Diffusion limit for the phenomenon of random genetic drift

Anna Marciniak

Applicationes Mathematicae (2000)

  • Volume: 27, Issue: 1, page 81-101
  • ISSN: 1233-7234

Abstract

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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.

How to cite

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Marciniak, Anna. "Diffusion limit for the phenomenon of random genetic drift." Applicationes Mathematicae 27.1 (2000): 81-101. <http://eudml.org/doc/219261>.

@article{Marciniak2000,
abstract = {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.},
author = {Marciniak, Anna},
journal = {Applicationes Mathematicae},
keywords = {genetic drift; telegraph equations; singularly perturbed systems; diffusion equation; random walk; modified Chapman-Enskog procedure},
language = {eng},
number = {1},
pages = {81-101},
title = {Diffusion limit for the phenomenon of random genetic drift},
url = {http://eudml.org/doc/219261},
volume = {27},
year = {2000},
}

TY - JOUR
AU - Marciniak, Anna
TI - Diffusion limit for the phenomenon of random genetic drift
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 1
SP - 81
EP - 101
AB - 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.
LA - eng
KW - genetic drift; telegraph equations; singularly perturbed systems; diffusion equation; random walk; modified Chapman-Enskog procedure
UR - http://eudml.org/doc/219261
ER -

References

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  2. [2] J. Banasiak, Singularly perturbed linear and semilinear hyperbolic systems: kinetic theory approach to some folk's theorems, Acta Appl. Math. (in print). Zbl0893.35009
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  7. [7] M. Iosifescu, Finite Markov Processes and Their Applications, Wiley, 1980. 
  8. [8] M. Kimura, Diffusion Models in Population Genetics, Harper&Row, 1970. Zbl0134.38301
  9. [9] D. Ludwig, Stochastic Population Theories, Lecture Notes in Math. 3, Springer, Berlin, 1974. 
  10. [10] B. P. Mikhailov, Partial Differential Equations, Nauka, Moscow, 1976 (in Russian). Zbl0342.35052
  11. [11] J. R. Mika and J. Banasiak, Singularly Perturbed Evolution Equations with Applications to Kinetic Theory, World Sci., Singapore, 1995. Zbl0948.35500
  12. [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1982. 
  13. [13] S. Wright, Evolution in Mendelian population, Genetics 16 (1931), 97-159. 
  14. [14] E. Zauderer, Partial Differential Equations of Applied Mathematics, Wiley, 1988. Zbl0551.35002

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