Noise-induced stabilization in a chemical reaction network without boundary effects

TL;DR

This paper demonstrates how stochastic noise can stabilize an otherwise unstable chemical reaction network, preventing finite-time blow-up through noise perturbations, independent of boundary effects.

Contribution

It introduces a chemical reaction network that is unstable deterministically but stabilized by stochastic noise, with a novel analysis using Lyapunov functions and state space decomposition.

Findings

Stochastic noise induces stabilization of the reaction network.

The system remains positive recurrent despite deterministic instability.

Stability is achieved without boundary effects influence.

Abstract

We present a chemical reaction network that is unstable under deterministic mass action kinetics, exhibiting finite-time blow-up of trajectories in the interior of the state space, but whose stochastic counterpart is positive recurrent. This provides an example of noise-induced stabilization of the model's dynamics arising due to noise perturbing transversally the divergent trajectories of the system that is independently of boundary effects. The proof is based on a careful decomposition of the state space and the construction of suitable Lyapunov functions in each region.