  This article is dedicated to the study and comparison of two chemostat-like
competition models in a heterogeneous environment. The first model is a
probabilistic model where we build a PDMP simulating the effect of the temporal
heterogeneity of an environment over the species in competition. Its study uses
classical tools in this field. The second model is a gradostat-like model
simulating the effect of the spatial heterogeneity of an environment over the
same species. Despite the fact that the nature of the two models is very
different, we will see that their long time behavior is globally very similar.
We define for both model quantities called invasion rates which model the
growth rate of a species when it is near to extinction. We show that the signs
of these invasion rates essentially determine the long time behavior for both
systems. In particular, we exhibit a new example of bistability between a
coexistence steady state and a semi-trivial steady state.
