We develop a framework for analysing the outcome of resource competition based on bifurcation theory. We elaborate our methodology by readdressing the problem of competition of two species for two resources in a chemostat environment. In the case of perfect-essential resources it has been extensively discussed using Tilman's representation of resource quarter plane plots. Our mathematically rigorous analysis yields bifurcation diagrams with a striking similarity to Tilman's method including the interpretation of the consumption vector and the resource supply vector. However, our approach is not restricted to a particular class of models but also works with other trophic interaction formulations. This is illustrated by the analysis of a model considering interactively-essential or complementary resources instead of prefect-essential resources. Additionally, our approach can also be used for other ecosystem compositions: multiple resources-multiple species communities with equilibrium or oscillatory dynamics. Hence, it gives not only a new interpretation of Tilman's graphical approach, but it constitutes an extension of competition analyses to communities with many species as well as non-equilibrium dynamics. © 2013 EDP Sciences.