We investigate the long-time dynamics of a competition-selection model in the case of a continuously distributed initial population. We introduce an epidemic model of pathogen species competing for a single resource, whose efficiency is encoded by a continuous variable (the ``trait variable'') living in a Euclidean space. The differential equation is solved in a space of measures to allow the observation of the natural concentration of the distribution on specific traits. We show the concentration of the distribution on the maximizing set of the fitness function in the sense of the Kantorovitch-Rubinstein metric. When the initial mass of the maximal fitness set is positive, we give a precise description of the convergence of the orbit, including a formula for the asymptotic distribution. We also investigate precisely the case of a finite number of regular global maxima and show that the initial distribution may have an influence on the support of the eventual distribution. In particular, the natural process of competition is not always selecting a unique species, but several species may coexist as long as they maximize the fitness function. In many cases it is possible to compute the eventual distribution of the surviving competitors. In some configurations, species that maximize the fitness may still get extinct depending on the shape of the initial distribution and some other parameter of the model, and we provide a way to characterize when this unexpected extinction happens.