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

Abstract: 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, Zhou and Hickey, 2008, and Nath and Griffiths, 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, Krone and Lascoux, 2001. We also provide a complete boundary classification for this two-dimensional SDE using martingale-based reasoning known as McKean's argument.