Dynamics of the Selkov oscillator

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 Poincaré compactification of the system is given. It is proved that for any values of the parameters there are solutions which tend to infinity for large times and are eventually monotone. It is shown that when the unique steady state is stable any bounded solution converges to it for large times. When the steady state is unstable and a periodic solution exists it is shown that the periodic solution is unique and that all bounded solutions other than the steady state converge to the periodic solution for large times. When the steady state is unstable and no periodic solution exists it is shown that all solutions other than the steady state are unbounded. In this case each solution which tends to infinity is eventually monotone and each solution other than the steady state which does not tend to infinity exhibits unbounded oscillations.