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Self-organization, the ability of a system of microscopically interacting entities to shape macroscopically ordered structures, is ubiquitous in Nature. Spatio-temporal patterns are abundantly observed in a large plethora of applications, encompassing different fields and scales. Examples of emerging patterns are the spots and stripes on the coat or skin of animals, the spatial distribution of vegetation in arid areas, the organization of the colonies of insects in host-parasitoid systems and the architecture of large complex ecosystems. Spatial self-organization can be described following the visionary intuition of Alan Turing, who showed how non-linear interactions between slow diffusing activators and fast diffusing inhibitors could induce patterns. The Turing instability, as the mechanism described is universally referred to, was raised to paradigm status in those realms of investigations where microscopic entities are subject to diffusion, from small biological systems to large ecosystems. Requiring a significant ratio of the assigned diffusion constants however is a stringent constraint, which limited the applicability of the theory. Building on the observation that spatial interactions are usually direction biased, and often strongly asymmetric, we here propose a novel framework for the generation of short wavelength patterns which overcomes the limitation inherent in the Turing formulation. In particular, we will prove that patterns can always set in when the system is composed by sufficiently many cells - the units of spatial patchiness - and for virtually any ratio of the diffusivities involved. Macroscopic patterns that follow the onset of the instability are robust and show oscillatory or steady-state behavior.
We report a new mechanism for the formation of localized states, which takes place without front propagation. Correspondingly, localized structures appear as solitary states, displaying a behavior of single independent cells. The phenomenon is observ
We explain some pde2path setups for pattern formation in 1D, 2D and 3D. A focus is on new pde2path functions for branch switching at steady bifurcation points of higher multiplicity, typically due to discrete symmetries, but we also review general co
Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near
Driven many-body systems have been shown to exhibit discrete time crystal phases characterized by broken discrete time-translational symmetry. This has been achieved generally through a subharmonic response, in which the system undergoes one oscillat
Morphogenetic patterns are highly sophisticated dissipative structures. Are they governed by the same general mechanisms as chemical and hydrodynamic patterns? Turings symmetry breaking and Wolperts signalling provide alternative mechanisms. The curr