Theorems on chemical network stability

Abstract

The stability of steady states of a chemical reaction system is considered within a diagrammatic formulation of the problem. The system’s stability depends upon the kinds of cycles that can be constructed from a set of arrows. The following theorems are proven. (1) A chemical network has no unstable steady states if the set of cycles which can be constructed contains only certain 2−cycles or is empty; (2) an m−cycle, which passes through a reactant whole self−vertex is exactly cancelled by autocatalysis destabilizes the network in certain restrictive circumstances; and (3) a 3−cycle as in (2) destabilizes the network under broader circumstances. The restrictive circumstances of the second theorem do not appear to be capable of being broadened in general because of complexities that can be understood within the full diagrammatic formulation of the problem.