Cycle structure in SR and DSR graphs: implications for multiple equilibria and stable oscillation in chemical reaction networks

TL;DR

This paper explores how the structure of SR and DSR graphs associated with chemical reaction networks can predict the systems' long-term behavior, including the absence of multiple equilibria and stable oscillations.

Contribution

It introduces new theorems and methods for analyzing SR and DSR graphs to determine dynamical properties of chemical reaction networks.

Findings

Theorems for ruling out multiple equilibria.

Criteria for excluding stable oscillations.

Illustrative examples demonstrating the methods.

Abstract

Associated with a chemical reaction network is a natural labelled bipartite multigraph termed an SR graph, and its directed version, the DSR graph. These objects are closely related to Petri nets. The construction of SR and DSR graphs for chemical reaction networks is presented. Conclusions about asymptotic behaviour of the associated dynamical systems which can be drawn easily from the graphs are discussed. In particular, theorems on ruling out the possibility of multiple equilibria or stable oscillation in chemical reaction networks based on computations on SR/DSR graphs are presented. These include both published and new results. The power and limitations of such results are illustrated via several examples.