Qualitative study of the Selkov model

The Selkov oscillator was formulated in 1968 and now it is a classical model for studying the glycolysis. It is a differential system of two equations depending on two parameters in dimensionless variables. When the two equations are polynomials we prove that the Selkov system is not Liouvillian integrable. Additionally, we prove that the polynomial Selkov system for any integer n≥1 has nine distinct phase portraits in the Poincaré disk.