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On the survival of a gene in a subdivided population

  • Edward Pollak (a1)

Extract

A classical type of problem in population genetics is that of calculating the probability that a line descended from a particular gene will become extinct. In one problem of this sort, dealt with by Fisher (1922) and Haldane (1927), it is assumed that the population being studied is very large and that initially the number of genes of a particular type, say type A, is small. These authors obtained the solution by the use of the theory of branching processes.

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References

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[1] Fisher, R. A. (1922) On the dominance ratio. Proc. Roy. Soc. Edinburgh 42, 321341.
[2] Fisher, R. A. (1930) The distribution of gene ratios for rare mutations. Proc. Roy. Soc. Edinburgh 50, 204219.
[3] Gantmacher, F. R. (1959) Applications of the Theory of Matrices. Translated and revised by Brenner, J. L. with the assistance of Bushaw, D. W. and Evanusa, S. Interscience Publishers, Inc., New York.
[4] Haldane, J. B. S. (1927) A mathematical theory of natural and artificial selection. Part V : Selection and mutation. Proc. Cambridge Philos. Soc. 23, 838844.
[5] Harris, T. E. (1951) Some mathematical models for branching processes. Proc. Second Berkeley Symposium on Math. Statistics and Probability, 305328.
[6] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin, Göttingen, Heidelberg.
[7] Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. D. Van Nostrand Company, Inc., Princeton, New Jersey.
[8] Kimura, M. (1957) Some problems of stochastic processes in genetics. Ann. Math. Statist. 28, 882901.
[9] Malecot, G. (1952) Les processus stochastiques et la méthode des fonctions génératrices ou caractéristiques. Publ. Inst. Stat. Univ. Paris, 1, (fasc. 3).
[10] Moran, P. A. P. (1960) The survival of a mutant gene under selection. II Austral. J. Math. 1, 485491.
[11] Moran, P. A. P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.
[12] Sevastyanov, B. A. (1948) On the theory of branching random processes (in Russian). Doklady Acad. Nauk SS SR, (n.s.) 59, 14071410.
[13] Sevastyanov, B. A. (1951) Theory of branching stochastic processes (in Russian). Uspehi Mat. Nauk. (n. s.) 6, 4799.
[14] Wright, S. (1931) Evolution in Mendelian populations. Genetics 16, 97159.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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