Separation of timescales for the seed bank diffusion and its jump-diffusion limit.

We investigate scaling limits of the seed bank model when migration (to and from the seed bank) is 'slow' compared to reproduction. This is motivated by models for bacterial dormancy, where periods of dormancy can be orders of magnitude larger than reproductive times. Speeding up time, we encounter a separation of timescales phenomenon which leads to mathematically interesting observations, in particular providing a prototypical example where the scaling limit of a continuous diffusion will be a jump diffusion. For this situation, standard convergence results typically fail. While such a situation could in principle be attacked by the sophisticated analytical scheme of Kurtz (J Funct Anal 12:55-67, 1973), this will require significant technical efforts. Instead, in our situation, we are able to identify and explicitly characterise a well-defined limit via duality in a surprisingly non-technical way. Indeed, we show that moment duality is in a suitable sense stable under passage to the limit and allows a direct and intuitive identification of the limiting semi-group while at the same time providing a probabilistic interpretation of the model. We also obtain a general convergence strategy for continuous-time Markov chains in a separation of timescales regime, which is of independent interest.

Statistical tools for seed bank detection.

We derive statistical tools to analyze the patterns of genetic variability produced by models related to seed banks; in particular the Kingman coalescent, its time-changed counterpart describing so-called weak seed banks, the strong seed bank coalescent, and the two-island structured coalescent. As (strong) seed banks stratify a population, we expect them to produce a signal comparable to population structure. We present tractable formulas for Wright's FST and the expected site frequency spectrum for these models, and show that they can distinguish between some models for certain ranges of parameters. We then use pseudo-marginal MCMC to show that the full likelihood can reliably distinguish between all models in the presence of parameter uncertainty under moderate stratification, and point out statistical pitfalls arising from stratification that is either too strong or too weak. We further show that it is possible to infer parameters, and in particular determine whether mutation is taking place in the (strong) seed bank.

Structural properties of the seed bank and the two island diffusion.

We investigate various aspects of the (biallelic) Wright-Fisher diffusion with seed bank in conjunction with and contrast to the two-island model analysed e.g. in Kermany et al. (Theor Popul Biol 74(3):226-232, 2008) and Nath and Griffiths (J Math Biol 31(8):841-851, 1993), including moments, stationary distribution and reversibility, for which our main tool is duality. Further, we show that the Wright-Fisher diffusion with seed bank can be reformulated as a one-dimensional stochastic delay differential equation, providing an elegant interpretation of the age structure in the seed bank also forward in time in the spirit of Kaj et al. (J Appl Probab 38(2):285-300, 2001). We also provide a complete boundary classification for this two-dimensional SDE using martingale-based reasoning known as McKean's argument.