  In recent years there has been a growing interest in the study of the
dynamics of stochastic populations. A key question in population biology is to
understand the conditions under which populations coexist or go extinct.
Theoretical and empirical studies have shown that coexistence can be
facilitated or negated by both biotic interactions and environmental
fluctuations. We study the dynamics of $n$ populations that live in a
stochastic environment and which can interact nonlinearly (through competition
for resources, predator-prey behavior, etc.). Our models are described by
$n$-dimensional Kolmogorov systems with white noise (stochastic differential
equations - SDE). We give sharp conditions under which the populations converge
exponentially fast to their unique stationary distribution as well as
conditions under which some populations go extinct exponentially fast.
  The analysis is done by a careful study of the properties of the invariant
measures of the process that are supported on the boundary of the domain. To
our knowledge this is one of the first general results describing the
asymptotic behavior of stochastic Kolmogorov systems in non-compact domains.
  We are able to fully describe the properties of many of the SDE that appear
in the literature. In particular, we extend results on two dimensional
Lotka-Volterra models, two dimensional predator-prey models, $n$ dimensional
simple food chains, and two predator and one prey models. We also show how one
can use our methods to classify the dynamics of any two-dimensional stochastic
Kolmogorov system satisfying some mild assumptions.
