  We study a mathematical model describing the growth process of a population
structured by age and a phenotypical trait, subject to aging, competition
between individuals and rare mutations. Our goals are to describe the
asymptotic behaviour of the solution to a renewal type equation, and then to
derive properties that illustrate the adaptive dynamics of such a population.
We begin with a simplified model by discarding the effect of mutations, which
allows us to introduce the main ideas and state the full result. Then we
discuss the general model and its limitations. Our approach uses the
eigenelements of a formal limiting operator, that depend on the structuring
variables of the model and define an effective fitness. Then we introduce a new
method which reduces the convergence proof to entropy estimates rather than
estimates on the constrained Hamilton-Jacobi equation. Numerical tests
illustrate the theory and show the selection of a fittest trait according to
the effective fitness. For the problem with mutations, an unusual Hamiltonian
arises with an exponential growth, for which we show existence of a global
viscosity solution, using an uncommon a priori estimate and a new uniqueness
result.
