A Stratum Approach to Global Stability of Complex Balanced Systems

TL;DR

This paper introduces a stratum-based approach to prove the global stability of complex balanced systems, extending known results from detailed balanced systems and addressing a long-standing conjecture.

Contribution

It develops a novel methodology dividing the positive orthant into strata to analyze global stability of complex balanced systems, generalizing previous results from detailed balanced systems.

Findings

Generalized global stability results to complex balanced systems

Demonstrated trajectories are repelled from boundary faces within strata

Extended the methodology from detailed to complex balanced systems

Abstract

It has long been known that complex balanced mass-action systems exhibit a restrictive form of behaviour known as locally stable dynamics. This means that within each compatibility class $\mathcal{C}_{\mathbf{x}_0}$---the forward invariant space where solutions lies---there is exactly one equilibrium concentration and that this concentration is locally asymptotically stable. It has also been conjectured that this stability extends globally to $\mathcal{C}_{\mathbf{x}_0}$. That is to say, all solutions originating in $\mathcal{C}_{\mathbf{x}_0}$ approach the unique positive equilibrium concentration rather than $\partial \mathcal{C}_{\mathbf{x}_0}$ or $\infty$. To date, however, no general proof of this conjecture has been found. In this paper, we approach the problem of global stability for complex balanced systems through the methodology of dividing the positive orthant into regions called strata. This methodology has been previously applied to detailed balanced systems---a proper subset of complex balanced systems---to show that, within a stratum, trajectories are repelled from any face of $\mathbb{R}_{\geq 0}^m$ adjacent to the stratum. Several known global stability results for detailed balanced systems are generalized to complex balanced systems.