  Motivated by issues arising in population dynamics, we consider the problem
of iterating a given analytic function a number of times. We use the celebrated
technique known as Carleman linearization that turns (for a certain class of
functions) this problem into simply taking the power of a real number. We
expand this method, showing in particular that it can be used for population
models with immigration, and we also apply it to the famous logistic map. We
also are able to give a number of results for the invariant density of this
map, some being related to the Carleman linearization.
