Quasi-stationary distributions of a pair of markov chains related to time evolution of a dna locus

We consider a pair of Markov chains representing statistics of the Fisher-Wright-Moran model with mutations and drift. The chains have absorbing state at 0 and are related by the fact that some random time + ago they were identical, evolving as a single Markov chain with values in {0, 1, ...J; from that time on they began to evolve independently, conditional on a state at the time of split, according to the same transition probabilities. The distribution of + is a function of deterministic effective population size 2N (-). We study the impact of demographic history on the shape of the quasi-stationary distribution, conditional on nonabsorption at the margin (where one of the chains is at 0), and on the speed with which the probability mass escapes to the margin.