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Kingman’s model with random mutation probabilities: convergence and condensation I

Published online by Cambridge University Press:  25 February 2022

Linglong Yuan*
Affiliation:
University of Liverpool and Xi’an Jiaotong-Liverpool University
*
*Postal address: University of Liverpool, Department of Mathematical Sciences, Peach Street, L69 7ZL, Liverpool, UK. Email address: [email protected]

Abstract

For a one-locus haploid infinite population with discrete generations, the celebrated model of Kingman describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability. This paper generalises Kingman’s model by using independent and identically distributed random mutation probabilities, to reflect the influence of a random environment. The weak convergence of fitness distributions to the globally stable equilibrium is proved. Condensation occurs when almost surely a positive proportion of the population travels to and condenses at the largest fitness value. Condensation may occur when selection is favoured over mutation. A criterion for the occurrence of condensation is given.

Type
Original Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Aldous, D. J. and Bandyopadhyay, A. (2005). A survey of max-type recursive distributional equations. Ann. Prob. 15, 10471110.Google Scholar
Betz, V., Dereich, S. and Mörters, P. (2018). The shape of the emerging condensate in effective models of condensation. Ann. Inst. H. Poincaré Prob. Statist. 19, 18691889.10.1007/s00023-018-0673-7CrossRefGoogle Scholar
Bürger, R. (1986). On the maintenance of genetic variation: global analysis of Kimurabs continuum-of-alleles model. J. Math. Biol. 24, 341351.10.1007/BF00275642CrossRefGoogle ScholarPubMed
Bürger, R. (1989). Mutation–selection balance and continuum-of-alleles models. Math. Biosci. 12, 6783.Google Scholar
Bürger, R. (1998). Mathematical properties of mutation–selection models. Genetica 102, 279298.10.1023/A:1017043111100CrossRefGoogle Scholar
Bürger, R. (2000). The Mathematical Theory of Selection, Recombination, and Mutation. John Wiley, Chichester, New York.Google Scholar
Crow, J. F. and Kimura, M. (1970). An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
Dereich, S. and Mörters, P. (2013). Emergence of condensation in Kingmanbs model of selection and mutation. Acta Appl. Math. 127, 1726.10.1007/s10440-012-9790-3CrossRefGoogle Scholar
Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41, 4576.10.1137/S0036144598338446CrossRefGoogle Scholar
Evans, S. N., Steinsaltz, D. and Wachter, K. W. (2013). A Mutation–Selection Model with Recombination for General Genotypes. American Mathematical Society, Providence, RI.Google Scholar
Ewens, W. J. (1979). Mathematical Population Genetics. Springer, New York.Google Scholar
Gonzalez-Casanova, A., Kurt, N., Wakolbinger, A. and Yuan, L. (2016). An individual-based model for the Lenski experiment, and the deceleration of the relative fitness. Stoch. Process. Appl. 126, 22112252.10.1016/j.spa.2016.01.009CrossRefGoogle Scholar
Grange, P. (2019). Steady states in a non-conserving zero-range process with extensive rates as a model for the balance of selection and mutation. J. Phys. A 52, 365601.10.1088/1751-8121/ab3370CrossRefGoogle Scholar
Haldane, J. B. S. (1937). The effect of variation on fitness. Amer. Naturalist 71, 337349.10.1086/280722CrossRefGoogle Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, Cham.Google Scholar
Kallenberg, O. (2017). Random Measures: Theory and Applications. Springer, Cham.10.1007/978-3-319-41598-7CrossRefGoogle Scholar
Kimura, M. (1965). A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Nat. Acad. Sci. USA 54, 731736.10.1073/pnas.54.3.731CrossRefGoogle Scholar
Kingman, J. F. C. (1977). On the properties of bilinear models for the balance between mutation and selection. Proc. Camb. Phil. Soc. 80, 443453.10.1017/S0305004100053512CrossRefGoogle Scholar
Kingman, J. F. C. (1978). A simple model for the balance between selection and mutation. J. Appl. Prob. 15, 112.10.2307/3213231CrossRefGoogle Scholar
Kingman, J. F. C. (1980). Mathematics of Genetic Diversity. Society for Industrial and Applied Mathematics, Philadelphia.10.1137/1.9781611970357CrossRefGoogle Scholar
Steinsaltz, D., Evans, S. N. and Wachter, K. W. (2005). A generalized model of mutation–selection balance with applications to aging. Adv. Appl. Math. 35, 1633.10.1016/j.aam.2004.09.003CrossRefGoogle Scholar
Yuan, L. (2017). A generalization of Kingmanbs model of selection and mutation and the Lenski experiment. Math. Biosci. 285, 6167.10.1016/j.mbs.2016.12.007CrossRefGoogle ScholarPubMed
Yuan, L. (2020). Kingman’s model with random mutation probabilities: convergence and condensation II. J. Statist. Phys. 181, 870896.10.1007/s10955-020-02609-wCrossRefGoogle Scholar