem, there is a matrix function with bounded variation
components ηθ, μ, θ ∈ −1, 0 such that
Lμ φ 0
−1
dη θ, μ φθ,
for φ ∈ C.
3.5
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In fact, we can choose
η θ, μ ⎧
τ10 μ B C D,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ τ10 μ C D,
⎪
⎪
⎪
⎪
⎪
⎪ τ10 μ D,
⎪
⎪
⎪
⎩
0,
θ 0,
∗ τ
θ ∈ − 2 ,0 ,
τ10
τ2∗
θ ∈ −1, −
,
τ10
3.6
θ −1.
For φ ∈ C−1, 0, R2 , define
⎧
dφθ
⎪
⎪
,
−1 ≤ θ < 0,
⎪
⎨
dθ
A μ φ 0
⎪
⎪
⎪
⎩
dη s, μ φs, θ 0,
3.7
−1
⎧
⎨0,
Rφ ⎩F μ, φ,
−1 ≤ θ < 0,
θ 0.
Then, 3.1 is equivalent to the abstract differential equation
u̇t A μ ut R μ ut ,
3.8
where ut θ ut θ, θ ∈ −1, 0.
For ψ ∈ C0, 1, R2 ∗ , define
⎧
dψs
⎪
⎪
⎪
⎨− ds ,
A∗ ψs 0
⎪
⎪
⎪
dηT t, 0ψ−t,
⎩
−1
s ∈ 0, 1,
3.9
s 0.
For φ ∈ C−1, 0, R2 and ψ ∈ C0, 1, R2 ∗ , define the bilinear form
!
"
ψ, φ ψ0φ0 −
0 θ
−1
3.10
ψ T ξ − θdηθφξdξ,
ξ0
where ηθ ηθ, 0, the A A0 and A∗ are adjoint operators. By the discussions in
Section 2, we know that ±iω∗ τ10 are eigenvalues of A0, and they are also eigenvalues of A∗
corresponding to iω∗ τ10 and −iω∗ τ10 , respectively. By direct computation, we can obtain
qθ 1, αT eiω
∗
τ10 θ
,
∗
q∗ s M1, α∗ eiω τ10 s ,
M
1
,
K
3.11
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where
α
∗
iω∗ − m1 − m3 e−iω
m2
α −
∗
τ10
iω∗ m1 m3 e−iω
n2 e
−iω∗ τ2∗ /τ10
∗
,
τ10
3.12
,
τ∗
τ∗
∗
∗ ∗
∗ ∗
K 1 αα∗ τ10 m3 eiω τ10 2 n2 αeiω τ2 /τ10 αα∗ n3 2 eiω τ2 /τ10 .
τ10
τ10
Furthermore, q∗ s, qθ
1 and q∗ s, qθ
0.
Next, we use the same notations as those in Hassard 7 and we first compute the coordinates to describe the center manifold C0 at μ 0. Let ut be the solution of 3.1 when μ 0.
Define
"
!
zt q∗ , ut ,
#
$
Wt, θ ut θ − 2 Re ztqθ ,
3.13
on the center manifold C0 , and we have
Wt, θ Wzt, zt, θ,
3.14
where
Wzt, zt, θ Wz, z W20
z2
z2
W11 zz W02 · · · ,
2
2
3.15
and z and z are local coordinates for center manifold C0 in the direction of q∗ and q∗ . Noting
that W is also real if ut is real, we consider only real solutions. For solutions ut ∈ C0 of 3.1,
#
$ def
żt iω∗ τ10 z q∗ θF0, Wz, z, θ 2 Re zqθ iω∗ τ10 z q∗ 0F0 .
3.16
That is
żt iω∗ τ10 z gz, z,
3.17
where
gz, z g20
z2
z2
g11 zz g02 · · · .
2
2
3.18
Hence, we have to obtain the expression of gz, z see Appendix B. Then, it is easy to obtain
the expression of g20 , g11 , g02 , g21 see Appendix B.
Discrete Dynamics in Nature and Society
i
11
i
For unknown W20 θ, W11 θ, i 1, 2 in g21 , we still need to compute them.
From 3.8, 3.13, we have
⎧
⎨AW − 2 Re q∗ 0fqθ ,
−1 ≤ θ < 0,
W ⎩AW − 2 Re q∗ 0fqθ F, θ 0,
3.19
def
AW Hz, z, θ,
where
Hz, z, θ H20 θ
z2
z2
H11 θzz H02 θ · · · .
2
2
3.20
Comparing the coefficients, we obtain
A − 2iτ10 ω∗ W20 −H20 θ,
3.21
AW11 θ −H11 θ,
3.22
Hz, z, θ −q∗ 0f0 qθ − q∗ 0f 0 qθ −gz, zqθ − gz, zqθ.
3.23
and we know that, for θ ∈ −1, 0,
Comparing the coefficients of 3.23 with 3.20 gives that
H20 θ −g20 qθ − g 02 qθ,
3.24
H11 θ −g11 qθ − g 11 qθ.
3.25
From 3.21, 3.24, and the definition of A, we get
Ẇ20 θ 2iω∗ τ10 W20 θ g20 qθ g 02 qθ.
3.26
∗
Noting that qθ q0eiω τ10 θ , we have
W20 θ 1
ig 02
ig20
∗
∗
∗
q0eiω τ10 θ q0e−iω τ10 θ E1 e2iω τ10 θ ,
ω∗ τ10
3ω∗ τ10
3.27
2
where E1 E1 , E1 T ∈ R2 is a constant vector.
Similarly, from 3.22, 3.25, and the definition of A, we have
Ẇ11 θ g11 qθ g 11 qθ,
W11 θ −
1
2
ig
ig11
∗
∗
q0eiω τ10 θ ∗ 11 q0e−iω τ10 θ E2 ,
ω∗ τ10
ω τ10
where E2 E2 , E2 T ∈ R2 is a constant vector
3.28
3.29
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In what follows, we shall seek appropriate E1 , E2 in 3.27, 3.29, respectively. It follows from the definition of A and 3.24, 3.25 that
0
3.30
dηθW20 θ 2iω∗ τ10 W20 0 − H20 0,
−1
0
−1
3.31
dηθW11 θ −H11 0,
where ηθ η0, θ.
From 3.21, we have
H20 0 −g20 q0 − g 02 q0 2τ10 H1 , H2 T ,
3.32
H11 0 −g11 q0 − g 11 0q0 2τ10 P1 , P2 T ,
3.33
where
∗
H1 −e−iω τ10 l1 l2 α2 l3 α,
∗ ∗
∗ ∗
H2 k1 α k2 α2 e−iω τ2 k3 k4 αe−2iω τ2 ,
1 −iω∗ τ1
iω∗ τ10
0 e
l1 l2 |α|2 2 Re{α},
e
2
∗ ∗
∗ ∗
∗ ∗
P2 k1 Re αeiω τ2 k2 |α|2 e−iω τ2 eiω τ2 k3 k4 Re{α}.
P1 −
Noting that
∗
iω τ10 I −
−iω∗ τ10 I −
0
−1
0
−1
e
iω∗ τ10 θ
dηθ q0 0,
3.35
∗
e−iω τ10 θ dηθ q0 0,
and substituting 3.27 and 3.32 into 3.30, we have
0
2iω∗ τ10 I −
That is
2iω∗ − m3 e−2iω
∗ ∗
−n2 e−2iω τ2
∗
τ10
−1
e2iω
∗
3.34
τ10 θ
dηθ E1 2τ10 H1 , H2 T .
−m2
∗
∗ ∗
2iω∗ − n1 e−2iω τ10 − n3 e−2iω τ2
3.36
E1 2H1 , H2 T .
3.37
It follows that
1
E1 Δ11
,
Δ1
2
E1 Δ12
,
Δ1
3.38
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13
where
∗
∗
∗ ∗
∗ ∗
Δ1 2iω∗ − m3 e−2iω τ10 2iω∗ − n1 e−2iω τ10 − n3 e−2iω τ2 − m2 n2 e−2iω τ2 ,
∗
∗ ∗
Δ11 2H1 2iω∗ − n1 e−2iω τ10 − n3 e−2iω τ2 2H2 m2 ,
3.39
∗
∗ ∗
Δ12 2H2 2iω∗ − m3 e−2iω τ10 2H1 n2 e−2iω τ2 .
Similarly, substituting 3.28 and 3.33 into 3.31, we have
0
−1
dηθ E2 2τ10 P1 , P2 T .
3.40
That is
m1 m3
m2
n2
n1 n3
E2 2−P1 , −P2 T .
3.41
It follows that
Δ21
,
Δ2
1
E2 2
E2 Δ22
,
Δ2
3.42
where
Δ2 m1 m3 n1 n3 − m2 n2 ,
Δ21 2m2 P2 − 2P1 n1 n3 ,
3.43
Δ22 2n2 P1 − 2P2 m1 m3 .
From 3.27, 3.29, 3.38, 3.42, we can calculate g21 and derive the following values:
c1 0 i
2ω∗ τ10
2 g02 2
g21
g20 g11 − 2 g11 −
,
3
2
μ2 −
Re{c1 0}
,
Re{λ τ10 }
3.44
β2 2 Rec1 0,
T2 −
Im{c1 0} μ2 Im{λ τ10 }
.
ω∗ τ10
These formulas give a description of the Hopf bifurcation periodic solutions of 3.1 at τ τ10
on the center manifold. From the discussion above, we have the following result.
14
Discrete Dynamics in Nature and Society
0.55
0.65
0.5
0.6
0.45
0.55
0.5
u2 (t)
u1 (t)
0.4
0.35
0.3
0.45
0.4
0.35
0.25
0.3
0.2
0.25
0
200
400
600
800
0.2
1000
200
0
400
600
800
1000
t
t
a
b
0.65
0.6
0.7
0.55
0.6
u 2 (t)
u2 (t)
0.5
0.45
0.4
0.5
0.4
0.3
0.35
0.2
0.8
0.3
0.25
0.2
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
c
0.6
u1 (
0.4
t)
0.2
0
0
600
400
t
200
800
1000
d
Figure 1: Trajectory portrait and phase portrait of system 4.1 with τ1 0, τ2 1.5 < τ20 ≈ 1.52. The
positive equilibrium E0 0.3038, 0.4230 is asymptotically stable. The initial value is 0.2,0.2.
Theorem 3.1. The periodic solution is forward (backward) if μ2 > 0μ2 < 0; the bifurcating periodic
solutions are orbitally asymptotically stable with asymptotical phase (unstable) if β2 < 0β2 > 0; the
periods of the bifurcating periodic solutions increase (decrease) if T2 > 0T2 < 0.
4. Numerical Examples
In this section, we present some numerical results of system 1.1 to verify the analytical predictions obtained in the previous section. From Section 3, we may determine the direction of
a Hopf bifurcation and the stability of the bifurcation periodic solutions. Let us consider the
following system:
u1 tu2 t
,
0.05 u1 t 0.6u2 t
u1 t − τ2 u̇2 t u2 t −0.5 ,
0.05 u1 t − τ2 0.6u2 t − τ2 u̇1 t u1 t1 − u1 t − τ1 −
4.1
Discrete Dynamics in Nature and Society
15
0.7
0.55
0.65
0.5
0.6
0.45
0.55
u2 (t)
u1 (t)
0.4
0.35
0.3
0.5
0.45
0.4
0.35
0.25
0.3
0.2
0.25
0
200
400
t
600
800
0.2
1000
0
200
400
600
800
1000
t
a
b
0.7
0.65
0.6
0.7
0.6
0.5
u2 (t)
u2 (t)
0.55
0.45
0.4
0.5
0.4
0.35
0.3
0.3
0.2
0.8
0.25
0.2
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
u1 (t)
0.6
u1 0.4
(t)
0.2
c
0
0
600
400
200
t
800
1000
d
Figure 2: Trajectory portrait and phase portrait of system 4.1 with τ1 0, τ2 1.6 > τ20 ≈ 1.52. Hopf
bifurcation occurs from the positive equilibrium E0 0.3038, 0.4230. The initial value is 0.2,0.2.
which has a positive equilibrium E0 u∗1 , u∗2 ≈ 0.3038, 0.4230. When τ1 0, then we can easily
obtain that H1 and H3 are satisfied. Take n 0, for example, by some computation by
means of Matlab 7.0, we get ω0 ≈ 0.1342, τ20 ≈ 1.52. From Lemma 2.2, we know that the transversal condition is satisfied. Thus, the positive equilibrium E0 ≈ 0.3038, 0.4230 is asymptotically stable for τ2 < τ20 ≈ 1.52 and unstable for τ2 > τ20 ≈ 1.52 which is shown in Figure 1.
When τ2 τ20 ≈ 1.52, 4.1 undergoes a Hopf bifurcation at the positive equilibrium E0 ≈
0.3038, 0.4230, that is, a small amplitude periodic solution occurs around E0 ≈ 0.3038,
0.4230 when τ1 0 and τ2 is close to τ20 1.52 which is shown in Figure 2.
Let τ2 1.5 ∈ 0, 1.52 and choose τ1 as a parameter. We have τ10 ≈ 1.2930. Then, the
positive equilibrium is asymptotically when τ1 ∈ 0, τ10 . The Hopf bifurcation value of 4.1
is τ10 ≈ 1.2930. By the algorithm derived in Section 3, we can obtain
λ τ10 ≈ 0.5018 − 7.2021i,
μ2 ≈ 0.2646,
c1 0 ≈ −2.0231 − 3.3225i,
β2 ≈ −4.2342,
T2 ≈ 8.3701.
4.2
Furthermore, it follows that μ2 > 0 and β2 < 0. Thus, the positive equilibrium E0 ≈ 0.3038,
0.4230 is stable when τ1 < τ10 as is illustrated by the computer simulations see Figure 3.
16
Discrete Dynamics in Nature and Society
0.8
0.65
0.6
0.7
0.55
0.5
0.5
u2 (t)
u1 (t)
0.6
0.4
0.4
0.35
0.3
0.3
0.2
0.1
0.45
0.25
0
100
200
300
t
400
500
0.2
600
0
100
200
a
300
t
400
500
600
b
0.65
0.6
0.55
0.7
0.6
0.45
u2 (t)
u2 (t)
0.5
0.4
0.35
0.4
0.3
0.2
0.8
0.3
0.25
0.2
0.1
0.5
0.2
0.3
0.4 0.5
u1 (t)
c
0.6
0.7
0.8
0.6
u1 (
0.4
t)
0.2
0
0
200
t
400
600
d
Figure 3: Trajectory portrait and phase portrait of system 4.1 with τ2 0.5, τ1 1.2 < τ10 ≈ 1.2930. The
positive equilibrium E0 0.3038, 0.4230 is asymptotically stable. The initial value is 0.2,0.2.
When τ1 passes through the critical value τ10 , the positive equilibrium E0 ≈ 0.3038, 0.4230
loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcations from the positive equilibrium E0 ≈ 0.3038, 0.4230. Since μ2 > 0 and β2 < 0, the direction of the Hopf bifurcation is τ1 > τ10 , and these bifurcating periodic solutions from E0 ≈
0.3038, 0.4230 at τ10 are stable, which are depicted in Figure 4.
5. Conclusions
In this paper, we have investigated local stability of the positive equilibrium E0 u∗1 , u∗2 and
local Hopf bifurcation of a Beddington-DeAngelis functional response predator-prey model
with two delays. We have showed that, if one of the conditions a, b, c, and d holds and
H1, H3, and H5 are satisfied, and τ2 ∈ 0, τ20 , then the positive equilibrium E0 u∗1 , u∗2 is asymptotically stable when τ1 ∈ 0, τ10 , as the delay τ1 increases, the positive equilibrium
E0 u∗1 , u∗2 loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium E0 u∗1 , u∗2 , that is, a family of periodic orbits bifurcates from the the positive equilibrium E0 u∗1 , u∗2 . At last, the direction of Hopf bifurcation and the stability of the bifurcating
periodic orbits are discussed by applying the normal form theory and the center manifold
Discrete Dynamics in Nature and Society
17
0.65
0.8
0.6
0.7
0.55
0.5
0.5
u2 (t)
u1 (t)
0.6
0.4
0.4
0.35
0.3
0.3
0.2
0.1
0.45
0.25
0
200
400
600
800
0.2
1000
0
200
400
600
800
1000
t
t
a
b
0.65
0.6
0.55
0.7
0.6
0.45
u2 (t)
u2 (t)
0.5
0.4
0.35
0.4
0.3
0.3
0.2
0.8
0.25
0.2
0.1
0.5
0.2
0.3
0.4 0.5
u1 (t)
c
0.6
0.7
0.8
0.6
u1 ( 0.4
t)
0.2
0 0
600
400
t
200
800
1000
d
Figure 4: Trajectory portrait and phase portrait of system 4.1 with τ2 0.5, τ1 1.35 > τ10 ≈ 1.2930. Hopf
bifurcation occurs from the positive equilibrium E∗ 0.32, 0.96. The initial value is 0.2,0.2.
theorem. A numerical example verifying our theoretical results is carried out. In addition, we
must point out that, although Gakkhar et al. 4 have also investigated the existence of Hopf
bifurcation for system 1.1 with respect to positive equilibrium E0 u∗1 , u∗2 , it is assumed that
τ1 τ2 . For τ1 /
τ2 , only numerical simulations are carried out to discuss the existence of Hopf
τ2 , we investigate the existence of Hopf bifurbifurcation. In this paper, under the case τ1 /
cation quantitatively. Our work generalizes the known results of Sunita Gakkhar et al. 4.
Similarly, we can investigate the Hopf bifurcation of system 1.2 by choosing the delay τ2 as
bifurcation parameter. We will further investigate the topic elsewhere in the near future.
Appendices
A.
The expressions of mi , ni i 1, 2, 3, and Fi i 1, 2 are as follows:
u∗1 u∗2
a1
∗
∗
u2 −
,
m 1 1 − u1 −
a u∗1 bu∗2
a u∗1 bu∗2
u∗1 u∗2
a1
∗
m2 −
u1 −
, m3 −u∗1 ,
a u∗1 bu∗2
a u∗1 bu∗2
18
Discrete Dynamics in Nature and Society
n1 −dd1 d1 u∗1
,
a u∗1 bu∗2
n2 d1 u∗1 u∗2
,
a u∗1 bu∗2
d1 bu∗1 u∗2
n3 − a u∗1 bu∗2
2 ,
F1 −u1 tu1 t − τ1 l1 u21 t l2 u22 t l3 u1 tu2 t l4 u31 t
l5 u32 t l6 u21 tu2 t l7 u1 tu22 t h.o.t.,
F2 k1 u1 t − τ2 u2 t k2 u2 tu2 t − τ2 k3 u21 t − τ2 k4 u1 t − τ2 u2 t − τ2 e1 u21 t − τ2 u2 t e2 u1 t − τ2 u2 t − τ2 u2 t
e3 u31 t − τ2 e4 u1 t − τ2 u22 t − τ2 e5 u32 t − τ2 h.o.t.,
A.1
where
%
&
u∗1 u∗2
−u∗2
a1
,
l1 −
a u∗1 bu∗2 a u∗1 bu∗2 a u∗ bu∗ 2
2
1
%
&
−bu∗1
b2 u∗1 u∗2
a1
l2 −
,
a u∗1 bu∗2 a u∗1 bu∗2 a u∗ bu∗ 2
2
1
%
&
∗
∗
∗ ∗
u1 bu2
2bu1 u2
a1
l3 −
1−
,
a u∗1 bu∗2
a u∗1 bu∗2 a u∗ bu∗ 2
2
1
%
&
u∗1 u∗2
u∗2
a1
l4 −
,
−
a u∗1 bu∗2 a u∗ bu∗ 2 a u∗ bu∗ 3
2
2
1
1
%
&
b2 u∗1
a1
b3
l5 −
,
−
a u∗1 bu∗2 a u∗ bu∗ 2 a u∗ bu∗ 3
2
2
1
1
%
&
u∗1 2bu∗2
3bu∗1 u∗2
−1
a1
l6 −
,
−
a u∗1 bu∗2 a u∗1 bu∗2 a u∗ bu∗ 2 a u∗ bu∗ 3
2
2
1
1
%
&
b2 u∗2 2bu∗1
3b2 u∗1 u∗2
−b
a1
l7 −
,
−
a u∗1 bu∗2 a u∗1 bu∗2 a u∗ bu∗ 2 a u∗ bu∗ 3
2
2
1
1
%
&
∗
u1
1
d1 b
, k2 − k1 d1
−
2 ,
a u∗1 bu∗2 a u∗ bu∗ 2
a u∗1 bu∗2
2
1
%
&
u∗1
1
∗
k3 d1 u2 − 2 3 ,
a u∗1 bu∗2
a u∗1 bu∗2
%
&
2bu∗1
b
∗
k4 d1 u2 − 2 3 ,
a u∗1 bu∗2
a u∗1 bu∗2
%
&
u∗1
1
∗
e 1 d 1 u2 3 − 4 ,
a u∗1 bu∗2
a u∗1 bu∗2
Discrete Dynamics in Nature and Society
%
&
2bu∗1
b
e 2 d1 − 2 3 ,
a u∗1 bu∗2
a u∗1 bu∗2
%
&
u∗1
1
∗
e 3 d 1 u2 3 − 4 ,
a u∗1 bu∗2
a u∗1 bu∗2
e4 d1 u∗2 b2
3 ,
a u∗1 bu∗2
d1 u∗1 u∗2 b3
e5 − a u∗1 bu∗2
19
4 .
A.2
B.
The expressions of gz, z, g20 , g11 , g02 , and g21 are as follows:
gz, z q∗ 0F0 z, z F0, ut ∗
Mτ10 −e−iω τ10 l1 l2 α2 l3 α α∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
× k1 αe−iω τ2 k2 α2 e−iω τ2 k3 e−2iω τ2 k4 αe−2iω τ2 z2
∗
∗
Mτ10 −eiω τ10 − e−iω τ10 2l1 2l2 |α|2 2l3 Re{α} α∗
∗ ∗
∗ ∗
∗ ∗
× 2k1 Re αeiω τ2 k2 |α|2 eiω τ2 e−iω τ2 2k3 2k4 Re{α} zz
∗
Mτ10 −eiω τ10 l1 l2 α2 l3 α α∗
2
∗ ∗
∗ ∗
∗ ∗
∗ ∗
× k1 αeiω τ2 k2 |α|2 eiω τ2 k3 e2iω τ2 k4 αeiω τ2 z
1 1
1 1
1
1
iω∗ τ10
−iω∗ τ10
Mτ10 − W20 0e
W20 0 W11 0e
W11 0
2
2
1
1
2
2
l1 W20 0 2W11 0 l2 αW20 0 2αW11 0
1 2
1
2
1
1
l3 W11 0 W20 0 αW20 0 αW11 0
2
2
3l4 3l5 α2 α l6 2α α l7 2|α|2 α2 α∗
∗
τ
1 2
1
∗ ∗
∗ ∗
∗ ∗
2
1
2
× k1 W11 0e−iω τ2 αW20 − 2 W20 0eiω τ2 W11 0e−iω τ2
2
τ10
2
∗
∗
τ
τ
1
1
∗ ∗
∗ ∗
2
2
2
2
k2 αW11 − 2 αW20 − 2 αW20 0eiω τ2 αW11 0e−iω τ2
τ10 2
τ10 2
∗
∗
τ2
τ2
1
1
−iω∗ τ2∗
iω∗ τ2∗
k3 2W11 −
W20 −
e
e
τ10
τ10
20
Discrete Dynamics in Nature and Society
∗
∗
τ
τ
1 2
∗ ∗
∗ ∗
2
k4 W11 − 2 e−iω τ2 W20 − 2 eiω τ2
τ10
2
τ10
∗
∗
τ
τ
1
∗ ∗
∗ ∗
∗ ∗
1
1
αW20 − 2 eiω τ2 αW11 − 2 e−iω τ2 e1 2α αe−2iω τ2
2
τ10
τ10
∗ ∗
∗ ∗
∗ ∗
∗ ∗
2
2
e2 α2 |α| |α| e−2iω τ2 3e3 e−iω τ2 e4 α2 e−iω τ2 2|α|2 e−iω τ2
∗ ∗
∗ ∗
z2 z h.o.t.,
e5 α2 e−iω τ2 2|α|2 αe−iω τ2
∗
g20 2Mτ10 −e−iω τ10 l1 l2 α2 l3 α α∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
× k1 αe−iω τ2 k2 α2 e−iω τ2 k3 e−2iω τ2 k4 αe−2iω τ2 ,
∗
∗
g11 Mτ10 −eiω τ10 − e−iω τ10 2l1 2l2 |α|2 2l3 Re{α}
∗ ∗
∗ ∗
∗ ∗
α∗ 2k1 Re αeiω τ2 k2 |α|2 eiω τ2 e−iω τ2 2k3 2k4 Re{α} ,
∗
g02 2Mτ10 −eiω τ10 l1 l2 α2 l3 α
∗ ∗
∗ ∗
∗ ∗
∗ ∗
α∗ k1 αeiω τ2 k2 |α|2 eiω τ2 k3 e2iω τ2 k4 αeiω τ2 ,
1 1
1 1
∗
∗
1
1
g21 2Mτ10 − W20 0eiω τ10 W20 0 W11 0e−iω τ10 W11 0
2
2
1
1
2
2
l1 W20 0 2W11 0 l2 αW20 0 2αW11 0
1 2
1
2
1
1
l3 W11 0 W20 0 αW20 0 αW11 0
2
2
3l4 3l5 α2 α l6 2α α l7 2|α|2 α2
∗
τ2
1
1 2
2
1
2
−iω∗ τ2∗
iω∗ τ2∗
−iω∗ τ2∗
αW20 −
W11 0e
α k1 W11 0e
W20 0e
2
τ10
2
∗
∗
τ
τ
1
1
∗ ∗
∗ ∗
2
2
2
2
k2 αW11 − 2 αW20 − 2 αW20 0eiω τ2 αW11 0e−iω τ2
τ10
2
τ10
2
∗
∗
τ
τ
∗ ∗
∗ ∗
1
1
k3 2W11 − 2 e−iω τ2 W20 − 2 eiω τ2
τ10
τ10
∗
∗
∗
τ
τ
τ
1 2
1
∗ ∗
∗ ∗
∗ ∗
2
1
k4 W11 − 2 e−iω τ2 W20 − 2 eiω τ2 αW20 − 2 eiω τ2
τ10
2
τ10
2
τ10
∗
τ2
1
−iω∗ τ2∗
e
αW11 −
τ10
∗ ∗
∗ ∗
∗ ∗
e1 2α αe−2iω τ2 e2 α2 |α|2 |α|2 e−2iω τ2 3e3 e−iω τ2
∗
2 −iω∗ τ2∗
e4 α e
2 −iω∗ τ2∗
2|α| e
2
2 −iω∗ τ2∗
−iω∗ τ2∗
e5 α e
2|α| αe
.
B.1
Discrete Dynamics in Nature and Society
21
Acknowledgment
This work is supported by National Natural Science Foundation of China no. 10771215,
Doctoral Foundation of Guizhou College of Finance and Economics 2010, the soft
Science and Technology Program of Guizhou Province no. 2011LKC2030, and Nature and
Technology Foundation of Guizhou Province J20122100
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Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
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Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
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Hindawi Publishing Corporation
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International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Optimization
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014