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.
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