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

Comparison of the global dynamics for two chemostat-like models: random temporal variation versus spatial heterogeneity