Stochastic persistence in degenerate stochastic Lotka-Volterra food chains

<p style='text-indent:20px;'>We consider a Lotka-Volterra food chain model with possibly intra-specific competition in stochastic environment represented by differential equations. In the non-degenerate setting, this has already been studied A. Hening and D. Nguyen [<xref ref-type="bibr" rid="b9">9</xref>, <xref rid="b10">10</xref>] where they provided conditions for persistence extinction. paper, we extend their results to degenerate situation which top or bottom species is subject random perturbations. Under condition, there exists unique invariant probability measure supported interior of <inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb R}}_+^n $\end{document}</tex-math></inline-formula> having smooth density.</p><p style='text-indent:20px;'>Moreover, study more general model, give new make it possible characterize convergence semi-group towards either at an exponential rate polynomial one. This will be used see that if occurs all species, while other cases polynomial.</p>