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

Article Contents

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

This issue Previous Article Epidemic threshold conditions for seasonally forced SEIR models Next Article A nonlinear $L^2$-stability analysis for two-species population dynamics with dispersal

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: January 31, 2005

Revised: February 28, 2005

Published: October 2005

Abstract Related Papers Cited by

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.

Keywords:

Mathematics Subject Classification: 34D35.

Citation:

Article Metrics

HTML views() PDF downloads(67) Cited by(0)

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

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

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

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content