Relaxing Turing’s Idealization: Heterogeneous and Evolving Domains in Reaction-Diffusion Pattern Formation
Motivated by a range of problems in embryology and ecology, I will present recent extensions to Turing’s classical reaction-diffusion paradigm for pattern formation. This will start by reviewing Turing-type instabilities and their analysis via classical linear instability theory. I will discuss two cases where this approach must be modified to account for complex properties of the medium. Firstly are models of heterogeneous or pre-patterned domains. Under the approximation of a sufficiently smooth heterogeneity relative to small diffusion coefficients, we can use WKB asymptotics to derive a localisation of the classical Turing conditions, and hence a localisation of patterns. This leads into a discussion of designing patterning systems, as well as questions of phenomenology. We next consider pattern formation on evolving domains, and demonstrate some analytical progress using tools from non-autonomous dynamical systems theory. Numerical simulations highlight a range of rich phenomena not captured by the linear analysis, especially in cases of morphogen-regulated domain evolution. In both scenarios, we find both physically and mathematically rich theories, with numerous open questions remaining. Throughout we relate our modelling and analysis back to biological questions of form and function, and suggest key areas of exploration not present in even these more complicated models.