Derivation of replicator–mutator equations from a model in population genetics
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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.
KeywordsAdaptive dynamics replicator–mutator equation Population genetic model Scaling limits
Mathematics Subject Classification60H30 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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