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Freely forming groups: Trying to be rare

Published online by Cambridge University Press:  17 February 2009

Michael Baake ,
Uwe Grimm  and
Harald Jockusch
Michael Baake
Affiliation:
Fakultät für Mathematik, Univinersität Bielefeld, Box 100131, 33501 Bielefeld, Germany; e-mail: mbaake@math.uni-bielefeld.de.
Uwe Grimm
Affiliation:
Department of Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK; e-mail: u.g.grimm@open.ac.uk.
Harald Jockusch
Affiliation:
Fakultät für Biologie, Univinersität Bielefeld, Box 100131, 33501 Bielefeld, Germany, e-mail: h.jockusch@uni-bielefeld.de.
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Abstract

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A simple weakly frequency dependent model for the dynamics of a population with a finite number of types is proposed, based upon an advantage of being rare. In the infinite population limit, this model gives rise to a non-smooth dynamical system that reaches its globally stable equilibrium in finite time. This dynamical system is sufficiently simple to permit an explicit solution, built piecewise from solutions of the logistic equation in continuous time. It displays an interesting tree-like structure of coalescing components.

Keywords

population dynamicsnon-smooth dynamical systemsstable equilibrialogistic equation.
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 1 , July 2006 , pp. 1 - 10
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Amann, H., Ordinary Differential Equations (de Gruyter, Berlin, 1990).CrossRefGoogle Scholar
[2]Baake, M. and Baake, E., “An exactly solved model for mutation, recombination and selection”, Can. J. Math. 55 (2003) 3–41.CrossRefGoogle Scholar
[3]Bürger, R., The Mathematical Theory of Selection, Recombination, and Mutation (Wiley, Chichester, 2000).Google Scholar
[4]Cressman, R., Křivan, V. and Garay, J., “Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments”, Amer. Natur. 164 (2004) 473–489.CrossRefGoogle ScholarPubMed
[5]Deimling, K., Multivalued Differential Equations (de Gruyter, Berlin, 1992).CrossRefGoogle Scholar
[6]Ethier, S. N. and Kurtz, T. G., Markov Processes – Characterization and Convergence (Wiley, New York, 1986).CrossRefGoogle Scholar
[7]Ewens, W. J., Mathematical Population Genetics. I. Theoretical Introduction, 2nd ed. (Springer, New York, 2004).CrossRefGoogle Scholar
[8]Filippov, A. F., Differential Equations with Discontinuous Righthand Sides (Kluwer, Dordrecht, 1988).CrossRefGoogle Scholar
[9]Fretwell, S. D. and Lucas, H. L., “On territorial behavior and other factors influencing habitat distribution in birds. I. Theoretical development”, Acta Biotheor. 19 (1970) 16–36.CrossRefGoogle Scholar
[10]Garske, T. and Grimm, U., “A maximum principle for the mutation-selection equilibrium of nucleotide sequences”, Bull. Math. Biol. 66 (2004) 397–421.CrossRefGoogle ScholarPubMed
[11]Hofbauer, J. and Sigmund, K., Evolutionary Games and Population Dynamics (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
[12]Hofbauer, J. and Sigmund, K., “Evolutionary game dynamics”, Bull. Amer. Math. Soc. (N.S.) 40 (2003) 479–519.CrossRefGoogle Scholar
[13]Kunze, M., Non-Smooth Dynamical Systems, Springer, Berlin (Springer, Berlin, 2000).CrossRefGoogle Scholar
[14]Redner, O. and Baake, M., “Unequal crossover dynamics in discrete and continuous time”, J. Math. Biol. 49 (2004) 201–226.CrossRefGoogle ScholarPubMed