In this paper we study a Lotka-Volterra predator-prey system with prey logistic
growth under the telegraph noise. The telegraph noise switches at random two prey-predator models. The aim of this work is to determine
the subset of omega-limit set of the system and show out the existence of a stationary
distribution. We also focus on persistence of the predator and thus we look for
conditions that allow persistence of the predator and prey community. We show
that the asymptotic behaviour highly depends on the value of some constant
$\lambda$ which is useful to make suitable predictions about
the persistence of the system.