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Mathematical Biosciences and Engineering

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: 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:

    (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 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:

    (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 becomes large.


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