On the Karlin-Kimura approaches to the Wright-Fisher diffusion with fluctuating selection

Journal of Statistical Mechanics: Theory and Experiment

Volumn 2011, Issue 2, 2011, Pages

  Huillet, Thierry ^a  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

Author keywords

models for evolution (theory); mutational and evolutionary processes (theory); population dynamics (theory); stochastic processes

Indexed keywords

EID: 79952730789     PISSN: 17425468     EISSN: 17425468     Source Type: Journal    
DOI: 10.1088/1742-5468/2011/02/P02016     Document Type: Article

References (36)

1. Asmussen S and Hering H 1976 Strong limit theorems for general supercritical branching processes with applications to branching diffusions Z. Wahrscheinlichkeitstheor. Verwandte. Geb. 36 195
2. Asmussen S and Hering H 1977 Some modified branching diffusion models Math. Biosci. 35 281
3. Crow J and Kimura M 1970 An Introduction to Population Genetics Theory (New York: Harper and Row)
4. Crow J and Kimura M 1956 Some genetic problems in natural populations Proc. 3rd Berkeley Symp. on Mathematical Statistics and Probability, 1954-1955 vol IV (Berkeley, CA: University of California Press) pp1-22
5. Durrett R 2008 Probability Models for DNA Sequence Evolution (Probability and its Applications) 2nd edn (New York: Springer)
6. Dynkin E B 1965 Markov Processes vols I, II (New York: Academic) (Berlin: Springer) transl. J. Fabius, V.Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121, 122
7. Ethier S N and Kurtz T G 1986 Markov Processes. Characterization and Convergence (Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics) (New York: Wiley)
8. Engländer J and Kyprianou A E 2004 Local extinction versus local exponential growth for spatial branching processes Ann. Probab. 32 78 1A
9. Ewens W J 2004 Mathematical Population Genetics. I. Theoretical Introduction (Interdisciplinary Applied Mathematics vol 27) 2nd edn (New York: Springer)
10. Feller W 1952 The parabolic differential equations and the associated semi-groups of transformations Ann. Math. 55 468
11. Gillespie J H 1972 The effects of stochastic environments on allele frequencies in natural populations Theor. Popul. Biol. 3 241
12. Gillespie J H 1991 The Causes of Molecular Evolution (New York: Oxford University Press)
13. Griffiths R C 2003 The frequency spectrum of a mutation, and its age, in a general diffusion model Theor. Popul. Biol. 64 241
14. Huerta-Sanchez E, Durrett R and Bustamante C D 2008 Population genetics of polymorphism and divergence under fluctuating selection Genetics 178 325
15. Huillet T 2007 On Wright-Fisher diffusion and its relatives J. Stat. Mech. P11006
16. It K 1951 On stochastic differential equations Mem. Am. Math. Soc. 4 51
17. Jensen L 1973 Random selective advantages of genes and their probabilities of fixation Genet. Res. 21 215
18. Karlin S and Levikson B 1974 Temporal fluctuations in selection intensities: case of small population size Theor. Popul. Biol. 6 383
19. Karlin S and Liberman U 1975 Random temporal variation in selection intensities: one-locus two-allele model J. Math. Biol. 2 1
20. Karlin S and Taylor H M 1981 A Second Course in Stochastic Processes (New York: Academic)
21. Kimura M 1954 Process leading to quasi-fixation of genes in natural populations due to random fluctuations of selection intensities Genetics 39 280
22. Kimura M 1955 Stochastic processes and distribution of gene frequencies under natural selection Cold Spring Harbor Symposia on Quantitative Biology vol 20 pp33-53 Also, contribution 114 of the National Institute of Genetics, Mishima-shi, Japan
23. Kimura M 1962 On the probability of fixation of mutant genes in a population Genetics 47 713
24. Kussell E and Leibler S 2005 Phenotypic diversity, population growth, and information in fluctuating environments Science 309 2075
25. Mandl P 1968 Analytical Treatment of One-Dimensional Markov Processes (New York: Springer) Die Grundlehren der mathematischen Wissenschaften, Band 151 Academia Publishing House of the Czechoslovak Academy of Sciences, Prague
26. Maruyama T 1977 Stochastic Problems in Population Genetics (Springer Lecture Notes in Biomathematics vol 17) (Berlin: Springer)
27. May R M 1971 Stability in model ecosystems Proc. Ecol. Soc. Aust. 6 18
28. Mustonen V and Lssig M 2008 Molecular evolution under fitness fluctuations Phys. Rev. Lett. 100 108101
29. Mortensen R E 1969 Mathematical problems of modeling stochastic nonlinear dynamic systems J. Stat. Phys. 1 271
30. Nagylaki T 1989 Anecdotal, historical and critical commentaries on Genetics. Gustave Malécot and the transition from classical to modern population genetics. Edited by James F Crow and William F Dove Genetics 122 253
31. Pinsky R G 1995 Positive Harmonic Functions and Diffusion (Cambridge Studies in Advanced Mathematics vol 45) (Cambridge: Cambridge University Press)
32. Steinsaltz D and Evans S N 2007 Quasistationary distributions for one-dimensional diffusions with killing Trans. Am. Math. Soc. 359 1285 (electronic)
33. Stratonovich R L 1966 A new representation for stochastic integrals and equations SIAM J. Control 4 362
34. Takahata N, Ishii K and Matsuda H 1975 Effect of temporal fluctuation of selection coefficient on gene frequency in a population Proc. Nat. Acad. Sci. 72 4541
35. Tavaré S 2004 Ancestral inference in population genetics Lectures on Probability Theory and Statistics (Saint-Flour 2001) (Lecture Notes in Math. vol 1837) (Berlin: Springer) pp1-188
36. Tuckwell H C 1976 The effects of random selection on gene frequency Math. Biosci. 30 113