  Populations of competing biological species exhibit a fascinating interplay
between the nonlinear dynamics of evolutionary selection forces and random
fluctuations arising from the stochastic nature of the interactions. The
processes leading to extinction of species, whose understanding is a key
component in the study of evolution and biodiversity, are influenced by both of
these factors.
  In this paper, we investigate a class of stochastic population dynamics
models based on generalized Lotka-Volterra systems. In the case of neutral
stability of the underlying deterministic model, the impact of intrinsic noise
on the survival of species is dramatic: it destroys coexistence of interacting
species on a time scale proportional to the population size. We introduce a new
method based on stochastic averaging which allows one to understand this
extinction process quantitatively by reduction to a lower-dimensional effective
dynamics. This is performed analytically for two highly symmetrical models and
can be generalized numerically to more complex situations. The extinction
probability distributions and other quantities of interest we obtain show
excellent agreement with simulations.
