A journey in the zoo of Turing patterns

Self-organized phenomena are widespread in Nature and have been studied for long time in various domains, be it physics, chemistry, biology, ecology, neurophysiology, to name a few. Despite the rich literature on the subject, there is still need for understanding, analyzing and predicting their emergence and behavior. Patterns are commonly based on local interaction rules that determine the creation and destruction of the basic entities - species - at spatial locations, upon which the action of a diffusion process determines the migration of the species. For this reason reaction-diffusion systems are a common framework for modeling such systems. In a pioneer article, Turing considered a two-species model of morphogenesis. For the first time, he established the conditions for a stable spatially homogeneous state, to migrate towards a new heterogeneous, spatially patched, equilibrium under the driving effect of diffusion, at odd with the idea that diffusion is a source of homogeneity. Even though the explanation for morphogenesis has evolved and now relies more on genetic programming, many actual results are grounded on this pioneering work. Nowadays, Turing instability goes beyond this initial framework and it can be used to explain emergence of self-organized collective patterns. The geometry of the underlying support where the reaction-diffusion acts, plays a relevant role in the patterned outcome, it can be because of the non-flat geometry (possibly growing) or because of its anisotropy. In several applications the underlying domain can be supposed to be divided into local patches where reactions occur and diffusion across patches is realized via the links existing among the latter; this framework leads naturally to the introduction of reactiondiffusion systems defined on complex networks. The aim of this talk is to introduce some of the recent developments we obtained with my collaborators concerning the emergence of Turing patterns on complex networks and their generalization, such as multiplex, time varying networks, or higher-order structures such as hypergraphs or simplicial complexes.

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