Electronic Journal of Probability

The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns

Raphaël Forien

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We study a one-dimensional spatial population model where the population sizes of the subpopulations are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away from 0 and infinity. We suppose that the frequencies of a particular genetic type in the colonies evolve according to a system of interacting diffusions, following the stepping stone model of Kimura. We show that, over large spatial and temporal scales, this model behaves like the solution to a stochastic heat equation with Wright-Fisher noise with constant coefficients. These coefficients are the effective diffusion rate of genes within the population and the effective local population density. We find that, in our model, the local heterogeneity leads to a slower effective diffusion rate and a larger effective population density than in a uniform population. Our proof relies on duality techniques, an invariance principle for reversible random walks in a random environment and a convergence result for a system of coalescing random walks in a random environment.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 57, 35 pp.

Received: 5 July 2018
Accepted: 2 May 2019
First available in Project Euclid: 13 June 2019

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60G99: None of the above, but in this section 60H15: Stochastic partial differential equations [See also 35R60] 60J25: Continuous-time Markov processes on general state spaces

random environment population genetics coalescing random walks local central limit theorem scaling limits of measure-valued Markov processes

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Forien, Raphaël. The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns. Electron. J. Probab. 24 (2019), paper no. 57, 35 pp. doi:10.1214/19-EJP314. https://projecteuclid.org/euclid.ejp/1560391565

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  • [BCD16] Matthias Birkner, Jiri Cerny, and Andrej Depperschmidt, Random walks in dynamic random environments and ancestry under local population regulation, Electronic Journal of Probability 21 (2016).
  • [BCDG13] Matthias Birkner, Jiri Cerny, Andrej Depperschmidt, and Nina Gantert, Directed random walk on the backbone of an oriented percolation cluster, Electronic Journal of Probability 18 (2013).
  • [BDE02] Nick H. Barton, Frantz Depaulis, and Alison M. Etheridge, Neutral evolution in spatially continuous populations, Theoretical Population Biology 61 (2002), no. 1, 31–48.
  • [BEKV13] Nick H. Barton, Alison M. Etheridge, Jerome Kelleher, and Amandine Véber, Inference in two dimensions: Allele frequencies versus lengths of shared sequence blocks, Theoretical population biology 87 (2013), 105–119.
  • [BGS18] Matthias Birkner, Nina Gantert, and Sebastian Steiber, Coalescing directed random walks on the backbone of a 1 +1-dimensional oriented percolation cluster converge to the Brownian web, arXiv:1812.03733 [math] (2018).
  • [DD09] Jérôme Depauw and Jean-Marc Derrien, Variance limite d’une marche aléatoire réversible en milieu aléatoire sur Z, Comptes Rendus Mathematique 347 (2009), no. 7, 401–406.
  • [DEF$^{+}$00] Peter Donnelly, Steven N. Evans, Klaus Fleischmann, Thomas G. Kurtz, and Xiaowen Zhou, Continuum-Sites Stepping-Stone Models, Coalescing Exchangeable Partitions and Random Trees, The Annals of Probability 28 (2000), no. 3, 1063–1110.
  • [Der15] Jean-Marc Derrien, A local limit theorem in stationary random environment of conductances on Z, Bulletin de la Société Mathématique de France 143 (2015), no. 3, 467–488.
  • [DMFGW89] Anna De Masi, Pablo A. Ferrari, Sheldon Goldstein, and William David Wick, An invariance principle for reversible Markov processes. Applications to random motions in random environments, Journal of Statistical Physics 55 (1989), no. 3-4, 787–855.
  • [DR08] Richard Durrett and Mateo Restrepo, One-dimensional stepping stone models, sardine genetics and Brownian local time, The Annals of Applied Probability 18 (2008), no. 1, 334–358.
  • [EK86] Stewart N. Ethier and Thomas G. Kurtz, Markov processes: Characterization and convergence, John Wiley & Sons, Inc., New York, 1986.
  • [Eth11] Alison Etheridge, Some mathematical models from population genetics, Lecture Notes in Mathematics, vol. 2012, Springer, Heidelberg, 2011, Lectures from the 39th Probability Summer School held in Saint-Flour, 2009, Ecole d’Eté de Probabilités de Saint-Flour.
  • [EVY18] Alison Etheridge, Amandine Véber, and Feng Yu, Rescaling limits of the spatial Lambda-Fleming-Viot process with selection, arXiv preprint arXiv:1406.5884 (2018).
  • [FINR04] L. R. G. Fontes, Marco Isopi, Charles M. Newman, and Krishnamurthi Ravishankar, The Brownian web: Characterization and convergence, The Annals of Probability 32 (2004), no. 4, 2857–2883.
  • [GdHK18] Andreas Greven, Frank den Hollander, and Anton Klimovsky, The hierarchical Cannings process in random environment, Latin American Journal of Probability and Mathematical Statistics XV (2018), 295–351.
  • [GKW01] Andreas Greven, Achim Klenke, and Anton Wakolbinger, Interacting Fisher–Wright diffusions in a catalytic medium, Probability Theory and Related Fields 120 (2001), no. 1, 85–117 (en).
  • [GLW05] Andreas Greven, Vlada Limic, and Anita Winter, Representation theorems for interacting moran models, interacting fisher-wrighter diffusions and applications, Electronic Journal of Probability 10 (2005), 1286–1358.
  • [GSW16] Andreas Greven, Rongfeng Sun, and Anita Winter, Continuum space limit of the genealogies of interacting Fleming-Viot processes on Z, Electronic Journal of Probability 21 (2016) (EN).
  • [Kim53] Motoo Kimura, Stepping-stone model of population, Annual Report of the National Institute of Genetics 3 (1953), 62–63.
  • [Koz85] Sergei Mikhailovich Kozlov, The method of averaging and walks in inhomogeneous environments, Russian Mathematical Surveys 40 (1985), no. 2, 73–145.
  • [KW64] Motoo Kimura and George H. Weiss, The stepping stone model of population structure and the decrease of genetic correlation with distance, Genetics 49 (1964), no. 4, 561.
  • [Lam12] Hoang Chuong Lam, Les théoremes limites pour des processus stationnaires, Ph.D. thesis, Université de Tours, 2012.
  • [Lam14] Hoang-Chuong Lam, A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z, Journal of Applied Probability 51 (2014), no. 4, 1051–1064 (en).
  • [Lia09] Richard Hwa Liang, Two continuum-sites stepping stone models in population genetics with delayed coalescence, Ph.D. thesis, University of California, Berkeley, 2009.
  • [Mal48] Gustave Malécot, Les Mathématiques de l’Hérédité, Masson et Cie., Paris, 1948.
  • [Mar71] Takeo Maruyama, The rate of decrease of heterozygosity in a population occupying a circular or a linear habitat, Genetics 67 (1971), no. 3, 437–454.
  • [Mil16] Katja Miller, Random walks on weighted, oriented percolation clusters, Latin American Journal of Probability and Mathematical Statistics 13 (2016), 53–77.
  • [MT95] Carl Mueller and Roger Tribe, Stochastic pde’s arising from the long range contact and long range voter processes, Probability theory and related fields 102 (1995), no. 4, 519–545.
  • [RCB17] Harald Ringbauer, Graham Coop, and Nicholas H. Barton, Inferring Recent Demography from Isolation by Distance of Long Shared Sequence Blocks, Genetics 205 (2017), no. 3, 1335–1351 (en).
  • [Reb80] Rolando Rebolledo, Central limit theorems for local martingales, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 51 (1980), no. 3, 269–286.
  • [Rou97] François Rousset, Genetic differentiation and estimation of gene flow from F-statistics under isolation by distance, Genetics 145 (1997), no. 4, 1219–1228.
  • [Saw76] Stanley Sawyer, Results for the Stepping Stone Model for Migration in Population Genetics, The Annals of Probability 4 (1976), no. 5, 699–728.
  • [Saw77] Stanley Sawyer, Asymptotic properties of the equilibrium probability of identity in a geographically structured population, Advances in Applied Probability 9 (1977), no. 2, 268–282.
  • [Shi88] Tokuzo Shiga, Stepping Stone Models in Population Genetics and Population Dynamics, Stochastic Processes in Physics and Engineering, Mathematics and Its Applications, Springer, 1988, pp. 345–355.
  • [SSS15] Emmanuel Schertzer, Rongfeng Sun, and Jan Swart, The Brownian web, the Brownian net, and their universality, Advances in Disordered Systems, Random Processes and Some Applications (2015), 270–368.
  • [Wri43] Sewall Wright, Isolation by distance, Genetics 28 (1943), no. 2, 114.