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On the properties of bilinear models for the balance between genetic mutation and selection

Published online by Cambridge University Press:  24 October 2008

J. F. C. Kingman
J. F. C. Kingman
Affiliation:
Mathematical Institute, University of Oxford
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Extract

Moran (7) has shown that some models in population genetics lead to a bilinear recurrence relation which, for a finite number of alleles, has a globally stable equilibrium. When the possible alleles are infinite in number, non-trivial problems of existence and stability of the equilibrium arise, which Moran has resolved in special cases. In this paper a powerful sufficient condition is established for the existence of a globally stable equilibrium, and its consequences are explored for cases of genetical interest. A more speculative final section describes a variant of Moran's model which may possibly have some relevance to the assessment of experimental evidence for or against selective neutrality.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 3 , May 1977 , pp. 443 - 453
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Billingsley, P.Convergence of probability measures (New York, Wiley, 1968).Google Scholar
(2)Ewens, W. J.The sampling theory of selectively neutral alleles. Theor. Pop. Biol. 3 (1972), 87–112.CrossRefGoogle ScholarPubMed
(3)Foster, F. G.On the stochastic matrices associated with certain queueing processes. Ann. Math. Statist. 24 (1953), 355–360.CrossRefGoogle Scholar
(4)Foster, F. G. and Stuart, A.Distribution-free tests in time-series based on the breaking of records. J. Roy. Statistic. Soc. Ser. B 16 (1954), 1–22.Google Scholar
(5)Kingman, J. F. C.Coherent random walks arising in some genetical models. Proc. Roy. Soc. Ser. A 351 (1976), 19–31.Google Scholar
(6)Moran, P. A. P.Wandering distributions and the electrophoretic profile. Theor. Pop. Biol. 8 (1975), 318–330.CrossRefGoogle ScholarPubMed
(7)Moran, P. A. P.Global stability of genetic systems governed by mutation and selection. Math. Proc. Cambridge Philos. Soc. 80 (1976), 331–336.CrossRefGoogle Scholar
(8)Moran, P. A. P.A selective model for electrophoretic profiles in protein polymorphisms. Genet. Res. 28 (1976), 47–54.CrossRefGoogle ScholarPubMed
(9)Rényi, A. Théorie des éléments saillants d'une suite d'observations. Colloquium on combinatorial methods in probability theory (Aarhus, 1962), 104–115.Google Scholar
(10)Seneta, E.Non-negative matrices (London, Allen & Unwin, 1973).Google Scholar
(11)Watterson, G. A.The sampling theory of selectively neutral alleles. Adv. App. Prob. 6 (1974), 463–488.CrossRefGoogle Scholar
(12)Watterson, G. A. Heterosis or neutrality. (To appear.)Google Scholar