Stability of topologically similar chemical networks

Abstract

Theorems are proven which support the view that chemical reaction network topology determines which networks can have unstable steady states and when the steady states are unstable. These theorems make it possible to determine the stability of large networks in certain circumstances by determining the stability of simpler networks with identical topology. Chemical networks are classified as ’’qualitatively vertex stable’’ (QVS), ’’qualitatively vertex marginally stable’’ (QVM), and ’’qualitatively vertex unstable’’ (QVU). Roughly speaking, QVS networks always have stable steady states, QVM networks have linearized dynamics which is marginally stable for all rate constants and constrained concentrations, and QVU networks have unstable steady states for specific parameter ranges. The sense in which the preceding sentence is rough is discussed in the text. It is shown that the linear steady state stability analysis problems for topological similar networks are closely related. A geometrical and graph theoretical approach to the stability problem previously developed by the author is then used to prove a number of theorems establishing in certain cases that topologically similar networks possess identical qualitative vertex stability classifications.