Tyson-Hong-Thron-Novak昼夜节律振荡器模型的动力学
Dynamics of the Tyson-Hong-Thron-Novak circadian oscillator model
摘要：
本文研究了Tyson，Hong，Thron和Novak提出的一个昼夜节律振荡器模型的动力学行为。该模型表明果蝇昼夜节律的分子机制。在详细研究了平衡后，我们进一步研究了mRNA降解和合成速率的影响。当mRNA降解速率较快时，我们证明了该模型不存在周期轨道。当mRNA降解速率足够慢时，该模型转化为慢-快系统。然后基于几何奇异扰动理论，证明了随着mRNA降解和合成速率的变化，系统存在鸭式爆炸、松弛振荡、同宿/异宿轨道和鞍结分岔.最后，我们给出了所得结果的生物学解释，并指出该模型可以转化为Liénard型方程，这可能有助于研究一般情况下的动力学。
英文摘要：
We study the dynamics of a circadian oscillator model which was proposed by Tyson, Hong, Thron and Novak. This model indicates a molecular mechanism for the circadian rhythm in Drosophila. After giving a detailed study of the equilibria, we further investigate the effects of the rates of mRNA degradation and synthesis. When the rate of mRNA degradation is rather fast, we prove that there are no periodic orbits in this model. When the rate of mRNA degradation is slow enough, this model is transformed into a slow-fast system. Then based on the geometric singular perturbation theory, we prove the existence of canard explosion, relaxation oscillations, homoclinic/heteroclinic orbits and saddle-node bifurcations as the rates of mRNA degradation and synthesis change. Finally, we give the biological interpretation of the obtained results and point out that this model can be transformed into a Liénard-like equation, which could be helpful to investigate the dynamics of the general case.
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