In this talk, we will present a mathematical model which describes the time and space evolution of the gut microbiota population. This model couples a system of population dynamics describing the metabolism of bacterial populations during the fibre degradation with a fluid dynamics model for the gut content. A simplified version of the model, which is based on mixture theory, will be derived and numerical simulations will be presented. This work has been done in collaboration with S. Labarthe, B. Polizzi, B. Laroche, T. Phan and T. Goudon.