Applied Mathematics, 2012, 3, 652-661 http://dx.doi.org/10.4236/am.2012.36099 Published Online June 2012 (http://www.SciRP.org/journal/am)

Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

Xinhui Wang, Haihong Liu Department of Mathematics, Yunnan Normal University, Kunming, China

Email: wxhwj2005@163.com, liuwang.2011@yahoo.cn

Received December 29, 2011; revised May 2, 2012; accepted May 9, 2012

ABSTRACT

A delayed Lotka-Volterra two-species predator-prey system of population allelopathy with discrete delay is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymp-totic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direc-tion of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, some numerical simulations are carried out for illustrating the theoretical results. Keywords: Lotka-Volterra Predator-Prey System; Discrete Delay; Allelopathy; Stability; Hopf Bifurcation; Periodic

Solution

1. Introduction

In recent years, the Lotka-Volterra predator-prey models modeled by ordinary differential equations (ODEs) have been proposed and studied extensively since the pio-neering theoretical works by Lotka [1] and Volterra [2]. With the modification of Brelot [3], the model has the form

1 11 12

2 21 22

d ,

d

t

t

x t x t r a x t a F t s y s s

y t y t r a G t s x s s a y t

.

,

(1)

where , , and 0ir 0ija , 1, 2i j 0

d 1tF s s

. Models such as (1) with various delay 0

tG s ds 1

kernels and delayed Intraspecific competetions have been investigated extensively by many researchers; see refer-ence [4-12] for detail. For example, when F(s) = δ(s − τ) (τ ≥ 0), then system (1) is reduced to the following Lotka-Volterra two-species predator-prey system with a discrete delay and a distributed delay:

1 11 12

2 21 22

,

d .t

x t x t r a x t a y t

y t y t r a G t s x s s a y t

(2)

The delay kernel function G(s) may take the so-called “weak” generic kernel function G(s) = se

2 (α > 0)

and “strong” generic kernel function G(s) = se (α >

0), where the “weak” generic kernel implies that the im-portance of events in the past simply decreases exponen-tially the further one looks into the past while the “strong” generic kernel implies that a particular time in the past is more iportant than any other [13]. When G(s) takes the “weak” generic kernel function and the “strong” generic kernel function G(s) = 2 se

12a y t

t

(α > 0), proper-ties of the stability of the positive equilibrium of system (2) and Hopf bifurcations of nonconstant periodic solu-tions have been investigated respectively by using the normal form theory and the center manifold reduction for FDEs [14,15]. See [5,16] for details.

When F(s) = δ(s − τ) (τ ≥ 0) and G(s) = δ(s − η) (η ≥ 0) where δ denotes Dirac delta function. Then system (1) is transformed into the following form with two different discrete delays

1 11

2 21 22

,

.

x t x t r a x t

y t y t r a x a y t

(3)

He [17] and Lu andWang [18] investigated the stabil-ity of the positive equilibrium of the system, and they found that the positive equilibrium is globally asymp-totically stable for any values of delays τ and η when the coefficients of the system satisfy the condition 11 22a a

12 21 > 0 when η > 0, and consider η or the sum of two delays τ and η as the bifurcation parameter, one can see [6-8] for details. Yan and Zhang [9] studied the effect of delay on the dynamics of system (3) when τ = η. Fur-thermore, for the study of system (3) with delayed in-

a a

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X. H. WANG, H. H. LIU 653

tra-specific competitions, one can refer to [10,11]. For Latka-Volterra two species competition model, an

important observation made by many works is that in-creased population of one species might affect the growth of another species by the production of allelo-pathic toxins or stimulators, thus influencing seasonal succession [19]. For instance, Maynard Smith [20] in-corporated the effect of toxic substances in a two species Lotka-Volterra competitive system by considering that each species produce a substance toxic to the other but only when the other is present. Then the Lotka-Volterra competitive system was modified into the form:

1 1 1 1 1 12 2

1 1 2

2 2 2 2 2 21 1

2 1 2

,

.

N t N t k N t N t

N t N t

N t N t k N t N t

N t N t

(4)

Chattopadyay [21] studied the stability properties of the above system, although the study contains the flaw of ignoring an important delay factor in the system. [22] suggested “In reality, a species needs sometime for ma-turity to produce a substance which will be toxic (or sti-mulatory) to the other and hence a delay term in the sys-tem arises”. Mukhopadhyay, Chattpadhyay and Tapaswi [22] introduced a delay in system (4), which leads the following form

1 1 1 1 1 12 2

1 1 2

2 2 2 2 21 1

2 1 2

,

( ) .

N t N t k N t N t

N t N t

N t N t k x t N t

N t N t

(5)

When γ1 > 0, γ2 > 0, the model system (5) represents an allelopathic inhibitory system, each species producing a substance toxic to the other; when γ1 < 0, γ2 < 0, (5) repr esents an allelopathic stimulatory system, each spe-cies producing a substance stimulatory to the growth of the other species.

Similar phenomenon also exist in predator-prey model. Rice [19] has suggested that “all meaningful, functional ecological models will eventually have to include a category on allelopathic and other allelochemic effects”. To our knowledge, such viewpoint haven’t been investi-gated in predator-prey model so far. On the other hand, some species in the real nature world may produce sub-stances which are toxic or stimulatory to the others while they themselves do not experience any reciprocal effects during the process of predation. For example, some spe-cies of poisonous snake release toxic substance to control prey. The production of toxic substance by the predator species will not be instantaneous, but mediated by some time lag, see [7,19-22]. From this viewpoint and com-

bining the factors appeared above of different type of time delay and allelopathic effect in predator-prey model, we have modified the model of (3). Therefore, by con-sidering that one species produced a substance toxic to the other during the process of predation, but only when the other is present. Then the system (3) can be written as

1 1 12

1

2 2 21

,

.

x t x t k x t y t

x t y t

y t y t k x t y t

(6)

where , 0ik 0i , 0ij (i, j = 1, 2), 1 0 e have investigated the bifurcation behavior on time delay of this modified dynamical system (6). It has also been observed that time delay can drive the competitive sys-tem to sustained oscillations, as shown by Hopf bifurca-tion analysis and limit cycle stability. Hence interaction between the time delay effect produced by delayed toxin can regulate the densities of different competing species in the aquatic ecosystem, thus influencing seasonal suc-cesssion, blooms and pulses. To the best of our knowl-edge no such attempts have been taken to include inter-action between the time delay effect produced by delayed toxin in a predator-prey system. Therefore, this research might behelpful to the study of predator-prey model and related problem in biological system.

. W

This paper is organized as follows. In Section 2, by linearizing the resulting two-dimensional system at the positive equilibrium and analyzing the associated char-acteristic equation, it is found that under suitable condi-tions on the parameters the positive equilibrium is as-ymptotically stable when the delay is less than a certain critical value and unstable when the delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for FDEs, we find that the system can also undergo a Hopf bifurcation of nonconstant pe-riodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. In Section 3, to determine the direction of the Hopf bifurcations and the stability of bifurcated periodic solutions occurring through Hopf bifurcations, an explicit algorithm is given by applying the normal form theory and the center mani-fold reduction for FDEs developed by Hassard, Kaza-rinoff and Wan [23]. To verify our theoretical predictions, some numerical simulations are also included in Section 4.

2. Stability of Equilibria and Existence of Hopf Bifurcations

The state of equilibria of the system (6) for τ = 0 are as follows:

0 0,0 ,E 21

21

0, ,k

E

1

21

,0 ,k

E

, ,E x y

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X. H. WANG, H. H. LIU 654

where

1 3 2 3

2 1 21 1

1 1 21 2 12 1 2 2 1 21 2 12 1 2

2

3 1 21 2 12 1 2 2 1 12 2 21 1

, ,2 2

, ,

4 .

l l l lx y

l k l

l k k

k

k

,E x y is a unique positive equilibrium when the condition (H1) 1 2 2 1 0k k holds. Throughout this section, we always assume that the condition H(1) holds.

Clearly, the characteristic equation of the linearized system of system (6) at the equilibrium is 0 0,0E

1 2 0k k ,

which has two real roots, 1 , 2 . Therefore, the equilibrium is unstable and is a saddle point of system (6). The linearized system of system (6) at the

0k 0k 0 0,0E

equilibrium 21

21

0, ,k

E

is

12 2 21 12

21

0k k

k

,

which has two real roots, , 2 0k 12 2 21 1

21

0k k

.

Therefore, the equilibrium 21 0,

kE

21

, is an unstable

node of system (6). The characteristic equation at the

equilibrium 12

1

,0k

E

resulting from the linear system

(6) has the form 2 1

1 2

11

0k k

k

. Under

the condition (H1), Equation (7) has a negative real root

and a positive real root 1k 2 1

1

1 2k k

. Therefore, the

equilibrium 12

1

,0k

E

is unstable and is also a saddle

point of system (6) when the condition (H1) is satisfied. In what follows, we investigate the stability of the

positive equilibrium of system (6). ,E x y

Underthe assumption (H1), let 1x t x t x , 2x t y t y . Then system (6) is equivalent to the

following two dimensional system:

21 1 2 11 1

212 1 2 13 1 2

22 1 2 11 2

12 1 2

,

,

x t Mx t Nx t a x t

a x t x t a x t x t

x t Dx t Ex t b x t

b x t x t

where

.

(8)

2* * * * *1 1 12 1

* * *21 2 11 1 1

*12 12 1 13 1 11 21 12 2

, ,

, , ,

2 , , ,

M x x y N x x

E y D y a y

a x a b b

2

and the positive equilibrium of system (6) is transformed into the zero equilibrium (0, 0) of system (8). It is easy to see that the characteristic equation of the linearized system of system (8) at the zero equilibrium (0, 0) is

,E x y

21 0 0 0,a a b e (9)

where b0 = −ND, a0 = ME, a1 = −(M + E). It is well known that the stability of the zero equilib-

rium (0, 0) of system (8) is determined by the real parts of the roots of Equation (9). If all roots of Equation (9) locate the left-half complex plane, then the zero equilib-rium (0, 0) of system (8) is asymptotically stable. If Equation (9) has a root with positive real part, then the zero solution is unstable. Therefore, to study the stability of the zero equilibrium (0, 0) of system (8), an important problem is to investigate the distribution of roots in the complex plane of the characteristic Equation (9).

For Equation (9), according to the Routh-Hurwitz cri-terion, we have the following result.

Lemma 2.1. The two roots of Equation (9) with τ = 0 have always negative real parts, the zero equilibrium (0,0) of system (8) with τ = 0 is asymptotically stable.

Next, we consider the effects of a positive delay τ on the stability of the zero equilibrium (0, 0) of system (8). Since the roots of the characteristic Equation (9) depend continuously on τ, a change of τ must lead to a change of the roots of Equation (9). If there is a critical value of τ such that a certain root of (9) has zero real part, then at this critical value the stability of the zero equilibrium (0, 0) of system (8) will switch, and under certain conditions a family of small amplitude periodic solutions can bifur-cate from the zero equilibrium (0, 0); that is, a Hopf bi-furcation occurs at the zero equilibrium (0, 0).

Now, we look for the conditions under which the cha-racteristic Equation (9) has a pair of purely imaginary roots, see [24]. Clearly, iω(ω > 0) is a root of Equation (9) if and only if ω satisfies the following equation:

21 0 0 cos 2 sin 2 0ia a b

Separating the real and imaginary parts of the above equation yields the following equations:

20 0

1 0

cos 2 ,

sin .

a b

a b

(10)

Adding up the squares of the corresponding sides of the above equations yields equations with respect to ω:

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X. H. WANG, H. H. LIU 655

4 2 2 21 0 0 02a a a b 0

. (11)

22 2 21 0 1 0 0 02

2 2 4

2

a a a a a b

2

0.

,

Since

22 2 21 0 0 0

4 2

2 4

4 4

a a a b

M E ME M E ND

If , Equation (11) has no positive real root. Otherwise (11) has an unique positive root, sign it as

2 20 0 0a b

0 .

1

222 2 21 0 1 0 0 0

0

2 2 4

2

a a a a a b

2

Suppose (H2) in the following. From the first equation of (10), we know that the value of τ associ-ated with

2 20 0 0a b

0 should satisfy 20 0

00

cos 2a

b

(12)

If we define

20 0

0 0

1arccos 2 π , 0,1,

2j

aj j

b

, (13)

then when Equation (9) has a pair of purely imaginary roots ±

0,1, 2, ,j j i 0 .

Let λ(τ) = α(τ) + iω(τ) be a root of Equation (9) near τ = j satisfying α( j ) = 0 and ω( j ) = 0 . For this pair of conjugate complex roots,we have the following result.

Lemma 2.2.

d0, 0,1, .

dj

j

Proof. Differentiating both sides of Equation (9) with respect to τ, and noticing that λ is a function with respect

to τ, we have 21 0 0

d2 2 2

da b e b e 2 .

From the above equation, one can easily obtain

1 21

0

2d.

d 2

a e

b

It follows easily from λ( j ) = 0i that

21 1 2

0

1 1 0 0

2 1 0 0

2

2 c

cos 2 2 sin 2 ,

sin 2 2 cos 2 .

j

Thus, we have

11 0 0

0

sin 2 2 cos 2d.

d cos 2j j

j

a

Combining (10) and some simple computations show that

1 21 0

20

2 2d0.

d 2

a a

b

This completes the proof. From the above discussion and the Hopf bifurcation

theorem of FDEs [14,23], we can obtain the following results on the stability of the zero equilibrium of system (8); that is, the stability of the positive equilibrium

,E x y of system (6).

Theorem 2.3. Suppose that the coefficients i , k i (i = 1, 2) in system (6) satisfy the condition (H1) and 0 ,

0 satisfies the condition (H2); then the following re-sults hold.

ab

1) The positive equilibrium is asymptoti-cally stable when

,E x y 00 and unstable when 0 .

2) When τ crosses through each j (j = 0, 1, 2, 3, ···), system (6) can undergo a Hopf bifurcation at the positive equilibrium ,E x y ; that is, a family of nonconstant periodic solutions can bifurcate from the positive equi-

librium ,E x y when τ crosses through each critical

value j (j = 0, 1, 2, 3, ···).

3. Properties of Hopf Bifurcations

In the previous section, we studied mainly the stability of the positive equilibrium of system (6) and the existence of Hopf bifurcations at the positive equilib-rium

,E x y ,E x y .

0os 2 j

j j

j j

a e s ise

b i

s a

s i a

In this section, we shall study the properties of the Hopf bifurcations obtained by Theorem 2.3 and the sta-bility of bifurcated periodic solutions occurring through Hopf bifurcations by using the normal form theory and the center manifold reduction for retarded functional dif-ferential equations (RFDEs) due to Hassard, Kazarinoff and Wan [23]. To guarantee the existence of the above Hopf bifurcations, throughout this section, we always assume that the conditions (H1) and (H2). Under these conditions, for fixed 0,1,2,j , let τ = j + μ; then μ = 0 is the Hopf bifurcation value of system (6) at the positive equilibrium ,E x y . Since system (6) is equivalent to system (8), in the following discussion we shall consider mainly system (8).

In system (8), let k k x t x t and drop the bars for simplicity of notation. Then system (8) can be rewrit-ten as a system of RFDEs in C([−1, 0], R2) of the form

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X. H. WANG, H. H. LIU 656

1 1 2 11

2 21 12 1 2 13 1 2

2 1 2 11

22 12 1 2

1

1 1

1

1 .

j

j

x t Mx t Nx t a

x t a x t x t a x t x t

x t Dx t Ex t b

x t b x t x t

,

(14)

Define the linear operator and the nonlinear operator

2:L C R 2 , :f C R by

1

2

1

2

00

00

10

10

j

j

ML

E

N

D

1

0

(15)

and

211 1 13 1

22 13 1 2

211 2 12 1 2

0 0

, 1 0

0 1

j

a a

f a

b b

(16)

respectively, where , and let 1 2,T

C 1 2,x x x .

By the Riesz representation theorem, there exists a 2 × 2 matrix function η(θ, μ), −1 ≤ θ ≤ 0, whose elements are of bounded variation such that

0

1d ,L

for 21,0 , .C R

In fact, we can choose

0 1, 1j j ,

where

0 1

0 0,

0 0

M N

E D

.

For 1 1,0 , ,C R 2 define

0

1

d, 1

d

d , , 0.

A

s s

,0 ,

t

(17)

and

0, 1,0 ,

, , 0.R

f

(18)

Then system (14) is equivalent to

t tx A x R x . (19)

For , define 1 0,1 ,C R

*

0

1

d, 0

d

d ,0 , 0.

ss

sA

t t s

,1 , (20)

and a bilinear inner product

0

1 0

, 0 0

d d

s

,

(21)

where η(θ) = η(θ, 0). Then A(0) and A* are adjoint op-erators. In addition, from Section 2 we know that ± 0 ji are eigenvalues of A(0). Thus, they are also eigenvalues of A*. Let q(θ) is the eigenvector of A(0) corresponding to 0 ji and *q s

0

is the eigenvector of A* corre-sponding to ji .

Let 011, jT i

q v e and 11, 0 ji s q s . v e

From the above discussion, it is easy to know that 00 0 0jA q i q and 00 0 jA q i q 0

0jq

0jq

. That is

0

0 00 1

0 0j j

M Nq q i

E D

and

* *0

0 00 1

0 0j j

M Dq q i

E N

Thus, we can easily obtain

0

0

0

1, ,j

j

TiiDe

q ei E

0

0*

0

1,j

j

ii sNe

q s G ei E

.

Since

0 0

0

0

0

0* *

1 0

*

0 ( )*1 11 0

0* *

1

* *

* *1 1 1 1

,

0 0 ( )d d

0 0

1, d 1, d

0 0 0 d 0

00 0 0 0

0

1 .

j j

j

j

j

Ti i

i

ij

ij

q s q

q q q q

q q

G v e v e

q q q e q

Nq q q e q

D

G v v e Dv Nv

2We may choose G and G as

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X. H. WANG, H. H. LIU

Copyright © 2012 SciRes. AM

657

in the direction of q and q . Note that W is real if tx is real. We consider only real solutions. For solution

0tx C of (14), since μ = 0,

0

0

* *1 1 1 1

* *1 1 1 1

1,

1

1.

(1 )

j

j

ij

ij

Gv v e Dv Nv

Gv v e Dv Nv

(22)

*0

*0 0

0 0, , , 2

0 ,

j

j

z t i z q f W z z zq

i z q f

(25)

which assures that * , 1q s q . Using the same notations as in Hassard, Kazarinoff,

and Wan [23], we first compute the coordinates to de-scribe the center manifold 0 at μ = 0. Let C tx be the solution of Equation (14) when μ = 0. Define

that is

0 , ,jz t i z t g z z (26)

where

* , ,

, 2Re

t

t

z t q x

W t x z t q

(23)

2 2

20 11 02 21,2 2 2

z z zg z z g g zz g g

z (27)

Then it follows from (23) that On the center manifold we have 0C , ( , , )W t W z z , where

0 0

2 2

20 11 02

1 1

, 2

2 2

1, 1,j j

t

i i

x W t z t q

z zW W zz W

v e z v e z

(28)

2

20 11

2 3

02 30

, ,2

.2 6

zW z z W W zz

z zW W

(24)

z and z are local coordinates for center manifold 0C It follows together with (16) that

0 0

0 0

0 0

* *0

21 1 2

11 12 1 1

2* 1 (2)13 1 1

22 1 2

11 1 1 12 1 2

11 12 1

, 0 , 0 0,

0 0 1

0 0 1

0 1 0

2

j j

j j

j j

t

i i

i ij

i i

ij

g z z q f z z q f x

a W z z a W z z W v e z v e z

q a W z z W v e z v e z

b W v z v z b W e z e z W v z v z

G a a v e

0 0

0 0

0 0

0 0

2* 2 *

11 1 1 12 1 1

* *11 12 1 11 1 1 1 12 1 1

2* 2 *

11 12 1 11 1 1 12 1 1

21 * * * (1)11 12 1 20 11 1 1 12 1 20 12 1 1 20

2

2

22

2 0 2 0 1

j j

j j

j j

j j

i

i i

i i

i i

zb v v b v v e

a a v e b v v v b v v e zz

za a v e b v v b v v e

a a v e W b v v b v e W b v v W

a

0 0

0 0

2 1 * * (2)12 20 11 12 1 11 11 1 1 12 1 11

21 2*

12 1 1 11 12 11 13 1 13 1

( 1) 4 2 0 4 2 0

2 1 2 1 2 42

j j

j j

i i

i i

W a a v e W b v v b v e W

z zb v v W a W a v e a v e

Comparing the coefficients with (27), we obtain

0 0* 2 *20 11 12 1 11 1 1 12 1 12 j ji i

jg G a a v e b v v b v v e ,

0 0* *11 11 12 1 11 1 1 1 12 1 12 j ji i

jg G a a v e b v v v b v v e ,

0 0* 2 *02 11 12 1 11 1 1 12 1 12 ,j ji i

jg G a a v e b v v b v v e

X. H. WANG, H. H. LIU 658

0 0

0 0

0 0

2 11 * * *21 11 12 1 20 11 1 1 12 1 20 12 1 1 20

2 1 * *12 20 11 12 1 11 11 1 1 12 1 11

21 2*

12 1 1 11 12 11 13 1 13 1

2 0 2 0

1 4 2 0 4 2 0

2 1 2 1 2 42

j j

j j

j j

i ij

i i

i i

g G a a v e W b v v b v e W b v v W

a W a a v e W b v v b v e W

z zb v v W a W a v e a v e

2

1

Since there are 20W and 11W in 21g , we still

need to compute them. From (19) and (23), we have

*0

*0 0

2 ,

2 ,

, , ,

tW x zq zq

AW q f q

AW q f q f

AW H z z

1,0

0 (29)

where

2 2

20 11 02, ,2 2

z zH z z H H zz H (30)

Substituting the corresponding series into (29) and comparing the coefficients, we obtain

0 20 20

11 11

2 ,

,

jA i W H

AW H

(31)

From (29), we know that for 1,0 ,

0 0, , 0 0

, 0 , .

H z z q f q q f q

g z z q g z z q

(32)

Comparing the coefficients with (30) gives that

20 20 02H g q g q (33)

and

11 11 11H g q g q (34)

From (31) and (33), we get

20 0 20 20 022 .jW i W g q g q

Note that 00 jiq q e

, hence

0

0

20 2020

0 0

21

0 03

.

0j j

j

i

j j

i

iigW q e q e

E e

ig

(35)

Similarly, from (31) and (34), we have

11 11 11W g q g q

and

0 011 1111 2

0 0

0 0 . (36)

In what follows we shall seek appropriate and .

1E

2EFrom the definition of A and (31) that

j ji i

j j

ig igW q e q e E

0

0 ,

0

20 0 20 201d 2 0jW i W H

, (37)

and

0

11 111d W H

(38)

where 0, . From (29), we have

0

0

20 20 02

11 12 1

211 1 12 1

0 0 0

2j

j

i

j i

H g q g q

a a v e

b v b v e

(39)

0

0

11 11 11

11 12 1

11 1 1 12 1

0 0

2 .

j

j

i

j i

H g q g q

a a v e

b v v b v e

(40)

Substituting (35) and (39) into (37), and noticing that

00

0 1d 0ji

ji I e q

0

and

00

0 1d 0ji

ji I e q

0 ,

We obtain

0

0

0

0 20 11

11 12 1

211 1 12 1

2 d

2 .

j

j

j

ij

i

j i

i I e E

a a v e

b v b v e

12 1

1

(41)

which leads to

0 0

0 0

20 11

12 20 11 1 12 1

22 .

2

j j

j j

i i

i i

i M Ne a a v eE

De i E b v b v e

(42)

Therefore,

0 0

0 0

120 11 12

1 2 20 11 1 12 1

22

2

j j

j j

i i

i i

i M Ne a a v eE

De i E b v b v e

Similarly, substituting (35) and (40) into (38), we get

Copyright © 2012 SciRes. AM

X. H. WANG, H. H. LIU 659

0

0

11 12 10

21

11 1 1 12 1

d 2

j

j

i

j i

a a v eE

b v v b v e

(43)

0

0

111 12 1

2

11 1 1 12 1

2

j

j

i

i

a a v eM NE

D E b v v b v e

It follows from (35), (36), (42), and (43) that g21 can be expressed. Thus, we can compute the following val-ues:

2

2 02 211 11 20 11

0

0 22 3j

g,

2

gic g g g

12

0

0,

j

c

2 12 0c ,

1 2 0

20

0.

jcT

which determine the quantities of bifurcating periodic solutions at the critical value j . That is, 2 deter-mines the directions of the Hopf bifurcation: If 2 0 ( 2 0 ), then the Hopf bifurcation is supercritical (sub-critical) and the bifurcating periodic solutions exist for

j ( j ); 2 determines the stability of the bi-furcating periodic solutions: the bifurcating periodic so-lutions in the center manifold are stable (unstable) if

2 0 ( 2 0

T

); and 2 determines the period of the bifurcating periodic solutions: the period increase (de-crease) if 2 ( 2T ). Further, it follows from Lemma 2.2 and (44) that the following results about the direction of the Hopf bifurcations hold.

T

00

Theorem 3.1. Suppose that (H1), (H2) hold. If ( ), then system (6) can un-

dergo a supercritical (subcritical) Hopf bifurcation at the positive equilibrium when τ crosses through the critical values

1 0 0c 1 0c

,E x y

0

j . In addition, the bifurcated periodic

solutions occurring through Hopf bifurcations are orbi-tally asymptotically stable on the center manifold if

and unstable if 1 0c 0 1 0 0c .

4. Numerical Simulations

In this section, we give some numerical simulations for a special case of system (6) to support our analytical re-sults in this paper. As an example, we consider system (6) with the coefficients , 11 2k 1 , 12 1 , 1 2.6 ,

, 2 1k 2 1.2 , 21 1.2 ; that is

2 2.6

1 1.2 .

x t x t x t y t x t y t

y t y t x t y t

,

(45) Figure 1. The numerical approximations of system (45) when τ = 0. The positive equilibrium E(1.04678, 0.25613) is asymptotically stable.

Copyright © 2012 SciRes. AM

X. H. WANG, H. H. LIU 660

Figure 2. The numerical approximations of system (45) when τ = 1.35. The positive equilibrium E(1.04678, 0.25613) is asymptotically stable.

Figure 3. The numerical approximations of system (45) when τ = 1.4. The positive equilibrium E(1.04678, 0.25613) is unstable and a stable periodic solution bifurcates from E.

Copyright © 2012 SciRes. AM

X. H. WANG, H. H. LIU

Copyright © 2012 SciRes. AM

661

Obviously, (H1) 2 1 1 2k k holds; therefore, system (45) has a unique positive equilibrium E(1.04678, 0.25613). From Lemma 2.1, we know that the positive equilibrium E(1.04678, 0.25613) system (45) is asymp-totically stable when τ = 0; see Figure 1.

On the other hand, since (H1) 2 1 1 2k k , (H2) = −1.2342 < 0, from Theorem2.3, we know that

the positive equilibrium E(1.04678, 0.25613) of system (45) is asymptotically stable when 0 ≤ τ < τ0 = 1.3788 and unstable when τ > τ0 = 1.3788, and system (45) can also undergo a Hopf bifurcation at the positive equilib-rium E(1.04678, 0.25613) when τ crosses through the critical values

20 0a b 2

j = 1.3788 + 3.3502jπ (j = 0, 1, 2, ···), i.e., a family of periodic solutions bifurcate from E(1.04678, 0.25613) see Figures 2 and 3.

5. Acknowledgements

The authors of this paper express their grateful gratitude for any helpful suggestions from reviewers and the par-tial support of Yunnan Provience science fundation 2011FZ086.

REFERENCES [1] A. J. Lotka, “Elements of Physical Biology,” Williams

and Wilkins, New York, 1925.

[2] V. Volterra, “Variazionie Fluttuazioni del Numero d’Individui in Specie Animali Conviventi,” Mem. Acad. Licei, Vol. 2, 1926, pp. 31-113.

[3] M. Brelot, “Sur le Probleme Biologique Hereditaire de Deux Especes Devorante et Devore,” Annali di Mate-matica Pura ed Applicata, Vol. 9, No. 1, 1931, pp. 58-74. doi:10.1007/BF02414092

[4] L. Chen, “Mathematical Models and Methods in Ecol-ogy,” Science Press, Beijing, 1988.

[5] Y. Song and S. Yuan, “Bifurcation Analysis in a Preda-tor-Prey System with Delay,” Nonlinear Analysis: Real World Applications, Vol. 7, 2006, pp. 265-284. doi:10.1016/j.nonrwa.2005.03.002

[6] T. Faria, “Stability and Bifurcation for a Delayed Preda-tor-Prey Model and the Effect of Diffusion,” Journal of Mathematical Analysis and Applications, Vol. 254, No. 2, 2001, pp. 433-463.

[7] S. Ruan, “Absolute Stability, Conditional Stability and Bifurcation in Kolmogorov-Type Predator-Prey System with Discrete Delays,” Quarterly of Applied Mathematics, Vol. 59, 2001, pp. 159-172.

[8] X. P. Yan and Y. D. Chu, “Stability and Bifurcation Analysis for a Delayed Lotka-Volterra Predator-Prey Sys-tem,” Journal of Computational and Applied Mathemat-ics, Vol. 196, No. 1, 2006, pp. 198-210.

[9] X. P. Yan and C. H. Zhang, “Hopf Bifurcation in a De-

layed Lokta-Volterra Predator-Prey System,” Nonlinear Analysis, Vol. RWA 9, 2008, pp. 114-127. doi:10.1016/j.nonrwa.2006.09.007

[10] Y. Song and J. Wei, “Local Hopf Bifurcation and Global Periodic Solutions in a Delayed Predator-Prey System,” Journal of Mathematical Analysis and Applications, Vol. 301, 2005, pp. 1-21. doi:10.1016/j.jmaa.2004.06.056

[11] X. P. Yan and W. T. Li, “Hopf Bifurcation and Global Periodic Solutions in a Delayed Predatorprey System,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, pp. 427-445.

[12] J. Zhang and B. Feng, “Geometric Theory and Bifurca-tion Problems of Ordinary Differential Equations,” Bei-jing University Press, Beijing, 2000.

[13] S. A. Gourley, “Travelling Fronts in the Diffusive Nichol-sons Blowflies Equation with Distributed Delays,” Ma- thematical and Computer Modelling, Vol. 32, 2000, pp. 843-853. doi:10.1016/S0895-7177(00)00175-8

[14] J. K. Hale, “Theory of Functional Differential Equations,” Spring-Verlag, New York, 1977. doi:10.1007/978-1-4612-9892-2

[15] J. Wu, “Theory and Applications of Partial Functional Differential Equations,” Springer-Verlag, New York, 1996. doi:10.1007/978-1-4612-4050-1

[16] C. H. Zhang, X. P. Yan and G. H. Cui, “Hopf Bifurca-tions in a Predator-Prey System with a Discrete Delay and a Distributed Delay,” Nonlinear Analysis, Vol. RWA 11, 2010, pp. 4141-4153. doi:10.1016/j.nonrwa.2010.05.001

[17] X. Z. He, “Stability and Delays in a Predator-Prey Sys-tem,” Journal of Mathematical Analysis and Applications, Vol. 198, No. 2, 1996, pp. 355-370.

[18] Z. Lu and W. Wang, “Global Stability for Two-Species Lotka-Volterra Systems with Delay,” Journal of Mathe-matical Analysis and Applications, Vol. 208, No. 1, 1997, pp. 277-280.

[19] E. L. Rice, “Allelopathy,” 2nd Edition, Academic Press, New York, 1984.

[20] J. M. Smith, “Models in Ecology,” Cambridge University, Cambridge, 1974.

[21] J. Chattopadhyay, “Effects of toxic Substance on a Two- Species Competitive System,” Ecological Modelling, Vol. 84, 1996, pp. 287-289. doi:10.1016/0304-3800(94)00134-0

[22] A. Mukhopadhyay, J. Chattopadhyay and P. K. Tapaswi, “A Delay Differential Equations Model of Plankton Al-lelopathy,” Mathematical Biosciences, Vol. 149, No. 2, 1998, pp. 167-189.

[23] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, “Theory and Applications of Hopf Bifurcation,” Cambridge Uni-versity Press, Cambridge, 1981.

[24] Y. Song, M. Han and J. Wei, “Stability and Hopf Bifur-cation on a Simplified BAM Neural Network with De-lays,” Physica D, Vol. 200, 2005, pp. 185-204. doi:10.1016/j.physd.2004.10.010