## The Higgins-Selkov oscillator, part 3

Here I report on some further progress in understanding the dynamics of mathematical models for glycolysis. In a previous post I described some results on the Selkov model obtained in a paper by Pia Brechmann and myself. The main question left open there is whether this model admits unbounded oscillations. We have now been able to resolve this issue in a new paper, showing that there is precisely one value of the parameter $\alpha$ for which solutions of this type exist. This progress was stimulated by a paper of Merkin, Needham and Scott (SIAM J. Appl. Math. 47, 1040). It was a piece of luck that we found this paper since its title gives no indication that it has any relation to the problem we were interested in. It concerns a model for certain chemical reactors which turns out to be mathematically identical to the Selkov model with the parameter $\gamma$ equal to two. The authors claim that there are unbounded solutions for exactly one value of $\alpha$ and present a derivation of this which is an intricate argument involving matched asymptotic expansions. I find this argument impressive but I have no idea how it could be transformed into a rigorous proof. Despite this it helped us indirectly. We had previously tried, without success, to find some way to transform the equations so as to obtain a well-behaved limit $\alpha\to\infty$. Combining some transformations in the paper of Merkin et al. with some we had used in our previous paper allowed this goal to be achieved. Once this has been done the dynamics is under control for $\alpha$ sufficiently large. The next step is to do a shooting argument to show that there is a parameter value $\alpha_1$ for which a heteroclinic cycle at infinity exists. It had already been shown in our first paper that this is enough to conclude the occurrence of unbounded oscillations. The paper of Merkin et al. also helped us to understand what was the right set-up for the shooting argument. The uniqueness of $\alpha_1$ was obtained using a monotonicity property of the parameter dependence which appears to be new. At this point I think it is justified to say that the analysis of the main qualitative features of solutions of the Selkov oscillator is essentially complete. There is just one more thing I would like to know although I see it just as a curiosity. We showed that for $\alpha$ a bit less than $\alpha_1$ there exists a stable periodic solution and that its diameter tends to infinity as $\alpha\to\alpha_1$. Merkin et al. give an expression for the leading order contribution to the diameter in this limit for $\gamma=2$. How could this be proved rigorously? How could an analogous expression be derived for other values of $\gamma$?

In the original paper of Selkov the system we have been discussing up to now is derived from a system with kinetics of Michaelis-Menten type by letting a parameter $\nu$ tend to zero. He suggests that in the system with $\nu>0$ the unbounded oscillations are eliminated so that they could be thought of as an artefact of setting $\nu=0$. In our new paper we investigated this question although we were unfortunately only able to get partial results. One feature of the Selkov model not mentioned by Selkov in the original paper is that there are solutions which are unbounded and tend to infinity in a monotone manner. We showed that solutions of this type also exist for the Michaelis-Menten model and that they have exactly the same leading order asymptotics as in the case $\nu=0$. The periodic solutions of the Selkov oscillator are stable and arise in a supercritical Hopf bifurcation. That system admits no unstable periodic solutions whatsoever. By contrast, we found that the Michaelis-Menten system also exhibits subcritical Hopf bifurcations and correspondingly unstable periodic solutions.

I see a number of interesting directions in which this work could be extended but we will not attempt to do so in the foreseeable future since we have too many other projects to work on. I would only return to this if there turned out to be intriguing connections to other research projects or to the themes of master theses I was supervising.