On the dependence of the existence of the positive steady states on the rate coefficients for deficiency-one mass action systems: single linkage class

TL;DR

This paper characterizes when deficiency-one mass action systems with a single linkage class have positive steady states regardless of rate coefficients, extending the understanding beyond weakly reversible systems.

Contribution

It provides a characterization of deficiency-one, single linkage class mass action systems for which positive steady states exist independently of rate coefficients.

Findings

Identifies conditions ensuring positive steady states regardless of rate coefficients.

Provides an equivalent condition for the existence of rate coefficients leading to positive steady states.

Extends the deficiency-one theorem to non-weakly reversible systems with one linkage class.

Abstract

The Deficiency-One Theorem states that there exists a unique positive steady state in each positive stoichiometric class for weakly reversible deficiency-one mass action systems with one linkage class (regardless of the values of the rate coefficients). The non-emptiness of the set of positive steady states does not remain valid if we omit the weak reversibility. A recently published paper provided an equivalent condition to the existence of a positive steady state for deficiency-one mass action systems that are not weakly reversible, but still has only one linkage class. Based on that result, we characterise in this paper those of these mass action systems for which the non-emptiness of the set of positive steady states holds regardless of the values of the rate coefficients. Also, we provide an equivalent condition to the existence of rate coefficients such that the set of positive steady states is nonempty for the resulting mass action system.