Dynamical failure of Turing patterns

Abstract

The emergence of stable disordered patterns in reactive system on spatially homogenous substrate is studied in the context of vegetation patterns in the semi-arid climatic zone. It is shown that reaction-diffusion systems that allow for Turing instability may exhibit heterogeneous "glassy" steady state, with no characteristic wavelength, if the diffusion rate associated with the "slow" reactant is very small. Upon decreasing the diffusion constant of the slow reactant three phases are identified: strong diffusion yields a stable homogenous phase, intermediate diffusion implies Turing (crystal like) patterns while in the slow diffusion limit the glassy state is the generic stable solution. In this disordered phase the dynamics is of crucial importance, with strong differences between local and global initiation.