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STABILITY AND BIFURCATION IN A DELAYED RATIO-DEPENDENT PREDATOR–PREY SYSTEM

Published online by Cambridge University Press:  27 January 2003

Dongmei Xiao  and
Wenxia Li
Dongmei Xiao
Affiliation:
Department of Mathematics, Central China Normal University, Wuhan 430079, People's Republic of China
Wenxia Li
Affiliation:
Department of Mathematics, Central China Normal University, Wuhan 430079, People's Republic of China
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Abstract

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Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate forpredator–prey interactions where predation involves serious searching processes. Due to the fact that every populationgoes 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, thequalitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in theinterior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable andthere exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcationparameter we study the Hopf bifurcation and the stability of the periodic solutions.

AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14

Keywords

stabilitybifurcationpredator–prey systemratio-dependent responsetime delay
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 1 , February 2003 , pp. 205 - 220
Copyright
Copyright © Edinburgh Mathematical Society 2002