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Parametrized polynomial ordinary differential equation systems are broadlyused for modeling, specially in the study of biochemical reaction networksunder the assumption of mass-action kinetics. Understanding the qualitativebehavior of the solutions with respect to the parameter values gives rise tocomplex problems within real algebraic geometry, concerning the study of thesigns of multivariate polynomials over the positive orthant. In this work weprovide further insight into the number of positive steady states of abenchmark model, namely the two-site phosphorylation cycle. In particular, weprovide new conditions on the reaction rate constants for the existence of oneor three positive steady states, partially filling a gap left in previousworks.

On the reaction rate constants that enable multistationarity in the two-site phosphorylation cycle