Structural obstruction to the simplicity of the eigenvalue zero in chemical reaction networks

Abstract

Multistationarity is the property of a system to exhibit two distinct equilibria (steady‐states) under otherwise identical conditions, and it is a phenomenon of recognized importance for biochemical systems. Multistationarity may appear in the parameter space as a consequence of saddle‐node bifurcations, which necessarily require an algebraically simple eigenvalue zero of the Jacobian, at the bifurcating equilibrium. Matrices with a simple eigenvalue zero are generic in the set of singular matrices. Thus, one would expect that in applications singular Jacobians are always with a simple eigenvalue zero. However, chemical reaction networks typically consider a fixed network structure, while the freedom rests with the various choices of kinetics. Here, we present an example of a chemical reaction network, whose Jacobian is either nonsingular or has an algebraically multiple eigenvalue zero. This in particular constitutes an obstruction to standard saddle‐node bifurcations. The presented structural obstruction is based on the network structure alone, and it is independent of the value of the positive concentrations and the choice of kinetics.