# Stochastic fluctuations and quasi-pattern formation in reaction-diffusion systems with anomalous transport

## TL;DR

This paper develops a Gaussian approximation for intrinsic noise in reaction-subdiffusion systems, revealing how stochastic fluctuations and anomalous transport can induce and enhance pattern formation.

## Contribution

It introduces a novel analytical framework to incorporate intrinsic noise into fractional reaction-diffusion equations, advancing understanding of noise-driven pattern formation in subdiffusive systems.

## Key findings

- Subdiffusion can promote both deterministic and stochastic pattern formation.
- Intrinsic noise reduces the diffusion ratio threshold for pattern emergence.
- The combined effect of subdiffusion and stochasticity enhances pattern formation more than either factor alone.

## Abstract

Many approaches to modelling reaction-diffusion systems with anomalous transport rely on deterministic equations and ignore fluctuations arising due to finite particle numbers. Starting from an individual-based model we use a generating-functional approach to derive a Gaussian approximation for this intrinsic noise in subdiffusive systems. This results in corrections to the deterministic fractional reaction-diffusion equations. Using this analytical approach, we study the onset of noise-driven quasi-patterns in reaction-subdiffusion systems. We find that subdiffusion can be conducive to the formation of both deterministic and stochastic patterns. Our analysis shows that the combination of subdiffusion and intrinsic stochasticity can reduce the threshold ratio of the effective diffusion coefficients required for pattern formation to a greater degree than either effect on its own.