利用中心流形和正规型理论研究了一类由常微分方程组来刻画的基因调控模型,得到该系统局部稳定性和出现Hopf分支的一些充分条件,通过数值模拟验证了所得结论的正确性.
The main purpose of this paper is to investigate a class of genetic regulator model which was described by ordinary differential equations using center manifold theory and normal form theorem. Some sufficient conditions which guarantee the local stability and Hopf bifurcation are obtained,and some numerical simulation for justifying the theoretical results is also provided.
[1] SMOLEN P,BAXTER D A,BYRNE J H. Frequency selectivity,multistability,and oscillations emerge from models of genetic regulatory systems[J].Amer J Phys,1998,274:C531-C542.
[2] WAN A Y,ZOU X F.Hopf bifurcation analysis for a model of genetic regulatory system with delay[J].J Math Anal Appl,2009,365:464-476.DOI:10.1016/j.jmaa.2009.03.037.
[3] CAO J Z,JIANG H J.Stability and Hopf bifurcation analysis on Goodwin model with three delays[J]. Chaos Solitons Fractals,2011,44:613-618.DOI:10.1016/j.chaos.2011.05.010.
[4] CAO J Z,JIANG H J.Hopf bifurcation analysis for a model of single genetic negative feedback autoregulatory system with delay[J].Neurocomputing,2013,99:381-389.DOI:10.1016/j.neucom.2012.07.021.
[5] YU T T,ZHANG X,ZHANG G D,et al.Hopf bifurcation analysis for genetic regulatory networks with two delays[J].Neurocomputing,2015,164:190-200.DOI:10.1016/j.neucom.2015.02.070.
[6] 陆启韶.分岔与奇异性[M].上海:上海科技教育出版社,1995.
[7] WIGGINS S.Introduction to applied nonlinear dynamical systems and chaos[M].New York:Springer,2003.
[8] 唐三一,肖燕妮.单种群生物动力系统[M].北京:科学出版社,2008.
