Turing instabilities are not enough to ensure pattern formation

TL;DR

This paper shows that Turing instabilities alone do not guarantee pattern formation, as simple models can satisfy instability conditions yet only produce transient patterns, challenging traditional assumptions.

Contribution

It demonstrates that Turing-like instabilities are not sufficient for sustained pattern formation in systems with multistability and nonlinearity.

Findings

Turing instability conditions can be met without resulting in stable patterns.

Transient patterns can occur even when classical instability criteria are satisfied.

Multistability can cause failures of linear stability analysis in predicting pattern formation.

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. \ak{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 \ak{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.