Global asymptotic stability for a class of nonlinear chemical equations

TL;DR

This paper proves that certain weakly reversible, deficiency-zero chemical reaction systems with mass action kinetics are globally asymptotically stable under specific boundary intersection conditions, extending previous local stability results.

Contribution

It extends existing global stability results to cases where boundary intersections are discrete, broadening the class of chemical systems known to be globally stable.

Findings

Global asymptotic stability proven for specified chemical systems

Boundary intersection conditions are relaxed from previous requirements

Extends the class of systems with known global stability

Abstract

We consider a class of nonlinear differential equations that arises in the study of chemical reaction systems that are known to be locally asymptotically stable and prove that they are in fact globally asymptotically stable. More specifically, we will consider chemical reaction systems that are weakly reversible, have a deficiency of zero, and are equipped with mass action kinetics. We show that if for each $c \in \R_{> 0}^m$ the intersection of the stoichiometric compatibility class $c + S$ with the subsets on the boundary that could potentially contain equilibria, $L_W$, are at most discrete, then global asymptotic stability follows. Previous global stability results for the systems considered in this paper required $(c + S) \cap L_W = \emptyset$ for each $c \in \R^m_{> 0}$, and so this paper can be viewed as an extension of those works.