A mean field model for species abundance

In this paper, we use the multitype mean field voter model as a model of species interaction, to obtain results about species abundance. Briefly, we start with the complete graph on n vertices, Cn, with each site occupied by a particle. Particles are represented by a value in (0,1), where distinct values represent different species. Particles then undergo mutation at rate α, and are relabelled with a value chosen uniformly from (0,1). Particles also give birth at rate 1, and invade any of the other n sites randomly. This process has a unique stationary distribution denoted by ξ∞n, which is given by the Ewens sampling formula. For each value in (0,1) that is present in ξ∞n, we count the number of particles represented by the same value, and call that the patch size of the species. Let Kn[a,b] denote the number of species with patch size in [a,b]. We study the limiting distribution of Kn[a,b], for certain values of a and b, as the mutation rate α tends to 0, which will in turn force n→∞.