In a toy model of binary splitting branching processes with population size dependence (supercritical below and subcritical above a threshhold, the carrying capacity) the chance of a little population establishing itself in the sense of reaching a band around the carrying capacity is determined, and so is the persistence time of the population. Mutations and competition between morphs are introduced, and it turns out that the resulting processes exhibit evolutionary branchings which occur in a manner slightly different from that predicted by established deterministic theory. The validity of conclusions is discussed in terms of more general branching processes. (Joint work with Serik Sagitov, Fima Klebaner et al.)