  We are interested in a stochastic model of trait and age-structured
population undergoing mutation and selection. We start with a continuous time,
discrete individual-centered population process. Taking the large population
and rare mutations limits under a well-chosen time-scale separation condition,
we obtain a jump process that generalizes the Trait Substitution Sequence
process describing Adaptive Dynamics for populations without age structure.
Under the additional assumption of small mutations, we derive an age-dependent
ordinary differential equation that extends the Canonical Equation. These
evolutionary approximations have never been introduced to our knowledge. They
are based on ecological phenomena represented by PDEs that generalize the
Gurtin-McCamy equation in Demography. Another particularity is that they
involve a fitness function, describing the probability of invasion of the
resident population by the mutant one, that can not always be computed
explicitly. Examples illustrate how adding an age-structure enrich the
modelling of structured population by including life history features such as
senescence. In the cases considered, we establish the evolutionary
approximations and study their long time behavior and the nature of their
evolutionary singularities when computation is tractable. Numerical procedures
and simulations are carried.
