The Annals of Applied Probability

One-dimensional stepping stone models, sardine genetics and Brownian local time

Richard Durrett and Mateo Restrepo

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Abstract

Consider a one-dimensional stepping stone model with colonies of size M and per-generation migration probability ν, or a voter model on ℤ in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that if /L and L/K2 converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 1 (2008), 334-358.

Dates
First available in Project Euclid: 9 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1199890025

Digital Object Identifier
doi:10.1214/07-AAP451

Mathematical Reviews number (MathSciNet)
MR2380901

Zentralblatt MATH identifier
1145.92024

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
Stepping stone model voter model Brownian local time coalescent

Citation

Durrett, Richard; Restrepo, Mateo. One-dimensional stepping stone models, sardine genetics and Brownian local time. Ann. Appl. Probab. 18 (2008), no. 1, 334--358. doi:10.1214/07-AAP451. https://projecteuclid.org/euclid.aoap/1199890025


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References

  • Aldous, D. and Fill, J. A. (2002). Chapter 3 in Reversible Markov Chains and Random Walks on Graphs. Available at http://www.stat.berkeley.edu/users/aldous/RWG/book.html.
  • Billingsley, P. (1971). Weak Convergence of Measures: Applications in Probability. SIAM, Philadelphia.
  • Borodin, A. N. (1981). On the asymptotic behavior of local times of recurrent random walks with finite variance. Theory Probab. Appl. 26 758--772.
  • Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion---Facts and Formulae. Birkhäuser, Basel.
  • Bowen, B. W. and Grant, W. S. (1997). Phylogeography of the sardines (Sardinops spp.): Assessing biogeographic models and population histories in temperate upwelling regions. Evolution 51 1601--1610.
  • Cox, J. T. and Durrett, R. (2002). The stepping stone model: New formulas expose old myths. Ann. Appl. Probab. 12 1348--1377.
  • Durrett, R. (2005). Stochastic Calculus, 3rd ed. CRC Press, Boca Raton, FL.
  • Durrett, R. (2004). Probability: Theory and Examples, 3rd Ed. Duxbury Press, Belmont, CA.
  • Itô, K. and McKean, H. P. (1974). Diffusion Processes and Their Sample Paths, 2nd ed. Springer, Berlin.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • Maruyama, T. (1970). Stepping stone models of finite length. Adv. in Appl. Probab. 2 229--258.
  • Maruyama, T. (1971). The rate of decrease of heterozygosity in population occupying a circular or linear habitat. Genetics 67 437--454.
  • Matsen, F. A. and Wakeley, J. (2005). Convergence to Island model coalescent process in populations with restricted migration. Genetics published online October 11, 2005.
  • Wilkins, J. F. and Wakeley, J. (2002). The coalescent in a continuous, finite, linear population. Genetics 161 873--888.
  • Zähle, I., Cox, T. and Durrett, R. (2005). The stepping stone model. II. Genealogies and the infinite sites model. Ann. Appl. Probab. 15 671--699.