Abstract and Applied Analysis

Volume 2013 (2013), Article ID 605471, 8 pages

http://dx.doi.org/10.1155/2013/605471

Research Article

Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

Xia Liu and Jinling Wang

College of Mathematics and Information Science, Henan Normal University, 453007, China

Received 20 July 2013; Revised 4 September 2013; Accepted 4 September 2013

Academic Editor: Yanni Xiao

Copyright © 2013 Xia Liu and Jinling Wang. 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.

Abstract.

A delayed modified Leslie-Gower predator prey system with nonlinear harvesting is considered. The existence conditions that an equilibrium is Bogdanov-Takens (BT) or triple zero singularity of the system are given. By using the center manifold reduction, the normal form theory, and the formulae developed by Xu and Huang, 2008 and Qiao et al., 2010, the normal forms and the versal unfoldings for this singularity are presented. The Hopf bifurcation of the system at another interior equilibrium is analyzed by taking delay (small or large) as bifurcation parameter.

1. Introduction

For a more detailed study on the properties of the predator prey systems, the multiple bifurcations for some systems (ODE) with more interior equilibria are investigated by many authors, see [1–4] for example. To deal with this type of systems, the more difficult problem is how to obtain the normal form of the system at its degenerate equilibrium, that is, BT bifurcation.

When introducing the time delay into this type of systems, using the methods developed by [5], the authors in [6, 7] have researched the BT bifurcation of some predator prey systems (DDE). Their results show that time delay may have an effect or not on the BT bifurcation.

Recently, papers [8–10] have considered the triple zero bifurcation of some delay differential equations, depending on the parameters in the original system; some interesting bifurcation results are obtained. But we find that there are few results about the triple zero bifurcation for predator prey systems.

Summarizing the above references, we will consider the following predator prey system with Michaelis-Menten type (nonlinear) prey harvesting:

where and denote the prey and predator populations, respectively. represents the negative feedback of the predator’s density. For complete reason, we give the biological meaning of the parameters, one can be seeing them in [11]. and are intrinsic growth rate and environmental carrying capacity for the prey, respectively. is the maximum value of the per capita reduction rate of prey, measures the extent to which the environment provides protection to prey and predator respectively, measures the growth rate of the predator species, and is the maximum value of the per capita reduction rate of predator. represents Michaelis-Menten type harvesting, is the catchability coefficient, is the effort applied to harvest the prey species, and and are suitable constants.

The authors in [12, 13] have studied system (1) without prey harvesting , respectively, in [12], the global stability and persistence of the system are investigated. In [13], by using the Hopf bifurcation theorem and taking the delay as a parameter of bifurcation for small and large cases, the existence of the bifurcated limit cycle around a boundary equilibrium or an interior equilibrium is mainly considered.

For system (1) with , the authors in [11] have given detailed analysis about the existence of the multiple bifurcations (including BT bifurcation) depending on the parameters of the system.

For computation simplicity, we first rescale system (1).

Let , , , and  ; then dropping the bars we obtain

where , , , , , and .

System (2) with initial conditions is

where and are all continuous bounded functions in the interval .

From [11], we know that (2) has interior equilibrium if

where , and interior equilibrium if , where , , and .

In this paper, for system (2), we will mainly consider the BT and triple zero bifurcations at and the Hopf bifurcation at . It is easy to see that this system is with six parameters which will let our work become more challenging. When dealing with the BT and triple bifurcations of the delay systems, the core problem is to change delay systems as ordinary differential systems (ODEs).

The concrete organization of the paper is as follows: in Section 2, we will give the conditions under which the equilibrium is a BT singularity, and a universal unfolding will be exhibited; in Section 3, when is a triple zero, the universal unfolding will be presented, and in Section 4, some Hopf bifurcation results at will be obtained.

2. Bogdanov-Takens Bifurcation

System (2) also can be written as

Linearizing system (5) at yields the following linear system

the corresponding characteristic equation is

Evidently, is a double zero eigenvalue if

is a triple zero eigenvalue if

It is easy to prove that in the above two cases the rest eigenvalues all have negative real parts.

Under the conditions (4) and (8), let , and let ; then the Taylor expansion system (5) at is

where ,  , , , .

In the following, we first give the normal form of the system (10) at the singularity . Reference [6], we first rewrite system (10) as , here ,  , and . By the normal form theory developed by Faria and Magalhaes [5], one can obtain the center manifold of this system at the origin which is two-dimensional and system can be reduced to an ODE in the plane.

Define to be the infinitesimal generator of system. Consider and let denote the invariant space of associated with the eigenvalue , using the formal adjoint theory in [5], the phase space can be decomposed by as . Let and be the bases for and , respectively, and be let them normalized such that , ,and , where and are matrices, where .

Next, we will find the and based on the techniques developed by [14].

Lemma 1 (see Xu and Huang [14]). The bases of and their dual space have the following representations:

where , , and , , , which satisfy

(1), (2), (3), (4), (5), (6).

By system (6), we know that

By Lemma 1, we have

where , ,

therefore,

Taking a similar theory in [6], system (6) in the center manifold is

where

Using (10) and (15), system (16) also can be written as

where , , . Through a series of transformation, system (18) becomes

where and . If , we have the following theorem.

Theorem 2. Let (4), (8), and hold. Then, the equilibrium of system (5) is a BT singularity.

Next, we are interested in giving a versal unfolding for system (5) at BT singularity. Choosing and as bifurcation parameters and incorporating and to system (5), where and vary in a small neighborhood of , we obtain

Let and . Then, system (20) becomes

Then, system (21) can be decomposed as

where

Then, the normal form of system (21) at is ; that is,

where

Following the normal form formula in Kuznetsov [15], system (24) can be reduced to

where

Then, system (5) exists in the following bifurcation curves in a small neighborhood of the origin in the plane.

Theorem 3. Let (4), (8), and hold. System (5) admits the following bifurcations:

(i)a saddle-node bifurcation curve ;(ii)a Hopf bifurcation curve ;(iii)a homoclinic bifurcation curve .

3. Triple-Zero Bifurcation

From Section 2, we know that under the conditions (4) and (9) the equilibrium of system (5) is a triple zero singularity. In the following reference, from the work of [8, 9] we will give the triple zero bifurcation at .

Let and ; then, system (5) becomes

To determine a versal unfolding for the original system (28) at , we choose , , and as bifurcation parameters, and let them become , and , respectively, where , and vary in a small neighborhood of ; then (28) can be written as

Next, we need to find the expressions of and based on the techniques developed by [9].

Lemma 4 (see Qiao et al. [9]). The bases of and their dual space have the following representations:

where , , , , and , , , , which satisfy

(1), (2), (3), (4), (5), (6), (7), (8), (9)   + + .

For system (29), one can see that

By Lemma 4, we can obtain

such that , , and , where is given by .

Using the similar methods as used in [9], system (29) can be rewritten as

where is a parameter vector, . Let

then, we can expand , , + + + + + + + + + + + + , where

Following the formula of Theorem 3.1 in [9], the normal form with versal unfolding of system (29) on the center manifold takes the following form:

where

Referring [8], we know that if

that is,

for the unfolding normal form (36) then there exist the following results.

Theorem 5. Let (4), (9), and (39) hold. For the parameters , and are sufficiently small,

(i)system (36) undergoes a transcritical bifurcation at the origin on the curve

(ii)system (36) undergoes a Hopf bifurcation at the origin on the curve

(iii)system (36) undergoes a Hopf-bifurcation at the nontrivial equilibrium point on the curve

(iv)system (36) undergoes a Bogdanov-Takens bifurcation at the origin on the curve

(v)system (36) undergoes a zero-Hopf bifurcation at the origin on the curve

4. Hopf Bifurcation

The Jacobian matrix of system (2) at takes the following form:

then, the characteristic equation is

When , by Routh-Hurwitz criterion, all roots of (46) have negative real part if

Using the formula in Theorem 2.4 of [13] or Lemma 2.2 of [16], we have the following theorem.

Theorem 6. Let (47) hold. Then, there exist

such that the positive equilibrium is asymptotically stable for , moreover, system (2) undergoes Hopf bifurcation at for , where and .

In the following, using the Hopf bifurcation theorem for a retarded differential system introduced by [17], the Hopf bifurcations at for small delay and large delay are presented.

Using the same methods as the ones used in [13], for small delay, let , together with (46), the following bifurcation results can be obtained.

Theorem 7. Let , , and . Then, there exists such that for each , system (2) near has a family or periodic solutions with period for such that , , and .

For large delay, by [13] we have the following results.

Theorem 8. Let (47) hold. Then there exists such that for each system (2) near has a family or periodic solutions with period for such that , and .

Remark. Because the proofs of Theorems 7 and 8 are the same as the proofs of Theorems 3.4 and 3.5 in [13], we omit them here.

Conflict of Interests

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

Acknowledgments

This paper is supported by NSFC (11226142), the Foundation of Henan Educational Committee (2012A110012), the Foundation of Henan Normal University (2011QK04, 2012PL03), and the Scientific Research Foundation for Ph.D. of Henan Normal University (no. 1001).

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