Abstract
We consider pattern formation in a chemotaxis model with a vanishing chemotaxis coefficient at high population densities. This model was developed in Hillen and Painter (2001, Adv. Appli. Math. 26(4), 280–301.) to model volume effects. The solutions show spatio-temporal patterns which allow for ultra-long transients and merging or coarsening. We study the underlying bifurcation structure and show that the existence time for the pseudo- structures exponentially grows with the size of the system. We give approximations for one-step steady state solutions. We show that patterns with two or more steps are metastable and we approximate the two-step interaction using asymptotic expansions. This covers the basic effects of coarsening/merging and dissolving of local maxima. These effects are similar to pattern dynamics in other chemotaxis models, in spinodal decomposition of Cahn–Hilliard models, or to metastable patterns in microwave heating models.
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Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday
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Potapov, A.B., Hillen, T. Metastability in Chemotaxis Models. J Dyn Diff Equat 17, 293–330 (2005). https://doi.org/10.1007/s10884-005-2938-3
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DOI: https://doi.org/10.1007/s10884-005-2938-3