Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system
We consider the following Lotka-Volterra predator-prey system with two delays:$x'(t) = x(t) [r_1 - ax(t- \tau_1) - by(t)]$$y'(t) = y(t) [-r_2 + cx(t) - dy(t- \tau_2)]$ (E)We show that a positive equilibrium of system (E) is globally asymptotically stable for small delays. Critical values...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2005-10-01
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| Series: | Mathematical Biosciences and Engineering |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.173 |
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| Summary: | We consider the following Lotka-Volterra predator-prey system with two delays:$x'(t) = x(t) [r_1 - ax(t- \tau_1) - by(t)]$$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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| ISSN: | 1551-0018 |