Our motivation in this work is to understand the influence of the spatial structure of a population on the efficacy of natural selection acting upon it. From a biological perspective, what is interesting is that whereas when population density is high, the probability that a selectively favoured genetic type takes over a population is independent of spatial dimension, when population density is low, this is no longer the case and spatial dimension plays an important role. The proofs are of independent mathematical interest: for example, in one dimension we find a new route to the Brownian net, from a continuum model of branching and coalescing lineages. In the biologically most interesting setting of two spatial dimensions, as we rescale our continuum model there is a finely balanced tradeoff between branching and coalescing lineages, eventually resulting in a branching Brownian motion.
