  We consider a continuum active polar fluid model for the spreading of
epithelial monolayers introduced by R. Alert, C. Blanch-Mercader, and J.
Casademunt, 2019. The corresponding free boundary problem possesses flat front
traveling wave solutions. Linear stability of these solutions under periodic
perturbations is considered. It is shown that the solutions are stable for
short-wave perturbations while exhibiting long-wave instability under certain
conditions on the model parameters (if the traction force is sufficiently
strong). Then, considering the prescribed period as the bifurcation parameter,
we establish the emergence of nontrivial traveling wave solutions with a
finger-like periodic structure (pattern). We also construct asymptotic
expansions of the solutions in the vicinity of the bifurcation point and study
their stability. We show that, depending on the value of the contractility
coefficient, the bifurcation can be a subcritical or a supercritical pitchfork.
