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Hopf bifurcation analysis for a model of genetic regulatory system with delay. (English) Zbl 1178.34104
The paper investigates a mathematical model that describe a genetic regulatory system. The model has a delay which affects the dynamics of the system. The authors first consider the local stability of the equilibrium and the existence of Hopf bifurcations. By using the normal form theory and center manifold reduction, they derive explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support the analytic results.
##### MSC:
 34K60 Qualitative investigation and simulation of models 34K18 Bifurcation theory of functional differential equations 34K20 Stability theory of functional-differential equations 34K13 Periodic solutions of functional differential equations 92D15 Problems related to evolution
##### References:
 [1] Hassard, B.; Kazarinoff, N.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981) [2] Ruan, S.; Wei, J.: On the zeros of transcendental functions to stability of delay differential equations with two delays, Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 10, 863-874 (2003) · Zbl 1068.34072 [3] Smolen, P.; Baxter, D. A.; Byrne, J. H.: Frequency selectivity, multistability, and oscillations emerge from models of genetic regulatory systems, Amer. J. Phys. 277, C777-C790 (1998) [4] Smolen, P.; Baxter, D. A.; Byrne, J. H.: Modeling transcriptional control in gene networks-methods, recent results, and future, Bull. math. Biol. 62, 247-292 (2000)