Modern Perspectives on NearEquilibrium Analysis of Turing Systems
Abstract
In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reactiondiffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations, and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical twospecies reactiondiffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reactiontransport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of `trivial' base states. We emphasise important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.08375
 Bibcode:
 2021arXiv210608375K
 Keywords:

 Nonlinear Sciences  Pattern Formation and Solitons;
 Quantitative Biology  Cell Behavior;
 92C15 (primary) 35B36;
 35K57 (secondary)
 EPrint:
 21 pages, 6 figures