We consider spatial stochastic population models, such as those modelling biological systems, which exhibit bistability, including several interacting particle systems and a variant of the Spatial Lambda Fleming-Viot process. In such models, narrow interfaces tend to form between regions dominated by one of the two stable states. To understand how the population evolves, we may study the dynamics of these interfaces in time. For several bistable population models, it is known from recent work that the limiting interfaces, under certain rescalings, converge to a geometric evolution called mean curvature flow. Surfaces evolving by mean curvature flow develop singularities in finite time, which imposes a short-time constraint and regularity assumptions on the previous convergence results.

In this talk, I will first discuss some models which exhibit this phenomenon, and results concerning their interfaces. I will then discuss a recent work which uses tools from analysis, in particular level-set methods and the theory of viscosity solutions, to improve upon recent interface convergence results for a broad class of bistable stochastic population models. In particular, we give checkable conditions on an ancestral dual process which guarantee that the interfaces converge globally in time to a generalized mean curvature flow.

This is joint work with Jessica Lin (McGill).