  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 system was
originally derived from another system with Michaelis-Menten kinetics. It is
shown 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 in the
literature.
