The ancestral process of long term seed bank models

Details The ancestral process of long term seed bank models The ancestral process of long term seed bank models

2012-03-23

arxiv.org/abs/1203.5202v1

Mathematics

Probability

23 pages, 2 figures

Scientific paper

We present a new model for the evolution of genetic types in the presence of so-called seed banks, i.e., where individuals may obtain their genetic type from ancestors which have lived in the near as well as the very far past. The classical Wright-Fisher model, as well as a seed bank model with bounded age distribution considered by Kaj, Krone and Lascoux (2001) are special cases of our model. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. We prove that for age distributions with finite mean, the rescaled ancestral process converges to a time-changed Kingman coalescent, while in the case of infinite mean, ancestral lineages might not merge at all with positive probability. Further, we present a construction of the forward in time process in equilibrium. The mathematical methods are based on renewal theory, the urn process introduced by Kaj et. al., as well as on a Gibbsian approach introduced by Hammond and Sheffield (2011) in a different context. Our model has already drawn interest by biologists, who suggest that it can explain, at least on a principal level, increased levels of genetic diversity in a bacterial species, Azotobacter vinelandii (see Gonz\'alez Casanova et. al., (2012)).

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