  In this note, we characterize the solution of a system of elliptic
integro-differential equations describing a phe-notypically structured
population subject to mutation, selection and migration. Generalizing an
approach based on Hamilton-Jacobi equations, we identify the dominant terms of
the solution when the mutation term is small (but nonzero). This method was
initially used, for different problems from evolutionary biology, to identify
the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses.
A key point is a uniqueness property related to the weak KAM theory. This
method allows to go further than the Gaussian approximation commonly used by
biologists and is an attempt to fill the gap between the theories of adaptive
dynamics and quantitative genetics.
