  A Hamilton-Jacobi formulation has been established previously for
phenotypically structured population models where the solution concentrates as
Dirac masses in the limit of small diffusion. Is it possible to extend this
approach to spatial models? Are the limiting solutions still in the form of
sums of Dirac masses? Does the presence of several habitats lead to polymorphic
situations? We study the stationary solutions of a structured population model,
while the population is structured by continuous phenotypical traits and
discrete positions in space. The growth term varies from one habitable zone to
another, for instance because of a change in the temperature. The individuals
can migrate from one zone to another with a constant rate. The mathematical
modeling of this problem, considering mutations between phenotypical traits and
competitive interaction of individuals within each zone via a single resource,
leads to a system of coupled parabolic integro-differential equations. We study
the asymptotic behavior of the stationary solutions to this model in the limit
of small mutations. The limit, which is a sum of Dirac masses, can be described
with the help of an effective Hamiltonian. The presence of migration can modify
the dominant traits and lead to polymorphic situations.
