Travelling Turing patterns with anomalous diffusion

We investigate the formation of Turing patterns (arising from a diffusion driven instability) when diffusion is anomalous. We analyse a model that contains the important features of Turing systems and that has been extensively used in the past to model interesting biological patterns. We concentrate on the case of asymmetric anomalous diffusion, and we cast a version of the model, suitable for numerical calculations, using a discrete version of the fractional derivatives defining the anomalous diffusion operator. The results are interesting in the sense that patterns are no longer stationary but acquire a velocity that depends on the exponent of the fractional derivatives. Extensive numerical calculations in one and two dimensions exhibit many of the features predicted by the analysis of the equations.