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arXiv · 1705.10352

Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility

Leonid Berlyand·Jan Fuhrmann·Volodymyr Rybalko
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Abstract

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a free boundary setting.

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Leonid Berlyand, Jan Fuhrmann, Volodymyr Rybalko. 2017-07-10. Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility. https://arxiv.org/abs/1705.10352

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