Hopf Bifurcation Analysis for a Modified Time-Delay Predator-Prey System with Harvesting

Yang Ni,Yiming Ding,Yan Meng

doi:10.4236/jamp.2015.37094

Yang Ni, Yiming Ding + Show 1 more

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https://doi.org/10.4236/jamp.2015.37094

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Abstract

In this paper, we consider the direction and stability of time-delay induced Hopf bifurcation in a delayed predator-prey system with harvesting. We show that the positive equilibrium point is asymptotically stable in the absence of time delay, but loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold. Furthermore, using the norm form and the center manifold theory, we investigate the stability and direction of the Hopf bifurcation.

Highlights

Due to its universal existence and importance, the study on the dynamics of predator-prey systems is one of the dominant subjects in ecology and mathematical ecology since Lotka [1] and Volterra [2] proposed the wellknown predator-prey model [3]-[6]
The purpose of this paper is to investigate the effect of time-delay on a modified predator-prey model with harvesting
We discussed the existence of Hopf bifurcation of system (1) and the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are given

Summary

Introduction

Due to its universal existence and importance, the study on the dynamics of predator-prey systems is one of the dominant subjects in ecology and mathematical ecology since Lotka [1] and Volterra [2] proposed the wellknown predator-prey model [3]-[6]. A new method of central manifold has been developed to study the stability of delay induced bifurcation. Where dot means differentiation with respect to time t , x(t) and y(t) are the prey and predator population densities, respectively. Parameter r > 0 is the specific growth rate of prey in the absence of predation and without environment limitation. (2015) Hopf Bifurcation Analysis for a Modified Time-Delay PredatorPrey System with Harvesting. The purpose of this paper is to investigate the effect of time-delay on a modified predator-prey model with harvesting. We discussed the existence of Hopf bifurcation of system (1) and the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are given

Positive Equilibrium and Locally Asymptotically Stabiliy

Hopf Bifurcaion

The Direction and Stability of the Hopf Bifurcation

Conclusion