The lognormal distribution has been fit to data on the abundance of species for geographically diverse communities of birds, intertidal organisms, insects and plants. However, it fails miserably on the following tropical data from Hubbell (1996).
Here, following up our research on species-area curves, we describe a new approach to the abundance of species based on the voter model with mutation. In this system each site x in the square lattice (the points in the plane with integer coordinates) is assigned a value in the interval (0,1) which describes the type of the individual there. The model has two mechanisms, invasion and mutation, that are described by the following rules:
(i) Each site x at rate 1 invades one of its four nearest neighbors y chosen at random and changes the value at y to the value at x.
(ii) Each site x at rate a mutates, changing to a new type w, chosen uniformly on (0,1).
It has been known for some time that this system has a unique stationary distribution.
In most situations, the rate at which new species enter the system through migration or genetic mutation will be small, so we will investigate the limiting behavior of the species abundance distribution as a approaches 0.