Hopf Bifurcation Control in a Delayed Predator-Prey System with Prey Infection and Modified Leslie-Gower Scheme

Zizhen Zhang,Huizhong Yang

doi:10.1155/2013/704320

Abstract

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.

Highlights

The dynamics of epidemiological models have been investigated by many scholars [1–7] since Kermack and McKendrick [8] proposed the classical SIR model
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 [19]: g20 = 2τ0󸀠D [a13 + a14q(2) (0) + q∗2 (a24q(2) (0) + a25(q(2) (0))[2]
For system (21), when τ = τ0󸀠, the direction of the Hopf bifurcation and stability of periodic solutions are determined by the formulas (51), and the following results hold

Summary

Introduction

The dynamics of epidemiological models have been investigated by many scholars [1–7] since Kermack and McKendrick [8] proposed the classical SIR model. Based on the classical SIR model, Chattopadhyay and Arino [9] proposed a predator-prey epidemiological model with disease spreading in the prey, and they studied the boundedness of the solutions and the existence of Hopf bifurcation for the model. C1 represents the maximum value of the per capita rate of the infected prey due to the predator. C2 represents the maximum value of the per capita rate of the predator due to the infected prey population. Hu and Li [15] considered a delayed predator-prey system with disease in prey, and they studied Hopf bifurcation and the stability of the periodic solutions induced by the time delay. In order to delay the onset of Hopf bifurcation, we will incorporate the state feedback and parameter perturbation into system (2).

Stability of Positive Equilibrium and Hopf Bifurcation

Hopf Bifurcation Control

Numerical Simulation Examples

Conclusions