Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system
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Department of Systems Engineering, Faculty of Engineering, Shizuoka University, Johoku 3-5-1, Hamamatsu, Shizuoka 432-8561
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Received:
01 February 2005
Accepted:
29 June 2018
Published:
01 November 2005
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MSC :
34D35.
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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.
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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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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