Seemingly stable chemical kinetics can be stable, marginally stable, or unstable

TL;DR

This paper demonstrates that chemically similar reaction networks can exhibit vastly different stochastic behaviors, including stability, instability, and marginal stability, despite having nearly identical deterministic limits.

Contribution

It introduces a set of Lyapunov functions to analyze and distinguish the stochastic stability properties of reaction networks with similar deterministic dynamics.

Findings

One network is positive recurrent (stable)

One network is transient (unstable)

One network is null recurrent (marginally stable)

Abstract

We present three examples of chemical reaction networks whose ordinary differential equation scaling limit are almost identical and in all cases stable. Nevertheless, the Markov jump processes associated to these reaction networks display the full range of behaviors: one is stable (positive recurrent), one is unstable (transient) and one is marginally stable (null recurrent). We study these differences and characterize the invariant measures by Lyapunov function techniques. In particular, we design a natural set of such functions which scale homogeneously to infinity, taking advantage of the same scaling behavior of the reaction rates.