Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Hopf bifurcation analysis in a model of oscillatory gene expression with delay

Junjie Weia1 and Chunbo Yua2

a1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People's Republic of China (weijj@hit.edu.cn)

a2 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People's Republic of China

Abstract

The dynamics of a gene expression model with time delay are investigated. The investigation confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. An explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions has been derived by using the theory of the centre manifold and the normal forms method. The global existence of periodic solutions has been established using a global Hopf bifurcation result by Wu and a Bendixson criterion for higher-dimensional ordinary differential equations due to Li and Muldowney.

(Received January 12 2007)

(Accepted August 15 2008)