Stability analysis of a model reaction network using graph theory

Abstract

The stability of a model chemical reaction network which is unstable with respect to both homogeneous and inhomogeneous perturbations far from equilibrium is analyzed using a graph theoretic approach. Equations for the boundary of stability and the frequency and wavelength of the unstable perturbation at the boundary are obtained by interpreting the dominant graphs in the stability inequality geometrically. It is found that perturbations within a broad of wave vectors become unstable together at the point of marginal stability and that usually across this band ω(k) ∝ k⁻¹. Of the hundreds of graphs in the stability inequality only a very few play a role in the instability. Most of these are abnormal cycle graphs through the autocatalytic vertex and the reminder are maximum overlap graphs from the diagonal term of the Hurwitz determinant. The three destabilizing graphs give insight into the normal modes associated with the three unstable regions. Comparison of this analysis with an earlier computer analysis of the same model network clearly demonstrates the power of the graph theoretical approach.