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

  • Received: 01 February 2005 Accepted: 29 June 2018 Published: 01 November 2005
  • MSC : 34D35.

  • 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.

    Citation: S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system[J]. Mathematical Biosciences and Engineering, 2006, 3(1): 173-187. doi: 10.3934/mbe.2006.3.173

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  • 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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    © 2006 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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