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An integro-differential equation from population genetics and perturbations of differentiable semigroups in Fréchet spaces

Published online by Cambridge University Press:  14 November 2011

Reinhard Bürger
Reinhard Bürger
Affiliation:
Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
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Synopsis

Existence and uniqueness of solutions of an integro-differential equation that arises in population genetics are proved. This equation describes the evolution of type densities in a population that is subject to mutation and directional selection on a quantitative trait. It turns out that a certain Fréchet space is the natural framework to show existence and uniqueness. One of the main steps in the proof is the investigation of perturbations of generators of differentiable semigroups in Fréchet spaces.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 118 , Issue 1-2 , 1991 , pp. 63 - 73
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Bürger, R.. Perturbations of positive semigroups and applications to population genetics. Math. Z. 197 (1988), 259–272.CrossRefGoogle Scholar
2Bürger, R.. The dynamics of directional selection I. Cycles, spirals and travelling waves (preprint, 1989).Google Scholar
3Choe, Y. H.. C 0-semigroups on a locally convex space. J. Math. Anal. Appl. 106 (1985), 293–320.CrossRefGoogle Scholar
4Davies, E. B.. One-Parameter Semigroups (London: Academic Press, 1980).Google Scholar
5Dembart, B.. On the theory of semigroups of operators on locally convex spaces. J. Fund. Anal. 16 (1974), 123–160.CrossRefGoogle Scholar
6Fleming, W. H.. Equilibrium distributions of continuous polygenic traits. SIAM J. Appl. Math. 36 (1979), 148–168.CrossRefGoogle Scholar
7Hegner, S. J.. Linear decomposable systems in continuous time. SIAM J. Math. Anal. 12 (1981), 243–273.CrossRefGoogle Scholar
8Kimura, M.. A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 731–736.CrossRefGoogle ScholarPubMed
9Köthe, G.. Topologische Vektorräume I. 2nd edn. (Berlin: Springer, 1965).Google Scholar
10Lande, R.. The response to selection on major and minor mutation affecting a metrical trait. Heredity 50 (1983), 47–65.CrossRefGoogle Scholar
11Reed, M. and Simon, B.. Methods of Modern Mathematical Physics I. Functional Analysis, 2nd edn. (Orlando: Academic Press, 1980).Google Scholar
12Thompson, C. J. and McBride, J. L.. On Eigen's theory of self-organization of matter and the evolution of biological macromolecules. Math. Biosci. 21 (1974), 127–142.CrossRefGoogle Scholar
13Turelli, M. and Barton, N.. Dynamics of polygenic characters under selection. Theoret. Pop. Biol. 38 (1990), 1–57.CrossRefGoogle Scholar