On the lifetime of a size-dependent branching process

We discuss lifetimes for a family of population-dependent branching processes. The attenuation factor (due to environment or competition, for example) is of Ricker type, i.e., the probability of an individual having offspring at all is of the form if the total population is n. Equivalently we can write the probability as where the carrying capacity K is the inverse of the attenuating factor. It is well known that the expected lifetime of such a process is exponential in K. If the carrying capacities vary much over time, for instance, if they are i.i.d. with a heavy-tailed distribution, the extinction scenario may change to a growth-catastrophe one with expected lifetimes much shorter. In addition to Ricker's model, production functions of the Beverton-Holt and Hassell types are also discussed.