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Journal of Mathematical Biology

, Volume 50, Issue 1, pp 83–114 | Cite as

An asymptotic maximum principle for essentially linear evolution models

  • Ellen BaakeEmail author
  • Michael Baake
  • Anton Bovier
  • Markus Klein
Article

Abstract.

Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional maximum principle in the limit N→∞ (where N, or N d with d≥1, is proportional to the number of types). In order to extend this variational principle to a larger class of models, we consider here a family of reversible matrices of asymptotic dimension N d and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order 1/N. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types.

Keywords

Asymptotics of leading eigenvalue Reversibility Mutation-selection models Ancestral distribution Lumping 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Ellen Baake
    • 1
    Email author
  • Michael Baake
    • 2
  • Anton Bovier
    • 3
    • 5
  • Markus Klein
    • 4
  1. 1.Technische FakultätUniversität BielefeldBielefeldGermany
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  3. 3.Weierstrass-Institut für Angewandte Analysis und StochastikBerlinGermany
  4. 4.Institut für MathematikUniversität PotsdamPotsdamGermany
  5. 5.Institut für MathematikTechnische Universität BerlinBerlinGermany

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