  Phase-field models have recently had great success in describing the dynamic
morphologies and motility of eukaryotic cells. In this work we investigate the
minimal phase-field model introduced in [Berlyand, Potomkin, Rybalko (2017)].
Rigorous analysis of its sharp interface limit dynamics was completed in
[Mizuhara, Berlyand, Rybalko, Zhang (2016); Mizuhara, Zhang (2019)], where it
was observed that persistent cell motion was not stable. In this work we
numerically study the pre-limiting phase-field model near the sharp interface
limit, to better understand this lack of persistent motion. We find that
immobile, persistent, and rotating states are all exhibited in this minimal
model, and investigate the loss of persistent motion in the sharp interface
limit. In addition we study cell speed as a function of biophysical parameters.
