The island model with stochastically variable migration rate and immigrant
gene frequency is investigated. It is supposed that the migration rate and the
immigrant gene frequency are independent of each other in each generation,
and each of them is independently and identically distributed in every generation. The treatment is confined to a single diallelic locus without mutation.
If the diploid population is infinite, selection is absent and the immigrant gene
frequency is fixed, then the gene frequency on the island converges to the
immigrant frequency, and the logarithm of the absolute value of its deviation
from it is asymptotically normally distributed. Assuming only neutrality, the
evolution of the exact mean and variance of the gene frequency are derived
for an island with finite population. Selection is included in the diffusion
approximation: if all evolutionary forces have comparable roles, the gene frequency will be normally distributed at all times. All results in the paper are
We investigated various cases of the island model with stochastic migration. If the population is infinite, the immigrants have a fixed gene frequency and the alleles are neutral, the gene frequency on the island converges to that of the immigrants.
What this means is that the genes initially on the model island, in effect, disappeared.