Lotka.Volterra with randomly fluctuating environments or .how switching between beneficial environments can make survival harder.

We consider two-dimensional Lotka.Volterra systems in a fluctuating environment. Relying on recent results on stochastic persistence and piecewise deterministic Markov processes, we show that random switching between two environments that are both favorable to the same species can lead to the extinction of this species or coexistence of the two competing species.