The study of pattern formation in far from equilibrium systems is a topic of interest to many areas of science and technology including biology, chemistry, fluids, plasmas, and nonlinear optics. In recent years there have been significant advances in the understanding of pattern forming systems. These advances have been driven by laboratory experiments that have shown the formation of patterns in relatively well-controlled settings, and the study of reduced mathematical models that regardless of their relative simplicity have proved to capture some of the basic elements of pattern formation. Among the best known mathematical models of pattern formation are reaction diffusion models. The main lesson learned from these models is that diffusion can give rise to the spontaneous formation of spatio-temporal patterns. At first hand this seems surprising because diffusion is typically viewed as an smoothing process that wipes out structures. However, since the seminal work of A.Turing in 1954, many systems have been found in which the competition of two or more fields coupled to diffusion give rise to patterns. Our interest on reaction-diffusion dynamics has its origins on plasmas physics. However, as discussed in the following links, the models studied and the results obtained have a wider range of applicability that goes beyond plasma physics. We are particularly interested in the role of cross diffusion and fractional diffusion in reaction diffusion systems. In reaction-diffusion systems involving two or more fields, cross-diffusion is present whenever the flux of one of the fields is driven by the gradients of another field. On the hand, fractional diffusion is believed to play an important role whenever the underlying diffusive process is not Gaussian. A problem of particular interest to us is the case of anomalous, superdiffusion caused by Levy flights.
