Influence of temperature on Turing pattern formation

The Turing instability is one of the most commonly studied mechanisms leading to pattern formation in reaction-diffusion systems, yet there are still many open questions on the applicability of the Turing mechanism. Although experiments on pattern formation using chemical systems have shown that temperature differences play a role in pattern formation, there is far less theoretical work concerning the interplay between temperature and spatial instabilities. We consider a thermodynamically extended reaction-diffusion system, consisting of a pair of reaction-diffusion equations coupled to an energy equation for temperature, and use this to obtain a natural extension of the Turing instability accounting for temperature. We show that thermal contributions can restrict or enlarge the set of unstable modes possible under the instability, and in some cases may be used to completely shift the set of unstable modes, strongly modifying emergent Turing patterns. Spatial heterogeneity plays a role under several experimentally feasible configurations, and we give particular consideration to scenarios involving thermal gradients, thermodynamics of chemicals transported within a flow, and thermodiffusion. Control of Turing patterns is also an area of active interest, and we also demonstrate how patterns can be modified using time-dependent control of the boundary temperature.