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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, Japan, Japan, Japan |
$x'(t) = x(t) [r_1 - ax(t- \tau_1) - by(t)]$
$y'(t) = y(t) [-r_2 + cx(t) - dy(t- \tau_2)]$ (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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