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)]$$y'(t) = y(t) [-r_2 + cx(t) - dy(t- \tau_2)]$ (E)We show that a positive equilibrium of system (E) is globally asymptotically stable for small delays. Critical values...
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AIMS Press
2005-10-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.173 |
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| author | S. Nakaoka Y. Saito Y. Takeuchi |
| author_facet | S. Nakaoka Y. Saito Y. Takeuchi |
| author_sort | S. Nakaoka |
| collection | DOAJ |
| description | We consider the following Lotka-Volterra predator-prey system with two delays:$x'(t) = x(t) [r_1 - ax(t- \tau_1) - by(t)]$$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. |
| format | Article |
| id | doaj-art-ebbb2fabccee4d0999f2dc524fd8dc8c |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2005-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-ebbb2fabccee4d0999f2dc524fd8dc8c2025-01-24T01:51:12ZengAIMS PressMathematical Biosciences and Engineering1551-00182005-10-013117318710.3934/mbe.2006.3.173Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey systemS. Nakaoka0Y. Saito1Y. Takeuchi2Department of Systems Engineering, Faculty of Engineering, Shizuoka University, Johoku 3-5-1, Hamamatsu, Shizuoka 432-8561Department of Systems Engineering, Faculty of Engineering, Shizuoka University, Johoku 3-5-1, Hamamatsu, Shizuoka 432-8561Department of Systems Engineering, Faculty of Engineering, Shizuoka University, Johoku 3-5-1, Hamamatsu, Shizuoka 432-8561We 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.https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.173subcritical hopf bifurcationnonlinear dynamics.chaotic behaviorpredator-preymathematical model |
| spellingShingle | S. Nakaoka Y. Saito Y. Takeuchi Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system Mathematical Biosciences and Engineering subcritical hopf bifurcation nonlinear dynamics. chaotic behavior predator-prey mathematical model |
| title | Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system |
| title_full | Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system |
| title_fullStr | Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system |
| title_full_unstemmed | Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system |
| title_short | Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system |
| title_sort | stability delay and chaotic behavior in a lotka volterra predator prey system |
| topic | subcritical hopf bifurcation nonlinear dynamics. chaotic behavior predator-prey mathematical model |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.173 |
| work_keys_str_mv | AT snakaoka stabilitydelayandchaoticbehaviorinalotkavolterrapredatorpreysystem AT ysaito stabilitydelayandchaoticbehaviorinalotkavolterrapredatorpreysystem AT ytakeuchi stabilitydelayandchaoticbehaviorinalotkavolterrapredatorpreysystem |