Turing Instabilities in Homogeneous Systems

Front propagation and Turing instabilities are two emblematic phenomena displayed by nonlinear reaction-diffusion systems. In the former, a stable state invades an unstable or less stable state. In the latter, diffusion couples with the local nonlinear transformations and drives the uniform steady state of the system unstable. We focus on the effect of deviations from standard diffusion and spatial homogeneity of the medium on these two signature phenomena. 

The DIFICI is closely related to the Turing instability (TI) [4], In both instabilities, the homogeneous and stable state of a system composed of an activator (a species that tends to grow autocatalytically) and an inhibitor (which suppresses the growth) may lose its stability through their differential transport by a differential bulk flow in the case of the DIFICI and by differential diffusivity in the TI. The ultimate cause in both instabilities is the inherent tendency of the activator to grow locally when decoupled spatially from the inhibitor response. 

Over the past several years there have been many experimental and theoretical studies aimed at developing a better understanding of pattern formation in reaction-diffusion systems. The focus of recent studies has been on more complex behavior away from the onset of instability. For some parameter values, spatiotemporal chaos may occur near the boundary between the Turing region and the region of homogeneous oscillations (Figure 12). 

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