By Sepideh Mirrahimi (CNRS)
Joint work with Nicolas Champagnat, Sylvie Méléard and Chi Tran.
Abstract: An approach based on Hamilton-Jacobi equations has been developed during the last two
decades to study quantitative genetics models, leading to an analytical description of the
phenotypic density. Such Hamilton-Jacobi equations are derived, in the regime of small
mutational variance, from integro-di erential models, which are themselves derived from
stochastic individual based models in the limit of large populations. These equations are
hence derived in two steps, each of them being an asymptotic derivation, considering  rst
large populations and next small mutational effects.
In this work, we derive such a Hamilton-Jacobi equation, directly from a stochastic individual
based model. This derivation allows a better understanding of the results obtained
by the Hamilton-Jacobi approach and would lead to a recti cation of the approach taking
into account possible extinctions of sub-populations.
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