Research Article Bogdanov-Takens Bifurcation of a Delayed ...Bogdanov-Takens Bifurcation of a Delayed Ratio-Dependent Holling-Tanner Predator Prey System XiaLiu, 1 YanweiLiu, 2 andJinlingWang

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 898015, 5 pageshttp://dx.doi.org/10.1155/2013/898015

Research ArticleBogdanov-Takens Bifurcation of a Delayed Ratio-DependentHolling-Tanner Predator Prey System

Xia Liu,1 Yanwei Liu,2 and Jinling Wang1

1 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China2 Institute of Systems Biology, Shanghai University, Shanghai 200444, China

Correspondence should be addressed to Xia Liu; [email protected]

Received 3 July 2013; Accepted 7 August 2013

Academic Editor: Luca Guerrini

Copyright © 2013 Xia Liu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A delayed predator prey systemwith refuge and constant rate harvesting is discussed by applying the normal form theory of retardedfunctional differential equations introduced by Faria and Magalhães. The analysis results show that under some conditions thesystemhas a Bogdanov-Takens singularity. A versal unfolding of the system at this singularity is obtained. Ourmain results illustratethat the delay has an important effect on the dynamical behaviors of the system.

1. Introduction

It is well known that the multiple bifurcations will occurwhen a predator prey system (ODE) with more interior pos-itive equilibria, such as Bogdanov-Takens bifurcation, Hopfbifurcation, and backward bifurcation; see [1–5] for example.However, when the predator prey systems with delay andBogdanov-Takens bifurcation are researched relative few (see[6–8] and the reference therein) using similar methods as [6–8], the authors of [9–11] consider the Bogdanov-Takens bifur-cation of some delayed single inertial neuron or oscillatormodels.

Motivated by the works of [5, 6], we mainly consider theBogdanov-Takens bifurcation of the following system:

�̇� = 𝑟𝑥 (1 −

𝑥

𝐾

) −

𝛼𝑦 (𝑥 − 𝑚)

𝐴𝑦 + 𝑥 − 𝑚

− ℎ,

̇𝑦 = 𝑠𝑦(1 −

𝑏𝑦 (𝑡 − 𝜏)

𝑥 (𝑡 − 𝜏) − 𝑚

) ,

(1)

where 𝑥 and 𝑦 stand for prey and predator population (ordensities) at time 𝑡, respectively. The predator growth is oflogistic type with growth rate 𝑟 and carrying capacity 𝐾 inthe absence of predation; 𝛼 and 𝐴 stand for the predatorcapturing rate and half saturation constant, respectively;

𝑠 is the intrinsic growth rate of predator; however, carryingcapacity 𝑥/𝑏 (𝑏 is the conversion rate of prey into predators)is the function on the population size of prey.The parameters𝛼,𝐴,𝑚, ℎ, 𝑠, 𝑏, and 𝜏 are all positive constants.𝑚 is a constantnumber of prey using refuges; ℎ is the rate of prey harvesting.

For simplicity, we first rescale system (1). Let 𝑋 = 𝑥 − 𝑚,𝑌 = 𝑦; system (1) can be written as (still denoting 𝑋, 𝑌 as𝑥, 𝑦)

�̇� = 𝑟 (𝑥 + 𝑚)(1 −

𝑥 + 𝑚

𝐾

) −

𝛼𝑥𝑦

𝐴𝑦 + 𝑥

− ℎ,

̇𝑦 = 𝑠𝑦(1 −

𝑏𝑦 (𝑡 − 𝜏)

𝑥 (𝑡 − 𝜏)

) .

(2)

Next, let 𝑡 = 𝑟𝑡, 𝑋(𝑡) = 𝑥(𝑡)/𝐾, and 𝑌(𝑡) = 𝛼𝑦(𝑡)/𝑟𝐾;then system (2) takes the form (still denoting 𝑋, 𝑌, 𝑡 as 𝑥, 𝑦,𝑡)

�̇� = (𝑥 + 𝑚) (1 − 𝑥 − 𝑚) −

𝑥𝑦

𝑎𝑦 + 𝑥

− ℎ,

̇𝑦 = 𝛿𝑦(𝛽 −

𝑦 (𝑡 − 𝜏)

𝑥 (𝑡 − 𝜏)

) ,

(3)

where𝑚 = 𝑚/𝐾, 𝑎 = 𝐴𝑟/𝛼, 𝛿 = 𝑠ℎ/𝛼, 𝛽 = 𝛼/𝑏𝑟, ℎ = ℎ/𝑟, and𝜏 = 𝑟𝜏.

2 Abstract and Applied Analysis

When 𝜏 = 0, we have known that for some parametervalues system (3) exhibits Bogdanov-Takens bifurcation (see[5]). Summarizing the methods used by [6] and the formulaein [12], the sufficient conditions which depend on delay toguarantee that system (3) has a Bogdanov-Takens singularitywill be given.Therefore, the delay has effect on the occurrenceof Bogdanov-Takens bifurcation.

In the next section we will compute the normal formand give the versal unfolding of system (3) at the degenerateequilibrium.

2. Bogdanov-Takens Bifurcation

System (3) can also be written as

�̇� = 𝜏 ((𝑥 + 𝑚) (1 − 𝑥 − 𝑚) −

𝑥𝑦

𝑎𝑦 + 𝑥

− ℎ) ,

̇𝑦 = 𝜏𝛿𝑦(𝛽 −

𝑦 (𝑡 − 1)

𝑥 (𝑡 − 1)

) .

(4)

Let

0 < 𝑚 <

1

2

(1 −

𝛽

𝑎𝛽 + 1

) ,

ℎ =

̃

ℎ =

1

4

(

𝛽

𝑎𝛽 + 1

− 1)

2

+

𝑚𝛽

𝑎𝛽 + 1

.

(5)

Then 𝑃∗= (𝑥

∗, 𝑦

∗) is an interior positive equilibrium of

systems (3) and (4), where 𝑥∗= −(1/2)((𝛽/(𝑎𝛽+1))+2𝑚−1),

𝑦

∗= 𝛽𝑥

∗.

In order to discuss the properties of system (4) in theneighborhood of the equilibrium𝑃

∗= (𝑥

∗, 𝑦

∗), let𝑥 = 𝑥−𝑥

∗,

𝑦 = 𝑦 − 𝑦

∗; then 𝑃

∗is translated to (0, 0), and system (4)

becomes (still denoting 𝑥, 𝑦 as 𝑥, 𝑦)

�̇� =

𝜏𝛽

(𝑎𝛽 + 1)

2𝑥 −

𝜏

(𝑎𝛽 + 1)

2𝑦 + 𝑔

1𝑥

2

+ 𝑔

2𝑥𝑦 + 𝑔

3𝑦

2+ h.o.t,

̇𝑦 = 𝜏𝛿𝛽

2𝑥 (𝑡 − 1) − 𝜏𝛿𝛽𝑦 (𝑡 − 1) + 𝑔

4𝑥

2(𝑡 − 1)

+ 𝑔

5𝑥 (𝑡 − 1) 𝑦 (𝑡 − 1) + 𝑔

6𝑥 (𝑡 − 1) 𝑦

+ 𝑔

7𝑦 (𝑡 − 1) 𝑦 + h.o.t,

(6)

where h.o.t denotes the higher order terms and

𝑔

1= −(

2𝑎𝛿𝛽

2𝜏

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

+ 𝜏) ,

𝑔

2=

4𝑎𝛿𝛽𝜏

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

,

𝑔

3=

−2𝑎𝛿𝜏

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

,

𝑔

4=

2𝜏𝛿𝛽

2(𝑎𝛽 + 1)

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

,

𝑔

5=

−2𝜏𝛿𝛽 (𝑎𝛽 + 1)

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

,

𝑔

6= −

2𝜏𝛿𝛽 (𝑎𝛽 + 1)

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

,

𝑔

7=

2𝜏𝛿 (𝑎𝛽 + 1)

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

.

(7)

The characteristic equation of the linearized part ofsystem (6) is

𝐹 (𝜆) = 𝜆

2+ [𝜏𝛿𝛽𝑒

−𝜆−

𝜏𝛽

(𝑎𝛽 + 1)

2]𝜆. (8)

Clearly, if

𝛿 =

1

(𝑎𝛽 + 1)

2,

𝜏 ̸=

1

𝛿𝛽

, that is, 𝜏 ̸=(𝑎𝛽 + 1)

2

𝛽

;

(9)

then 𝜆 = 0 is double zero eigenvalue; if 𝛿 = 1/(𝑎𝛽 + 1)2 and𝜏 = 1/𝛿𝛽, that is, 𝜏 = (𝑎𝛽 + 1)2/𝛽, then 𝜆 = 0 is triple zeroeigenvalue. We will mainly discuss the first case in this paper.

According to the normal form theory developed by Fariaand Magalhães [13], first, rewrite system (4) as ̇𝑋(𝑡) = 𝐿(𝑋

𝑡),

where 𝑋(𝑡) = (𝑥1(𝑡), 𝑥

2(𝑡)), 𝐿(𝜙) = 𝐿 ( 𝜙1(−1)

𝜙2(0)

), and 𝜙 =(𝜙

1, 𝜙

2). Take 𝐴

0as the infinitesimal generator of system. Let

Λ = {0}, and denote by 𝑃 the invariant space of𝐴0associated

with the eigenvalue 𝜆 = 0; using the formal adjoint theoryof RFDE in [13], the phase space 𝐶

1can be decomposed by

Λ as 𝐶1= 𝑃 ⊕ 𝑄. Define Φ and Ψ as the bases for 𝑃 and

𝑃

∗, the space associated with the eigenvalue 𝜆 = 0 of theadjoint equation, respectively, and to be normalized such that(Ψ,Φ) = 𝐼, ̇Φ = Φ𝐽, and ̇Ψ = −𝐽Ψ, where Φ and Ψ are 2 × 2matrices.

Next we will find the Φ(𝜃) and Ψ(𝑠) based on the tech-niques developed by [14].

Lemma 1 (see Xu andHuang [14]). Thebases of 𝑃 and its dualspace 𝑃∗ have the following representations:

𝑃 = spanΦ,Φ (𝜃) = (𝜑1(𝜃) , 𝜑

2(𝜃)) , −1 ≤ 𝜃 ≤ 0,

𝑃

∗= spanΨ,Ψ (𝑠) = col (𝜓

1(𝑠) , 𝜓

2(𝑠)) , 0 ≤ 𝑠 ≤ 1,

(10)

where 𝜑1(𝜃) = 𝜑

0

1∈ 𝑅

𝑛\ {0}, 𝜑

2(𝜃) = 𝜑

0

2+ 𝜑

0

1𝜃, and 𝜑0

2∈ 𝑅

𝑛

and 𝜓2(𝑠) = 𝜓

0

2∈ 𝑅

𝑛∗\ {0}, and 𝜓

1(𝑠) = 𝜓

0

1− 𝑠𝜓

0

2, 𝜓

0

1∈ 𝑅

𝑛∗,which satisfy

(1) (𝐴 + 𝐵)𝜑01= 0,

(2) (𝐴 + 𝐵)𝜑02= (𝐵 + 𝐼)𝜑

0

1,

(3) 𝜓02(𝐴 + 𝐵) = 0,

(4) 𝜓01(𝐴 + 𝐵) = 𝜓

0

2(𝐵 + 𝐼),

Abstract and Applied Analysis 3

(5) 𝜓02𝜑

0

2− (1/2)𝜓

0

2𝐵𝜑

0

1+ 𝜓

0

2𝐵𝜑

0

2= 1,

(6) 𝜓01𝜑

0

2− (1/2)𝜓

0

1𝐵𝜑

0

1+ 𝜓

0

1𝐵𝜑

0

2+ (1/6)𝜓

0

2𝐵𝜑

0

1−

(1/2)𝜓

0

2𝐵𝜑

0

2= 0.

By (6), we have 𝐴 = ( 𝜏𝛿𝛽 −𝜏𝛿0 0

) and 𝐵 = ( 0 0𝜏𝛿𝛽2−𝜏𝛿𝛽

); usingLemma 1, we obtain

Φ (𝜃) = (

1

1

𝜏𝛿𝛽

+ 𝜃

𝛽 𝛽𝜃

) , −1 ≤ 𝜃 ≤ 0,

Ψ (𝑠) = (

𝑚

1+ 𝑠𝛽𝑛 𝑚

2− 𝑠𝑛

−𝛽𝑛 𝑛

) , 0 ≤ 𝑠 ≤ 1,

(11)

where 𝑚1= 𝜏𝛿𝛽(𝜏𝛿𝛽 − 2)/2(𝜏𝛿𝛽 − 1)

2= 𝜏𝛽[𝜏𝛽 − 2(𝑎𝛽 +

1)

2]/2[𝜏𝛽− (𝑎𝛽+ 1)

2]

2,𝑚2= (1 + (𝜏𝛿𝛽− 1)

2)/2𝛽(𝜏𝛿𝛽− 1)

2=

(1/2𝛽)(1+(𝑎𝛽+1)

4/(𝜏𝛽−(𝑎𝛽+1))

2), and 𝑛 = 𝜏𝛿/(𝜏𝛿𝛽−1) =

𝜏/(𝜏𝛽 − (𝑎𝛽 + 1)

2).

The matrix 𝐽 satisfying ̇Φ = Φ𝐽 is given by 𝐽 = ( 0 10 0

).System (6) in the center manifold is equivalent to the system�̇� = 𝐽𝑧 + Ψ(0)𝐹(Φ𝑧), where

𝐹 (𝜙) = (

𝑔

1𝜙

2

1(0) + 𝑔

2𝜙

1(0) 𝜙

2(0) + 𝑔

3𝜙

2

2(0) + h.o.t

𝑔

4𝜙

2

1(−1) + 𝑔

5𝜙

1(−1) 𝜙

2(−1) + 𝑔

6𝜙

1(−1) 𝜙

2(0) + 𝑔

7𝜙

2(−1) 𝜙

2(0) + h.o.t

) . (12)

It is easy to obtain

Φ (𝜃) 𝑧 = (

𝑧

1+ (

1

𝜏𝛿𝛽

+ 𝜃) 𝑧

2

𝛽 (𝑧

1+ 𝑥

2𝜃)

) ,

𝜙

1(0) = 𝑧

1+

1

𝜏𝛿𝛽

𝑧

2,

𝜙

1(−1) = 𝑧

1+ (

1

𝜏𝛿𝛽

− 1) 𝑧

2,

𝜙

2(0) = 𝛽𝑧

1, 𝜙

2(−1) = 𝛽 (𝑧

1− 𝑧

2) .

(13)

Hence (6) becomes

�̇�

1= 𝑧

2− 𝜏𝑚

1𝑧

2

1−

2𝑚

1

𝛿𝛽

𝑧

1𝑧

2+ 𝑙

1𝑧

2

2+ h.o.t,

�̇�

2= 𝑛𝜏𝛽𝑧

2

1+

2𝑛

𝛿

𝑧

1𝑧

2+ 𝑙

2𝑧

2

2+ h.o.t,

(14)

where 𝑙1= 𝑚

1𝑔

1/𝜏

2𝛿

2𝛽

2+2𝑚

2(𝛿+𝜇

1)(𝑎𝛽+1)(1−𝜏𝛿𝛽)/𝜏𝛿

2(𝛽+

(𝑎𝛽+1)(2𝑚−1)), 𝑙2= −(𝑛𝑔

1/𝛽𝜏

2𝛿

2) + 2𝑛(𝛿+𝜇

1)(𝑎𝛽+1)(1−

𝜏𝛿𝛽)/𝜏𝛿

2(𝛽 + (𝑎𝛽 + 1)(2𝑚 − 1)).

After a series of transformations we obtain

�̇�

1= 𝑧

2+ h.o.t,

�̇�

2= 𝑑

1𝑧

2

1+ 𝑑

2𝑧

1𝑧

2+ h.o.t,

(15)

where 𝑑1= 𝛽𝑛𝜏 = 𝛽𝜏

2/(𝜏𝛽 − (𝑎𝛽 + 1)

2), 𝑑2= −𝜏𝑚

1+ 𝑛/𝛿 =

−(𝜏

2𝛽

2+ 2(𝑎𝛽 + 1)

2[(𝑎𝛽 + 1)

2− 2𝜏𝛽])/(𝜏𝛽 − (𝑎𝛽 + 1)

2)

2; if𝜏 ̸= (2 ±

√2)/𝛿𝛽, then 𝑑

2̸= 0.

Hence, we have the following theorem.

Theorem 2. Let (5), (9), and 𝜏 ̸= (2 ± √2)/𝛿𝛽 hold. Then theequilibrium 𝑃

∗of system (4) is a Bogdanov-Takens singularity.

In the following, we will do versal unfolding of system (4)at 𝑃∗:

�̇� = 𝜏 ((𝑥 + 𝑚) (1 − 𝑥 − 𝑚) −

𝑥𝑦

𝑎𝑦 + 𝑥

− ℎ) ,

̇𝑦 = 𝜏 (𝛿 + 𝜇

1) 𝑦 (𝛽 + 𝜇

2−

𝑦 (𝑡 − 1)

𝑥 (𝑡 − 1)

) .

(16)

When 𝜇1= 𝜇

2= 0, system (16) has a Bogdanov-Takens

singularity 𝑃∗and a two-dimensional center manifold exists.

The Taylor expansion of system (16) at 𝑃∗takes the form

�̇� = 𝜏𝛿𝛽𝑥 − 𝜏𝛿𝑦 + 𝑔

1𝑥

2+ 𝑔

2𝑥𝑦 + 𝑔

3𝑦

2+ h.o.t,

̇𝑦 = 𝑤

0𝜇

2+ 𝜏 (𝛿 + 𝜇

1) 𝛽

2𝑥 (𝑡 − 1) − 𝜏 (𝛿 + 𝜇

1) 𝛽𝑦 (𝑡 − 1)

+ 𝜏𝛿𝜇

2𝑦 + 𝑤

4𝑥

2(𝑡 − 1) + 𝑤

5𝑥 (𝑡 − 1) 𝑦 (𝑡 − 1)

+ 𝑤

6𝑥 (𝑡 − 1) 𝑦 + 𝑤

7𝑦 (𝑡 − 1) 𝑦 + h.o.t,

(17)

where

𝑤

0= −

𝜏𝛿𝛽 [𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)]

2 (𝑎𝛽 + 1)

,

𝑤

4=

2𝜏 (𝛿 + 𝜇

1) 𝛽

2(𝑎𝛽 + 1)

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

,

𝑤

5= −

2𝜏 (𝛿 + 𝜇

1) 𝛽 (𝑎𝛽 + 1)

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

,

𝑤

6= −

2𝜏 (𝛿 + 𝜇

1) (𝑎𝛽 + 1)

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

,

𝑤

7=

2𝜏 (𝛿 + 𝜇

1) (𝑎𝛽 + 1)

𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)

.

(18)

4 Abstract and Applied Analysis

We decompose the enlarged phase space BC of system asBC = 𝑃⊕Ker𝜋. Then 𝑦 in system can be decomposed as 𝑦 =Φ𝑧+𝑢with 𝑧 ∈ 𝑅2 and 𝑢 ∈ 𝑄. Hence, system is decomposedas

�̇� = 𝐵

1+ 𝐵

2𝑧 + Ψ (0) 𝐺 (Φ𝑧 + 𝑢) ,

�̇� = 𝐴

𝑄𝑢 + (𝐼 − 𝜋)𝑋

0

× [𝐵

0+ 𝐵

2(Φ

0𝑧 + 𝑢 (0)) + 𝐺 (Φ𝑧 + 𝑢)] ,

(19)

where

𝑋

0(𝜃) =

{

{

{

𝐼, 𝜃 = 0

0, −1 ≤ 𝜃 < 0,

𝐵

0= (

0

𝑤

0𝜇

2

) , 𝐵

1= Ψ (0) 𝐵

0= (

𝑤

0𝑚

2𝜇

2

𝑤

0𝑛𝜇

2

) ,

𝐵

2= Ψ (0)(

𝜏𝛿𝛽𝜙

1(0) − 𝜏𝛿𝜙

2(0)

𝜏 (𝛿 + 𝜇

1) 𝛽

2𝜙

1(−1) − 𝜏 (𝛿 + 𝜇

1) 𝛽𝜙

2(−1) + 𝜏𝛿𝜇

2𝜙

2(0)

) ,

𝐺 (𝜙) = (

𝑔

1𝜙

2

1(0) + 𝑔

2𝜙

1(0) 𝜙

2(0) + 𝑔

3𝜙

2

2(0) + h.o.t

𝑤

4𝜙

2

1(−1) + 𝑤

5𝜙

1(−1) 𝜙

2(−1) + 𝑤

6𝜙

1(−1) 𝜙

2(0) + 𝑤

7𝜙

2(−1) 𝜙

2(0) + h.o.t

) .

(20)

To compute the normal form of system at 𝑃∗, consider �̇� =

𝐵

1+ 𝐵

2𝑧 + Ψ(0)𝐺(Φ𝑧); together with (13) we obtain

�̇�

1= 𝑤

0𝑚

2𝜇

2+ 𝑧

2+ 𝑚

2𝜏𝛿𝛽𝜇

2𝑧

1+

𝑚

2𝛽𝜇

1

𝛿

𝑧

2

− 𝜏𝑚

1𝑧

2

1−

2𝑚

1

𝛿𝛽

𝑧

1𝑧

2+ 𝑙

1𝑧

2

2+ h.o.t,

�̇�

2= 𝑤

0𝑛𝜇

2+ 𝑛𝜏𝛿𝛽𝜇

2𝑧

1+

𝑛𝛽𝜇

1

𝛿

𝑧

2

+ 𝑛𝜏𝛽𝑧

2

1+

2𝑛

𝛿

𝑧

1𝑧

2+ 𝑙

2𝑧

2

2+ h.o.t.

(21)

Following the normal form formula in Kuznetsov [12],system (21) can be reduced to

�̇�

1= 𝑧

2+ h.o.t,

�̇�

2= 𝛾

1+ 𝛾

2𝑧

1+ 𝑑

1𝑧

2

1+ 𝑑

2𝑧

1𝑧

2+ h.o.t,

(22)

where

𝛾

1= −

𝜏𝛿𝛽 [𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)]

2 (𝑎𝛽 + 1)

𝑛𝜇

2

=

𝜏

2𝛽 [𝛽 + (2𝑚 − 1) (𝑎𝛽 + 1)]

2(𝑎𝛽 + 1)

3

[(𝑎𝛽 + 1)

2

− 𝜏𝛽]

𝜇

2,

𝛾

2=

𝛽

2𝑛

2𝜏

𝑛 − 𝜏𝑚

1𝛿

𝜇

1

+ (𝑛𝜏𝛿𝛽 −

2𝑤

0𝑛

𝛿𝛽

−

𝑤

0𝑛 (𝑙

2𝛿𝛽 − 2 + 2 𝑚

2𝛽)

𝛿𝛽

+ (𝑛𝜏 (𝑚

2𝜏 𝛿

2𝛽

2+ 2𝑙

1𝑤

0𝑛𝛿𝛽 + 2𝑤

0𝑚

2

− 2𝑤

0𝑚

2

2𝛽 − 2𝑤

0𝑚

2𝑙

2𝛿𝛽))

× (−𝜏𝛿 + 𝑚

2𝜏𝛿𝛽 + 𝑛)

−1

)𝜇

2.

(23)

Hence system (4) exist the following bifurcation curves ina small neighborhood of the origin in the (𝜇

1, 𝜇

2) plane.

Theorem3. Let (5), (9), and 𝜏 ̸= (2±√2)/𝛿𝛽 hold.Then system(4) admits the following bifurcations:

(i) a saddle-node bifurcation curve 𝑆𝑁 = {(𝜇1, 𝜇

2) : 𝛾

1=

(1/4𝑑

1)𝛾

2

2};

(ii) a Hopf bifurcation curve 𝐻 = {(𝜇1, 𝜇

2) : 𝜇

2= 0, 𝛾

2<

0};

(iii) a homoclinic bifurcation curve 𝐻𝐿 = {(𝜇1, 𝜇

2) : 𝛾

1=

−(6/25𝑑

1)𝛾

2

2, 𝛾

2< 0}.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Abstract and Applied Analysis 5

Acknowledgments

This paper is supported by NSFC (11226142), Foundation ofHenan Educational Committee (2012A110012), and Founda-tion of Henan Normal University (2011QK04, 2012PL03, and1001).

References

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[8] J.-F. Zhang, W.-T. Li, and X.-P. Yan, “Multiple bifurcations ina delayed predator-prey diffusion system with a functionalresponse,” Nonlinear Analysis. Real World Applications, vol. 11,no. 4, pp. 2708–2725, 2010.

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Research Article Bogdanov-Takens Bifurcation of a Delayed ...Bogdanov-Takens Bifurcation of a Delayed Ratio-Dependent Holling-Tanner Predator Prey System XiaLiu, 1 YanweiLiu, 2 andJinlingWang