Turing instability conditions for growing domains with divergence free mesh velocity

Abstract

Questions have been asked from a theoretical point of view about the effects of domain growth on Turing [A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Lond. B 237 (1952) 37–72] diffusion-driven instability analysis for reaction–diffusion systems on continuously growing domains. The increase in computational power and tools has given the scientific community an almost immediate vast amount of computational data on pattern transitions (peak splitting, peak insertion, period doubling, circular patterns etc) on growing domains but with little knowledge of the theoretical justifications of the role played by domain growth in pattern formation. This paper derives the conditions for Turing diffusion-driven instability on growing domains whose domain velocity is divergence free. In particular we establish the result that the conditions for diffusion-driven instability growing domains whose domain velocity is divergence free are equivalent to those derived on a fixed domain. For this type of growth, the advection terms have no effect on the standard diffusion-driven instability conditions. Under these conditions, travelling wave solutions are obtained. This result has implications in the selection of parameter values as these are selected such that they belong to the Turing space. Our results do not however generalise to arbitrary growth protocols such as exponential, linear or logistic growths since the divergence of the mesh velocity is time-dependent.