Spatial Moran models with local interactions

We begin by considering a genetic stepping stone model with sites on the one-dimensional lattice n−1$Z$. Within each site, particles are subject to Moran model interactions with selection. Between interactions, they mutate and migrate independently. If the population density is held constant while the lattice density n increases, then—under suitable parameter scalings—the migration random walks converge to Brownian motions and the limiting interactions are determined by Poisson counting processes driven by clocks proportional to the local times at zero of the distances between pairs. The result is a finite-density collection of Brownian motions with local-time Moran interactions.; We study the limiting behavior of these models as the population density increases to infinity by ordering the particles with randomly assigned “levels” in the non-negative reals $R+$. If neutral interactions are restricted to occur in only one direction, so that the higher-level particle changes its type to that of the lower-level particle, the result is an ordered model that can be extended to infinite densities.; Restricted to a given maximum level, these infinite-density ordered models have the same empirical location/type distributions as the original, symmetric Moran models. In the stepping-stone case, we establish this by means of a generator argument. In the Brownian case, we establish it through a more direct coupling.; Under appropriate initial conditions, these ordered models have a simple Poisson structure. In the Brownian case, for each t, there exists a measure-valued diffusion ν_t such that the point process consisting of the location, type, and level of each particle is conditionally Poisson with mean measure ν _t × $\ell R+$.; We study this diffusion process for the Brownian case with selection, showing that it almost surely has continuous paths and giving a martingale characterization.