  This survey concerns the study of quasi-stationary distributions with a
specific focus on models derived from ecology and population dynamics. We are
concerned with the long time behavior of different stochastic population size
processes when 0 is an absorbing point almost surely attained by the process.
The hitting time of this point, namely the extinction time, can be large
compared to the physical time and the population size can fluctuate for large
amount of time before extinction actually occurs. This phenomenon can be
understood by the study of quasi-limiting distributions. In this paper, general
results on quasi-stationarity are given and examples developed in detail. One
shows in particular how this notion is related to the spectral properties of
the semi-group of the process killed at 0. Then we study different stochastic
population models including nonlinear terms modeling the regulation of the
population. These models will take values in countable sets (as birth and death
processes) or in continuous spaces (as logistic Feller diffusion processes or
stochastic Lotka-Volterra processes). In all these situations we study in
detail the quasi-stationarity properties. We also develop an algorithm based on
Fleming-Viot particle systems and show a lot of numerical pictures.
