  We introduce a two-dimensional Hele-Shaw 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 gel (active gel)
coupled with advection-diffusion equation for myosin density leading to
elliptic-parabolic Keller-Segel system. This system is supplemented with
Hele-Shaw type boundary conditions: Young-Laplace equation for pressure and
continuity of velocities. We first show that radially symmetric stationary
solutions become unstable and bifurcate to traveling wave solutions at a
critical value of the total myosin mass. Next we perform linear stability
analysis of these traveling wave solutions and identify the type of bifurcation
(sub- or supercritical). Our study sheds light on the mathematics underlying
instability/stability transitions in this model. Specifically, we show that
these transitions occur via generalized eigenvectors of the linearized
operator.
