The impact of selection in the $\Lambda$-Wright-Fisher model

Clément Foucart (Humboldt-Universität zu Berlin)


The purpose of this article is to study some asymptotic properties of the $\Lambda$-Wright-Fisher process with selection. This process represents the frequency of a disadvantaged allele. The resampling mechanism is governed by a finite measure $\Lambda$ on $[0,1]$ and selection by a parameter $\alpha$. When the measure $\Lambda$ obeys $\int_{0}^{1}-\log(1-x)x^{-2}\Lambda(dx)<\infty$, some particular behaviour in the frequency of the allele can occur. The selection coefficient $\alpha$ may be large enough to override the random genetic drift. In other words, for certain selection pressure, the disadvantaged allele will vanish asymptotically with probability one. This phenomenon cannot occur in the classical Wright-Fisher diffusion. We study the dual process of the $\Lambda$-Wright-Fisher process with selection and prove this result through martingale arguments.

There is an Erratum in ECP volume 19 paper 15 (2014).

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-10

Publication Date: August 24, 2013

DOI: 10.1214/ECP.v18-2838


  • B. Bah and E. Pardoux. λ-look-down model with selection. Preprint, 2012. arXiv.1303.1953.
  • Berestycki, Nathanaël. Recent progress in coalescent theory. Ensaios Matemáticos [Mathematical Surveys], 16. Sociedade Brasileira de Matemática, Rio de Janeiro, 2009. 193 pp. ISBN: 978-85-85818-40-1 MR2574323
  • Bertoin, Jean; Le Gall, Jean-François. Stochastic flows associated to coalescent processes. II. Stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 3, 307--333. MR2139022
  • Birkner, Matthias; Blath, Jochen. Measure-valued diffusions, general coalescents and population genetic inference. Trends in stochastic analysis, 329--363, London Math. Soc. Lecture Note Ser., 353, Cambridge Univ. Press, Cambridge, 2009. MR2562160
  • Dawson, Donald A.; Li, Zenghu. Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2012), no. 2, 813--857. MR2952093
  • R. Der, C. Epstein, and J. Plotkin. Dynamics of neutral and selected alleles when the offspring diffusion is skewed. Genetics, May 2012.
  • R. Der, C. L. Epstein, and J. B. Plotkin. Generalized population models and the nature of genetic drift. Theoretical Population Biology, 80(2):80 -- 99, 2011.
  • Etheridge, Alison. Some mathematical models from population genetics. Lectures from the 39th Probability Summer School held in Saint-Flour, 2009. Lecture Notes in Mathematics, 2012. Springer, Heidelberg, 2011. viii+119 pp. ISBN: 978-3-642-16631-0 MR2759587
  • A. M. Etheridge, R. C. Griffiths, and J. E. Taylor. A coalescent dual process in a moran model with genic selection, and the lambda coalescent limit. Theoretical Population Biology, 78(2):77 -- 92, 2010.
  • Foucart, Clément. Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration. Adv. in Appl. Probab. 43 (2011), no. 2, 348--374. MR2848380
  • Gnedin, Alexander; Iksanov, Alexander; Marynych, Alexander. On $\Lambda$-coalescents with dust component. J. Appl. Probab. 48 (2011), no. 4, 1133--1151. MR2896672
  • S. Jansen and N. Kurt. On the notion(s) of duality for markov processes. 2012. arXiv.1210.7193.
  • C. Krone and S. Neuhauser. Ancestral processes with selection. Theoretical Population Biology, 51(3):210--37, 1997.
  • Lagerås, Andreas Nordvall. A population model for $\Lambda$-coalescents with neutral mutations. Electron. Comm. Probab. 12 (2007), 9--20 (electronic). MR2284043
  • M. Möhle and P. Herriger. Conditions for exchangeable coalescents to come down from infinity. ALEA Lat. Am. J. Probab. Math. Stat., 9:637--665, 2012.
  • Pitman, Jim. Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, 1870--1902. MR1742892
  • Schweinsberg, Jason. A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Comm. Probab. 5 (2000), 1--11 (electronic). MR1736720

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.