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

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.