  We introduce two stochastic chemostat models consisting in a coupled
population-nutrient process reflecting the interaction between the nutrient and
the bacterias in the chemostat with finite volume. The nutrient concentration
evolves continuously but depending on the population size, while the population
size is a birth and death process with coefficients depending on time through
the nutrient concentration. The nutrient is shared by the bacteria and creates
a regulation of the bacterial population size. The latter and the fluctuations
due to the random births and deaths of individuals make the population go
almost surely to extinction. Therefore, we are interested in the long time
behavior of the bacterial population conditioned to the non-extinction. We
prove the global existence of the process and its almost sure extinction. The
existence of quasi-stationary distributions is obtained based on a general
fixed point argument. Moreover, we prove the absolute continuity of the
nutrient distribution when conditioned to a fixed number of individuals and the
smoothness of the corresponding densities.
