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Technical Brief

Stability and Hopf Bifurcation in a Three-Species Food Chain System With Harvesting and Two Delays

[+] Author and Article Information
Zizhen Zhang

Key Laboratory of Advanced Process Control
for Light Industry (Ministry of Education),
Jiangnan University,
Wuxi 214122, China
School of Management Science and Engineering,
Anhui University of Finance and Economics,
Bengbu 233030, China
e-mail: zzzhaida@163.com

Huizhong Yang

Key Laboratory of Advanced Process Control
for Light Industry (Ministry of Education),
Jiangnan University,
Wuxi 214122, China
e-mail: yhz@jiangnan.edu.cn

1Corresponding author.

Contributed by Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 30, 2012; final manuscript received September 18, 2013; published online January 9, 2014. Assoc. Editor: Henryk Flashner.

J. Comput. Nonlinear Dynam 9(2), 024501 (Jan 09, 2014) (5 pages) Paper No: CND-12-1233; doi: 10.1115/1.4025670 History: Received December 30, 2012; Revised September 18, 2013

In this paper, we analyze the dynamics of a delayed food chain system with harvesting. Sufficient conditions for the local stability of the positive equilibrium and for the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation. Formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, numerical simulation results are presented to validate the theoretical analysis.

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Figures

Grahic Jump Location
Fig. 4

E* is unstable when τ2 = 2.1500  >  1.7628 = τ2* and τ1 = 0.1500

Grahic Jump Location
Fig. 3

E* is locally asymptotically stable when τ2 = 1.6000<  1.7628 = τ2* and τ1 = 0.1500

Grahic Jump Location
Fig. 2

E* is unstable when τ1 = 0.4150  >  0.4043 = τ10

Grahic Jump Location
Fig. 1

E* is locally asymptotically stable when τ1 = 0.3750< 0.4043 = τ10

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