A New Model for Evolution in a Spatial Continuum

Nick H Barton (Institute of Science and Technology)
Alison M Etheridge (University of Oxford)
Amandine Véber (Université Paris 11)

Abstract


We investigate a new model for populations evolving in a spatial continuum. This model can be thought of as a spatial version of the Lambda-Fleming-Viot process. It explicitly incorporates both small scale reproduction events and large scale extinction-recolonisation events. The lineages ancestral to a sample from a population evolving according to this model can be described in terms of a spatial version of the Lambda-coalescent. Using a technique of Evans (1997), we prove existence and uniqueness in law for the model. We then investigate the asymptotic behaviour of the genealogy of a finite number of individuals sampled uniformly at random (or more generally `far enough apart') from a two-dimensional torus of sidelength L as L tends to infinity. Under appropriate conditions (and on a suitable timescale) we can obtain as limiting genealogical processes a Kingman coalescent, a more general Lambda-coalescent or a system of coalescing Brownian motions (with a non-local coalescence mechanism).

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

Pages: 162-216

Publication Date: February 3, 2010

DOI: 10.1214/EJP.v15-741

References

  1. Barton, N.H., Depaulis, F., and Etheridge, A.M. (2002). Neutral evolution in spatially continuous populations. Theor. Pop. Biol., 61:31--48.
  2. Barton, N.H., Kelleher, J., and Etheridge, A.M. (2009). A new model for large-scale population dynamics: quantifying phylogeography. Preprint.
  3. Berestycki, N., Etheridge, A.M., and Hutzenthaler, M. (2009). Survival, extinction and ergodicity in a spatially continuous population model. Markov Process. Related Fields, 15:265--288. MR2554364
  4. Bertoin, J. (1996). Lévy Processes. Cambridge University Press. MR1406564 (98e:60117)
  5. Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Probab. Theory Related Fields, 126:261--288. MR1990057 (2004f:60080)
  6. Bhattacharya, R.N. (1977). Refinements of the multidimensional central limit theorem and applications. Ann. Probab., 5:1--27. MR0436273 (55 #9220)
  7. Billingsley, P. (1995). Probability and Measure. Wiley. MR1324786 (95k:60001)
  8. Birkner, M., Blath, J., Capaldo, M., Etheridge, A.M., M\"ohle, M., Schweinsberg, J., and Wakolbinger, A. (2005). Alpha-stable branching and Beta-coalescents. Electron. J. Probab., 10:303--325. MR2120246 (2006c:60100)
  9. Cox, J.T. (1989). Coalescing random walks and voter model consensus times on the torus in Z^d. Ann. Probab., 17:1333--1366. MR1048930 (91d:60250)
  10. Cox, J.T. and Durrett, R. (2002). The stepping stone model: new formulas expose old myths. Ann. Appl. Probab., 12:1348--1377. MR1936596 (2003j:60138)
  11. Cox, J.T. and Griffeath, D. (1986). Diffusive clustering in the two-dimensional voter model. Ann. Probab., 14:347--370. MR0832014 (87j:60146)
  12. Cox, J.T. and Griffeath, D. (1990). Mean field asymptotics for the planar stepping stone model. Proc. London Math. Soc., 61:189--208. MR1051103 (92b:60098)
  13. Donnelly, P.J. and Kurtz, T.G. (1999). Particle representations for measure-valued population models. Ann. Probab., 27:166--205. MR1681126 (2000f:60108)
  14. Eller, E., Hawks, J., and Relethford, J.H. (2004). Local extinction and recolonization, species effective population size, and modern human origins. Human Biology, 76(5):689--709.
  15. Etheridge, A.M. (2008). Drift, draft and structure: some mathematical models of evolution. Banach Center Publ., 80:121--144. MR2433141 (2009m:60237)
  16. Ethier, S.N. and Kurtz, T.G. (1986). Markov processes: characterization and convergence. Wiley. MR0838085 (88a:60130)
  17. Evans, S.N. (1997). Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types. Ann. Inst. H. Poincaré Probab. Statist., 33:339--358. MR1457055 (98m:60126)
  18. Felsenstein, J. (1975). A pain in the torus: some difficulties with the model of isolation by distance. Amer. Nat., 109:359--368.
  19. Kimura, M. (1953). Stepping stone model of population. Ann. Rep. Nat. Inst. Genetics Japan, 3:62--63.
  20. Kingman, J.F.C. (1982). The coalescent. Stochastic Process. Appl., 13:235--248. MR0671034 (84a:60079)
  21. Limic, V. and Sturm, A. (2006). The spatial Lambda-coalescent. Electron. J. Probab., 11(15):363--393. MR2223040 (2007b:60183)
  22. Malécot, G. (1948). Les Mathématiques de l'hérédité. Masson et Cie, Paris. MR0027490 (10,314c)
  23. Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Probab., 29:1547--1562. MR1880231 (2003b:60134)
  24. Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab., 27:1870--1902. MR1742892 (2001h:60016)
  25. Ridler-Rowe, C.J. (1966). On first hitting times of some recurrent two-dimensional random walks. Z. Wahrsch. verw. Geb., 5:187--201. MR0199901 (33 #8041)
  26. Rogers, L.C.G. and Williams, D. (1987). Diffusions, Markov processes, and martingales: Itô calculus. Wiley. MR1780932 (2001g:60189)
  27. Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab., 26:1116--1125. MR1742154 (2001f:92019)
  28. Sawyer, S. and Fleischmann, J. (1979). The maximal geographical range of a mutant allele considered as a subtype of a Brownian branching random field. Proc. Natl. Acad. Sci. USA, 76(2):872--875.
  29. Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab., 5:1--50. MR1781024 (2002g:60113)
  30. Wilkins, J.F. (2004). A separation of timescales approach to the coalescent in a continuous population. Genetics, 168:2227--2244.
  31. Wilkins, J.F. and Wakeley, J. (2002). The coalescent in a continuous, finite, linear population. Genetics, 161:873--888.
  32. Wright, S. (1931). Evolution in Mendelian populations. Genetics, 16:97--159.
  33. Wright, S. (1943). Isolation by distance. Genetics, 28:114--138.
  34. Zähle, I., Cox, J.T., and Durrett, R. (2005). The stepping stone model II: genealogies and the infinite sites model. Ann. Appl. Probab., 15:671--699. MR2114986 (2006d:60157)
Bookmark and Share


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