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This work is devoted to the study of scaling limits in small mutations andlarge time of the solutions u^$\epsilon$ of two deterministic models ofphenotypic adaptation, where the parameter $\epsilon$ > 0 scales the size ofmutations. The first model is the so-called Lotka-Volterra parabolic PDE in R dwith an arbitrary number of resources and the second one is an adaptation ofthe first model to a finite phenotype space. The solutions of such systemstypically concentrate as Dirac masses in the limit $\epsilon$ $\rightarrow$ 0.Our main results are, in both cases, the representation of the limits of$\epsilon$ log u^$\epsilon$ as solutions of variational problems and regularityresults for these limits. The method mainly relies on Feynman-Kac typerepresentations of u $\epsilon$ and Varadhan's Lemma. Our probabilisticapproach applies to multi-resources situations not covered by standardanalytical methods and makes the link between variational limit problems andHamilton-Jacobi equations with irregular Hamiltonians that arise naturally fromanalytical methods. The finite case presents substantial difficulties since therate function of the associated large deviation principle has non-compact levelsets. In that case, we are also able to obtain uniqueness of the solution ofthe variational problem and of the associated differential problem which can beinterpreted as a Hamilton-Jacobi equation in finite state space.

A probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resources