Dynamics of the Selkov oscillator

TL;DR

This paper provides a comprehensive rigorous analysis of the dynamics of the Selkov oscillator, a mathematical model for glycolysis, including stability, periodic solutions, and behavior of unbounded solutions.

Contribution

It offers the first complete analysis of the Selkov oscillator's dynamics, including stability, periodicity, and unbounded solution behavior, using Poincaré compactification.

Findings

Stable steady state implies convergence of bounded solutions.

Unstable steady state leads to either periodic solutions or unbounded solutions.

Solutions tend to infinity monotonically or exhibit large oscillations.

Abstract

A classical example of a mathematical model for oscillations in a biological system is the Selkov oscillator, which is a simple description of glycolysis. It is a system of two ordinary differential equations which, when expressed in dimensionless variables, depends on two parameters. Surprisingly it appears that no complete rigorous analysis of the dynamics of this model has ever been given. In this paper several properties of the dynamics of solutions of the model are established. With a view to studying unbounded solutions a thorough analysis of the Poincar\'e compactification of the system is given. It is proved that for any values of the parameters there are solutions which tend to infinity at late times and are eventually monotone. It is shown that when the unique steady state is stable any bounded solution converges to the steady state at late times. When the steady state is unstable it is shown that for given values of the parameters either there is a unique periodic solution to which all bounded solutions other than the steady state converge at late times or there is no periodic solution and all solutions other than the steady state are unbounded. In the latter case each unbounded solution which tends to infinity is eventually monotone and each unbounded solution which does not tend to infinity has the property that each variable takes on arbitrarily large and small values at arbitrarily late times.