Bifurcation of finger-like structures in traveling waves of epithelial tissues spreading

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.