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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Blath J, González Casanova A, Eldon B, Kurt N, Wilke-Berenguer M (2015) Genetic variability under the seedbank coalescent. Genetics 200(3):921–934
Blath J, González Casanova A, Kurt N, Wilke-Berenguer M (2016) A new coalescent for seed-bank models. Ann Appl Probab 26(2):857–891
den Hollander F, Pederzani G (2017) Multi-colony Wright–Fisher with seed-bank. Indag Math 28(3):637–669
Etheridge A (2011) Some mathematical models from population genetics, volume 2012 of lecture notes in mathematics. Springer, Heidelberg. Lectures from the 39th Probability Summer School held in Saint-Flour, 2009. École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School]
Etheridge AM, Griffiths RC (2009) A coalescent dual process in a Moran model with genic selection. Theor Popul Biol 75(4):320–330 (Sam Karlin: Special Issue)
Etheridge AM, Griffiths RC, Taylor JE (2010) A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit. Theor Popul Biol 78(2):77–92
Ethier SN, Kurtz TG (1992) On the stationary distribution of the neutral diffusion model in population genetics. Ann Appl Probab 2(1):24–35
Filipović D, Larsson M (2016) Polynomial diffusions and applications in finance. Finance Stoch 20(4):931–972
Fu R, Gelfand AE, Holsinger KE (2003) Exact moment calculations for genetic models with migration, mutation, and drift. Theor Popul Biol 63(3):231–243 (Uses of DNA and genetic markers for forensics and population studies)
González Casanova A, Spanò D (2018) Duality and fixation for \(\Xi \)-Wright-Fisher processes with frequency-dependent selection. Ann Appl Probab 28(1):250–284
Griffiths RC, Jenkins PA, Lessard S (2016) A coalescent dual process for a Wright–Fisher diffusion with recombination and its application to haplotype partitioning. Theor Popul Biol 112(Supplement C):126–138
Hildebrandt TH, Schoenberg IJ (1933) On linear functional operations and the moment problem for a finite interval in one or several dimensions. Ann Math 34(2):317–328
Jansen S, Kurt N (2014) On the notion(s) of duality for Markov processes. Probab Surv 11:59–120
Kaj I, Krone SM, Lascoux M (2001) Coalescent theory for seed bank models. J Appl Probab 38(2):285–300
Kermany ARR, Zhou X, Hickey DA (2008) Joint stationary moments of a two-island diffusion model of population subdivision. Theor Popul Biol 74(3):226–232
Krone SM, Neuhauser C (1997) Ancestral processes with selection. Theor Popul Biol 51(3):210–237
Lambert A, Ma C (2015) The coalescent in peripatric metapopulations. J Appl Probab 52(2):538–557
Larsson M, Pulido S (2017) Polynomial diffusions on compact quadric sets. Stoch Process Appl 127(3):901–926
Lennon JT, Jones SE (2011) Microbial seed banks: the ecological and evolutionary implications of dormancy. Nat Rev Microbiol 9(2):119–130
Maisonneuve B (1977) Une mise au point sur les martingales locales continues définies sur un intervalle stochastique. In: Lecture notes in mathematics, vol 581, pp 435–445
Mayerhofer E, Pfaffel O, Stelzer R (2011) On strong solutions for positive definite jump diffusions. Stoch Process Appl 121(9):2072–2086
Moran PAP (1959) The theory of some genetical effects of population subdivision. Austral J Biol Sci 12(2):109–116
Nath HB, Griffiths RC (1993) The coalescent in two colonies with symmetric migration. J Math Biol 31(8):841–851
Revuz D, Yor M (1999) Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], 3rd edn. Springer, Berlin
Shiga T, Shimizu A (1980) Infinite-dimensional stochastic differential equations and their applications. J Math Kyoto Univ 20(3):395–416
Shoemaker WR, Lennon JT (2018) Evolution with a seed bank: the population genetic consequences of microbial dormancy. Evol Appl. 11(1):60–75
Wakeley J (ed) (2008) Coalescent theory: an introduction. Roberts and Company, Greenwood Village, CO
Wright S (1931) Evolution in Mendelian populations. Genetics 16(2):97–159
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.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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
- Wright–Fisher diffusion
- Seed bank coalescent
- Two island model
- Boundary classification
- Stochastic delay differential equation
Mathematics Subject Classification
- Primary 60K35
- Secondary 92D10