We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $\Lambda $-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.
"Branching Brownian motion, mean curvature flow and the motion of hybrid zones." Electron. J. Probab. 22 1 - 40, 2017. https://doi.org/10.1214/17-EJP127