Local and global stability of equilibria for a class of chemical reaction networks

TL;DR

This paper investigates the stability of equilibria in a broad class of chemical reaction networks, establishing conditions for local and global stability without relying on mass action kinetics, using monotone dynamical systems theory.

Contribution

It introduces a new class of reaction networks characterized by matrix factorization and graph connectivity, proving stability results under mild assumptions and extending local to global stability.

Findings

Positive equilibria are locally asymptotically stable within their stoichiometry class.

Under certain conditions, local stability extends to global stability.

Lyapunov functions are constructed based on monotonicity and integrals of the system.

Abstract

A class of chemical reaction networks is described with the property that each positive equilibrium is locally asymptotically stable relative to its stoichiometry class, an invariant subspace on which it lies. The reaction systems treated are characterised primarily by the existence of a certain factorisation of their stoichiometric matrix, and strong connectedness of an associated graph. Only very mild assumptions are made about the rates of reactions, and in particular, mass action kinetics are not assumed. In many cases, local asymptotic stability can be extended to global asymptotic stability of each positive equilibrium relative to its stoichiometry class. The results are proved via the construction of Lyapunov functions whose existence follows from the fact that the reaction networks define monotone dynamical systems with increasing integrals.