Demographic variation and survival in discrete population models

Persistence and extinction attributes of a discrete population model are explored on both a finite and an infinite time horizon. For a first-order autonomous nonlinear difference equation, a classification is found of when, for each positive integer N, trajectories go to extinction at time N. The dynamic complexity that is known to permeate difference-equation models is also present in the survival analyses developed here. Also found is an interesting decomposition of the continuum of initial population sizes into intervals where populations are persistent at time N and intervals leading to extinction at time n less than or equal to N.