Qualitative studies of some biochemical models

A computational approach to detect Andronov - Hopf bifurcations in polynomial systems of ordinary differential equations depending on parameters is proposed. It relies on algorithms of computational commutative algebra based on the Groebner bases theory. The approach is applied to the investigation of two models related to the MAPK (mitogen-activated protein kinases) double phosphorylation, a biochemical network that occurs in many cellular pathways. For the models we perform the analysis of roots of the characteristic polynomials of the Jacobians at the steady states and prove the absence of Andronov - Hopf bifurcations for biochemically relevant values of parameters. We also performed a search for algebraic invariant subspaces in the systems (which represent "weak" conservations laws) and find all subfamilies admitting linear invariant subspaces. The search is done using the Darboux method. That, is we look for Darboux polynomials and cofactors as polynomials with undetermined coefficients and then determine the coefficients using the algorithms of the elimination theory.