  Comprehensive models of stochastic, clonally reproducing populations are
defined in terms of general branching processes, allowing birth during maternal
life, as for higher organisms, or by splitting, as in cell division. The
populations are assumed to start small, by mutation or immigration, reproduce
supercritically while smaller than the habitat carrying capacity but
subcritically above it. Such populations establish themselves with a
probability wellknown from branching process theory. Once established, they
grow up to a band around the carrying capacity in a time that is logarithmic in
the latter, assumed large. There they prevail during a time period whose
duration is exponential in the carrying capacity. Even populations whose life
style is sustainble in the sense that the habitat carrying capacity is not
eroded but remains the same, ultimately enter an extinction phase, which again
lasts for a time logarithmic in the carrying capacity. However, if the habitat
can carry a population which is large, say millions of individuals, and it
manages to avoid early extinction, time in generations to extinction will be
exorbitantly long, and during it, population composition over ages, types,
lineage etc. will have time to stabilise. This paper aims at an exhaustive
description of the life cycle of such populations, from inception to
extinction, extending and overviewing earlier results. We shall also say some
words on persistence times of populations with smaller carrying capacities and
short life cycles, where the population may indeed be in danger in spite of not
eroding its environment.
