Abstract
We investigate a reaction–diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction–diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction–diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.
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Glimm and Zhang are joint first authors.
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Glimm, T., Zhang, J., Shen, YQ. et al. Reaction–Diffusion Systems and External Morphogen Gradients: The Two-Dimensional Case, with an Application to Skeletal Pattern Formation. Bull Math Biol 74, 666–687 (2012). https://doi.org/10.1007/s11538-011-9689-6
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DOI: https://doi.org/10.1007/s11538-011-9689-6