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Turing Instabilities are Not Enough to Ensure Pattern Formation

Krause, Andrew L.; Gaffney, Eamonn A.; Jewell, Thomas Jun; Klika, Václav; Walker, Benjamin J.

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Authors

Eamonn A. Gaffney

Thomas Jun Jewell

Václav Klika

Benjamin J. Walker



Abstract

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. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as 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.

Citation

Krause, A. L., Gaffney, E. A., Jewell, T. J., Klika, V., & Walker, B. J. (2024). Turing Instabilities are Not Enough to Ensure Pattern Formation. Bulletin of Mathematical Biology, 86(2), Article 21. https://doi.org/10.1007/s11538-023-01250-4

Journal Article Type Article
Acceptance Date Dec 22, 2023
Online Publication Date Jan 22, 2024
Publication Date Feb 1, 2024
Deposit Date Jan 29, 2024
Publicly Available Date Jan 29, 2024
Journal Bulletin of Mathematical Biology
Print ISSN 0092-8240
Electronic ISSN 1522-9602
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 86
Issue 2
Article Number 21
DOI https://doi.org/10.1007/s11538-023-01250-4
Keywords Multistability, Pattern formation, Turing instabilities
Public URL https://durham-repository.worktribe.com/output/2174487

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Publisher Licence URL
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Copyright Statement
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.






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