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Tuesday 26 Apr 2022Pattern Formation in Stratified and Heterogeneous Domains

Andrew Krause - Durham University

Harrison 101 13:30-14:30

We review Turing-type instabilities and their analysis via classical linear instability analysis. From here we discuss two cases where this approach must be modified to account for complex properties of the medium. Firstly we describe the case of a heterogeneous domain, where a pre-pattern or other spatial heterogeneity influences the reaction-diffusion dynamics. Under the approximation of a sufficiently smooth heterogeneity relative to small diffusion coefficients, we can use WKB asymptotics to show a localisation of the classical Turing conditions. In the opposite regime, when the medium is composed of two distinct stratified layers, we can also make some progress under suitable assumptions. In both scenarios we find much richer dispersion relations, as well as numerous open questions suitable for a range of mathematical techniques and new ideas. Throughout we relate our modelling and analysis back to biological questions of form and function, and suggest further areas of exploration not present in even these more complicated models.

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