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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 MagalhaÌes. 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 [1â5] for example.However, when the predator prey systems with delay andBogdanov-Takens bifurcation are researched relative few (see[6â8] and the reference therein) using similar methods as [6â8], the authors of [9â11] 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:
ï¿œÌï¿œ = ðð¥ (1 â
ð¥
ðŸ
) â
ðŒðŠ (ð¥ â ð)
ðŽðŠ + ð¥ â ð
â â,
ÌðŠ = ð ðŠ(1 â
ððŠ (ð¡ â ð)
ð¥ (ð¡ â ð) â ð
) ,
(1)
where ð¥ and ðŠ stand for prey and predator population (ordensities) at time ð¡, respectively. The predator growth is oflogistic type with growth rate ð and carrying capacity ðŸ inthe absence of predation; ðŒ and ðŽ stand for the predatorcapturing rate and half saturation constant, respectively;
ð is the intrinsic growth rate of predator; however, carryingcapacity ð¥/ð (ð is the conversion rate of prey into predators)is the function on the population size of prey.The parametersðŒ,ðŽ,ð, â, ð , ð, and ð are all positive constants.ð is a constantnumber of prey using refuges; â is the rate of prey harvesting.
For simplicity, we first rescale system (1). Let ð = ð¥ â ð,ð = ðŠ; system (1) can be written as (still denoting ð, ð asð¥, ðŠ)
ï¿œÌï¿œ = ð (ð¥ + ð)(1 â
ð¥ + ð
ðŸ
) â
ðŒð¥ðŠ
ðŽðŠ + ð¥
â â,
ÌðŠ = ð ðŠ(1 â
ððŠ (ð¡ â ð)
ð¥ (ð¡ â ð)
) .
(2)
Next, let ð¡ = ðð¡, ð(ð¡) = ð¥(ð¡)/ðŸ, and ð(ð¡) = ðŒðŠ(ð¡)/ððŸ;then system (2) takes the form (still denoting ð, ð, ð¡ as ð¥, ðŠ,ð¡)
ï¿œÌï¿œ = (ð¥ + ð) (1 â ð¥ â ð) â
ð¥ðŠ
ððŠ + ð¥
â â,
ÌðŠ = ð¿ðŠ(ðœ â
ðŠ (ð¡ â ð)
ð¥ (ð¡ â ð)
) ,
(3)
whereð = ð/ðŸ, ð = ðŽð/ðŒ, ð¿ = ð â/ðŒ, ðœ = ðŒ/ðð, â = â/ð, andð = ðð.
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2 Abstract and Applied Analysis
When ð = 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
ï¿œÌï¿œ = ð ((ð¥ + ð) (1 â ð¥ â ð) â
ð¥ðŠ
ððŠ + ð¥
â â) ,
ÌðŠ = ðð¿ðŠ(ðœ â
ðŠ (ð¡ â 1)
ð¥ (ð¡ â 1)
) .
(4)
Let
0 < ð <
1
2
(1 â
ðœ
ððœ + 1
) ,
â =
Ì
â =
1
4
(
ðœ
ððœ + 1
â 1)
2
+
ððœ
ððœ + 1
.
(5)
Then ðâ= (ð¥
â, ðŠ
â) is an interior positive equilibrium of
systems (3) and (4), where ð¥â= â(1/2)((ðœ/(ððœ+1))+2ðâ1),
ðŠ
â= ðœð¥
â.
In order to discuss the properties of system (4) in theneighborhood of the equilibriumð
â= (ð¥
â, ðŠ
â), letð¥ = ð¥âð¥
â,
ðŠ = ðŠ â ðŠ
â; then ð
âis translated to (0, 0), and system (4)
becomes (still denoting ð¥, ðŠ as ð¥, ðŠ)
ï¿œÌï¿œ =
ððœ
(ððœ + 1)
2ð¥ â
ð
(ððœ + 1)
2ðŠ + ð
1ð¥
2
+ ð
2ð¥ðŠ + ð
3ðŠ
2+ h.o.t,
ÌðŠ = ðð¿ðœ
2ð¥ (ð¡ â 1) â ðð¿ðœðŠ (ð¡ â 1) + ð
4ð¥
2(ð¡ â 1)
+ ð
5ð¥ (ð¡ â 1) ðŠ (ð¡ â 1) + ð
6ð¥ (ð¡ â 1) ðŠ
+ ð
7ðŠ (ð¡ â 1) ðŠ + h.o.t,
(6)
where h.o.t denotes the higher order terms and
ð
1= â(
2ðð¿ðœ
2ð
ðœ + (2ð â 1) (ððœ + 1)
+ ð) ,
ð
2=
4ðð¿ðœð
ðœ + (2ð â 1) (ððœ + 1)
,
ð
3=
â2ðð¿ð
ðœ + (2ð â 1) (ððœ + 1)
,
ð
4=
2ðð¿ðœ
2(ððœ + 1)
ðœ + (2ð â 1) (ððœ + 1)
,
ð
5=
â2ðð¿ðœ (ððœ + 1)
ðœ + (2ð â 1) (ððœ + 1)
,
ð
6= â
2ðð¿ðœ (ððœ + 1)
ðœ + (2ð â 1) (ððœ + 1)
,
ð
7=
2ðð¿ (ððœ + 1)
ðœ + (2ð â 1) (ððœ + 1)
.
(7)
The characteristic equation of the linearized part ofsystem (6) is
ð¹ (ð) = ð
2+ [ðð¿ðœð
âðâ
ððœ
(ððœ + 1)
2]ð. (8)
Clearly, if
ð¿ =
1
(ððœ + 1)
2,
ð Ìž=
1
ð¿ðœ
, that is, ð Ìž=(ððœ + 1)
2
ðœ
;
(9)
then ð = 0 is double zero eigenvalue; if ð¿ = 1/(ððœ + 1)2 andð = 1/ð¿ðœ, that is, ð = (ððœ + 1)2/ðœ, then ð = 0 is triple zeroeigenvalue. We will mainly discuss the first case in this paper.
According to the normal form theory developed by Fariaand MagalhaÌes [13], first, rewrite system (4) as Ìð(ð¡) = ð¿(ð
ð¡),
where ð(ð¡) = (ð¥1(ð¡), ð¥
2(ð¡)), ð¿(ð) = ð¿ ( ð1(â1)
ð2(0)
), and ð =(ð
1, ð
2). Take ðŽ
0as the infinitesimal generator of system. Let
Î = {0}, and denote by ð the invariant space ofðŽ0associated
with the eigenvalue ð = 0; using the formal adjoint theoryof RFDE in [13], the phase space ð¶
1can be decomposed by
Î as ð¶1= ð â ð. Define Ί and Κ as the bases for ð and
ð
â, the space associated with the eigenvalue ð = 0 of theadjoint equation, respectively, and to be normalized such that(Κ,Ί) = ðŒ, ÌΊ = Ίðœ, and ÌΚ = âðœÎš, where Ί and Κ are 2 à 2matrices.
Next we will find the Ί(ð) and Κ(ð ) based on the tech-niques developed by [14].
Lemma 1 (see Xu andHuang [14]). Thebases of ð and its dualspace ðâ have the following representations:
ð = spanΊ,Ί (ð) = (ð1(ð) , ð
2(ð)) , â1 †ð †0,
ð
â= spanΚ,Κ (ð ) = col (ð
1(ð ) , ð
2(ð )) , 0 †ð †1,
(10)
where ð1(ð) = ð
0
1â ð
ð\ {0}, ð
2(ð) = ð
0
2+ ð
0
1ð, and ð0
2â ð
ð
and ð2(ð ) = ð
0
2â ð
ðâ\ {0}, and ð
1(ð ) = ð
0
1â ð ð
0
2, ð
0
1â ð
ðâ,which satisfy
(1) (ðŽ + ðµ)ð01= 0,
(2) (ðŽ + ðµ)ð02= (ðµ + ðŒ)ð
0
1,
(3) ð02(ðŽ + ðµ) = 0,
(4) ð01(ðŽ + ðµ) = ð
0
2(ðµ + ðŒ),
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Abstract and Applied Analysis 3
(5) ð02ð
0
2â (1/2)ð
0
2ðµð
0
1+ ð
0
2ðµð
0
2= 1,
(6) ð01ð
0
2â (1/2)ð
0
1ðµð
0
1+ ð
0
1ðµð
0
2+ (1/6)ð
0
2ðµð
0
1â
(1/2)ð
0
2ðµð
0
2= 0.
By (6), we have ðŽ = ( ðð¿ðœ âðð¿0 0
) and ðµ = ( 0 0ðð¿ðœ2âðð¿ðœ
); usingLemma 1, we obtain
Ί (ð) = (
1
1
ðð¿ðœ
+ ð
ðœ ðœð
) , â1 †ð †0,
Κ (ð ) = (
ð
1+ ð ðœð ð
2â ð ð
âðœð ð
) , 0 †ð †1,
(11)
where ð1= ðð¿ðœ(ðð¿ðœ â 2)/2(ðð¿ðœ â 1)
2= ððœ[ððœ â 2(ððœ +
1)
2]/2[ððœâ (ððœ+ 1)
2]
2,ð2= (1 + (ðð¿ðœâ 1)
2)/2ðœ(ðð¿ðœâ 1)
2=
(1/2ðœ)(1+(ððœ+1)
4/(ððœâ(ððœ+1))
2), and ð = ðð¿/(ðð¿ðœâ1) =
ð/(ððœ â (ððœ + 1)
2).
The matrix ðœ satisfying ÌΊ = Ίðœ is given by ðœ = ( 0 10 0
).System (6) in the center manifold is equivalent to the systemï¿œÌï¿œ = ðœð§ + Κ(0)ð¹(Ίð§), where
ð¹ (ð) = (
ð
1ð
2
1(0) + ð
2ð
1(0) ð
2(0) + ð
3ð
2
2(0) + h.o.t
ð
4ð
2
1(â1) + ð
5ð
1(â1) ð
2(â1) + ð
6ð
1(â1) ð
2(0) + ð
7ð
2(â1) ð
2(0) + h.o.t
) . (12)
It is easy to obtain
Ί (ð) ð§ = (
ð§
1+ (
1
ðð¿ðœ
+ ð) ð§
2
ðœ (ð§
1+ ð¥
2ð)
) ,
ð
1(0) = ð§
1+
1
ðð¿ðœ
ð§
2,
ð
1(â1) = ð§
1+ (
1
ðð¿ðœ
â 1) ð§
2,
ð
2(0) = ðœð§
1, ð
2(â1) = ðœ (ð§
1â ð§
2) .
(13)
Hence (6) becomes
ï¿œÌï¿œ
1= ð§
2â ðð
1ð§
2
1â
2ð
1
ð¿ðœ
ð§
1ð§
2+ ð
1ð§
2
2+ h.o.t,
ï¿œÌï¿œ
2= ðððœð§
2
1+
2ð
ð¿
ð§
1ð§
2+ ð
2ð§
2
2+ h.o.t,
(14)
where ð1= ð
1ð
1/ð
2ð¿
2ðœ
2+2ð
2(ð¿+ð
1)(ððœ+1)(1âðð¿ðœ)/ðð¿
2(ðœ+
(ððœ+1)(2ðâ1)), ð2= â(ðð
1/ðœð
2ð¿
2) + 2ð(ð¿+ð
1)(ððœ+1)(1â
ðð¿ðœ)/ðð¿
2(ðœ + (ððœ + 1)(2ð â 1)).
After a series of transformations we obtain
ï¿œÌï¿œ
1= ð§
2+ h.o.t,
ï¿œÌï¿œ
2= ð
1ð§
2
1+ ð
2ð§
1ð§
2+ h.o.t,
(15)
where ð1= ðœðð = ðœð
2/(ððœ â (ððœ + 1)
2), ð2= âðð
1+ ð/ð¿ =
â(ð
2ðœ
2+ 2(ððœ + 1)
2[(ððœ + 1)
2â 2ððœ])/(ððœ â (ððœ + 1)
2)
2; ifð Ìž= (2 ±
â2)/ð¿ðœ, then ð
2Ìž= 0.
Hence, we have the following theorem.
Theorem 2. Let (5), (9), and ð Ìž= (2 ± â2)/ð¿ðœ hold. Then theequilibrium ð
âof system (4) is a Bogdanov-Takens singularity.
In the following, we will do versal unfolding of system (4)at ðâ:
ï¿œÌï¿œ = ð ((ð¥ + ð) (1 â ð¥ â ð) â
ð¥ðŠ
ððŠ + ð¥
â â) ,
ÌðŠ = ð (ð¿ + ð
1) ðŠ (ðœ + ð
2â
ðŠ (ð¡ â 1)
ð¥ (ð¡ â 1)
) .
(16)
When ð1= ð
2= 0, system (16) has a Bogdanov-Takens
singularity ðâand a two-dimensional center manifold exists.
The Taylor expansion of system (16) at ðâtakes the form
ï¿œÌï¿œ = ðð¿ðœð¥ â ðð¿ðŠ + ð
1ð¥
2+ ð
2ð¥ðŠ + ð
3ðŠ
2+ h.o.t,
ÌðŠ = ð€
0ð
2+ ð (ð¿ + ð
1) ðœ
2ð¥ (ð¡ â 1) â ð (ð¿ + ð
1) ðœðŠ (ð¡ â 1)
+ ðð¿ð
2ðŠ + ð€
4ð¥
2(ð¡ â 1) + ð€
5ð¥ (ð¡ â 1) ðŠ (ð¡ â 1)
+ ð€
6ð¥ (ð¡ â 1) ðŠ + ð€
7ðŠ (ð¡ â 1) ðŠ + h.o.t,
(17)
where
ð€
0= â
ðð¿ðœ [ðœ + (2ð â 1) (ððœ + 1)]
2 (ððœ + 1)
,
ð€
4=
2ð (ð¿ + ð
1) ðœ
2(ððœ + 1)
ðœ + (2ð â 1) (ððœ + 1)
,
ð€
5= â
2ð (ð¿ + ð
1) ðœ (ððœ + 1)
ðœ + (2ð â 1) (ððœ + 1)
,
ð€
6= â
2ð (ð¿ + ð
1) (ððœ + 1)
ðœ + (2ð â 1) (ððœ + 1)
,
ð€
7=
2ð (ð¿ + ð
1) (ððœ + 1)
ðœ + (2ð â 1) (ððœ + 1)
.
(18)
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4 Abstract and Applied Analysis
We decompose the enlarged phase space BC of system asBC = ðâKerð. Then ðŠ in system can be decomposed as ðŠ =Ίð§+ð¢with ð§ â ð 2 and ð¢ â ð. Hence, system is decomposedas
ï¿œÌï¿œ = ðµ
1+ ðµ
2ð§ + Κ (0) ðº (Ίð§ + ð¢) ,
ï¿œÌï¿œ = ðŽ
ðð¢ + (ðŒ â ð)ð
0
à [ðµ
0+ ðµ
2(Ί
0ð§ + ð¢ (0)) + ðº (Ίð§ + ð¢)] ,
(19)
where
ð
0(ð) =
{
{
{
ðŒ, ð = 0
0, â1 †ð < 0,
ðµ
0= (
0
ð€
0ð
2
) , ðµ
1= Κ (0) ðµ
0= (
ð€
0ð
2ð
2
ð€
0ðð
2
) ,
ðµ
2= Κ (0)(
ðð¿ðœð
1(0) â ðð¿ð
2(0)
ð (ð¿ + ð
1) ðœ
2ð
1(â1) â ð (ð¿ + ð
1) ðœð
2(â1) + ðð¿ð
2ð
2(0)
) ,
ðº (ð) = (
ð
1ð
2
1(0) + ð
2ð
1(0) ð
2(0) + ð
3ð
2
2(0) + h.o.t
ð€
4ð
2
1(â1) + ð€
5ð
1(â1) ð
2(â1) + ð€
6ð
1(â1) ð
2(0) + ð€
7ð
2(â1) ð
2(0) + h.o.t
) .
(20)
To compute the normal form of system at ðâ, consider ï¿œÌï¿œ =
ðµ
1+ ðµ
2ð§ + Κ(0)ðº(Ίð§); together with (13) we obtain
ï¿œÌï¿œ
1= ð€
0ð
2ð
2+ ð§
2+ ð
2ðð¿ðœð
2ð§
1+
ð
2ðœð
1
ð¿
ð§
2
â ðð
1ð§
2
1â
2ð
1
ð¿ðœ
ð§
1ð§
2+ ð
1ð§
2
2+ h.o.t,
ï¿œÌï¿œ
2= ð€
0ðð
2+ ððð¿ðœð
2ð§
1+
ððœð
1
ð¿
ð§
2
+ ðððœð§
2
1+
2ð
ð¿
ð§
1ð§
2+ ð
2ð§
2
2+ h.o.t.
(21)
Following the normal form formula in Kuznetsov [12],system (21) can be reduced to
ï¿œÌï¿œ
1= ð§
2+ h.o.t,
ï¿œÌï¿œ
2= ðŸ
1+ ðŸ
2ð§
1+ ð
1ð§
2
1+ ð
2ð§
1ð§
2+ h.o.t,
(22)
where
ðŸ
1= â
ðð¿ðœ [ðœ + (2ð â 1) (ððœ + 1)]
2 (ððœ + 1)
ðð
2
=
ð
2ðœ [ðœ + (2ð â 1) (ððœ + 1)]
2(ððœ + 1)
3
[(ððœ + 1)
2
â ððœ]
ð
2,
ðŸ
2=
ðœ
2ð
2ð
ð â ðð
1ð¿
ð
1
+ (ððð¿ðœ â
2ð€
0ð
ð¿ðœ
â
ð€
0ð (ð
2ð¿ðœ â 2 + 2 ð
2ðœ)
ð¿ðœ
+ (ðð (ð
2ð ð¿
2ðœ
2+ 2ð
1ð€
0ðð¿ðœ + 2ð€
0ð
2
â 2ð€
0ð
2
2ðœ â 2ð€
0ð
2ð
2ð¿ðœ))
à (âðð¿ + ð
2ðð¿ðœ + ð)
â1
)ð
2.
(23)
Hence system (4) exist the following bifurcation curves ina small neighborhood of the origin in the (ð
1, ð
2) plane.
Theorem3. Let (5), (9), and ð Ìž= (2±â2)/ð¿ðœ hold.Then system(4) admits the following bifurcations:
(i) a saddle-node bifurcation curve ðð = {(ð1, ð
2) : ðŸ
1=
(1/4ð
1)ðŸ
2
2};
(ii) a Hopf bifurcation curve ð» = {(ð1, ð
2) : ð
2= 0, ðŸ
2<
0};
(iii) a homoclinic bifurcation curve ð»ð¿ = {(ð1, ð
2) : ðŸ
1=
â(6/25ð
1)ðŸ
2
2, ðŸ
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).
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