Wavelength selection beyond Turing

Spatial patterns arising spontaneously due to internal processes are ubiquitous in nature, varying from regular patterns of dryland vegetation to complex structures of bacterial colonies. Many of these patterns can be explained in the context of a Turing instability, where patterns emerge due to two locally interacting components that diffuse with different speeds in the medium. Turing patterns are multistable, such that many different patterns with different wavelengths are possible for the same set of parameters, but in a given region typically only one such wavelength is dominant. In the Turing instability region, random initial conditions will mostly lead to a wavelength that is similar to that of the leading eigenvector that arises from the linear stability analysis, but when venturing beyond, little is known about the pattern that will emerge. Using dryland vegetation as a case study, we use different models of drylands ecosystems to study the wavelength pattern that is selected in various scenarios beyond the Turing instability region, focusing the phenomena of localized states and repeated local disturbances

Spatial pattern formation induced by Gaussian white noise

The ability of Gaussian noise to induce ordered states in dynamical systems is here presented in an overview of the main stochastic mechanisms able to generate spatial patterns. These mechanisms involve: (i) a deterministic local dynamics term, accounting for the local rate of variation of the field variable, (ii) a noise component (additive or multiplicative) accounting for the unavoidable environmental disturbances, and (iii) a linear spatial coupling component, which provides spatial coherence and takes into account diffusion mechanisms. We investigate these dynamics using analytical tools, such as mean-field theory, linear stability analysis and structure function analysis, and use numerical simulations to confirm these analytical results.Comment: 11 pages, 8 figure