  We consider a Lotka-Volterra food chain model with possibly intra-specific
competition in a stochastic environment represented by stochastic differential
equations. In the non-degenerate setting, this model has already been studied
by A. Hening and D. Nguyen. They provided conditions for stochastic persistence
and extinction. In this paper, we extend their results to the degenerate
situation in which the top or the bottom species is subject to random
perturbations. Under the persistence condition, there exists a unique invariant
probability measure supported by the interior of $\mathbb{R}_+^n$ having a
smooth density.
  Moreover, we study a more general model, in which we give new conditions
which make it possible to characterise the convergence of the semi-group
towards the unique invariant probability measure either at an exponential rate
or at a polynomial one. This will be used in the stochastic Lotka-Volterra food
chain to see that if intra-specific competition occurs for all species, the
rate of convergence is exponential while in the other cases it is polynomial.
