  Let $(X_t)_{t \geq 0}$ be a continuous time Markov process on some metric
space $M,$ leaving invariant a closed subset $M_0 \subset M,$ called the {\em
extinction set}. We give general conditions ensuring either "Stochastic
persistence" (Part I) : Limit points of the occupation measure are invariant
probabilities over $M_+ = M \setminus M_0;$ or "Extinction" (Part II) : $X_t
\rightarrow M_0$ a.s. In the persistence case we also discuss conditions
ensuring the a.s convergence (respectively exponential convergence in total
variation) of the occupation measure (respectively the distribution) of $(X_t)$
toward a unique probability on $M_+.$ These results extend and generalize
previous results obtained for various stochastic models in population dynamics,
given by stochastic differential equations, random differential equations, or
pure jump processes.
