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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Emergence of traveling waves and their stability in a free boundary model of cell motility
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by Volodymyr Rybalko and Leonid Berlyand HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 1799-1844 Request permission


We consider a 2D free boundary model of cell motility, inspired by the 1D contraction-driven cell motility model due to P. Recho, T. Putelat, and L. Truskinovsky [Phys. Rev. Lett. 111 (2013), p. 108102]. The key ingredients of the model are the Darcy law for overdamped motion of the acto-myosin network, coupled with the advection-diffusion equation for myosin density. These equations are supplemented with the Young-Laplace equation for the pressure and no-flux condition for the myosin density on the boundary, while evolution of the boundary is subject to the acto-myosin flow at the edge.

The focus of the work is on stability analysis of stationary solutions and translationally moving traveling wave solutions. We study stability of radially symmetric stationary solutions and show that at some critical radius a pitchfork bifurcation occurs, resulting in emergence of a family of traveling wave solutions. We perform linear stability analysis of these latter solutions with small velocities and reveal the type of bifurcation (sub- or supercritical). The main result of this work is an explicit asymptotic formula for the stability determining eigenvalue in the limit of small traveling wave velocities.

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Additional Information
  • Volodymyr Rybalko
  • Affiliation: B. Verkin Institute for Low Temperature Physics and Engineering of NASU, 47 Nauky ave, Khariv 61103, Ukraine
  • Address at time of publication: Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg 412 96, Sweden
  • MR Author ID: 650362
  • ORCID: 0000-0002-1024-4127
  • Email:
  • Leonid Berlyand
  • Affiliation: Department of Mathematics, Huck Institutes of Life Sciences and Materials Research Institute at the Penn State University, University Park, Pennsylvania 16802
  • MR Author ID: 196680
  • Email:
  • Received by editor(s): April 5, 2021
  • Received by editor(s) in revised form: April 22, 2022, and August 2, 2022
  • Published electronically: December 16, 2022
  • Additional Notes: The visits of the first author to Penn State University were supported by NSF grants DMS-1405769 and DMS-2005262. The work of the second author was partially supported by NSF grant DMS-2005262
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 1799-1844
  • MSC (2020): Primary 35R35, 35B32, 92C17
  • DOI:
  • MathSciNet review: 4549692