Reaction-diffusion problems : from front dynamics to genealogies (6)

The last lecture dealt with genealogies in models of sexual and asexual evolution. In the case of sexual reproduction, the tree of ancestors very quickly reveals loops when we go back into the past. In the case of neutral evolution, for a population ofconstant size  N, you need to go back some 1.44 log N generations to find an ancestor common to the whole population, and 2.55 log N generations for the whole population to have all its ancestors in common. In the case of asexual reproduction, the Wright-Fisher model, which describes neutral evolution, predicts much longer coalescence times, of the order  N, the size of the population. These times fluctuate even when the population size is large, and the trees have a statistic given by Kingman's coalescent. In the case of asexual evolution in the presence of selection, as in N-BBM, several approaches, either numerical or analytical, indicate that tree statistics are modified by selection and are given by the Bolthausen-Sznitman coalescent. Coalescence times in the presence of selection are much shorter, on the order of (log N ) ³.