The ancestral process of long term seed bank models

TL;DR

This paper introduces a new seed bank model incorporating ancestral lineages from both recent and distant past, revealing different genetic behaviors depending on the age distribution's properties.

Contribution

It generalizes existing seed bank models by allowing for a broader age distribution and characterizes the ancestral process in different regimes, including convergence to a time-changed Kingman coalescent.

Findings

Finite mean age distribution leads to convergence to a time-changed Kingman coalescent.

Infinite mean age distribution may prevent ancestral lineages from merging.

Different parameter regimes significantly affect genetic variability and fixation times.

Abstract

We present a new model for seed banks, where direct ancestors of individuals may 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 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 paper by Hammond and Sheffield (2011).