The Fleming-Viot limit of an interacting spatial population with fast density regulation

Ankit Gupta (ETH Zurich)


We consider population models in which the individuals reproduce, die and also migrate in space. The population size scales according to some parameter $N$, which can have different interpretations depending on the context. Each individual is assigned a mass of $1/N$ and the total mass in the system is called population density. The dynamics has an intrinsic density regulation mechanism that drives the population density towards an equilibrium. We show that under a timescale separation between the slow migration mechanism and the fast density regulation mechanism, the population dynamics converges to a Fleming-Viot process as the scaling parameter $N \to \infty$. We first prove this result for a basic model in which the birth and death rates can only depend on the population density. In this case we obtain a neutral Fleming-Viot process. We then extend this model by including position-dependence in the birth and death rates, as well as, offspring dispersal and immigration mechanisms. We show how these extensions add mutation and selection to the limiting Fleming-Viot process. All the results are proved in a multi-type setting, where there are $q$ types of individuals reproducing each other. To illustrate the usefulness of our convergence result, we discuss certain applications in ecology and cell biology.

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

Pages: 1-55

Publication Date: December 18, 2012

DOI: 10.1214/EJP.v17-1964


  • W. Allee. Animal aggregations : A study in general sociology. University of Chicago Press, Chicago, USA, 1931.
  • S. J. Altschuler, S. B. Angenent, Y. Wang, and L. F. Wu. On the spontaneous emergence of cell polarity. Nature, 454:886--889, 2008.
  • Ball, Karen; Kurtz, Thomas G.; Popovic, Lea; Rempala, Greg. Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Probab. 16 (2006), no. 4, 1925--1961. MR2288709
  • A. Butty, N. Perrinjaquet, A. Petit, M. Jaquenoud, J. Segall, K. Hofmann, C. Zwahlen, and M. Peter. A positive feedback loop stabilizes the guanine-nucleotide exchange factor cdc24 at sites of polarization. EMBO Journal, 21:1565--1576, 2002.
  • Y. Cao, D. T. Gillespie, and L. R. Petzold. The slow-scale stochastic simulation algorithm. The Journal of Chemical Physics, 122(1), Jan. 2005.
  • Chicone, Carmen. Ordinary differential equations with applications. Texts in Applied Mathematics, 34. Springer-Verlag, New York, 1999. xvi+561 pp. ISBN: 0-387-98535-2 MR1707333
  • Dawson, D. A.; Maisonneuve, B.; Spencer, J. École d'Été de Probabilités de Saint-Flour XXI—1991. [Saint-Flour Summer School on Probability Theory XXI—1991] Papers from the school held in Saint-Flour, August 18–September 4, 1991. Edited by P. L. Hennequin. Lecture Notes in Mathematics, 1541. Springer-Verlag, Berlin, 1993. viii+352 pp. ISBN: 3-540-56622-8 MR1242574
  • Donnelly, Peter; Kurtz, Thomas G. Genealogical processes for Fleming-Viot models with selection and recombination. Ann. Appl. Probab. 9 (1999), no. 4, 1091--1148. MR1728556
  • Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models. Ann. Probab. 27 (1999), no. 1, 166--205. MR1681126
  • D. G. Drubin and W. J. Nelson. Origins of cell polarity. Cell, 84:335--344, 1996.
  • Ethier, S. N.; Kurtz, Thomas G. The infinitely-many-neutral-alleles diffusion model. Adv. in Appl. Probab. 13 (1981), no. 3, 429--452. MR0615945
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085
  • Ethier, S. N.; Kurtz, Thomas G. Fleming-Viot processes in population genetics. SIAM J. Control Optim. 31 (1993), no. 2, 345--386. MR1205982
  • Ewens, Warren J. Mathematical population genetics. I. Theoretical introduction. Second edition. Interdisciplinary Applied Mathematics, 27. Springer-Verlag, New York, 2004. xx+417 pp. ISBN: 0-387-20191-2 MR2026891
  • Feng, Shui. The Poisson-Dirichlet distribution and related topics. Models and asymptotic behaviors. Probability and its Applications (New York). Springer, Heidelberg, 2010. xiv+218 pp. ISBN: 978-3-642-11193-8 MR2663265
  • Fife, Paul C. Mathematical aspects of reacting and diffusing systems. Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin-New York, 1979. iv+185 pp. ISBN: 3-540-09117-3 MR0527914
  • Fleming, Wendell H.; Viot, Michel. Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28 (1979), no. 5, 817--843. MR0542340
  • A. Gierer and H. Meinhardt. A theory of biological pattern formation. Kybernetik, 12:30--39, 1972.
  • Gupta, Ankit. Stochastic model for cell polarity. Ann. Appl. Probab. 22 (2012), no. 2, 827--859. MR2953570
  • J. E. Irazoqui, A. S. Gladfelter, and D. J. Lew. Scaffold-mediated symmetry breaking by cdc42p. Nature Cell Biology, 5:1062--1070, 2003.
  • Joffe, A.; Métivier, M. Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. in Appl. Probab. 18 (1986), no. 1, 20--65. MR0827331
  • H.-W. Kang and T. G. Kurtz. Separation of time-scales and model reduction for stochastic reaction networks. The Annals of Applied Probability (to appear), 2012.
  • Katzenberger, G. S. Solutions of a stochastic differential equation forced onto a manifold by a large drift. Ann. Probab. 19 (1991), no. 4, 1587--1628. MR1127717
  • Khalil, Hassan K. Nonlinear systems. Macmillan Publishing Company, New York, 1992. xii+564 pp. ISBN: 0-02-363541-X MR1201326
  • M. Kimura. Solution of a process of random genetic drift with a continuous model. Proceedings of the National Academy of Sciences, 41(3):144--150, 1955.
  • M. Kimura and J. Crow. The number of alleles that can be maintained in a finite population. Genetics, 49:725--738, 1964.
  • Kingman, J. F. C.; Taylor, S. J.; Hawkes, A. G.; Walker, A. M.; Cox, David Roxbee; Smith, A. F. M.; Hill, B. M.; Burville, P. J.; Leonard, T. Random discrete distribution. With a discussion by S. J. Taylor, A. G. Hawkes, A. M. Walker, D. R. Cox, A. F. M. Smith, B. M. Hill, P. J. Burville, T. Leonard and a reply by the author. J. Roy. Statist. Soc. Ser. B 37 (1975), 1--22. MR0368264
  • A. Lotka. Elements of Physical Biology. The Williams and Watkins company, Baltimore, 1925.
  • Moran, P. A. P. Random processes in genetics. Proc. Cambridge Philos. Soc. 54 1958 60--71. MR0127989
  • R. M. Nisbet and W. S. C. Gurney. Modeling fluctuating populations. Wiley, 1982.
  • Oelschläger, Karl. On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes. Probab. Theory Related Fields 82 (1989), no. 4, 565--586. MR1002901
  • Seneta, E. Non-negative matrices and Markov chains. Revised reprint of the second (1981) edition [Springer-Verlag, New York; MR0719544]. Springer Series in Statistics. Springer, New York, 2006. xvi+287 pp. ISBN: 978-0387-29765-1; 0-387-29765-0 MR2209438
  • M. Sohrmann and M. Peter. Polarizing without a c(l)ue. Trends Cell Biology, 13:526--533, 2003.
  • T. Takaku, K. Ogura, H. Kumeta, N. Yoshida, and F. Inagaki. Solution structure of a novel cdc42 binding module of bem1 and its interaction with ste20 and cdc42. Journal of Biological Chemistry, 285(25):19346--19353, 2010.
  • Thieme, Horst R. Mathematics in population biology. Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton, NJ, 2003. xx+543 pp. ISBN: 0-691-09290-7; 0-691-09291-5 MR1993355
  • Varah, J. M. A lower bound for the smallest singular value of a matrix. Linear Algebra and Appl. 11 (1975), 3--5. MR0371929
  • P. Verhulst. Notice sur la loi que la population poursuit dans son accroissement. Correspondance Mathématique et Physique, 10:113--121, 1838.
  • V. Volterra. Fluctuations in the abundance of a species considered mathematically. Nature, 118:558--560, 1926.
  • O. Weiner, P. Neilsen, G. Prestwich, M. Kirschner, L. Cantley, and H. Bourne. A ptdinsp(3)- and rho gtpase-mediated positive feedback loop regulates neutrophil polarity. Nature Cell Biology, 4(5):509--13, 2002.
  • S. Wright. Evolution in Mendelian populations. Genetics, 16(2):97--159, 1931.

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