Modification of Turing patterns through the use of time-varying anisotropic diffusion

Abstract

The Turing mechanism underpinning pattern formation in reaction–diffusion relies on the interplay between diffusion parameters and reaction kinetics. Diffusion is typically assumed isotropic, however anisotropic diffusion is known to arise in both nature and laboratory conditions. We study how the Turing instability and resulting Turing patterns are modified when the underlying diffusion tensor is both anisotropic and time-dependent, modelling, for instance, chemical species reacting and diffusing through time-varying anisotropic media. We show that the set of unstable wavenumber vectors corresponding to Turing modes evolve in a spatially biased manner under this anisotropy, thereby modifying the spatial scale and structure of any resulting Turing patterns differently along each spatial coordinate. We employ this spatial bias to develop control strategies to modify the shape or even structure of Turing patterns over time. We are able to make minor changes to the aspect ratio of Turing patterns, such as morphing circular Schnakenberg spots into elliptical spots, as well as more major changes to the structure of patterns, for instance converting Gierer–Meinhardt spots into stripes or FitzHugh–Nagumo labyrinthine patterns into target patterns. Our results suggest that time-varying anisotropic media may be used as a tool by which to modify and even control Turing patterns.