# Turing conditions for pattern forming systems on evolving manifolds

## TL;DR

This paper develops general, practical conditions for predicting pattern formation in reaction-diffusion systems on evolving domains, extending classical Turing instability analysis to dynamic, non-autonomous settings.

## Contribution

It introduces versatile differential inequalities for Turing instability onset on time-dependent manifolds, generalizing existing static domain criteria and applicable to various domain evolutions.

## Key findings

- Derived sufficient conditions for diffusion-driven instabilities on evolving domains.
- Validated the conditions on different domain growth laws and oscillatory states.
- Extended the framework to higher-order spatial systems.

## Abstract

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.