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Hopf bifurcation of a delayed predator-prey system with prey infection and the modified Leslie-Gower scheme is investigated. The conditions for the stability and existence of Hopf bifurcation of the system are obtained. The state feedback and parameter perturbation are used for controlling Hopf bifurcation in the system. In addition, direction of Hopf bifurcation and stability of the bifurcated periodic solutions of the controlled system are obtained by using normal form and center manifold theory. Finally, numerical simulation results are presented to show that the hybrid controller is efficient in controlling Hopf bifurcation.

The dynamics of epidemiological models have been investigated by many scholars [

However, an important aspect which should be kept in mind while formulating an epidemiological system is the fact that it is often necessary to incorporate time delays into the system in order to reflect the dynamical behaviors of the system depending on the past history of the system, and epidemiological systems with delay have been studied extensively [

The initial conditions for system (

This paper is organized as follows. In Section

According to [

For

Obviously,

For the polynomial equation (

if

if

if

Suppose that the coefficients in

If the condition

Next, we give the transversality condition by the following Lemma.

Suppose that

Taking the derivative of

Obviously, if

Thus, the proof is completed.

By Lemmas

For system (

the positive equilibrium

the positive equilibrium

if

In this section, we will incorporate the state feedback and parameter perturbation into system (

Similar as in Section

Using Taylor expansion to expand the right-hand side of system (

Obviously, the characteristic equation of system (

In the following, we will use the normal form method and center manifold theorem introduced by Hassard et al. [

Let

By the previous discussions, we know that

By a simple computation, we can get

Next, we can get the coefficients determining the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions by the algorithms given in [

For system (

The Hopf bifurcation is supercritical (subcritical) if

In this section, we give some numerical simulations to illustrate our theoretical analysis in Sections

Next, we choose

Comparing Figures

In addition, from (

A delayed predator-prey system with prey infection and the modified Leslie-Gower scheme is investigated. Regarding the negative feedback delay of the predator as a parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are analyzed. The results show that, when the delay crosses a critical value, the system will lose its stability and a Hopf bifurcation occurs. To delay the onset of the Hopf bifurcation, we incorporate the state feedback and parameter perturbation into the system, and simulation results show the effectiveness of the controller. In addition, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions for the controlled system are also determined by the normal form theory and the center manifold argument.

This work is supported by the National Natural Science Foundation (NNSF) of China under Grant 61273070, a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and Doctor Candidate Foundation of Jiangnan University (JUDCF12030).