The Annals of Probability

Genealogical constructions of population models

Alison M. Etheridge and Thomas G. Kurtz

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Abstract

Representations of population models in terms of countable systems of particles are constructed, in which each particle has a “type,” typically recording both spatial position and genetic type, and a level. For finite intensity models, the levels are distributed on $[0,\lambda]$, whereas in the infinite intensity limit $\lambda\rightarrow\infty$, at each time $t$, the joint distribution of types and levels is conditionally Poisson, with mean measure $\Xi(t)\times\ell$ where $\ell$ denotes Lebesgue measure and $\Xi(t)$ is a measure-valued population process. The time-evolution of the levels captures the genealogies of the particles in the population.

Key forces of ecology and genetics can be captured within this common framework. Models covered incorporate both individual and event based births and deaths, one-for-one replacement, immigration, independent “thinning” and independent or exchangeable spatial motion and mutation of individuals. Since birth and death probabilities can depend on type, they also include natural selection. The primary goal of the paper is to present particle-with-level or lookdown constructions for each of these elements of a population model. Then the elements can be combined to specify the desired model. In particular, a nontrivial extension of the spatial $\Lambda$-Fleming–Viot process is constructed.

Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 1827-1910.

Dates
Received: July 2016
Revised: March 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1562205692

Digital Object Identifier
doi:10.1214/18-AOP1266

Mathematical Reviews number (MathSciNet)
MR3980910

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 92D10: Genetics {For genetic algebras, see 17D92} 92D15: Problems related to evolution 92D25: Population dynamics (general) 92D40: Ecology
Secondary: 60F05: Central limit and other weak theorems 60G09: Exchangeability 60G55: Point processes 60G57: Random measures 60H15: Stochastic partial differential equations [See also 35R60] 60J68: Superprocesses

Keywords
Population model Moran model lookdown construction genealogies voter model generators stochastic equations Lambda Fleming–Viot process stepping stone model

Citation

Etheridge, Alison M.; Kurtz, Thomas G. Genealogical constructions of population models. Ann. Probab. 47 (2019), no. 4, 1827--1910. doi:10.1214/18-AOP1266. https://projecteuclid.org/euclid.aop/1562205692


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References

  • Barton, N. H., Etheridge, A. M. and Véber, A. (2010). A new model for evolution in a spatial continuum. Electron. J. Probab. 15 162–216.
  • Barton, N. H., Etheridge, A. M. and Véber, A. (2013). Modelling evolution in a spatial continuum. J. Stat. Mech. Theory Exp. 2013 P01002, 38.
  • 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.
  • Blackwell, D. and Dubins, L. E. (1983). An extension of Skorohod’s almost sure representation theorem. Proc. Amer. Math. Soc. 89 691–692.
  • Bolker, B. M. and Pacala, S. W. (1999). Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal. Amer. Nat. 153 575–602.
  • Buhr, K. A. (2002). Spatial Moran Models with Local Interactions. ProQuest LLC, Ann Arbor, MI. Ph.D. thesis, Univ. Wisconsin–Madison.
  • Dawson, D. A. and Hochberg, K. J. (1982). Wandering random measures in the Fleming–Viot model. Ann. Probab. 10 554–580.
  • Donnelly, P. and Kurtz, T. G. (1996). A countable representation of the Fleming–Viot measure-valued diffusion. Ann. Probab. 24 698–742.
  • Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166–205.
  • Donnelly, P., Evans, S. N., Fleischmann, K., Kurtz, T. G. and Zhou, X. (2000). Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees. Ann. Probab. 28 1063–1110.
  • Etheridge, A. M. (2000). An Introduction to Superprocesses. University Lecture Series 20. Amer. Math. Soc., Providence, RI.
  • Etheridge, A. M. (2008). Drift, draft and structure: Some mathematical models of evolution. In Stochastic Models in Biological Sciences. Banach Center Publ. 80 121–144. Polish Acad. Sci. Inst. Math., Warsaw.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Forien, R. and Penington, S. (2017). A central limit theorem for the spatial $\Lambda$-Fleming–Viot process with selection. Electron. J. Probab. 22 Paper No. 5, 68.
  • Greven, A., Limic, V. and Winter, A. (2005). Representation theorems for interacting Moran models, interacting Fisher–Wright diffusions and applications. Electron. J. Probab. 10 1286–1356.
  • Kliemann, W. H., Koch, G. and Marchetti, F. (1990). On the unnormalized solution of the filtering problem with counting observations. IEEE Trans. Inform. Theory 316 1415–1425.
  • Kurtz, T. G. (1998). Martingale problems for conditional distributions of Markov processes. Electron. J. Probab. 3 no. 9, 29.
  • Kurtz, T. G. (2000). Particle representations for measure-valued population processes with spatially varying birth rates. In Stochastic Models (Ottawa, ON, 1998). CMS Conf. Proc. 26 299–317. Amer. Math. Soc., Providence, RI.
  • Kurtz, T. G. (2011). Equivalence of stochastic equations and martingale problems. In Stochastic Analysis 2010 113–130. Springer, Heidelberg.
  • Kurtz, T. G. and Nappo, G. (2011). The filtered martingale problem. In The Oxford Handbook of Nonlinear Filtering 129–165. Oxford Univ. Press, Oxford.
  • Kurtz, T. G. and Rodrigues, E. R. (2011). Poisson representations of branching Markov and measure-valued branching processes. Ann. Probab. 39 939–984.
  • Kurtz, T. G. and Stockbridge, R. H. (2001). Stationary solutions and forward equations for controlled and singular martingale problems. Electron. J. Probab. 6 no. 17, 52.
  • Müller, C. and Tribe, R. (1995). Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probab. Theory Related Fields 102 519–545.
  • Rogers, L. C. G. and Pitman, J. W. (1981). Markov functions. Ann. Probab. 9 573–582.
  • Taylor, J. (2009). The genealogical consequences of fecundity variance polymorphism. Genetics 182 813–837.
  • Véber, A. and Wakolbinger, A. (2015). The spatial Lambda-Fleming–Viot process: An event-based construction and a lookdown representation. Ann. Inst. Henri Poincaré Probab. Stat. 51 570–598.
  • Zheng, J. and Xiong, J. (2017). Pathwise uniqueness for stochastic differential equations driven by pure jump processes. Statist. Probab. Lett. 130 100–104.