A Spatial Model for the Abundance of Species

The voter model, with mutations occurring at a positive rate α , has a unique equilibrium distribution. We investigate the logarithms of the relative abundance of species for these distributions in d ≥ 2. We show that, as α→ 0, the limiting distribution is right triangular in d = 2 and uniform in d ≥ 3. We also obtain more detailed results for the histograms that biologists use to estimate the underlying density functions.