Polynomial Dynamical Systems, Reaction Networks, and Toric Differential Inclusions

Abstract

Some of the most common mathematical models in biology, chemistry, physics, and engineering are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand sides. Inspired by notions and results that have been developed for the analysis of reaction networks in biochemistry and chemical engineering, we show that any polynomial dynamical system on the positive orthant $\mathbb{R}^n_{>0}$ can be regarded as being generated by an oriented graph embedded in $\mathbb{R}^n$, called a Euclidean embedded graph. This allows us to recast key conjectures about reaction network models (such as the Global Attractor Conjecture, or the Persistence Conjecture) into more general versions about some important classes of polynomial dynamical systems. Then, we introduce toric differential inclusions, which are piecewise constant autonomous dynamical systems with a remarkable geometric structure. We show that if a Euclidean embedded graph $\mathcal{G}$ has some reversibility properties, then any polynomial dynamical system generated by $\mathcal{G}$ can be embedded into a toric differential inclusion. We discuss how this embedding suggests an approach for the proof of the Global Attractor Conjecture and Persistence Conjecture.