Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model

Michael Y. Li; Xihui Lin; Hao Wang

doi:10.3934/dcdsb.2014.19.747

Article Contents

2014, Volume 19, Issue 3: 747-760. Doi: 10.3934/dcdsb.2014.19.747

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Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model

1.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1

Received: August 31, 2013

Revised: October 31, 2013

Published: April 2014

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Abstract

In recent studies, global Hopf branches were investigated for delayed model of HTLV-I infection with delay-independent parameters. It is shown in [8,9] that when stability switches occur, global Hopf branches tend to be bounded, and different branches can overlap to produce coexistence of stable periodic solutions. In this paper, we investigate global Hopf branches for delayed systems with delay-dependent parameters. Using a delayed predator-prey model as an example, we demonstrate that stability switches caused by varying the time delay are accompanied by bounded global Hopf branches. When multiple Hopf branches exist, they are nested and the overlap produces coexistence of two or possibly more stable limit cycles.

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Mathematics Subject Classification: Primary: 37Gxx, 34Kxx, 37-04; Secondary: 92Bxx, 93Dxx.

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