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  • 1 Department of Zoology, University of Toronto Toronto25 Harbord St., Toronto, Ontario, Canada M5S 3G5
  • | 2 Department of Zoology, University of Toronto Toronto25 Harbord St., Toronto, Ontario, Canada M5S 3G5
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The geometric mean of fitness is considered to be the main indicator of evolutionary change in stochastic models. However, this measure was initially derived for models with infinite population sizes, where the long-term evolutionary behavior can be described with certainty. In this paper we begin an exploration of the limitations and utility of this approach to evolution in finite populations and discuss alternate methods for predicting evolutionary dynamics. We reanalyze a model of lottery competition under environmental stochasticity by including population finiteness, and show that the geometric mean predictions do not always agree with those based on the fixation probability of rare alleles. Further, the fixation probability can be inserted into adaptive dynamics equations to derive the mean state of the population. We explore the effects of increasing population size on these conclusions through simulations. These simulations show that for small population sizes the fixation probability accurately predicts the course of evolution, but as population size becomes large the geometric mean predictions are upheld. The two approaches are reconciled because the time scale on which the fixation probability approach applies becomes very large as population size grows.

  • Chesson, P. L. and Ellner, S. (1989): Invasibility and stochastic boundedness in monotonic competition models. J. Math. Biol.27:117-138.

    'Invasibility and stochastic boundedness in monotonic competition models ' () 27 J. Math. Biol. : 117 -138.

    • Search Google Scholar
  • Chesson, P. L. (1985): Coexistence of competitors in spatially and temporally varying environments: a look at the combined effects of different sorts of variability. Theor. Popul. Biol.28: 263-287.

    'Coexistence of competitors in spatially and temporally varying environments: a look at the combined effects of different sorts of variability ' () 28 Theor. Popul. Biol. : 263 -287.

    • Search Google Scholar
  • Chesson, P. L. and Warner, R. R. (1981): Environmental variability promotes coexistence in lottery competitive systems. Amer. Natur.117: 923-943.

    'Environmental variability promotes coexistence in lottery competitive systems ' () 117 Amer. Natur. : 923 -943.

    • Search Google Scholar
  • Christiansen, F. B. (1991): On conditions for evolutionary stability for a continuously varying character. Amer. Natur.138: 37-50.

    'On conditions for evolutionary stability for a continuously varying character ' () 138 Amer. Natur. : 37 -50.

    • Search Google Scholar
  • Crow, J. F. and Kimura, M. (1970): An Introduction to Population Genetics Theory. Harper and Row, New York.

    An Introduction to Population Genetics Theory , ().

  • Dieckmann, U. and Law, R. (1996): The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol.34:579-612.

    'The dynamical theory of coevolution: a derivation from stochastic ecological processes ' () 34 J. Math. Biol. : 579 -612.

    • Search Google Scholar
  • Kisdi, E. and Meszena, G. (1995): Life histories with lottery competition in a stochastic environment: ESSs which do not prevail. Theor. Popul. Biol.47: 191-211.

    'Life histories with lottery competition in a stochastic environment: ESSs which do not prevail ' () 47 Theor. Popul. Biol. : 191 -211.

    • Search Google Scholar
  • Maynard Smith, J. (1982): Evolution and the Theory of Games. Cambridge University Press, Cambridge.

    Evolution and the Theory of Games , ().

  • Proulx, S. R. (2000): The ESS under spatial variation with applications to sex allocation. Theor. Popul. Biol.58: 33-47.

    'The ESS under spatial variation with applications to sex allocation ' () 58 Theor. Popul. Biol. : 33 -47.

    • Search Google Scholar
  • Turelli, M. (1978b): A reexamination of stability in randomly varying versus deterministic environments with comments on the stochastic theory of limiting similarity. Theor. Popul. Biol.13: 244-267.

    'A reexamination of stability in randomly varying versus deterministic environments with comments on the stochastic theory of limiting similarity ' () 13 Theor. Popul. Biol. : 244 -267.

    • Search Google Scholar
  • Turelli, M. (1978a): Does environmental variability limit niche overlap? Proc. Natl. Acad. Sci75: 5085-5089.

    'Does environmental variability limit niche overlap? ' () 75 Proc. Natl. Acad. Sci : 5085 -5089.

    • Search Google Scholar
  • Yoshimura, J. and Clark, C. W. (1991): Individual adaptations in stochastic environments. Evol. Ecol.5: 173-192.

    'Individual adaptations in stochastic environments ' () 5 Evol. Ecol. : 173 -192.

  • Ellner, S. P. (1996): You bet your life: Life-history strategies in fluctuating environments, In Othmer, H. G., Adler, H. G., Lewis, M. A. and Dallon, J. C. (eds), Case Studies in Mathematical Modeling: Ecology, Physiology and Cell Biology. Prentice Hall, Upper Saddle River, New Jersey, pp. 3-24.

    Case Studies in Mathematical Modeling: Ecology, Physiology and Cell Biology , () 3 -24.

    • Search Google Scholar
  • Eshel, I. (1983): Evolutionary and continuous stability. J. Theor. Biol.103: 99-112.

    'Evolutionary and continuous stability ' () 103 J. Theor. Biol. : 99 -112.

  • Ferriere, R. and Gatto, M. (1995): Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations. Theor. Popul. Biol.48: 126-171.

    'Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations ' () 48 Theor. Popul. Biol. : 126 -171.

    • Search Google Scholar
  • Geritz, S. A. H., Kisdi, E., Meszena, G. and Metz, J. A. J. (1998): Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol.12: 35-57.

    'Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree ' () 12 Evol. Ecol. : 35 -57.

    • Search Google Scholar
  • Gillespie, J. H. (1973): Natural selection with varying selection coefficients - a haploid model. Genet. Res.21:115-120.

    'Natural selection with varying selection coefficients - a haploid model ' () 21 Genet. Res. : 115 -120.

    • Search Google Scholar
  • Gillespie, J. H. (1974): Natural selection for within generation variance in offspring number. Genetics76: 601-606.

    'Natural selection for within generation variance in offspring number ' () 76 Genetics : 601 -606.

    • Search Google Scholar
  • Grant, A. (1997): Selection pressures on vital rates in density-dependent populations. Proc. R. Soc. Lond. B264: 303-306.

    'Selection pressures on vital rates in density-dependent populations ' () 264 Proc. R. Soc. Lond. B : 303 -306.

    • Search Google Scholar
  • Haccou, P. and Iwasa, Y. (1996): Establishment probability in fluctuating environments: a branching process model. Theor. Popul. Biol.50: 254-280.

    'Establishment probability in fluctuating environments: a branching process model ' () 50 Theor. Popul. Biol. : 254 -280.

    • Search Google Scholar
  • Heino, M., Metz, J. A. J. and Kaitala, V. (1998a): The enigma of frequency-dependent selection. Trends Ecol. Evol.13:367-370.

    'The enigma of frequency-dependent selection ' () 13 Trends Ecol. Evol. : 367 -370.

  • Heino, M., Metz, J. A. J. and Kaitala, V. (1998b): Reply. Trends Ecol. Evol.13: 509.

    'Reply ' () 13 Trends Ecol. Evol. : 509.

  • Iwasa, Y. (1988): Free fitness that always increases in evolution. J. Theor. Biol.135: 265-281.

    'Free fitness that always increases in evolution ' () 135 J. Theor. Biol. : 265 -281.

  • Iwasa, Y. and Levin, S. A. (1995): The timing of life history events. J. Theor. Biol.172: 33-42.

    'The timing of life history events ' () 172 J. Theor. Biol. : 33 -42.

  • Jensen, L. (1973): Random selective advantages of genes and their probabilities of fixation. Genet. Res.21: 215-219.

    'Random selective advantages of genes and their probabilities of fixation ' () 21 Genet. Res. : 215 -219.

    • Search Google Scholar
  • Metz, J. A. J., Geritz, S. A. H., Meszena, G., Jacobs, F. J. A. and van Heerwaarden, J. S. (1996a): Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In Strein, S. J. van and VerduynLunel, S. M. (eds), Stochastic and Spatial Structures of Dynamical Systems. North Holland, Amsterdam, pp. 183-213.

    Stochastic and Spatial Structures of Dynamical Systems , () 183 -213.

  • Metz, J. A. J., Mylius, S. D. and Diekmann, O. (1996b): When does evolution optimise? On the relation between types of density dependence and evolutionary stable life history parameters. Technical report, IIASA Working Papers WP-96-004.

  • Metz, J. A. J., Nisbet, R. M. and Geritz, S. A. H. (1992): How should we define 'fitness' for general ecological scenarios? TREE7:198-202.

    'How should we define 'fitness' for general ecological scenarios ' () 7 TREE : 198 -202.

  • Otto, S. P. and Whitlock, M. C. (1997): The probability of fixation in populations of changing size. Genetics146: 723-733.

    'The probability of fixation in populations of changing size ' () 146 Genetics : 723 -733.

    • Search Google Scholar
  • Rand, D. A., Wilson, H. B. and McGlade, J. M. (1994): Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. Philos. Trans. R. Soc. Lond. B343:261-283.

    'Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics ' () 343 Philos. Trans. R. Soc. Lond. B : 261 -283.

    • Search Google Scholar
  • Ranta, E., Tesar, D., Alaja, S. and Kaitala, V. (2000): Does evolution of iteroparous and semelparous reproduction call for spatially structured systems? Evolution54:145-150.

    'Does evolution of iteroparous and semelparous reproduction call for spatially structured systems ' () 54 Evolution : 145 -150.

    • Search Google Scholar
  • Real, L. A. and Ellner, S. (1992): Life history evolution in stochastic environments: A graphical approach. Ecology73: 1227-1236.

    'Life history evolution in stochastic environments: A graphical approach ' () 73 Ecology : 1227 -1236.

    • Search Google Scholar
  • Seger, J. and Brockmann, H. J. (1987): What is bet-hedging? In Oxford Surveys in Evolutionary Biology, Volume 4, pp. 182-211.

  • Taylor, P. D. (1989): Evolutionary stability in one-parameter models under weak selection. Theor. Popul. Biol.36:125-143.

    'Evolutionary stability in one-parameter models under weak selection ' () 36 Theor. Popul. Biol. : 125 -143.

    • Search Google Scholar
  • Abrams, P. A., Matsuda, H. and Harada, Y. (1993): Evolutionary unstable fitness maxima and stable fitness minima of continuous traits. Evol. Ecol.7: 465-487.

    'Evolutionary unstable fitness maxima and stable fitness minima of continuous traits ' () 7 Evol. Ecol. : 465 -487.

    • Search Google Scholar
  • Benton, T. B. and Grant, A. (1999): Optimal reproductive effort in stochastic, density-dependent environments. Evolution53: 677-688.

    'Optimal reproductive effort in stochastic, density-dependent environments ' () 53 Evolution : 677 -688.

    • Search Google Scholar
  • Benton, T. B. and Grant, A. (2000): Evolutionary fitness in ecology: comparing measures of fitness in stochastic, density-dependent environments. Evol. Ecol. Res.2: 769-789.

    'Evolutionary fitness in ecology: comparing measures of fitness in stochastic, density-dependent environments ' () 2 Evol. Ecol. Res. : 769 -789.

    • Search Google Scholar
  • Bulmer, M. G. (1985): Selection for iteroparity in a variable environment. Amer. Nat.126: 63-71.

    'Selection for iteroparity in a variable environment ' () 126 Amer. Nat. : 63 -71.

  • Karlin, S. and Levikson, B. (1974): Temporal fluctuations in selection intensities: case of small population size. Theor. Popul. Biol.6:383-412.

    'Temporal fluctuations in selection intensities: case of small population size ' () 6 Theor. Popul. Biol. : 383 -412.

    • Search Google Scholar
  • Ellner, S. (1984): Asymptotic behavior of some stochastic difference equation population models. J. Math. Biol.19:169-200.

    'Asymptotic behavior of some stochastic difference equation population models ' () 19 J. Math. Biol. : 169 -200.

    • Search Google Scholar
  • Kisdi, E. (1998): Frequency dependence versus optimization. Trends Ecol. Evol.13: 508.

    'Frequency dependence versus optimization ' () 13 Trends Ecol. Evol. : 508.

Selection
Language English
Year of
Foundation
2001
Publication
Programme
ceased
Publisher Akadémiai Kiadó
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 1585-1931 (Print)
ISSN 1588-287X (Online)

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