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.

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Acknowledgements

The authors would like to thank S. Pulido for pointing out the connection to polynomial diffusions and the reviewers for helpful suggestions. JB and MWB were supported by DFG Priority Programme 1590 “Probabilistic Structures in Evolution”, Project BL 1105/5-1, MWB also by Project KU 2886/1-1 awarded to N. Kurt. MWB would like to thank the TU Berlin for their hospitality. EB received support from the Berlin Mathematical School and the DFG RTG 1845 “Stochastic Analysis with applications in biology, finance and physics”. AGC was supported by CONACYT, Project FC-2016-1946.

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Correspondence to Maite Wilke-Berenguer.

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Blath, J., Buzzoni, E., González Casanova, A. et al. Structural properties of the seed bank and the two island diffusion. J. Math. Biol. 79, 369–392 (2019). https://doi.org/10.1007/s00285-019-01360-5

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Keywords

  • Wright–Fisher diffusion
  • Seed bank coalescent
  • Two island model
  • Boundary classification
  • Duality
  • Reversibility
  • Stochastic delay differential equation

Mathematics Subject Classification

  • Primary 60K35
  • Secondary 92D10