Branching Diffusion Processes in Population Genetics
S. Sawyer. Advances in Applied Probability8 (4):
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 in Brownian motion.