Advertisement

Journal of Mathematical Biology

, Volume 79, Issue 6–7, pp 2069–2110 | Cite as

Diffusion approximation for an age-class-structured population under viability and fertility selection with application to fixation probability of an advantageous mutant

  • Cíntia Dalila Soares
  • Sabin LessardEmail author
Article

Abstract

In this paper, we ascertain the validity of a diffusion approximation for the frequencies of different types under recurrent mutation and frequency-dependent viability and fertility selection in a haploid population with a fixed age-class structure in the limit of a large population size. The approximation is used to study, and explain in terms of selection coefficients, reproductive values and population-structure coefficients, the differences in the effects of viability versus fertility selection on the fixation probability of an advantageous mutant.

Keywords

Age-class-structured population Leslie matrix Frequency-dependent selection Diffusion approximation Fixation probability Two timescales Reproductive value Population-structure coefficient 

Mathematics Subject Classification

Primary 92D25 Secondary 60J70 

Notes

Acknowledgements

This research was supported in part by NSERC of Canada, Grant No. 8833, and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 0968-13-7. We are also grateful to two anonymous referees for helpful comments to improve the manuscript.

References

  1. Auger P, Bravo de la Parra R, Poggiale JC, Sànchez E, Nguyen Huu T (2008) Aggregation of variables and applications to population dynamics. In: Magal P, Ruan S (eds) Structured population models in biology and epidemiology, vol 1936. Lecture notes in mathematics. Springer, Berlin, pp 209–263CrossRefGoogle Scholar
  2. Chalub FA, Souza MO (2009) From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics. Theor Popul Biol 76:268–277CrossRefGoogle Scholar
  3. Chalub FA, Souza MO (2014) The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations. J Math Biol 68:1089–1133MathSciNetCrossRefGoogle Scholar
  4. Champagnat N, Ferrière R, Méléard S (2006) Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theor Popul Biol 69:297–321CrossRefGoogle Scholar
  5. Champagnat N, Ferrière R, Méléard S (2008) From individual stochastic processes to macroscopic models in adaptive evolution. Stoch Models 24:2–44MathSciNetCrossRefGoogle Scholar
  6. Charlesworth B (1980) Evolution in age-structured populations, vol 1. Cambridge studies in mathematical biology. Cambridge University Press, CambridgezbMATHGoogle Scholar
  7. Charlesworth B (2009) Effective population size and patterns of molecular evolution and variation. Nat Rev Genet 10:195–205CrossRefGoogle Scholar
  8. Chesson J (1976) A non-central multivariate hypergeometric distribution arising from biased sampling with application to selective predation. J Appl Probab 13:795–797MathSciNetCrossRefGoogle Scholar
  9. Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper and Row, New YorkzbMATHGoogle Scholar
  10. Cushing JM (1998) An introduction to structured population dynamics, CBMS-NSF regional conference series in applied mathematics, vol 71. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
  11. Emigh TH (1979a) The dynamics of finite haploid populations with overlapping generations. I. Moments, fixation probabilities and stationary distributions. Genetics 92:323–337MathSciNetGoogle Scholar
  12. Emigh TH (1979b) The dynamics of finite haploid populations with overlapping generations. II. The diffusion approximation. Genetics 92:339–351MathSciNetGoogle Scholar
  13. Emigh TH, Pollak E (1979) Fixation probabilities and effective population numbers in diploid populations with overlapping generations. Theor Popul Biol 15:86–107MathSciNetCrossRefGoogle Scholar
  14. Etheridge A (2011) Some mathematical models from population genetics. Springer, HeidelbergCrossRefGoogle Scholar
  15. Ethier SN (1976) A class of degenerate diffusion processes occurring in population genetics. Commun Pure Appl Math 29:483–493MathSciNetCrossRefGoogle Scholar
  16. Ethier SN, Nagylaki T (1980) Diffusion approximations of Markov chains with two time scales and applications to population genetics. Adv Appl Probab 12:14–49MathSciNetCrossRefGoogle Scholar
  17. Ewens WJ (2004) Mathematical population genetics 1: theoretical introduction, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  18. Felsenstein J (1971) Inbreeding and variance effective numbers in populations with overlapping generations. Genetics 68:581–597MathSciNetGoogle Scholar
  19. Fisher RA (1930) The genetical theory of natural selection. Clarendon, OxfordCrossRefGoogle Scholar
  20. Godement R (2005) Analysis II: differential and integral calculus, Fourier series, holomorphic functions. Springer, BerlinzbMATHGoogle Scholar
  21. Huang W, Hauert C, Traulsen A (2015) Stochastic game dynamics under demographic fluctuations. Proc Natl Acad Sci USA 112:9064–9069CrossRefGoogle Scholar
  22. Jones OR, Scheuerlein A, Salguero-Gómez R, Camarda CG, Schaible R, Casper BB, Dahlgren JP, Ehrlén J, García MB, Menges ES, Quintana-Ascencio PF, Caswell H, Baudisch A, Vaupel JW (2014) Diversity of ageing across the tree of life. Nature 505:169–173CrossRefGoogle Scholar
  23. Karlin S (1982) Classification of selection-migration structures and conditions for a protected polymorphism. In: Hecht MK, Wallace B, Prance GT (eds) Evolutionary biology. Plenum, New York, pp 61–204Google Scholar
  24. Karlin S, Taylor HM (1975) A first course in stochastic processes. Academic Press, New YorkzbMATHGoogle Scholar
  25. Kebir A, Miled SB, Hbid ML, Bravo de La Parra R (2010) Effects of density dependent sex allocation on the dynamics of a simultaneous hermaphroditic population: modelling and analysis. J Theor Biol 263:521–529MathSciNetCrossRefGoogle Scholar
  26. Kebir A, Fefferman NH, Miled SB (2015) Understanding hermaphrodite species through game theory. J Math Biol 71:1505–1524MathSciNetCrossRefGoogle Scholar
  27. Kimura M (1964) Diffusion models in population genetics. J Appl Probab 1:177–232MathSciNetCrossRefGoogle Scholar
  28. Kroumi D, Lessard S (2015) Strong migration limit for games in structured populations: applications to dominance hierarchy and set structure. Games 6:318–346MathSciNetCrossRefGoogle Scholar
  29. Leslie PH (1945) On the use of matrices in certain population mathematics. Biometrika 33:183–212MathSciNetCrossRefGoogle Scholar
  30. Lessard S (2009) Diffusion approximations for one-locus multi-allele kin selection, mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics. J Math Biol 59:659–696MathSciNetCrossRefGoogle Scholar
  31. Lessard S, Soares CD (2018) Frequency-dependent growth in class-structured populations: continuous dynamics in the limit of weak selection. J Math Biol 77:229–259MathSciNetCrossRefGoogle Scholar
  32. Li XY, Kurokawa S, Giaimo S, Traulsen A (2016) How life history can sway the fixation probability of mutants. Genetics 203:1297–1313CrossRefGoogle Scholar
  33. Marvà M, Moussaoui A, Bravo de la Parra R, Auger P (2013) A density-dependent model describing age-structured population dynamics using hawk-dove tactics. J Differ Equ Appl 19:1022–1034MathSciNetCrossRefGoogle Scholar
  34. Marvà M, San Segundo F (2018) Age-structure density-dependent fertility and individuals dispersal in a population model. Math Biosci 300:157–167MathSciNetCrossRefGoogle Scholar
  35. Nagylaki T (1980) The strong-migration limit in geographically structured populations. J Math Biol 9:101–114MathSciNetCrossRefGoogle Scholar
  36. Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428:646–650CrossRefGoogle Scholar
  37. Nowak MA, Tarnita CE, Antal T (2010) Evolutionary dynamics in structured populations. Philos Trans R Soc B 365:19–30CrossRefGoogle Scholar
  38. Soares CD (2019) Évolution dans des populations structurées en classes. Ph.D. Université de Montréal, MontréalGoogle Scholar
  39. Soares CD, Lessard S (2019) First-order effect of frequency-dependent selection on fixation probability in an age-structured population with application to a public goods game. Theor Popul Biol.  https://doi.org/10.1016/j.tpb.2019.05.001
  40. Stegan IA (1964) Handbook of mathematical functions: with formulas, graphs and mathematical tables. Dover, New YorkGoogle Scholar
  41. Wallenius KT (1963) Biased sampling; the non-central hypergeometric probability distribution. Ph.D. Stanford University, StanfordGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de mathématiques et de statistiqueUniversité de MontréalMontréalCanada

Personalised recommendations