  We study an equation structured by age and a phenotypic trait describing the
growth process of a population subject to aging, competition between
individuals, and mutations. This leads to a renewal equation which occurs in
many evolutionary biology problems. We aim to describe precisely the
asymp-totic behavior of the solution, to infer properties that illustrate the
concentration and adaptive dynamics of such a population. This work is a
continuation of [38] where the case without mutations is considered. When
mutations are taken into account, it is necessary to control the corrector
which is the main novelty of the present paper. Our approach consists in
defining, by the Hopf transform, a Hamilton-Jacobi equation with an effective
Hamiltonian as in homogenization problems. Its solution carries the singular
part of the limiting density (typically Dirac masses) and the corrector defines
the weights. The main new result of this paper is to prove that the corrector
is uniformly bounded, using only the global Lipschitz and semi-convexity
estimates for the viscosity solution of the Hamilton-Jacobi equation. We also
establish the limiting equation satisfied by the corrector. To the best of our
knowledge, this is the first example where such bounds can be proved in such a
context.
