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

TL;DR

This paper explores the properties of the Wright-Fisher diffusion model with seed bank, analyzing its moments, stationary distribution, and reversibility, and reformulates it as a stochastic delay differential equation to better understand its age structure.

Contribution

It introduces a novel reformulation of the seed bank diffusion as a stochastic delay differential equation and provides a comprehensive boundary classification for the model.

Findings

Reformulation as a stochastic delay differential equation.

Complete boundary classification using martingale methods.

Analysis of moments, stationary distribution, and reversibility.

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.