Turing pattern formation with fractional diffusion and fractional reactions

Abstract

We have investigated Turing pattern formation through linear stability analysis andnumerical simulations in a two-species reaction–diffusion system in which a fractional ordertemporal derivative operates on both species, and on both the diffusion term and thereaction term. The order of the fractional derivative affects the time onset of patterning inthis model system but it does not affect critical parameters for the onset of Turinginstabilities and it does not affect the final spatial pattern. These results contrast withearlier studies of Turing pattern formation in fractional reaction–diffusion systems with afractional order temporal derivative on the diffusion term but not the reaction term.In addition to elucidating differences between these two model systems, our studies providefurther evidence that Turing linear instability analysis is an excellent predictor of both theonset of and the nature of pattern formation in fractional nonlinear reaction–diffusionequations.