We study a model of the chemostat with several species in competition on a single resource. We take into account the intra-specific interactions between individuals of the same population of microorganisms and we assume that the growth rates are increasing and the dilution rates are distinct. Using the concept of steady-state characteristic, we present a geometric characterization of the existence and stability of all equilibria. Moreover, we give the necessary and sufficient condition on the parameters such that the system has a positive equilibrium. Using a Lyapunov function, we give a global asymptotic stability result for the competition model of several species. The operating diagram describes the asymptotic behavior of this model with respect to control parameters and illustrates the effect of the intra-specific competition on the region of coexistence of several species.