In this paper, we examine the dynamics of reaction-diffusion systems with fractional time derivatives. It is shown that in these conditions diffusion is anomalous, in the sense that the mean-square displacement $⟨r^2⟩\sim t^{\gamma }$, where $\gamma <1$, a situation known as subdiffusion. We study the conditions for the appearance of a diffusion-driven instability and show that the restrictive conditions for a Turing instability are relaxed. This implies that systems whose kinetics are not of the activator-inhibitor kind can have a Turing instability and a modulated final state. We demonstrate our results with numerical calculations in two dimensions using a generic Turing model.

• Received 18 September 2008

DOI:https://doi.org/10.1103/PhysRevE.79.026109