Journal of Mathematical Biology ( IF 1.9 ) Pub Date : 2021-01-19 , DOI: 10.1007/s00285-021-01560-y Pia Brechmann 1 , Alan D Rendall 1
The Selkov oscillator, a simple description of glycolysis, is a system of two ordinary differential equations with mass action kinetics. In previous work the authors established several properties of the solutions of this system. In the present paper we extend this to prove that this system has solutions which diverge to infinity in an oscillatory manner at late times. This is done with the help of a Poincaré compactification of the system and a shooting argument. This system was originally derived from another system with Michaelis–Menten kinetics. A Poincaré compactification of the latter system is carried out and this is used to show that the Michaelis–Menten system, like that with mass action, has solutions which diverge to infinity in a monotone manner. It is also shown to admit subcritical Hopf bifurcations and thus unstable periodic solutions. We discuss to what extent the unbounded solutions cast doubt on the biological relevance of the Selkov oscillator and compare it with other models for the same biological system in the literature.
中文翻译:
糖酵解模型的无限解
塞尔科夫振荡器是糖酵解的简单描述,是具有质量作用动力学的两个常微分方程组。在之前的工作中,作者建立了该系统解决方案的几个属性。在本文中,我们对此进行了扩展,以证明该系统具有在后期以振荡方式发散至无穷大的解。这是在系统的庞加莱紧凑化和射击论证的帮助下完成的。该系统最初源自另一个具有米氏动力学的系统。对后一个系统进行了庞加莱紧致化,这用于表明米氏系统与质量作用系统一样,具有以单调方式发散至无穷大的解。它还显示出允许亚临界 Hopf 分岔,从而允许不稳定的周期解。我们讨论了无界解在多大程度上对塞尔科夫振荡器的生物相关性产生了怀疑,并将其与文献中同一生物系统的其他模型进行了比较。