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

1. Introduction

It is well known that the multiple bifurcations will occur when a predator prey system (ODE) with more interior positive equilibria, such as Bogdanov-Takens bifurcation, Hopf bifurcation, and backward bifurcation; see [15] for example. However, when the predator prey systems with delay and Bogdanov-Takens bifurcation are researched relative few (see [68] and the reference therein) using similar methods as [68], the authors of [911] consider the Bogdanov-Takens bifurcation of some delayed single inertial neuron or oscillator models.

Motivated by the works of [5, 6], we mainly consider the Bogdanov-Takens bifurcation of the following system: where and stand for prey and predator population (or densities) at time , respectively. The predator growth is of logistic type with growth rate and carrying capacity in the absence of predation; and stand for the predator capturing rate and half saturation constant, respectively; is the intrinsic growth rate of predator; however, carrying capacity ( is the conversion rate of prey into predators) is the function on the population size of prey. The parameters , , , , , , and are all positive constants. is a constant number of prey using refuges; is the rate of prey harvesting.

For simplicity, we first rescale system (1). Let , ; system (1) can be written as (still denoting , as , )

Next, let , , and ; then system (2) takes the form (still denoting , , as , , ) where , , , , , and .

When , we have known that for some parameter values system (3) exhibits Bogdanov-Takens bifurcation (see [5]). Summarizing the methods used by [6] and the formulae in [12], the sufficient conditions which depend on delay to guarantee that system (3) has a Bogdanov-Takens singularity will be given. Therefore, the delay has effect on the occurrence of Bogdanov-Takens bifurcation.

In the next section we will compute the normal form and give the versal unfolding of system (3) at the degenerate equilibrium.

2. Bogdanov-Takens Bifurcation

System (3) can also be written as


Then is an interior positive equilibrium of systems (3) and (4), where , .

In order to discuss the properties of system (4) in the neighborhood of the equilibrium , let , ; then is translated to , and system (4) becomes (still denoting , as , ) where denotes the higher order terms and

The characteristic equation of the linearized part of system (6) is Clearly, if then is double zero eigenvalue; if and , that is, , then is triple zero eigenvalue. We will mainly discuss the first case in this paper.

According to the normal form theory developed by Faria and Magalhães [13], first, rewrite system (4) as , where , , and . Take as the infinitesimal generator of system. Let , and denote by the invariant space of associated with the eigenvalue ; using the formal adjoint theory of RFDE in [13], the phase space can be decomposed by as . Define and as the bases for and , the space associated with the eigenvalue of the adjoint equation, respectively, and to be normalized such that , , and , where and are matrices.

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

Lemma 1 (see Xu and Huang [14]). The bases of and its dual space have the following representations: where , , and and , and , , which satisfy(1),(2),(3),(4),(5),(6).

By (6), we have and ; using Lemma 1, we obtain where , , and .

The matrix satisfying is given by . System (6) in the center manifold is equivalent to the system , whereIt is easy to obtain Hence (6) becomes where , .

After a series of transformations we obtain where , ; if , then .

Hence, we have the following theorem.

Theorem 2. Let (5), (9), and hold. Then the equilibrium of system (4) is a Bogdanov-Takens singularity.

In the following, we will do versal unfolding of system (4) at : When , system (16) has a Bogdanov-Takens singularity and a two-dimensional center manifold exists.

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

We decompose the enlarged phase space of system as . Then in system can be decomposed as with and . Hence, system is decomposed as whereTo compute the normal form of system at , consider ; together with (13) we obtain

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

Hence system (4) exist the following bifurcation curves in a small neighborhood of the origin in the plane.

Theorem 3. Let (5), (9), and hold. Then system (4) admits the following bifurcations:(i)a saddle-node bifurcation curve ; (ii)a Hopf bifurcation curve ; (iii)a homoclinic bifurcation curve .

Conflict of Interests

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


This paper is supported by NSFC (11226142), Foundation of Henan Educational Committee (2012A110012), and Foundation of Henan Normal University (2011QK04, 2012PL03, and 1001).