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Stationary distributions under mutation-selection balance: structure and properties

Part of: Special classes of linear operators Markov processes Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  01 July 2016

Reinhard Bürger  and
Immanuel M. Bomze
Reinhard Bürger*
Affiliation:
Universität Wien
Immanuel M. Bomze*
Affiliation:
Universität Wien
*
* Postal address: Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria.
** Postal address: I.S.O.C., Universität Wien, Universitätsstraße 5, A-1010 Wien, Austria.
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Abstract

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.

Keywords

PROBABILITY MEASURESPERRÖN-FROBENIUS THEORYINTEGRO-DIFFERENTIAL EQUATIONSPOSITIVE OPERATORSCONTINUUM-OF-ALLELES

MSC classification

Primary: 92D10: Genetics
Secondary: 60J35: Transition functions, generators and resolvents 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 47B65: Positive operators and order-bounded operators 47D03: Groups and semigroups of linear operators

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 1 , March 1996 , pp. 227 - 251
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Aliprantis, C. D. and Burkinshaw, O. (1985) Positive Operators. Academic Press, New York.Google Scholar
[2] Bomze, I. M. (1990) Dynamical aspects of evolutionary stability. Monatsh. Math. 110, 189–206.Google Scholar
[3] Bomze, I. M. (1991) Cross entropy minimization in uninvadable states of complex populations. J. Math. Biol. 30, 73–87.CrossRefGoogle Scholar
[4] Bürger, R. (1986) On the maintenance of genetic variation: global analysis of Kimura's continuum-of-alleles model. J. Math. Biol. 24, 341–351.Google Scholar
[5] Bürger, R. (1988) Perturbations of positive semigroups and applications to population genetics. Math. Z. 197, 259–272.CrossRefGoogle Scholar
[6] Bürger, R. (1991) Unbounded one-parameter semigroups, Fréchet spaces and applications. In Semigroup Theory and Evolution Equations. ed. Clément, Ph., Mitidieri, E. and de Pagter, B. pp. 77–91. Marcel Dekker, New York.Google Scholar
[7] Crow, J. F. and Kimura, M. (1964) The theory of genetic loads. Proc. XI Int. Congress on Genetics. pp. 495–505. Pergamon, Oxford.Google Scholar
[8] Crow, J. F. and Kimura, M. (1970) An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
[9] Eshel, I. (1972) Evolution processes with continuity of types. Adv. Appl. Prob. 4, 475–507.Google Scholar
[10] Ethier, S. N. and Kurtz, G. (1993) Fleming-Viot processes in population genetics. SIAM J. Control Optim. 31, 345–386.CrossRefGoogle Scholar
[11] Ethier, S. N. and Nagylaki, T. (1988) Diffusion approximations of Markov chains with two time scales and applications to population genetics, II. Adv. Appl. Prob. 20, 525–545:Google Scholar
[12] Feichtinger, H. G. (1984) Compactness in translation invariant Banach spaces of distributions and compact multipliers. J. Math. Anal. Appl. 102, 289–327.Google Scholar
[13] Fleming, W. H. (1979) Equilibrium distributions of continuous polygenic traits. SIAM J. Appl. Math. 36, 148–168.CrossRefGoogle Scholar
[14] Fleming, W. H. and Viot, M. (1979) Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28, 817–843.Google Scholar
[15] Hofbauer, J. (1985) The selection mutation equation. J. Math. Biol. 23, 41–53.Google Scholar
[16] Jörgens, K. (1970) Lineare Integraloperatoren. Teubner, Stuttgart.CrossRefGoogle Scholar
[17] Karlin, S. (1988) Non-Gaussian phenotypic models of quantitative traits. In Proc. 2nd Int. Conf on Quantitative Genetics. ed. Weir, B. S., Eisen, E. J., Goodman, M. M. and Namkoong, G. pp. 123–144. Sinauer, Sunderland, MA.Google Scholar
[18] Kimura, M. (1965) A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Nat. Acad. Sci. USA 54, 731–736.Google Scholar
[19] Kingman, J. F. C. (1978) A simple model for the balance between selection and mutation. J. Appl. Prob. 15, 1–12.Google Scholar
[20] Lande, R. (1975) The maintenance of genetic variability by mutation in a polygenic character with linked loci. Genet. Res. 26, 221–235.Google Scholar
[21] Moran, P. A. P. (1976) Global stability of genetic systems governed by mutation and selection. Math. Proc. Camb. Phil. Soc. 80, 331–336; II (1977) Math. Proc. Camb. Phil. Soc. 81, 435-441.Google Scholar
[22] Nagel, R., (ed) (1986) One-Parameter Semigroups of Positive Operators. (Lecture Notes in Mathematics). Springer, Berlin.Google Scholar
[23] Nagylaki, T. (1984) Selection on a quantitative character. In Human Population Genetics: The Pittsburgh Symposium. ed. Chakravarti, A. Van Nostrand, New York.Google Scholar
[24] Nagylaki, T. (1990) Models and approximations for random genetic drift. Theoret. Pop. Biol. 37, 192–212.CrossRefGoogle Scholar
[25] Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin.CrossRefGoogle Scholar
[26] Schaefer, H. H. (1974) Banach Lattices and Positive Operators. Springer, Berlin.Google Scholar
[27] Turelli, M. (1984) Heritable genetic variation via mutation-selection balance: Lerch's zeta meets the abdominal bristle. Theoret. Pop. Biol. 25, 138–193.Google Scholar
[28] Wagner, G. P. and Krall, P. (1993) What is the difference between models of error thresholds and Muller's ratchet? J. Math. Biol. 32, 33–44.CrossRefGoogle Scholar
[29] Zaanen, A. C. (1983) Riesz spaces: II. North-Holland, Amsterdam.Google Scholar