Why Turing mechanism is an obstacle to stationary periodic patterns in bounded reaction-diffusion media with advection

Abstract

Formation of stationary periodic patterns is paramount to many chemical, biological, physical, and ecological media. One of the most subtle mechanisms was suggested by Turing, who highlighted the applicability of isotropic reaction-diffusion dynamics with at least two diffusing fields. However, on finite domains with the presence of a symmetry breaking differential advection, two diffusing fields are rather disadvantageous to formation of stationary periodic patterns. We show that the criterion to stationary periodic patterns in Turing type models requires non-periodic boundary conditions and tuning of two parameters (a co-dimension-2 bifurcation in space) whereas in systems with one diffusing field (non-Turing) the bifurcation is of co-dimension 1 and thus easier to satisfy. We demonstrate this general result using spatial dynamics methods and direct numerical simulations of the canonical FitzHugh-Nagumo model.