  This paper deals with a model of sympatric speciation, that is, speciation in
the absence of geographical separation, originally proposed by U. Dieckmann and
M. Doebeli in 1999. We modify their original model to obtain a Fleming--Viot
type model and study its stationary distribution. We show that speciation may
occur, that is, the stationary distribution puts most of the mass on a
configuration that does not concentrate on the phenotype with maximum carrying
capacity, if competition between phenotypes is intense enough. Conversely, if
competition between phenotypes is not intense, then speciation will not occur
and most of the population will have the phenotype with the highest carrying
capacity. The length of time it takes speciation to occur also has a delicate
dependence on the mutation parameter, and the exact shape of the carrying
capacity function and the competition kernel.
