Spatial stochastic mean-field models and applications

A longstanding challenge in ecology, and many other natural sciences, is to bridge the gap between detailed microscale descriptions of systems and more tractable models of their macroscale behavior. We will outline a stochastic mean-field framework for building spatial models in ecological and biological applications where nonlocal interactions play a crucial role. Our first application of this theory generalizes a well-known (non-spatial) model of forest-savanna dynamics and allows us to understand transient dynamics and heterogeneity-driven phenomena. As a second application, we provide a rigorous microscale derivation for a popular class of cell-to-cell adhesion models and illustrate this theory with concrete examples. Nonlocal models of cell-to-cell adhesion have helped explain experimental observations in embryogenesis, cell sorting, and wound healing problems.