Widening the criteria for emergence of Turing patterns

Abstract

The classical concept for emergence of Turing patterns in reaction–diffusion systems requires that a system should be composed of complementary subsystems, one of which is unstable and diffuses sufficiently slowly while the other one is stable and diffuses sufficiently rapidly. In this work, the phenomena of emergence of Turing patterns are studied and do not fit into this concept, yielding the following results. (1) The criteria are derived, under which a reaction–diffusion system with immobile species should spontaneously produce Turing patterns under any diffusion coefficients of its mobile species. It is shown for such systems that under certain sets of types of interactions between their species, Turing patterns should be produced under any parameter values, at least provided that the corresponding spatially non-distributed system is stable. (2) It is demonstrated that in a reaction–diffusion system, which contains more than two species and is stable in absence of diffusion, the presence of a sufficiently slowly diffusing unstable subsystem is already sufficient for diffusion instability (i.e., Turing or wave instability), while its complementary subsystem can also be unstable. (3) It is shown that the presence of an immobile unstable subsystem, which leads to destabilization of waves within an infinite range of wavenumbers, in a spatially discrete case can result in the generation of large-scale stationary or oscillatory patterns. (4) It is demonstrated that under the presence of subcritical Turing and supercritical wave bifurcations, the interaction of two diffusion instabilities can result in the spontaneous formation of Turing structures outside the region of Turing instability.

Progressive Changes in Brain Regional Homogeneity Induced by Electroconvulsive Therapy Among Patients With Schizophrenia

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General conditions for Turing and wave instabilities in reaction -diffusion systems

“…( 2000 ) to systems that have components with zero diffusion coefficients, and Hoang and Hwang ( 2013 ) that shows that linear instability implies a nonlinear bifurcation under certain technical assumptions. Finally, we mention the work of Kuznetsov and Polezhaev ( 2020 ) who find new ways of obtaining Turing bifurcations by considering what happens to dispersion relations in the limit that some diffusion coefficients tend to zero. Their results, which are mostly restricted to 3-component systems, provide a key motivation to the method introduced in this paper.…”

Turing patterns by supramolecular self-assembly of a single salphen building block

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