This thesis focuses on the mathematical analysis of models of several species in competition on a single resource in a chemostat. The objective is to model and show the coexistence of microbial species by different mechanisms to better realize biodiversity found in nature like in the bioreactors. We are interested mainly in three mechanisms of coexistence : 1. The inter-specific competition between populations of micro-organisms and intra-specific between individuals of the same species. 2. The flocculation where the species who wins the competition inhibits its growth by the formation of flocs to be able to coexist with the other species. In fact, these flocs consume less substrate than isolated bacteria since they have less access to the substrate, given that this access to the substrate is proportional to the outside surface of the floc. 3. The density-dependence which the model can be construct from the flocculation model by assuming that the dynamics of flocculation is faster than the growth of the species. In this density-dependent model, the growth rate depends not only on the density of substrate but also on the density of biomass, and the removal rate of biomass is not constant but depends also on the density of biomass. Finally, we studied a 3-step model of anaerobic digestion with enzymatic degradation of the substrate (organic matter) that can partly be under a solid form. The mathematical analysis shows that this model may exhibit the quadri-stability with washout of none, one or two species according to the initial condition. The mathematical study of the qualitative behavior of different models of the chemostat, allowed us to better understand the competition and the coexistence of microbial species.
