  This paper investigates the influence of environmental noise on the
characteristic timescale of the dynamics of density-dependent populations.
General results are obtained on the statistics of time spent in rarity and time
spent in commonness. The nonlinear stochastic models under consideration form a
class of Markov chains on the state space $]0, \infty[$ which are transient if
the intrinsic growth rate is negative and recurrent if it is positive or null.
In the recurrent case, we obtain a necessary and sufficient condition for
positive recurrence and precise estimates for the distribution of times of
rarity and commonness. In the null recurrent, critical case that applies to
ecologically neutral species, the distribution of rarity time is a universal
power law with exponent -3/2. These non- trivial results should be of interest
to biologists involved in the conservation of threatened populations, and to
epidemiologists facing the need to better understanding the dynamics of pest or
disease outbreaks.
