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2000 | 27 | 1 | 81-101
Tytuł artykułu

Diffusion limit for the phenomenon of random genetic drift

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The paper deals with mathematical modelling of population genetics processes. The formulated model describes the random genetic drift. The fluctuations of gene frequency in consecutive generations are described in terms of a random walk. The position of a moving particle is interpreted as the state of the population expressed as the frequency of appearance of a specific gene. This leads to a continuous model on the microscopic level in the form of two first order differential equations (known as the telegraph equations). Applying the modified Chapman-Enskog procedure we show the transition from this system to a macroscopic model which is a diffusion type equation. Finally, the error of approximation is estimated.
Rocznik
Tom
27
Numer
1
Strony
81-101
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-01-15
poprawiono
1999-11-09
Twórcy
  • Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
Bibliografia
  • [1] H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math. 45 (1983), 225-253.
  • [2] J. Banasiak, Singularly perturbed linear and semilinear hyperbolic systems: kinetic theory approach to some folk's theorems, Acta Appl. Math. (in print).
  • [3] J. Banasiak and J. R. Mika, Singularly perturbed telegraph equations with application to the random walk theory, J. Appl. Math. Stochast. Anal. (in print).
  • [4] J. M. Connor and M. A. Ferguson-Smith, Essential Medical Genetics, Blackwell Sci. Publ., Oxford, 1987.
  • [5] J. Friedman, Genetics, William & Wilkins, Baltimore, 1996.
  • [6] J. B. S. Haldane, Suggestions as to quantitative measurement of rates of evolution, Evolution 3 (1949), 51-56.
  • [7] M. Iosifescu, Finite Markov Processes and Their Applications, Wiley, 1980.
  • [8] M. Kimura, Diffusion Models in Population Genetics, Harper&Row, 1970.
  • [9] D. Ludwig, Stochastic Population Theories, Lecture Notes in Math. 3, Springer, Berlin, 1974.
  • [10] B. P. Mikhailov, Partial Differential Equations, Nauka, Moscow, 1976 (in Russian).
  • [11] J. R. Mika and J. Banasiak, Singularly Perturbed Evolution Equations with Applications to Kinetic Theory, World Sci., Singapore, 1995.
  • [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1982.
  • [13] S. Wright, Evolution in Mendelian population, Genetics 16 (1931), 97-159.
  • [14] E. Zauderer, Partial Differential Equations of Applied Mathematics, Wiley, 1988.
Typ dokumentu
Bibliografia
Identyfikatory
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bwmeta1.element.bwnjournal-article-zmv27i1p81bwm
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