A geometric approach to the Global Attractor Conjecture

TL;DR

This paper proves the global attractor conjecture for a new class of complex-balanced systems called strongly endotactic networks, using geometric and combinatorial methods, extending previous results and including power-law systems.

Contribution

It introduces strongly endotactic networks and proves the global attractor conjecture for these systems, broadening the class of systems known to satisfy this conjecture.

Findings

Global attractor conjecture holds for strongly endotactic networks

Results extend to systems with multiple linkage classes

Uses differential inclusions and geometric analysis techniques

Abstract

This paper introduces the class of "strongly endotactic networks", a subclass of the endotactic networks introduced by G. Craciun, F. Nazarov, and C. Pantea. The main result states that the global attractor conjecture holds for complex-balanced systems that are strongly endotactic: every trajectory with positive initial condition converges to the unique positive equilibrium allowed by conservation laws. This extends a recent result by D. F. Anderson for systems where the reaction diagram has only one linkage class (connected component). The results here are proved using differential inclusions, a setting that includes power-law systems. The key ideas include a perspective on reaction kinetics in terms of combinatorial geometry of reaction diagrams, a projection argument that enables analysis of a given system in terms of systems with lower dimension, and an extension of Birch's theorem, a well-known result about intersections of affine subspaces with manifolds parameterized by monomials.