STABILITY AND BIFURCATION IN A DELAYED RATIO-DEPENDENT PREDATOR–PREY SYSTEM
Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often i...
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| Published in | Proceedings of the Edinburgh Mathematical Society Vol. 46; no. 1; pp. 205 - 220 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.02.2003
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0013-0915 1464-3839 |
| DOI | 10.1017/S0013091500001140 |
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| Abstract | Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled. The presence of time delay often greatly complicates the analytical study of such models. In this paper, the qualitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in the interior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcation parameter we study the Hopf bifurcation and the stability of the periodic solutions. AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14 |
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| AbstractList | Recently, ratio-dependent predator-prey systems have been regarded by some researchers as being more appropriate for predator-prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled. The presence of time delay often greatly complicates the analytical study of such models. In this paper, the qualitative behaviour of a class of ratio-dependent predator-prey systems with delay at the equilibrium in the interior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcation parameter we study the Hopf bifurcation and the stability of the periodic solutions. AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14 [PUBLICATION ABSTRACT] Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled. The presence of time delay often greatly complicates the analytical study of such models. In this paper, the qualitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in the interior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcation parameter we study the Hopf bifurcation and the stability of the periodic solutions. AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14 Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled. The presence of time delay often greatly complicates the analytical study of such models. In this paper, the qualitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in the interior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcation parameter we study the Hopf bifurcation and the stability of the periodic solutions. AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14 |
| Author | Xiao, Dongmei Li, Wenxia |
| Author_xml | – sequence: 1 givenname: Dongmei surname: Xiao fullname: Xiao, Dongmei organization: Department of Mathematics, Central China Normal University, Wuhan 430079, People's Republic of China – sequence: 2 givenname: Wenxia surname: Li fullname: Li, Wenxia organization: Department of Mathematics, Central China Normal University, Wuhan 430079, People's Republic of China |
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| Copyright | Copyright © Edinburgh Mathematical Society
2002 2002 Edinburgh Mathematical Society |
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| Keywords | bifurcation ratio-dependent response time delay stability predator–prey system |
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| Snippet | Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation... Recently, ratio-dependent predator-prey systems have been regarded by some researchers as being more appropriate for predator-prey interactions where predation... |
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| Title | STABILITY AND BIFURCATION IN A DELAYED RATIO-DEPENDENT PREDATOR–PREY SYSTEM |
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