A limit theorem for patch sizes in a selectively-neutral migration model

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Abstract
Assume a population is distributed in an infinite lattice of colonies in a migration and random mating model in which all individuals are selectively equivalent. In one and two dimensions, the population tends in time to consolidate into larger and larger blocks, each composed of the descendants of a single initial individual. LetN(t) be the (random) size of a block intersecting a fixed colony at timet.ThenE[N(t)] grows like √tin one dimension,t/logtin two, andtin three or more dimensions. On the other hand, each block by itself eventually becomes extinct. In two or more dimensions, we prove thatN(t)/E[N(t)] has a limiting gamma distribution, and thus the mortality of blocks does not make the limiting distribution ofN(t) singular. Results are proven for discrete time and sketched for continuous time.If a mutation rateu> 0 is imposed, the ‘block structure' has an equilibrium distribution. IfN(u) is the size of a block intersecting a fixed colony at equilibrium, then asu→ 0N(u)/E[N(u)] has a limiting exponential distribution in two or more dimensions. In biological systemsu≈ 10–6is usually quite small.The proofs are by using multiple kinship coefficients for a stepping stone population.