A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics

TL;DR

This paper introduces vertexical families of differential inclusions to analyze mass-action kinetics, providing a new structural approach that advances understanding of the global attractor conjecture.

Contribution

It defines vertexical families and demonstrates their good behavior under projections, enabling progress on the global attractor conjecture in mass-action systems.

Findings

Vertexical families are structurally inductive.

Trajectories approach boundary only at vertices or in lower dimensions.

Progress made on the global attractor conjecture.

Abstract

Motivated by questions in mass-action kinetics, we introduce the notion of vertexical family of differential inclusions. Defined on open hypercubes, these families are characterized by particular good behavior under projection maps. The motivating examples are certain families of reaction networks -- including reversible, weakly reversible, endotactic, and strongly endotactic reaction networks -- that give rise to vertexical families of mass-action differential inclusions. We prove that vertexical families are amenable to structural induction. Consequently, a trajectory of a vertexical family approaches the boundary if and only if either the trajectory approaches a vertex of the hypercube, or a trajectory in a lower-dimensional member of the family approaches the boundary. With this technology, we make progress on the global attractor conjecture, a central open problem concerning mass-action kinetics systems. Additionally, we phrase mass-action kinetics as a functor on reaction networks with variable rates.