  In this work, we provide an asymptotic analysis of the solutions to an
elliptic integro-differential equation. This equation describes the
evolutionary equilibria of a phenotypically structured population, subject to
selection, mutation, and both local and non-local dispersion in a spatially
heterogeneous, possibly patchy, environment. Considering small effects of
mutations, we provide an asymptotic description of the equilibria of the
phenotypic density. This asymptotic description involves a Hamilton-Jacobi
equation with constraint coupled with an eigenvalue problem. Based on such
analysis, we characterize some qualitative properties of the phenotypic density
at equilibrium depending on the heterogeneity of the environment. In
particular, we show that when the heterogeneity of the environment is low, the
population concentrates around a single phenotypic trait leading to a unimodal
phenotypic distribution. On the contrary, a strong fragmentation of the
environment leads to multi-modal phenotypic distributions.
