Coalescent theory for age-structured random mating populations with two sexes

Consider a population that develops over units of time labelled by zero and the negative integers. It is assumed that at any time r≤0 there are respectively N(m,a) males of age a and N(f,b) of females of age b, where a=1,…,A(m) and b=1,…,A(f). At time 0 a sample of n copies of a gene are assumed to be observed, where n≪min(ab)(N(m,a),N(f,b)). It is assumed that at any particular time r any possible mating is equally probable and that numbers of gametes contributed to offspring of age 1 and sex u by parents of sex v are exchangeable within age groups and independently distributed among age-groups. Coalescent theory is then derived, with time measured in multiples 2N(eC) of the effective population size N(eC), which depends on a measure T of the generation interval. Theory is developed for both autosomal and sex-linked loci in two special cases.