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HOPF BIFURCATION ANALYSIS FOR A RATIO-DEPENDENT PREDATOR–PREY SYSTEM INVOLVING TWO DELAYS

Part of: Local and nonlocal bifurcation theory Functional-differential and differential-difference equations

Published online by Cambridge University Press:  05 June 2014

E. KARAOGLU  and
H. MERDAN
E. KARAOGLU
Affiliation:
TOBB University of Economics and Technology, Faculty of Arts and Sciences, Department of Mathematics, Söğütözü 06530, Ankara, Turkey email [email protected], [email protected]
H. MERDAN*
Affiliation:
TOBB University of Economics and Technology, Faculty of Arts and Sciences, Department of Mathematics, Söğütözü 06530, Ankara, Turkey email [email protected], [email protected]
*
✉[email protected]
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Abstract

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The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.

Keywords

Hopf bifurcationdelay differential equationtime delaystabilityperiodic solutionspopulation dynamics

MSC classification

Secondary: 34K18: Bifurcation theory 34K20: Stability theory 37G15: Bifurcations of limit cycles and periodic orbits 92D25: Population dynamics (general)
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 3 , January 2014 , pp. 214 - 231
Copyright
Copyright © 2014 Australian Mathematical Society 

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