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In this note, we study the long time behavior of Lotka-Volterra systems whosecoefficients vary randomly. Bena{\"i}m and Lobry (2015) recently establishedthat randomly switching between two environments that are both favorable to thesame species may lead to different regimes: extinction of one species or theother, or persistence of both species. Our purpose here is to describe moreaccurately the range of parameters leading to these regimes, and the support ofthe invariant probability measure in case of persistence.

On the persistence regime for Lotka-Volterra in randomly fluctuating environments