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

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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.
    Mathematics Subject Classification: 34D35.

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