  This work is devoted to the study of scaling limits in small mutations and
large time of the solutions u^$\epsilon$ of two deterministic models of
phenotypic adaptation, where the parameter $\epsilon$ > 0 scales the size of
mutations. The first model is the so-called Lotka-Volterra parabolic PDE in R d
with an arbitrary number of resources and the second one is an adaptation of
the first model to a finite phenotype space. The solutions of such systems
typically 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 regularity
results for these limits. The method mainly relies on Feynman-Kac type
representations of u $\epsilon$ and Varadhan's Lemma. Our probabilistic
approach applies to multi-resources situations not covered by standard
analytical methods and makes the link between variational limit problems and
Hamilton-Jacobi equations with irregular Hamiltonians that arise naturally from
analytical methods. The finite case presents substantial difficulties since the
rate function of the associated large deviation principle has non-compact level
sets. In that case, we are also able to obtain uniqueness of the solution of
the variational problem and of the associated differential problem which can be
interpreted as a Hamilton-Jacobi equation in finite state space.
