Turing pattern dynamics in an activator-inhibitor system with superdiffusion.

The fractional operator is introduced to an activator-inhibitor system to describe species anomalous superdiffusion. The effects of the superdiffusive exponent on pattern formation and pattern selection are studied. Our linear stability analysis shows that the wave number of the Turing pattern increases with the superdiffusive exponent. A weakly nonlinear analysis yields a system of amplitude equations and the analysis of these amplitude equations predicts parameter regimes where hexagons, stripes, and their coexistence are expected. Numerical simulations of the activator-inhibitor model near the stability boundaries confirm our analytical results. Since diffusion rate manifests in both diffusion constant and diffusion exponent, we numerically explore their interactions on the emergence of Turing patterns. When the activator and inhibitor have different superdiffusive exponents, we find that the critical ratio of the diffusion rate of the inhibitor to the activator, required for the formation of the Turing pattern, increases monotonically with the superdiffusive exponent. We conclude that small ratio (than unity) of anomalous diffusion exponent between the inhibitor and activator is more likely to promote the emergence of the Turing pattern, relative to the normal diffusion.