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 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.