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Mutation-Selection Models in Population Genetics and Evolutionary Game Theory

  • Reinhard Bürger
Conference paper

Abstract

The investigation of models including the effects of selection and mutation has a long history in population genetics, which can be traced back to Fisher, Haldane and Wright. The reason is that selection and mutation are important forces in evolution. Selection favours an optimal type and tends to eliminate genetic variation, in particular in quantitative traits. Mutation, on the other hand, is the ultimate source of genetic variability. Traditionally, models with only two alleles per locus have been treated. At the end of the fifties the first general results for multi-allele models with selection but without mutation were proved. In particular, conditions for the existence of a unique and stable interior equilibrium were derived and Fisher’s Fundamental Theorem of Natural Selection was proved to be valid (e.g. Mulholland and Smith, 1959; Scheuer and Mandel, 1959; Kingman, 1961a). It teils that in a one-locus multi-aliele diploid model mean fitness always increases. It is well known now that in models with two loci or more this is wrong in general.

AMS Subject Classification (1980)

92A.10 

Key words

mutation-selection equation continuum of alleles gradient systems 

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References

  1. Akin, E. 1979. The Geometry of Population Genetics. Lect. Notes Biomath. 31. Berlin-Heidelberg-New York. Springer Verlag.zbMATHGoogle Scholar
  2. Akin, E., Hofbauer, J. 1982. Recurrence of the unfit. Math. Biosci. 61, 51–63.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Barton, N. 1986. The maintenance of polygenic variation through a balance between mutation and stabilizing selection. Genet. Res. 47, 209–216.CrossRefGoogle Scholar
  4. Buhner, M.G. 1971. Protein polymorphism. Nature 234, 410–411.Google Scholar
  5. Bürger, R. 1983. Dynamics of the classical genetic model for the evolution of dominance. Math. Biosci. 67, 269–280.CrossRefGoogle Scholar
  6. Bürger, R. 1986. On the maintenance of genetic variation: Global analysis of Kimura’s continuum-of-alleles model. J. Math. Biol. 24, 341–351.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bürger, R. 1988a. Perturbations of positive semigroups and applications to population genetics. Math. Z. 197, 259–272.zbMATHCrossRefGoogle Scholar
  8. Bürger, R. 1988b. Mutation-selection balance and continuum-of-alleles models. Math. Biosci. To appear.Google Scholar
  9. Bürger, R. 1988c. Linkage and the maintenance of heritable variation by mutation-selection balance. Submitted.Google Scholar
  10. Bürger, R., Wagner, G., Stettinger, F. 1988. How much heritable variation can be maintained in finite populations by a mutation-selection balance? Submitted.Google Scholar
  11. Crow, J.F., Kimura, M. 1964. The theory of genetic loads. Proc. XI Int. Congr. Genet. pp. 495–505. Oxford: Pergamon Press.Google Scholar
  12. Crow, J.F., Kimura, M. 1970. An Introduction to Population Genetics. New York: Harper and Row.zbMATHGoogle Scholar
  13. Eigen, M. 1971. Selforganization of matter and the evolution of biological macro-molecules. Die Naturwissenschaften 58, 465–523.CrossRefGoogle Scholar
  14. Fleming, W.H. 1979. Equilibrium distributions of continuous polygenic traits. SIAM J. Appl. Math. 36, 148–168.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Hadeler, K.P. 1981. Stable polymorphisms in a selection model with mutation, SIAM J. Appl. Math. 41, 1–7.MathSciNetCrossRefGoogle Scholar
  16. Hofbauer, J. 1985. The selection mutation equation. J. Math. Biol. 23, 41–53.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Hofbauer, J., Sigmund, K. (1988). Dynamical Systems and the Theory of Evolution. Cambridge Univ. Press. In press.zbMATHGoogle Scholar
  18. Kimura, M. 1965. A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Natl. Acad. Sci. USA 54, 731–736.zbMATHCrossRefGoogle Scholar
  19. Kingman, J.F.C. 1961a. On an inequality in partial averages. Quart. J. Math. 12, 78–80.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Kingman, J.F.C. 1961b. A convexity property of positive matrices. Quart. J. Math. 12, 283–284.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Kingman, J.F.C. 1977. On the properties of bilinear models for the balance between genetic mutation and selection. Math. Proc. Camb. Phil. Soc. 81, 443–453.MathSciNetzbMATHCrossRefGoogle Scholar
  22. Kingman, J.F.C. 1978. A simple model for the balance between selection and mutation. J. Appl. Prob. 15, 1–12.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Losert, V., Akin, E. 1983. Dynamics of games and genes: discrete versus continuous time. J. Math. Biol. 17, 241–251.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Lyubich, Yu.I., Maistiovskii, G.D., Ol’klovski, Yu.G. 1980. Selection-induced convergence to equilibrium in a single-locus autosomal population. Problemy Peredachi Informatsii 16, 93–104 (engl. transi.).Google Scholar
  25. Moran, P.A.P. 1976. Global stability of genetic systems governed by mutation and selection. Math. Proc. Camb. Phil. Soc. 80, 331–336.zbMATHCrossRefGoogle Scholar
  26. Moran, P.A.P. 1977. Global stability of genetic systems governed by mutation and selection. II. Math. Proc. Camb. Phil. Soc. 81, 435–441.zbMATHCrossRefGoogle Scholar
  27. Mulholland, H.P., Smith, C.A.B. 1959. An inequality arising in genetic theory. Amer. Math. Monthly 66, 673–683.MathSciNetCrossRefGoogle Scholar
  28. Nagylaki, T. 1984. Selection on a quantitative character. In: Human Population Genetics: The Pittsburgh Symposium. (A. Chakravarti, Ed.) New York: Van Nostrand.Google Scholar
  29. Nagylaki, T., Crow, J.F. 1974. Continuous selective models. Theor. Pop. Biol. 5, 257–283.CrossRefGoogle Scholar
  30. Newburgh, J.D. 1951. The variation of spectra. Duke Math. J. 18, 165–176.MathSciNetzbMATHCrossRefGoogle Scholar
  31. O’Brien, P. 1985. A genetic model with mutation and selection. Math. Biosci. 73, 239–251.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Ohta, T., Kimura, M. 1973. A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population. Gen. Res. 22, 201–204.MathSciNetCrossRefGoogle Scholar
  33. Ohta, T., Kimura, M. 1975. Theoretical analysis of electrophoretically detectable polymorphisms: Models of very slightly deleterious mutations. Amer. Natur. 109, 137–145.CrossRefGoogle Scholar
  34. Scheuer, P., Mandel, S. 1959. An inequality in population genetics. Heredity 13, 519–524.CrossRefGoogle Scholar
  35. Sigmund, K. 1987. Game dynamics, mixed strategies, and gradient systems. Theor. Pop. Biol. 32, 114–126.MathSciNetzbMATHCrossRefGoogle Scholar
  36. Taylor, P., Jonker, L. 1978. Evoluticnarily stable strategies and game dynamics. Math. Biosci. 40, 145–156.MathSciNetzbMATHCrossRefGoogle Scholar
  37. Thomas, B. Evolutionary stable sets and mixed strategist models. Theor. Pop. Biol. 28, 332–341.Google Scholar
  38. Thompson, C.J., McBride, J.L. 1974. On Eigen’s theory of self-organization of matter and the evolution of biological macromolecules. Math. Biosci. 21, 127–142.MathSciNetzbMATHCrossRefGoogle Scholar
  39. Turelli, M. 1984. Heritable genetic variation via mutation-selection balance: Lerch’s zeta meets the abdominal bristle. Theor. Pop. Biol. 25, 138–193.zbMATHCrossRefGoogle Scholar
  40. Turelli, M. 1986. Gaussian versus non-Gaussian genetic analyses of polygenic mutation-selection balance, pp. 607–628. In: Evolutionary Processes and Theory (edt. by S. Karlin and E. Nevo). New York: Academic Press.Google Scholar

Copyright information

© Kluwer Academic Publishers, Dordrecht, Holland 1989

Authors and Affiliations

  • Reinhard Bürger
    • 1
  1. 1.Institut für MathematikUniversität WienWienAustria

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