  We study the persistence and extinction of species in a simple food chain
that is modelled by a Lotka-Volterra system with environmental stochasticity.
There exist sharp results for deterministic Lotka-Volterra systems in the
literature but few for their stochastic counterparts. The food chain we analyze
consists of one prey and $n-1$ predators. The $j$th predator eats the $j-1$th
species and is eaten by the $j+1$th predator; this way each species only
interacts with at most two other species - the ones that are immediately above
or below it in the trophic chain. We show that one can classify, based on an
explicit quantity depending on the interaction coefficients of the system,
which species go extinct and which converge to their unique invariant
probability measure. Our work can be seen as a natural extension of the
deterministic results of Gard and Hallam '79 to a stochastic setting.
  As one consequence we show that environmental stochasticity makes species
more likely to go extinct. However, if the environmental fluctuations are
small, persistence in the deterministic setting is preserved in the stochastic
system. Our analysis also shows that the addition of a new apex predator makes,
as expected, the different species more prone to extinction.
  Another novelty of our analysis is the fact that we can describe the behavior
the system when the noise is degenerate. This is relevant because of the
possibility of strong correlations between the effects of the environment on
the different species.
