Decomposable and essentially univariate mass-action systems: Extensions of the deficiency one theorem

TL;DR

This paper extends the classical deficiency one theorem for reaction networks by introducing a dependency concept that allows for subnetworks with arbitrary deficiency and terminal components, broadening the theorem's applicability.

Contribution

It introduces a dependency one theorem for essentially univariate, decomposable polynomial systems and extends the deficiency one theorem to more general mass-action systems.

Findings

Provides a dependency one theorem for parametrized polynomial systems

Extends the deficiency one theorem to systems with arbitrary deficiency

Derives the classical deficiency one theorem as a special case

Abstract

The classical and extended deficiency one theorems by Feinberg apply to reaction networks with mass-action kinetics that have independent linkage classes or subnetworks, each with a deficiency of at most one and exactly one terminal strong component. The theorems assume the existence of a positive equilibrium and guarantee the existence of a unique positive equilibrium in every stoichiometric compatibility class. In our work, we use the $\textit{monomial dependency}$ which extends the concept of deficiency. First, we provide a dependency one theorem for parametrized systems of polynomial equations that are essentially univariate and decomposable. As our main result, we present a corresponding theorem for mass-action systems, which permits subnetworks with arbitrary deficiency and arbitrary number of terminal strong components. Finally, to complete the picture, we derive the extended deficiency one theorem as a special case of our more general dependency one theorem.