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Journal of Probability and Statistics
Volume 2010, Article ID 580762, 27 pages
http://dx.doi.org/10.1155/2010/580762
Research Article

On Discrete-Time Multiallelic Evolutionary Dynamics Driven by Selection

Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089 et Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France

Received 20 October 2009; Revised 7 April 2010; Accepted 14 May 2010

Academic Editor: Rongling Wu

Copyright © 2010 Thierry E. Huillet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. J. Ewens, Mathematical Population Genetics. I. Theoretical Introduction, vol. 27 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 2004. View at Zentralblatt MATH · View at MathSciNet
  2. S. Martinez, G. Michon, and J. San Martin, “Inverse of strictly ultrametric matrices are of Stieltjes type,” SIAM Journal on Matrix Analysis and Applications, vol. 15, pp. 98–106, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  3. R. Nabben and R. S. Varga, “A linear algebra proof that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix,” SIAM Journal on Matrix Analysis and Applications, vol. 15, no. 1, pp. 107–113, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. T. Maruyama, Stochastic Problems in Population Genetics, vol. 17 of Lecture Notes in Biomathematics, Springer, Berlin, Germany, 1977. View at MathSciNet
  5. Y. M. Svirezhev, “Optimum principles in genetics,” in Studies on Theoretical Genetics, V. A. Ratner, Ed., pp. 86–102, USSR Academy of Science, Novosibirsk, Russia, 1972. View at Google Scholar
  6. S. Shahshahani, “A new mathematical framework for the study of linkage and selection,” Memoirs of the American Mathematical Society, vol. 17, no. 211, 1979. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. F. C. Kingman, “A mathematical problem in population genetics,” Proceedings of the Cambridge Philosophical Society, vol. 57, pp. 574–582, 1961. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. T. Nagylaki and Y. Lou, “Multiallelic selection polymorphism,” Theoretical Population Biology, vol. 69, no. 2, pp. 217–229, 2006. View at Publisher · View at Google Scholar · View at PubMed · View at Zentralblatt MATH
  9. T. E. Huillet, “Information and (co)variances in discrete evolutionary genetics involving solely selection,” Journal of Statistical Mechanics, vol. 2009, no. 9, article P9013, 2009. View at Publisher · View at Google Scholar
  10. R. C. Lewontin, L. R. Ginzburg, and S. D. Tuljapurkar, “Heterosis as an explanation for large amounts of genic polymorphism,” Genetics, vol. 88, no. 1, pp. 149–169, 1978. View at Google Scholar
  11. T. Huillet, “On Wright-Fisher diffusion and its relatives,” Journal of Statistical Mechanics, vol. 2007, no. 11, article P11006, 2007. View at Publisher · View at Google Scholar
  12. H. P. McKean, “Propagation of chaos for a class of nonlinear parabolic equations,” in Stochastic Differential Equations, A. K. Aziz, Ed., vol. 2 of Lecture Series in Differential Equations, pp. 177–193, Van Nostrand Reinhold, New York, NY, USA, 1969. View at Google Scholar
  13. A. M. Yaglom, “Certain limit theorems of the theory of branching random processes,” Doklady Akademii Nauk SSSR, vol. 56, pp. 795–798, 1947. View at Google Scholar · View at MathSciNet
  14. V. I. Romanovsky, Discrete Markov Chains, Wolters-Noordhoff Publishing, Groningen, The Netherlands, 1970. View at MathSciNet
  15. Z. Rached, F. Alajaji, and L. Campbell, “Rényi's divergence and entropy rates for finite alphabet Markov sources,” IEEE Transactions on Information Theory, vol. 47, no. 4, pp. 1553–1561, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Lambert, “Population dynamics and random genealogies,” Stochastic Models, vol. 24, supplement 1, pp. 45–163, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985. View at Zentralblatt MATH · View at MathSciNet
  18. M. Kimura, “On the probability of fixation of mutant genes in a population,” Genetics, vol. 47, pp. 713–719, 1962. View at Google Scholar
  19. G. Sella and A. E. Hirsh, “The application of statistical physics to evolutionary biology,” Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 27, pp. 9541–9546, 2005. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
  20. G. Sella, “An exact steady state solution of Fisher's geometric model and other models,” Theoretical Population Biology, vol. 75, no. 1, pp. 30–34, 2009. View at Publisher · View at Google Scholar · View at PubMed
  21. K. Khare and H. Zhou, “Rates of convergence of some multivariate Markov chains with polynomial eigenfunctions,” The Annals of Applied Probability, vol. 19, no. 2, pp. 737–777, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet