Stability of steady states and bifurcation to traveling waves in a free boundary model of cell motility

TL;DR

This paper introduces a free boundary model for cell motility, analyzes the stability of steady states, and demonstrates bifurcation to traveling wave solutions using both linear and nonlinear stability methods.

Contribution

It presents a novel two-dimensional Keller-Segel type model incorporating Darcy law and Hele-Shaw boundary conditions for cell motility, analyzing stability and bifurcation phenomena.

Findings

Radially symmetric steady states become unstable and bifurcate into traveling waves.

Linear stability analysis is inconclusive for steady states and waves.

Nonlinear stability of steady states is established using invariance properties.

Abstract

We introduce a two-dimensional Keller-Segel type free boundary model for motility of eukaryotic cells on substrates. The key ingredients of this model are the Darcy law for overdamped motion of the cytoskeleton (active) gel and Hele-Shaw type boundary conditions (Young-Laplace equation for pressure and continuity of velocities). We first show that radially symmetric steady state solutions become unstable and bifurcate to traveling wave solutions. Next we establish linear and nonlinear stability of the steady states. We show that linear stability analysis is inconclusive for both steady states and traveling waves. Therefore we use invariance properties to prove nonlinear stability of steady states.