  In a view for a simple model where natural selection at the individual level
is confronted to selection effects at the group level, we consider some
individual-based models of some large population subdivided into a large number
of groups. We then obtain the convergence to the law of a stochastic process
with some Feynman-Kac penalization. To analyze the limiting behavior of this
law, we exploit a recent approach, designed for the convergence to
quasi-stationary distributions. We are able to deal with the fixation of the
stochastic process and relate the convergence to equilibrium to the one where
fixation implies extinction. We notably establish different regimes of
convergence. Besides the case of an exponential rate (the rate being uniform
over the initial condition), critical regimes with convergence in 1/t are also
to notice. We finally address the relevance of such limiting behaviors to
predict the long-time behavior of the individual-based model and describe more
specifically the cases of weak selection. Consequences in term of evolutive
dynamics are also derived, where such competition is assumed to occur
repeatedly at each de novo mutation.
