Stochastic Turing Patterns for systems with one diffusing species

TL;DR

This paper explores how stochastic effects can induce Turing-like patterns in reaction-diffusion systems with only one diffusing species, where classical Turing instability is absent.

Contribution

It introduces a stochastic framework showing that finite size corrections can lead to pattern formation in single-diffusing-species systems, providing general conditions for stochastic Turing patterns.

Findings

Stochastic effects enable pattern formation without classical Turing instability.

Finite size corrections seed Turing-like patterns.

Theoretical predictions are validated through a specific case study.

Abstract

The problem of pattern formation in a generic two species reaction--diffusion model is studied, under the hypothesis that only one species can diffuse. For such a system, the classical Turing instability cannot take place. At variance, by working in the generalized setting of a stochastic formulation to the inspected problem, Turing like patterns can develop, seeded by finite size corrections. General conditions are given for the stochastic Turing patterns to occur. The predictions of the theory are tested for a specific case study.