Open AccessStochastic persistence in degenerate stochastic Lotka-Volterra food chains
Author(s) -
Michel Benaı̈m,
Antoine Bourquin,
Dang H. Nguyen
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2022023
Subject(s) - mathematics , degenerate energy levels , invariant measure , polynomial , exponential function , invariant (physics) , stochastic dominance , stochastic differential equation , combinatorics , pure mathematics , discrete mathematics , mathematical analysis , mathematical physics , mathematical optimization , physics , quantum mechanics , ergodic theory
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 in [ 9 , 10 ] where 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 \begin{document}$ {{\mathbb R}}_+^n $\end{document} having a smooth density. Moreover, we study a more general model, in which we give new conditions which make it possible to characterize 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.