On a Notion of Linear Replicator Equations

  • Nihat Ay
  • Ionas Erb


We show that replicator equations follow naturally from the exponential affine structure of the simplex known from information geometry. It is then natural to call replicator equations linear if their fitness function is affine. For such linear replicator equations an explicit solution can be found. The approach is also demonstrated for the example of Eigen’s hypercycle, where some new analytic results are obtained using the explicit solution.


Replicator equation exponential affine structure logarithmic linearity hypercycle system. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amari, S., Nagaoka, H. 2000Methods of Information Geometry, AMS Translations of Mathematical Monographs vol 191Oxford University PressOxfordGoogle Scholar
  2. 2.
    Arrigoni, M. 1990Modello, di iperciclo autonomo per la selezione di molecole biologicheSguardo matematico nellea biologiaMendrisio - ottobreGoogle Scholar
  3. 3.
    Bröcker, T. 1980Analysis in Mehreren VariablenB.G. TeubnerStuttgartGoogle Scholar
  4. 4.
    Eigen, M. 1971Selforganization of matter and the evolution of biological macromoleculesNaturwiss58465523CrossRefPubMedGoogle Scholar
  5. 5.
    Hofbauer, J., Mallet-Paret, J., Smith H., L. 1991Stable periodic solutions for the hypercycle systemJ. Dynm. Diff. Eqns.3423436CrossRefGoogle Scholar
  6. 6.
    Hofbauer, J., Sigmund, , K.,  1998Evolutionary Games and Population DynamicsCambridge University PressCambridgeGoogle Scholar
  7. 7.
    Jones, B. L. 1977A solvable selfreproductive hypercycle model for the selection of biological molecules. J. MathBiol.4187193Google Scholar
  8. 8.
    Robinson, C. 1994Dynamical SystemsCRC PressBoca RatonGoogle Scholar
  9. 9.
    Stadler, P. 1991Dynamics of small autocatalytic reaction network IV: inhomogeneous replicator equations. BioSystems26119Google Scholar
  10. 10.
    Steiner, A., Gander, M.J. 1995Le soluzioni delle equazioni preda-predatore, IIVolterriano5928Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Nihat Ay
    • 1
    • 3
    • 4
  • Ionas Erb
    • 2
    • 3
  1. 1.Santa Fe InstituteSanta FeUSA
  2. 2.Bioinformatik, Institut für InformatikUniversity of LeipzigLeipzigGermany
  3. 3.Max-Planck Institute for Mathematics in the SciencesLeipzigGermany
  4. 4.Mathematical InstituteFriedrich-Alexander-University Erlangen-NurembergErlangenGermany

Personalised recommendations