The distribution of surviving blocks of an ancestral genome

What is the chance that some part of a stretch of genome will survive? In a population of constant size, and with no selection, the probability of survival of some part of a stretch of map length y<1 approaches y/log( yt/2) for log( yt) 1. Thus, the whole genome is certain to be lost, but the rate of loss is extremely slow. This solution extends to give the whole distribution of surviving block sizes as a function of time. We show that the expected number of blocks at time t is 1+yt and give expressions for the moments of the number of blocks and the total amount of genome that survives for a given time. The solution is based on a branching process and assumes complete interference between crossovers, so that each descendant carries only a single block of ancestral material. We consider cases where most individuals carry multiple blocks, either because there are multiple crossovers in a long genetic map, or because enough time has passed that most individuals in the population are related to each other. For species such as ours, which have a long genetic map, the genome of any individual which leaves descendants ( 80% of the population for a Poisson offspring number with mean two) is likely to persist for an extremely long time, in the form of a few short blocks of genome.