Derivation of replicator–mutator equations from a model in population genetics

  • Joe Yuichiro WakanoEmail author
  • Tadahisa Funaki
  • Satoshi Yokoyama
Original Paper Area 1


We introduce a Markov chain model to study evolution of a continuous trait based on population genetics. It corresponds to individual-based model which includes frequency dependent selection caused by m-player game interactions and stochastic fluctuations due to random genetic drift and mutation. We prove that under a proper scaling limit as the population size increases the system converges to the solution of replicator–mutator equations. Our result establishes an affirmative mathematical base to the adaptive dynamics formulation employed in the theory of the mathematical biology.


Adaptive dynamics replicator–mutator equation Population genetic model Scaling limits 

Mathematics Subject Classification

60H30 91A06 92B05 92.D10 92.D15 



The authors thank Professors Masayasu Mimura, Toshiyuki Ogawa and Kenji Handa for helpful discussions. They also express great thanks to the referee for his/her careful reading and kind comments which improve the paper considerably.


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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK 2017

Authors and Affiliations

  • Joe Yuichiro Wakano
    • 1
    Email author
  • Tadahisa Funaki
    • 2
    • 3
  • Satoshi Yokoyama
    • 2
    • 3
  1. 1.Meiji Institute for Advanced Study of Mathematical SciencesMeiji UniversityNakanoJapan
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoKomabaJapan
  3. 3.Department of Mathematics, School of Fundamental Science and EngineeringWaseda UniversityShinjuku-kuJapan

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