Disguised toric dynamical systems

TL;DR

This paper explores disguised toric dynamical systems inspired by biochemical networks, revealing their properties and developing algorithms to identify their parameter spaces, which differ from classical toric systems.

Contribution

It introduces the concept of disguised toric dynamical systems, extending known properties of toric systems and providing algorithms to detect their parameter regions.

Findings

Disguised toric locus can have positive measure even when toric locus is measure zero.

Some reaction networks have an empty or measure-zero toric locus but a positive-measure disguised toric locus.

Algorithms are proposed to detect the disguised toric locus in parameter space.

Abstract

We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric. They are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. By means of (real) algebraic geometry we show that some reaction networks have an empty toric locus or a toric locus of Lebesgue measure zero in parameter space, while their disguised toric locus is of positive measure. We also propose some algorithms one can use to detect the disguised toric locus.