One of the oldest facts in the modern science of Ecology is that
that when one plots the number of species S versus area A on
log-log paper a straightline results. For a concrete example, consider the
following data from the Galapagos Islands. Here the slope (power) of the
fitted line is 0.336.
There have been many explanations proposed for this phenomenon. Here,
we suggest a new one based on a simple interacting particle system,
the voter model with mutation. In this model, each point in the two
dimensional integer lattice is in a state in (0,1) indicating the
type of the individual at x.
(i) At rate 1, a site x changes its value to the state of a site y chosen according to a probability distribution p(x,y) = f(|y-x|), where f has finite variance.
(ii) A state mutates to a new type chosen at random from (0,1) at rate alpha.
The local dynamics of this model are not very realisitic if one is thinking of the competition of individuals of a plant or animal species. However, it is an explicit spatial model that incorporates a balance between colonization and extinction, the key ingredients in MacArthur and Wilson's theory. The first result we must prove about this model is:
Proposition 1. The multitype voter model with mutation has a unique stationary distribution.
The next figure shows a simulation of the stationary distribution for
the model on a 100 by 100 lattice with alpha = 0.001. There are 117
colors in the equilibirium state but only 16 in our color map.
