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Tracing the Path of a Wright-Fisher Process with One-way Mutation in the Case of a Large Deviation

  • F. Papangelou
Part of the Trends in Mathematics book series (TM)

Abstract

The large deviations theory developed by Wentzell for discrete time Markov chains is used to show that if the state of a Wright-Fisher process modeling the frequency of an allele in a biological population undergoes a transition from one value to another over a number of generations which is large but much smaller than the size of the population, then there is a preferred path which the process follows closely with near certainty in the intervening time. This path was identified in [8]_in the case in which there is only random drift acting on the genes. The case in which one-way mutation is added to the drift was cursorily mentioned in [8]_and is fully treated here. The preferred path in this case is shown to be an exponential, a parabola, a hyperbolic cosine or a trigonometric cosine, depending on the mutation parameter and the boundary conditions involved.

Keywords

Wright-Fisher process mutation large deviations action functional calculus of variations 

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References

  1. [1]
    Deuschel, J.-D. and Stroock D.W.: Large Deviations. Academic Press, Boston, 1989.zbMATHGoogle Scholar
  2. [2]
    Ewens, W.J.: Mathematical Population Genetics. Biomathematics Vol. 9. Springer-Verlag, New York, 1979.Google Scholar
  3. [3]
    Preidlin, M.I. and Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer-Verlag, New York, 1984.CrossRefGoogle Scholar
  4. [4]
    Gelfand, I.M. and Fomin, S.V.: Calculus of Variations. Prentice-Hall, Englewood Cliffs, 1963.Google Scholar
  5. [5]
    Karlin, S. and Taylor, H.M.: A Second Course in Stochastic Processes. Academic Press, New York, 1981.zbMATHGoogle Scholar
  6. [6]
    Kimura, M.: Diffusion Models in Population Genetics. Methuen’s Review Series in Applied Probability, Methuen, London, 1964.Google Scholar
  7. [7]
    Natanson, LP.: Theorie der Punktionen einer Reellen Veränderlichen. Akademie-Verlag, Berlin, 1961.Google Scholar
  8. [8]
    Papangelou, F.: Large deviations of the Wright-Fisher process. Proceedings of the Athens Conference on Applied Probability and Time Series Analysis, Vol.1: Applied Probability (eds. C.C. Heyde, Yu. V. Prohorov, R. Pyke, S.T. Rachev). Lecture Notes in Statistics 114, 245–252, Springer-Verlag, New York, 1996.Google Scholar
  9. [9]
    Wentzell, A.D.: Rough limit theorems on large deviations for Markov processes, Th. Probab. Appl. 21 (1976), 227–242 and 499-512; 24 (1979), 675-692; 27 (1982), 215-234.Google Scholar
  10. [10]
    Wentzell, A.D.: Limit Theorems on Large Deviations for Markov Stochastic Processes. Kluwer Academic Publishers, Dordrecht, 1990.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • F. Papangelou
    • 1
  1. 1.Department of MathematicsUniversity of ManchesterManchesterUK

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