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Branching diffusion processes in population genetics

Published online by Cambridge University Press:  01 July 2016

Stanley Sawyer
Stanley Sawyer*
Affiliation:
Yeshiva University, New York
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Abstract

A branching random field is considered as a model of either of two situations in genetics in which migration or dispersion plays a role. Specifically we consider the expected number of individuals NA in a (geographical) set A at time t, the covariance of NA and NB for two sets A, B, and the probability I(x, y, u) that two individuals found at locations x, y at time t are of the same genetic type if the population is subject to a selectively neutral mutation rate u. The last also leads to limit laws for the average degree of relationship of individuals in various types of branching random fields. We also find the equations that the mean and bivariate densities satisfy, and explicit formulas when the underlying migration process is Brownian motion.

Keywords

BRANCHING PROCESSBRANCHING RANDOM FIELDIDENTITY BY DESCENTGENETICSMIGRATIONPOISSON RANDOM FIELDPOINT PROCESSSTEPPING-STONE MODEL
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 4 , December 1976 , pp. 659 - 689
Copyright
Copyright © Applied Probability Trust 1976 

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