Spread of an Advantageous Gene

We wish to model the spread of genetic innovation through a spatially structured population. As genetic models are naturally specified in terms of the interactions of individuals, we begin by constructing a discrete-site particle model with migration and Moran-like reproduction interactions between pairs of particles within each site. For many purposes, such a discrete model is intractable, so we explore a diffusion approximation that results as the limit of a suitably scaled sequence of such discrete models. In some sense, this diffusion limit represents the interaction of an infinitely dense collection of particles. By means of machinery developed by Donnelly and Kurtz in [1], [2], and [3], we can make this intuition precise, and a natural geometric interpretation of the diffusion process emerges: particles arranged in a homogeneous Poisson field on R × R evolve as independent horizontal Brownian motions and interact via Poisson counting processes (either with constant rate or driven by local times of functions of the particle locations).