  We consider a continuous time stochastic individual based model for a
population structured only by an inherited vector trait and with logistic
interactions. We consider its limit in a context from adaptive dynamics: the
population is large, the mutations are rare and we view the process in the
timescale of mutations. Using averaging techniques due to Kurtz (1992), we give
a new proof of the convergence of the individual based model to the trait
substitution sequence of Metz et al. (1992) first worked out by Dieckman and
Law (1996) and rigorously proved by Champagnat (2006): rigging the model such
that "invasion implies substitution", we obtain in the limit a process that
jumps from one population equilibrium to another when mutations occur and
invade the population.
