Authors:
N. Champagnat Department of Ecology, Unit of Mathematical Eco-Evolutionary Biology, Ecole Normale Supérieure 46 rue d'Ulm, 75230 Paris cedex 05, France

Search for other papers by N. Champagnat in
Current site
Google Scholar
PubMed
Close
,
R. Ferričre Department of Ecology, Unit of Mathematical Eco-Evolutionary Biology, Ecole Normale Supérieure 46 rue d'Ulm, 75230 Paris cedex 05, France

Search for other papers by R. Ferričre in
Current site
Google Scholar
PubMed
Close
, and
G. Ben Arous4 Department of Mathematics and Applications, EPFL Lausanne, Switzerland

Search for other papers by G. Ben Arous4 in
Current site
Google Scholar
PubMed
Close
Restricted access

The Darwinian evolution of a quantitative adaptive character is described as a jump process. As the variance of the distribution of mutation steps goes to zero, this process converges in law to the solution of an ordinary differential equation. In the case where the mutation step distribution is symmetrical, this establishes rigorously the so-called canonical equation first proposed by Dieckmann and Law (1996). Our mathematical approach naturally leads to extend the canonical equation to the case of biased mutations, and to seek ecological and genetic conditions under which evolution proceeds either through punctualism or through radiation.

  • Marrow, P., Law, R. and Cannings, C. (1992): The coevolution of predator-prey interactions: ESSs and Red Queen dynamics. Proc. R. Soc. Lond. B250:133-141.

    'The coevolution of predator-prey interactions: ESSs and Red Queen dynamics ' () 250 Proc. R. Soc. Lond. B : 133 -141 .

    • Search Google Scholar
  • Metz, J. A. J., Nisbet, R. M. and Geritz, S. A. H. (1992): How should we define 'fitness' for general ecological scenarios? Trends Ecol. Evolut. 7:198-202.

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

    • Search Google Scholar
  • Metz, J. A. J., Geritz, S. A. H., Meszéna, G., Jacobs, F. A. J. and van Heerwaarden, J. S. (1996): Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In van Strien, S. J., Verduyn Lunel, S. M. (eds): Stochastic and Spatial Structures of Dynamical Systems. North Holland, Amsterdam, pp. 183-231.

    Stochastic and Spatial Structures of Dynamical Systems , () 183 -231 .

  • Mukai, T. (1964): Polygenic mutation affecting quantitative character of Drosophila melanogaster. In Mutations in Quantitative Traits. Proceedings Gamma Field Symposium 3, Ministry of Agriculture, Japan, pp. 13-29.

    'Polygenic mutation affecting quantitative character of Drosophila melanogaster ' , , .

    • Search Google Scholar
  • Pomiankowski, A., Iwasa, Y. and Nee, S. (1991): The evolution of costly mate preferences I. Fisher and biased mutation. Evolution45:1422-1430.

    'The evolution of costly mate preferences I. Fisher and biased mutation ' () 45 Evolution : 1422 -1430 .

    • Search Google Scholar
  • Rand, D. A. and Wilson, H. B. (1993): Evolutionary catastrophes, punctuated equilibria and gradualism in ecosystem evolution. Proc. R. Soc. Lond. B253:137-141.

    'Evolutionary catastrophes, punctuated equilibria and gradualism in ecosystem evolution ' () 253 Proc. R. Soc. Lond. B : 137 -141 .

    • Search Google Scholar
  • Schluter, D. (2000): The Ecology of Adaptive Radiation oxford University Press, Oxford.

    The Ecology of Adaptive Radiation , ().

  • Stanley, S. M. (1979): Macroevolution: Pattern and Process. Freeman, San Francisco, Ca.

    Macroevolution: Pattern and Process , ().

  • Wentzel, A. D. (1976b): Rough limit theorems on large deviations for Markov random processes, II. TheoryProbab. Appl. 21:499-512.

    'Rough limit theorems on large deviations for Markov random processes, II ' () 21 Theory Probab. Appl : 499 -512 .

    • Search Google Scholar
  • Lai, C. and Mackay, T. (1990): Hybrid dysgenesis-induced quantitative variation the X chromosome of Drosophila melanogaster. Genetics124:627-636.

    'Hybrid dysgenesis-induced quantitative variation the X chromosome of Drosophila melanogaster ' () 124 Genetics : 627 -636 .

    • Search Google Scholar
  • Freidlin, M. I. and Wentzel, A. D. (1984): Random Perturbations of Dynamical Systems. Springer-Verlag, Berlin.

    Random Perturbations of Dynamical Systems , ().

  • Hofbauer, J. and Sigmund, R. (1990): Adaptive dynamics and evolutionary stability. Appl. Math. Lets3:75-79.

    'Adaptive dynamics and evolutionary stability ' () 3 Appl. Math. Lets : 75 -79 .

  • Kisdi, E. (1999): Evolutionary branching under asymmetric competition. J. Theor. Biol. 198:149-162.

    'Evolutionary branching under asymmetric competition ' () 198 J. Theor. Biol : 149 -162 .

    • Search Google Scholar
  • Doebeli, M. and Dieckmann, U. (2000): Evolutionary branching and sympatric speciation caused by different types of ecological interactions. Amer. Nat. 156:S77-S101.

    'Evolutionary branching and sympatric speciation caused by different types of ecological interactions ' () 156 Amer. Nat. : S77 -S101 .

    • Search Google Scholar
  • Ethier, S. and Kurtz, T. (1986): Markov Processes, Characterization and Convergence. John Wiley and Sons, New York.

    Markov Processes, Characterization and Convergence , ().

  • Wentzel, A. D. (1976a): Rough limit theorems on large deviations for Markov random processes, I. TheoryProbab. Appl. 21:227-242.

    'Rough limit theorems on large deviations for Markov random processes, I ' () 21 Theory Probab. Appl : 227 -242 .

    • Search Google Scholar
  • 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
  • Collapse
  • Expand

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)