Scaling Methods for Stochastic Chemical Reaction Networks

TL;DR

This paper investigates the asymptotic behavior of stochastic chemical reaction networks under a specific scaling regime where initial states grow large, without altering reaction rates or network structure, providing insights into their qualitative properties.

Contribution

It introduces a new scaling analysis for CRNs that preserves network structure and reaction rates, offering simplified stability criteria and detailed asymptotic insights.

Findings

Analysis of asymptotic properties of CRNs with large initial states

Identification of a bi-modal behavior in a specific CRN

Simplification of stability analysis using Filonov's criterion

Abstract

The asymptotic properties of some Markov processes associated to stochastic chemical reaction networks (CRNs) driven by the kinetics of the law of mass action are analyzed. The scaling regime introduced in the paper assumes that the norm of the initial state is converging to infinity. The reaction rate constants are kept fixed. The purpose of the paper is of showing, with simple examples, a scaling analysis in this context. The main difference with the scalings of the literature is that it does not change the graph structure of the CRN or its reaction rates. Several CRNs are investigated to illustrate the insight that can be gained on the qualitative properties of these networks. A detailed scaling analysis of a CRN with several interesting asymptotic properties, with a bi-modal behavior in particular, is worked out in the last section. Additionally, with several examples, we also show that a stability criterion due to Filonov for positive recurrence of Markov processes may simplify significantly the stability analysis of these networks.