Electronic Journal of Probability

A New Model for Evolution in a Spatial Continuum

Nick Barton, Alison Etheridge, and Amandine Véber

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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).

Article information

Electron. J. Probab. Volume 15 (2010), paper no. 7, 162-216.

Accepted: 3 February 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 92D10: Genetics {For genetic algebras, see 17D92} 92D15: Problems related to evolution

genealogy evolution multiple merger coalescent spatial continuum spatial Lambda-coalescent generalised Fleming-Viot process

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


Barton, Nick; Etheridge, Alison; Véber, Amandine. A New Model for Evolution in a Spatial Continuum. Electron. J. Probab. 15 (2010), paper no. 7, 162--216. doi:10.1214/EJP.v15-741. https://projecteuclid.org/euclid.ejp/1464819792.

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  • Barton, N.H., Depaulis, F., and Etheridge, A.M. (2002). Neutral evolution in spatially continuous populations. Theor. Pop. Biol., 61:31–48.
  • Barton, N.H., Kelleher, J., and Etheridge, A.M. (2009). A new model for large-scale population dynamics: quantifying phylogeography. Preprint.
  • 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.
  • Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
  • Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Probab. Theory Related Fields, 126:261–288.
  • Bhattacharya, R.N. (1977). Refinements of the multidimensional central limit theorem and applications. Ann. Probab., 5:1–27.
  • Billingsley, P. (1995). Probability and Measure. Wiley.
  • Birkner, M., Blath, J., Capaldo, M., Etheridge, A.M., Möhle, M., Schweinsberg, J., and Wakolbinger, A. (2005). Alpha-stable branching and Beta-coalescents. Electron. J. Probab., 10:303–325.
  • Cox, J.T. (1989). Coalescing random walks and voter model consensus times on the torus in Z^d. Ann. Probab., 17:1333–1366.
  • Cox, J.T. and Durrett, R. (2002). The stepping stone model: new formulas expose old myths. Ann. Appl. Probab., 12:1348–1377.
  • Cox, J.T. and Griffeath, D. (1986). Diffusive clustering in the two-dimensional voter model. Ann. Probab., 14:347–370.
  • Cox, J.T. and Griffeath, D. (1990). Mean field asymptotics for the planar stepping stone model. Proc. London Math. Soc., 61:189–208.
  • Donnelly, P.J. and Kurtz, T.G. (1999). Particle representations for measure-valued population models. Ann. Probab., 27:166–205.
  • 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.
  • Etheridge, A.M. (2008). Drift, draft and structure: some mathematical models of evolution. Banach Center Publ., 80:121–144.
  • Ethier, S.N. and Kurtz, T.G. (1986). Markov processes: characterization and convergence. Wiley.
  • 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.
  • Felsenstein, J. (1975). A pain in the torus: some difficulties with the model of isolation by distance. Amer. Nat., 109:359–368.
  • Kimura, M. (1953). Stepping stone model of population. Ann. Rep. Nat. Inst. Genetics Japan, 3:62–63.
  • Kingman, J.F.C. (1982). The coalescent. Stochastic Process. Appl., 13:235–248.
  • Limic, V. and Sturm, A. (2006). The spatial Lambda-coalescent. Electron. J. Probab., 11(15):363–393.
  • Malécot, G. (1948). Les Mathématiques de l'hérédité. Masson et Cie, Paris. (10,314c)
  • Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Probab., 29:1547–1562.
  • Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab., 27:1870–1902.
  • Ridler-Rowe, C.J. (1966). On first hitting times of some recurrent two-dimensional random walks. Z. Wahrsch. verw. Geb., 5:187–201.
  • Rogers, L.C.G. and Williams, D. (1987). Diffusions, Markov processes, and martingales: Itô calculus. Wiley.
  • Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab., 26:1116–1125.
  • 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.
  • Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab., 5:1–50.
  • Wilkins, J.F. (2004). A separation of timescales approach to the coalescent in a continuous population. Genetics, 168:2227–2244.
  • Wilkins, J.F. and Wakeley, J. (2002). The coalescent in a continuous, finite, linear population. Genetics, 161:873–888.
  • Wright, S. (1931). Evolution in Mendelian populations. Genetics, 16:97–159.
  • Wright, S. (1943). Isolation by distance. Genetics, 28:114–138.
  • 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.