A continuous migration model with stable demography

Abstract

A probability model of a population undergoing migration, mutation, and mating in a geographic continuum R is constructed, and an integro-differential equation is derived for the probability of genetic identity. The equation is solved in one case, and asymptotic analysis done in others. Individuals at x, y epsilon R in the model mate with probability V(x, y) dt in any time interval (t, t + dt). In two dimensions, if V(x, y) = V(x - y) where V(x) approximately V(x/beta)/beta2 approaches a delta function, the equilibrium probability of identity vanishes as beta Leads to 0. The asymptotic rate at which this occurs is discussed for mutation rates u = u0 Greater than 0 and for beta approximately cua, alpha Greater than 0, and u Leads to 0.