  Understanding under what conditions populations, whether they be plants,
animals, or viral particles, persist is an issue of theoretical and practical
importance in population biology. Both biotic interactions and environmental
fluctuations are key factors that can facilitate or disrupt persistence. One
approach to examining the interplay between these deterministic and stochastic
forces is the construction and analysis of stochastic difference equations
$X_{t+1}=F(X_t,\xi_{t+1})$ where $X_t \in \R^k$ represents the state of the
populations and $\xi_1,\xi_2,...$ is a sequence of random variables
representing environmental stochasticity. In the analysis of these stochastic
models, many theoretical population biologists are interested in whether the
models are bounded and persistent. Here, boundedness asserts that
asymptotically $X_t$ tends to remain in compact sets. In contrast, persistence
requires that $X_t$ tends to be "repelled" by some "extinction set" $S_0\subset
\R^k$. Here, results on both of these proprieties are reviewed for single
species, multiple species, and structured population models. The results are
illustrated with applications to stochastic versions of the Hassell and Ricker
single species models, Ricker, Beverton-Holt, lottery models of competition,
and lottery models of rock-paper-scissor games. A variety of conjectures and
suggestions for future research are presented.
