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.

Department of Mathematical Sciences Durham University Upper Mountjoy Campus Stockton Road Durham DH1 3LE UK

Department of Mathematical Sciences University of Bath Bath BA2 7AY UK

Department of Mathematics FNSPE Czech Technical University Prague Trojanova 13 120 00 Prague Czech Republic

Department of Mathematics University College London London WC1E 6BT UK

Wolfson Centre for Mathematical Biology Mathematical Institute University of Oxford Oxford OX2 6GG UK

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Turing Instabilities are Not Enough to Ensure Pattern Formation