Turing instabilities are not enough to ensure pattern formation

Andrew L. Krause, Eamonn A. Gaffney, Thomas Jun Jewell, Václav Klika, Benjamin J. Walker
Nonlinear Sciences, Pattern Formation and Solitons, Pattern Formation and Solitons (nlin.PS), Cell Behavior (q-bio.CB), Tissues and Organs (q-bio.TO)
2023-08-28 16:00:00
Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction--diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. Given that biological systems (including gene regulatory networks and spatially distributed ecosystems) often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.
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