Graphical criteria for positive solutions to linear systems

TL;DR

This paper establishes new graphical criteria based on labeled multidigraphs to determine when solutions to certain linear systems over partially ordered rings are nonnegative, with applications in biology, ecology, and chemical reaction networks.

Contribution

It introduces novel graphical conditions for nonnegativity of solutions in linear systems over partially ordered rings, including systems with block-structured matrices relevant to chemical networks.

Findings

Provided conditions guarantee nonnegative solutions for general linear systems.

Extended criteria to block-structured systems common in chemical reaction network theory.

Applicable to systems modeling biological concentrations and abundances.

Abstract

We study linear systems of equations with coefficients in a generic partially ordered ring $R$ and a unique solution, and seek conditions for the solution to be nonnegative, that is, every component of the solution is a quotient of two nonnegative elements in $R$. The requirement of a nonnegative solution arises typically in applications, such as in biology and ecology, where quantities of interest are concentrations and abundances. We provide novel conditions on a labeled multidigraph associated with the linear system that guarantee the solution to be nonnegative. Furthermore, we study a generalization of the first class of linear systems, where the coefficient matrix has a specific block form and provide analogous conditions for nonnegativity of the solution, similarly based on a labeled multidigraph. The latter scenario arises naturally in chemical reaction network theory, when studying full or partial parameterizations of the positive part of the steady state variety of a polynomial dynamical system in the concentrations of the molecular species.