A spatial model for the abundance of species

Abstract

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