Optimal multi-time-scale estimates for diluted autocatalytic chemical networks. (1) Introduction and $\sigma^*$-dominant case

TL;DR

This paper develops a new recursive method inspired by quantum field theory to accurately estimate long-term behavior of autocatalytic chemical networks, especially in the diluted regime, using Lyapunov eigenvalues.

Contribution

It introduces a novel, scale-based recursive algorithm for precise asymptotic analysis of autocatalytic networks, extending previous approaches with a focus on $\sigma^*$-dominant graphs.

Findings

Estimation of Lyapunov eigenvalues for specific graph classes

Validation of the method on simple examples

Preliminary results on $\sigma^*$-dominant graphs

Abstract

Autocatalytic chemical networks are dynamical systems whose linearization around zero has a positive Lyapunov exponent; this exponent gives the growth rate of the system in the diluted regime, i.e. for near-zero concentrations. The generator of the dynamics in the kinetic limit is then a Perron-Frobenius matrix, suggesting the use of Markov chain techniques to get long-time asymptotics. This series of works introduces a new, general procedure providing precise quantitative information about such asymptotics, based on estimates for the Lyapunov eigenvalue and eigenvector. The algorithm, inspired from Wilson's renormalization group method in quantum field theory, is based on a downward recursion on kinetic scales, starting from the fastest, and terminating with the slowest rates. Estimates take on the form of simple rational functions of kinetic rates. They are accurate under a separation of scales hypothesis, loosely stating that kinetic rates span many orders of magnitude. We provide here a brief general motivation and introduction to the method, present some simple examples, and derive a number of preliminary results, in particular the estimation of Lyapunov data for a subclass of so-called $\sigma^*$-dominant graphs.