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

Published online by Cambridge University Press:  14 July 2016

Stanley Sawyer
Stanley Sawyer*
Affiliation:
University of Washington
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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. Let N(t) be the (random) size of a block intersecting a fixed colony at time t. Then E[N(t)] grows like √t in one dimension, t/log t in two, and t in three or more dimensions. On the other hand, each block by itself eventually becomes extinct. In two or more dimensions, we prove that N(t)/E[N(t)] has a limiting gamma distribution, and thus the mortality of blocks does not make the limiting distribution of N(t) singular. Results are proven for discrete time and sketched for continuous time.

If a mutation rate u > 0 is imposed, the ‘block structure' has an equilibrium distribution. If N(u) is the size of a block intersecting a fixed colony at equilibrium, then as u → 0 N(u)/E[N(u)] has a limiting exponential distribution in two or more dimensions. In biological systems u ≈ 10–6 is usually quite small.

The proofs are by using multiple kinship coefficients for a stepping stone population.

Keywords

POPULATION GENETICSMIGRATIONPOINT PROCESSESRATE OF CONSOLIDATIONSTEPPING STONELIMIT LAWGAMMA DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 3 , September 1979 , pp. 482 - 495
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research partially supported by the National Science Foundation under grant number MCS 75–08098–A01.

∗

Present address: Purdue University, W. Lafayette, IN 47907, U.S.A.

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