Symbolic hunt of instabilities and bifurcations in reaction networks

The localization of bifurcations in large parametric systems is still a challenge where the combination of rigorous criteria and informal intuition is often needed. With this motivation, we address symbolically the Jacobian matrix of reaction networks with general kinetics. More specifically, we consider any nonzero partial derivative of a reaction rate as a free positive symbol. The main tool are the Child-Selections: injective maps that associate to a species $m$ a reaction $j$ where $m$ participates as reactant. Firstly, we employ a Cauchy-Binet analysis and we structurally express any coefficient of the characteristic polynomial of the Jacobian in terms of Child-Selections. In particular, we fully characterize sign-changes of any of the coefficients. Secondly, we prove that the (in)stability of the Jacobian is inherited from the (in)stability of simpler submatrices identified by the Child-Selections. Thirdly, we provide sufficient conditions for purely imaginary eigenvalues of the Jacobian, hinting at Hopf bifurcation and oscillatory behavior. All conditions are in terms of signs of integer stoichiometric submatrices identified by the Child-Selections {and do not require any Hurwitz-type computaton}. Finally, we focus on systems endowed with Michaelis-Menten kinetics and we show that any symbolic realization of the Jacobian matrix can be achieved at a fixed equilibrium by a proper choice of the kinetic constants.