# Timing and Shape of Stochastic Autocatalytic Burst Formation

## TL;DR

This paper investigates how stochastic noise influences the timing and shape of burst formation in autocatalytic systems near a saddle-node bifurcation, revealing pattern nucleation driven by microscopic fluctuations.

## Contribution

It applies Large Deviation Theory to characterize burst shape and timing in spatially extended stochastic autocatalytic systems near bifurcation points.

## Key findings

- Burst shape and timing scale with noise amplitude
- Spatial nucleation of peaks occurs randomly across the domain
- Microscopic noise induces macroscopic pattern formation

## Abstract

Chemical, physical and ecological systems passing through a saddle-node bifurcation will, momentarily, find themselves balanced at a semi-stable steady state. If perturbed by noise, such systems will escape from the zero-steady state, with escape time sensitive to noise. When the model is extended to include space, this leads to different points in space "escaping from zero" at different times, and uniform initial conditions nucleate into sharp peaks spread randomly across a nearly uniform background, a phenomena closely resembling nucleation during phase transition. We use Large Deviation Theory to determine burst shape and temporal scaling with respect to noise amplitude. These results give a prototype for a particular form of patternless symmetry breaking in the vicinity of a stability boundary, and demonstrates how microscopic noise can lead to macroscopic effects in such a region.