On the Establishment, Persistence, and Inevitable Extinction of   Populations
  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. 
Kais Hamza, Peter Jagers, and Fima C. Klebaner
q-bio.PE math.PR