Electron. J. Diff. Equ., Vol. 2013 (2013), No. 09, pp. 1-16.

Stability and Hopf bifurcation in a symmetric Lotka-Volterra predator-prey system with delays

Jing Xia, Zhixian Yu, Rong Yuan

This article concerns a symmetrical Lotka-Volterra predator-prey system with delays. By analyzing the associated characteristic equation of the original system at the positive equilibrium and choosing the delay as the bifurcation parameter, the local stability and Hopf bifurcation of the system are investigated. Using the normal form theory, we also establish the direction and stability of the Hopf bifurcation. Numerical simulations suggest an existence of Hopf bifurcation near a critical value of time delay.

Submitted September 28, 2012. Published January 9, 2013.
Math Subject Classifications: 34K18, 37G05, 37G10, 92D25.
Key Words: Predator-prey system; delay; stability; Hopf bifurcation; normal form.

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Jing Xia
School of Mathematical Sciences
Peking University, Beijing 100871, China
email: xiajing2005@mail.bnu.edu.cn
Zhixian Yu
College of Science
University of Shanghai for Science and Technology
Shanghai 200093, China
email: yuzx0902@yahoo.com.cn
Rong Yuan
School of Mathematical Sciences
Beijing Normal University
Beijing 100875, China
email: ryuan@bnu.edu.cn

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