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Hopf bifurcation analysis in a model of oscillatory gene expression with delay. (English) Zbl 1185.34124
This paper concerns a delayed gene expression model. The purpose is to demonstrate that transcriptional and translational delay has a significant impact on the dynamics of gene expression. First, the standard procedure for local Hopf bifurcation is applied to study the existence and direction of local Hopf bifurcation and the stability of the bifurcating periodic solutions. Then global Hopf bifurcations are established by using the global Hopf bifurcation theorem due to J. Wu [Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034)]. The obtained theoretical results were supported with numerical simulations.
MSC:
34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
92C37Cell biology
34K19Invariant manifolds (functional-differential equations)
34K17Transformation and reduction of functional-differential equations and systems; normal forms