Bulletin of Mathematical Biology

, Volume 75, Issue 11, pp 2003–2027 | Cite as

The Common Ancestor Process Revisited

  • Sandra Kluth
  • Thiemo Hustedt
  • Ellen BaakeEmail author
Original Article


We consider the Moran model in continuous time with two types, mutation, and selection. We concentrate on the ancestral line and its stationary type distribution. Building on work by Fearnhead (J. Appl. Probab. 39 (2002), 38–54) and Taylor (Electron. J. Probab. 12 (2007), 808–847), we characterise this distribution via the fixation probability of the offspring of all individuals of favourable type (regardless of the offspring’s types). We concentrate on a finite population and stay with the resulting discrete setting all the way through. This way, we extend previous results and gain new insight into the underlying particle picture.


Moran model Ancestral process with selection Ancestral line Common ancestor process Fixation probabilities 



It is our pleasure to thank Anton Wakolbinger for enlightening discussions, and for Fig. 2. We are grateful to Jay Taylor for valuable comments on the manuscript, and to Barbara Gentz for pointing out a gap in an argument at an earlier stage of the work. This project received financial support by Deutsche Forschungsgemeinschaft (DFG-SPP 1590), Grant no. BA2469/5-1.


  1. Baake, E., & Bialowons, R. (2008). Ancestral processes with selection: branching and Moran models. Banach Cent. Publ., 80, 33–52. MathSciNetCrossRefzbMATHGoogle Scholar
  2. Barton, N. H., Etheridge, A. M., & Sturm, A. K. (2004). Coalescence in a random background. Ann. Appl. Probab., 14, 754–785. MathSciNetCrossRefzbMATHGoogle Scholar
  3. Birkhoff, G., & Rota, G. (1969). Ordinary differential equations (2nd ed.). Lexington: Xerox College Publ. zbMATHGoogle Scholar
  4. Durrett, R. (2002). Probability models for DNA sequence evolution. New York: Springer. CrossRefzbMATHGoogle Scholar
  5. Durrett, R. (2008). Probability models for DNA sequence evolution (2nd ed.). New York: Springer. CrossRefzbMATHGoogle Scholar
  6. Etheridge, A. M., & Griffiths, R. C. (2009). A coalescent dual process in a Moran model with genic selection. Theor. Popul. Biol., 75, 320–330. CrossRefzbMATHGoogle Scholar
  7. Etheridge, A. M., Griffiths, R. C., & Taylor, J. E. (2010). A coalescent dual process in a Moran model with genic selection, and the Lambda coalescent limit. Theor. Popul. Biol., 78, 77–92. CrossRefGoogle Scholar
  8. Ewens, W. J. (2004). Mathematical population genetics. I. Theoretical introduction (2nd ed.). New York: Springer. CrossRefzbMATHGoogle Scholar
  9. Fearnhead, P. (2002). The common ancestor at a nonneutral locus. J. Appl. Probab., 39, 38–54. MathSciNetCrossRefzbMATHGoogle Scholar
  10. Ford, L. R. (1955). Differential equations (2nd ed.). New York: McGraw-Hill. zbMATHGoogle Scholar
  11. Karlin, S., & Taylor, H. M. (1981). A second course in stochastic processes. San Diego: Academic Press. zbMATHGoogle Scholar
  12. Kingman, J. F. C. (1982a). The coalescent. Stoch. Process. Appl., 13, 235–248. MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kingman, J. F. C. (1982b). On the genealogy of large populations. J. Appl. Probab., 19A, 27–43. MathSciNetCrossRefzbMATHGoogle Scholar
  14. Krone, S. M., & Neuhauser, C. (1997). Ancestral processes with selection. Theor. Popul. Biol., 51, 210–237. CrossRefzbMATHGoogle Scholar
  15. Mano, S. (2009). Duality, ancestral and diffusion processes in models with selection. Theor. Popul. Biol., 75, 164–175. CrossRefzbMATHGoogle Scholar
  16. Neuhauser, C., & Krone, S. M. (1997). The genealogy of samples in models with selection. Genetics, 145, 519–534. Google Scholar
  17. Norris, J. R. (1999). Markov chains. Cambridge: Cambridge University Press. zbMATHGoogle Scholar
  18. Pokalyuk, C., & Pfaffelhuber, P. (2013). The ancestral selection graph under strong directional selection. Theor. Popul. Biol. 87, 25–33. CrossRefzbMATHGoogle Scholar
  19. Stephens, M., & Donnelly, P. (2003). Ancestral inference in population genetics models with selection. Aust. N. Z. J. Stat., 45, 901–931. MathSciNetCrossRefzbMATHGoogle Scholar
  20. Taylor, J. E. (2007). The common ancestor process for a Wright-Fisher diffusion. Electron. J. Probab., 12, 808–847. MathSciNetCrossRefzbMATHGoogle Scholar
  21. Vogl, C., & Clemente, F. (2012). The allele-frequency spectrum in a decoupled Moran model with mutation, drift, and directional selection, assuming small mutation rates. Theor. Popul. Biol., 81, 197–209. CrossRefzbMATHGoogle Scholar
  22. Wakeley, J. (2008). Conditional gene genealogies under strong purifying selection. Mol. Biol. Evol., 25, 2615–2626. CrossRefGoogle Scholar
  23. Wakeley, J., & Sargsyan, O. (2009). The conditional ancestral selection graph with strong balancing selection. Theor. Popul. Biol., 75, 355–364. CrossRefzbMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Technische FakultätUniversität BielefeldBielefeldGermany

Personalised recommendations