Joint work with Nicolas Champagnat, Sylvie Méléard and Chi Tran.
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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