  The behavior of the Poisson-Dirichlet distribution with small mutation rate
is studied through large deviations. The structure of the rate function
indicates that the number of alleles is finite at the instant when mutation
appears. The large deviation results are then used to study the asymptotic
behavior of the homozygosity, and the Poisson-Dirichlet distribution with
symmetric selection. The latter shows that several alleles can coexist when
selection intensity goes to infinity in a particular way as the mutation rate
approaches zero.
