Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system

S. Nakaoka; Y. Saito; Y. Takeuchi

doi:10.3934/mbe.2006.3.173

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2006, Volume 3, Issue 1: 173-187. Doi: 10.3934/mbe.2006.3.173

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Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system

1.
Department of Systems Engineering, Faculty of Engineering, Shizuoka University, Johoku 3-5-1, Hamamatsu, Shizuoka 432-8561

Received: February 2005

Revised: March 2005

Published: October 2005

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Abstract

We consider the following Lotka-Volterra predator-prey system with two delays:
$x'(t) = x(t) [r_1 - ax(t- \tau_1) - by(t)]$
$y'(t) = y(t) [-r_2 + cx(t) - dy(t- \tau_2)]$ (E)
We show that a positive equilibrium of system (E) is globally asymptotically stable for small delays. Critical values of time delay through which system (E) undergoes a Hopf bifurcation are analytically determined. Some numerical simulations suggest an existence of subcritical Hopf bifurcation near the critical values of time delay. Further system (E) exhibits some chaotic behavior when $tau_2$ becomes large.

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Mathematics Subject Classification: 34D35.

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Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system

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