Stochastic persistence in degenerate stochastic Lotka-Volterra food chains

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.