On the Regulary of Reaction Systems

TL;DR

This paper investigates the regularity of reaction systems, both stochastic and deterministic, using Lyapunov functions, and applies findings to various models in sciences and engineering.

Contribution

It introduces a simple Lyapunov-based condition to establish regularity in reaction systems and proves regularity for broad classes like endotactic and weakly reversible systems.

Findings

Every second-order endotactic mass-action system is regular.

Every bimolecular weakly reversible mass-action system is regular.

Results apply to models in biochemistry, epidemiology, ecology, and synthetic biology.

Abstract

Reaction networks have been widely used as generic models in diverse areas of applied sciences, such as biology, chemistry, ecology, epidemiology, and computer science. A reaction network incorporating noisy effects is modeled as a continuous time Markov chain (CTMC) and is called a stochastic reaction system. In contrast, the mean field limit of a sequence of volume-scaled stochastic reaction systems as the volume tends to infinity is modeled as an ordinary differential equation (ODE) and is called a deterministic reaction system. Non-explosivity of CTMCs and global existence of solutions of ODEs capture the regularity of respective dynamical processes. In this paper, we study the regularity of reaction systems, in both stochastic and deterministic senses. By constructing a simple linear Lyapunov function, we obtain the regularity in both sense for a class of reaction systems in terms of a simple checkable condition. As an application, we prove that (i) every second-order endotactic mass-action system is regular, and hence (ii) every bimolecular weakly reversible mass-action system is regular. We apply our results to diverse models in biochemistry, epidemiology, ecology, synthetic biology, and natural computing in the literature.