We model the evolution of the Internet at the autonomous system level as a process of competition for users and adaptation of bandwidth capability. From a weighted network formalism, where both nodes and links are weighted, we find the exponent of the degree distribution as a simple function of the growth rates of the number of autonomous systems and connections in the Internet, both empirically measurable quantities. Our approach also accounts for a high level of clustering as well as degree-degree correlations, both with the same hierarchical structure present in the real Internet. Further, it also highlights the interplay between bandwidth, connectivity, and traffic of the network.