  Parametrized polynomial ordinary differential equation systems are broadly
used for modeling, specially in the study of biochemical reaction networks
under the assumption of mass-action kinetics. Understanding the qualitative
behavior of the solutions with respect to the parameter values gives rise to
complex problems within real algebraic geometry, concerning the study of the
signs of multivariate polynomials over the positive orthant. In this work we
provide further insight into the number of positive steady states of a
benchmark model, namely the two-site phosphorylation cycle. In particular, we
provide new conditions on the reaction rate constants for the existence of one
or three positive steady states, partially filling a gap left in previous
works.
