Branching diffusion processes in population genetics

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 individualsN_Ain a (geographical) setAat timet, the covariance ofN_AandN_Bfor two setsA, B, and the probabilityI(x, y, u) that two individuals found at locationsx, yat timetare of the same genetic type if the population is subject to a selectively neutral mutation rateu.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.