Extinction in neutrally stable stochastic Lotka-Volterra models

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. Here, 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.