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.

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 [

When introducing the time delay into this type of systems, using the methods developed by [

Recently, papers [

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

The authors in [

For system (

For computation simplicity, we first rescale system (

Let

System (

From [

In this paper, for system (

The concrete organization of the paper is as follows: in Section

System (

Under the conditions (

In the following, we first give the normal form of the system (

Define

Next, we will find the

The bases of

By system (

By Lemma

Taking a similar theory in [

Using (

Let (

Next, we are interested in giving a versal unfolding for system (

Let

Then, the normal form of system (

Following the normal form formula in Kuznetsov [

Then, system (

Let (

a saddle-node bifurcation curve

a Hopf bifurcation curve

a homoclinic bifurcation curve

From Section

Let

To determine a versal unfolding for the original system (

Next, we need to find the expressions of

The bases of

For system (

Using the similar methods as used in [

Following the formula of Theorem 3.1 in [

Let (

system (

system (

system (

system (

system (

The Jacobian matrix of system (

Let (

In the following, using the Hopf bifurcation theorem for a retarded differential system introduced by [

Using the same methods as the ones used in [

Let

For large delay, by [

Let (

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

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).