Weak Dynamic Non-Emptiability and Persistence of Chemical Kinetics Systems

TL;DR

This paper introduces the concept of weak dynamic non-emptiability in chemical kinetics systems, extending previous persistence results by utilizing Farkas' Lemma and analyzing complex balanced and facet systems.

Contribution

It generalizes existing persistence conditions by defining weakly dynamically non-emptiable semi-locking sets and linking them to known classes of chemical systems.

Findings

Persistence holds if all critical semi-locking sets are weakly dynamically non-emptiable

Introduces the notion of weak dynamic non-emptiability using Farkas' Lemma

Connects new results to complex balanced and facet systems

Abstract

A frequently desirable characteristic of chemical kinetics systems is that of persistence, the property that if all the species are initially present then none of them may tend toward extinction. It is known that solutions of deterministically modelled mass-action systems may only approach portions of the boundary of the positive orthant which correspond to semi-locking sets (alternatively called siphons). Consequently, most recent work on persistence of these systems has been focused on these sets. In this paper, we focus on a result which states that, for a conservative mass-action system, persistence holds if every critical semi-locking set is dynamically non-emptiable and the system contains no nested locking sets. We will generalize this result by introducing the notion of a weakly dynamically non-emptiable semi-locking set and making novel use of the well-known Farkas' Lemma. We will also connect this result to known results regarding complex balanced systems and systems with facets.