Time delays play important roles in genetic regulatory networks. In this paper, a gene regulatory network model with time delays and mutual inhibition is considered, where time delays are regarded as bifurcation parameters. In the first part of this paper, we analyze the associated characteristic equations and obtain the conditions for the stability of the system and the existence of Hopf bifurcations in five special cases. Explicit formulas are given to determine the direction and stability of the Hopf bifurcation by using the normal form method and the center manifold theorem. Numerical simulations are then performed to illustrate the results we obtained. In the second part of the paper, using time-delayed stochastic numerical simulations, we study the impact of biological fluctuations on the system and observe that, in modest noise regimes, unexpectedly, noise acts to stabilize the otherwise destabilized oscillatory system.

中文翻译：

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两个自调控和互抑制基因之间时滞遗传调控网络的动态分析

时间延迟在遗传调控网络中起着重要作用。在本文中，考虑了一个具有时滞和相互抑制的基因调控网络模型，其中时滞被视为分叉参数。在本文的第一部分，我们分析了相关的特征方程，得到了系统在五种特殊情况下稳定和存在Hopf分岔的条件。利用范式方法和中心流形定理给出了确定Hopf分岔方向和稳定性的显式公式。然后进行数值模拟以说明我们获得的结果。在论文的第二部分，使用时滞随机数值模拟，我们研究了生物波动对系统的影响，并观察到，在适度的噪声状态下，