Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T09:25:23.021Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  01 July 2016

W. G. S. Hines
W. G. S. Hines*
Affiliation:
University of Guelph
*
* Postal address: Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada, N1G 2W1.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

ANIMAL CONFLICT MODELSCOMPLEX POPULATIONSESS THEORYMEAN-COVARIANCE SUMMARIESSEXUAL POPULATIONSSIMPLEX ANATOMYSINGULAR SURFACES
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 2 , June 1994 , pp. 361 - 376
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by NSERC Operating Grant A6187.

References

Akin, E. (1979) The Geometry of Population Genetics. Lecture notes in Biomathematics 31, Springer-Verlag, Berlin.Google Scholar
Akin, E., (1982) Exponential families and game dynamics. Cdn. J. Math. 34, 374–405.Google Scholar
Akin, E. (1990) The differential geometry of population genetics and evolutionary games. In Mathematical and Statistical Developments of Evolutionary Theory, ed. Lessard, S., pp. 1–93. Kluwer, Dordrecht.Google Scholar
Antonelli, P. and Strobeck, C. (1977) The geometry of random drift I. Stochastic distance and diffusion. Adv. Appl. Prob. 9, 238–249.Google Scholar
Cressman, R. and Hines, W. G. S. (1984) Evolutionarily stable strategies of diploid populations with semi-dominant inheritance patterns. J. Appl. Prob. 21, 1–9.CrossRefGoogle Scholar
Healy, M. J. R. (1986) Matrices for Statistics. Clarendon Press, Oxford.Google Scholar
Hines, W. G. S. (1980) Strategy stability in complex populations. J. Appl. Prob. 17, 600–610.CrossRefGoogle Scholar
Hines, W. S. G. (1987) Evolutionarily stable strategies: a review of basic theory. Theoret. Popn. Biol. 31, 195–272.Google Scholar
Hines, W. G. S. (1994) ESS modelling of diploid populations I: anatomy of one-locus allelic frequency simplices. Adv. Appl. Prob. 26 (this issue).Google Scholar
Hines, W. G. S. and Bishop, D. T. (1983) Evolutionarily stable strategies in diploid populations with general inheritance patterns. J. Appl. Prob. 20, 395–399.Google Scholar
Hines, W. G. S. and Bishop, D. T. (1984a). Can and will a sexual population attain an ESS? J. Theoret. Biol. 111, 667–686.Google Scholar
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, 215–224.Google Scholar
Hofbauer, J. and Sigmund, K. (1988) The Theory of Evolution and Dynamical Systems. London Mathematical Society Student Texts 7, Cambridge University Press.Google Scholar
Maynard Smith, J. (1974) The theory of games and the evolution of animal conflicts. J. Theoret. Biol. 47, 209–221.CrossRefGoogle Scholar
Shahshahani, S. (1979) A New Mathematical Framework for the Study of Linkage and Selection. Trans. Amer. Math. Soc. 211, American Mathematical Society, Providence, RI.Google Scholar