Abstract
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.
Highlights
When trying to understand a biological system with the help of mathematical modelling it often happens that there are several different models for the same biological situation in the literature
With this strategy in mind, we look in detail at the dynamical properties of certain models for glycolysis
It has been shown that the basic Selkov system admits solutions with unbounded oscillations and that the diameter of the image of a periodic solution can tend to infinity as α approaches a finite limit, completing the results of Brechmann and Rendall (2018) on that system and rigorously confirming a claim made in Selkov (1968)
Summary
When trying to understand a biological system with the help of mathematical modelling it often happens that there are several different models for the same biological situation in the literature. In view of this it is important to have criteria for deciding between models. One strategy for identifying criteria of this type is to look at relatively simple examples in great detail. In order to do this effectively it is necessary to have a sufficiently comprehensive understanding of the properties of solutions of the
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