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ESS modelling of diploid populations II: stability analysis of possible equilibria

  • W. G. S. Hines (a1)

Abstract

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, a companion paper explored relevant features of the probability simplex of allelic frequencies for a population which is genetically homogeneous except possibly at a single locus.

The Shahshahani metric is modified in this paper to produce a measure of distance near an arbitrary frequency F in the allelic simplex which can be used when some alleles are given zero weight by F. The equation of evolution for the modified metric can then be used to show that certain sets of frequencies (corresponding to equilibrium mean strategies) act as local attractors, as long as the mean strategies corresponding to those sets are non-singular or even, in most cases, singular. We identify conditions under which the measure of distance from an initial frequency to a nearby set of equilibrium frequencies corresponding to exceptional mean strategies might increase, either temporarily or for a protracted length of time.

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* Postal address: Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada, N1G 2W1.

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Research supported by NSERC Operating Grant A6187.

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References

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Akin, E. (1979) The Geometry of Population Genetics. Lecture notes in Biomathematics 31, Springer-Verlag, Berlin.
Akin, E., (1982) Exponential families and game dynamics. Cdn. J. Math. 34, 374405.
Akin, E. (1990) The differential geometry of population genetics and evolutionary games. In Mathematical and Statistical Developments of Evolutionary Theory, ed. Lessard, S., pp. 193. Kluwer, Dordrecht.
Antonelli, P. and Strobeck, C. (1977) The geometry of random drift I. Stochastic distance and diffusion. Adv. Appl. Prob. 9, 238249.
Cressman, R. and Hines, W. G. S. (1984) Evolutionarily stable strategies of diploid populations with semi-dominant inheritance patterns. J. Appl. Prob. 21, 19.
Healy, M. J. R. (1986) Matrices for Statistics. Clarendon Press, Oxford.
Hines, W. G. S. (1980) Strategy stability in complex populations. J. Appl. Prob. 17, 600610.
Hines, W. S. G. (1987) Evolutionarily stable strategies: a review of basic theory. Theoret. Popn. Biol. 31, 195272.
Hines, W. G. S. (1994) ESS modelling of diploid populations I: anatomy of one-locus allelic frequency simplices. Adv. Appl. Prob. 26 (this issue).
Hines, W. G. S. and Bishop, D. T. (1983) Evolutionarily stable strategies in diploid populations with general inheritance patterns. J. Appl. Prob. 20, 395399.
Hines, W. G. S. and Bishop, D. T. (1984a). Can and will a sexual population attain an ESS? J. Theoret. Biol. 111, 667686.
Hines, W. G. S. and Bishop, D. T. (1984b). On the local stability of an evolutionarily stable strategy in a diploid population. J. Appl. Prob. 21, 215224.
Hofbauer, J. and Sigmund, K. (1988) The Theory of Evolution and Dynamical Systems. London Mathematical Society Student Texts 7, Cambridge University Press.
Maynard Smith, J. (1974) The theory of games and the evolution of animal conflicts. J. Theoret. Biol. 47, 209221.
Shahshahani, S. (1979) A New Mathematical Framework for the Study of Linkage and Selection. Trans. Amer. Math. Soc. 211, American Mathematical Society, Providence, RI.

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ESS modelling of diploid populations II: stability analysis of possible equilibria

  • W. G. S. Hines (a1)

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