  Understanding how stochastic and non-linear deterministic processes interact
is a major challenge in population dynamics theory. After a short review, we
introduce a stochastic individual-centered particle model to describe the
evolution in continuous time of a population with (continuous) age and trait
structures. The individuals reproduce asexually, age, interact and die. The
'trait' is an individual heritable property (d-dimensional vector) that may
influence birth and death rates and interactions between individuals, and vary
by mutation. In a large population limit, the random process converges to the
solution of a Gurtin-McCamy type PDE. We show that the random model has a long
time behavior that differs from its deterministic limit. However, the results
on the limiting PDE and large deviation techniques \textit{\`a la}
Freidlin-Wentzell provide estimates of the extinction time and a better
understanding of the long time behavior of the stochastic process. This has
applications to the theory of adaptive dynamics used in evolutionary biology.
We present simulations for two biological problems involving life-history trait
evolution when body size is plastic and individual growth is taken into
account.
