The dynamics of weakly reversible population processes near facets

TL;DR

This paper proves that facets of weakly reversible population processes are repelling and confirms the Global Attractor Conjecture for certain complex-balancing systems, advancing understanding of their long-term stability.

Contribution

It establishes the repelling nature of facets in weakly reversible systems and proves the Global Attractor Conjecture for systems with two-dimensional invariant manifolds.

Findings

Facets of weakly reversible systems are shown to be repelling.

The Global Attractor Conjecture is confirmed for systems with two-dimensional invariant manifolds.

The results apply to complex-balancing chemical reaction systems, ensuring stability of interior equilibria.

Abstract

This paper concerns the dynamical behavior of weakly reversible, deterministically modeled population processes near the facets (codimension-one faces) of their invariant manifolds and proves that the facets of such systems are "repelling." It has been conjectured that any population process whose network graph is weakly reversible (has strongly connected components) is persistent. We prove this conjecture to be true for the subclass of weakly reversible systems for which only facets of the invariant manifold are associated with semilocking sets, or siphons. An important application of this work pertains to chemical reaction systems that are complex-balancing. For these systems it is known that within the interior of each invariant manifold there is a unique equilibrium. The Global Attractor Conjecture states that each of these equilibria is globally asymptotically stable relative to the interior of the invariant manifold in which it lies. Our results pertaining to weakly reversible systems imply that this conjecture holds for all complex-balancing systems whose boundary equilibria lie in the relative interior of the boundary facets. As a corollary, we show that the Global Attractor Conjecture holds for those systems for which the associated invariant manifolds are two-dimensional.