  We consider stochastic dynamics of a population which starts from a small
colony on a habitat with large but limited carrying capacity. A common
heuristics suggests that such population grows initially as a Galton-Watson
branching process and then its size follows an almost deterministic path until
reaching its maximum, sustainable by the habitat. In this paper we put forward
an alternative and, in fact, more accurate approximation which suggests that
the population size behaves as a special nonlinear transformation of the Galton
Watson process from the very beginning.
