Dynamic changes in microbial communities play an important role in human health and disease. Specifically, deciphering how microbial species in a community interact with each other and their environment can elucidate mechanisms of disease, a problem typically investigated using tools from community ecology. Yet, such methods naively require measurements of absolute densities, whereas typical datasets are compositional and only provide relative abundances. Here, we systematically investigate models of microbial dynamics in the simplex of relative abundances. We derive a new nonlinear dynamical system for microbial dynamics, termed ``compositional'' Lotka-Volterra (LV), unifying approaches using generalized LV equations from community ecology and compositional data analysis. On three real datasets, we demonstrate that the compositional version recapitulates interactions between relative abundances implied by generalized LV. Moreover, we show that relative abundance trajectories predicted using compositional LV are as accurate or better than those predicted by generalized LV using absolute abundances. We further compare compositional LV to two other models of relative abundance dynamics motivated by common assumptions in the literature---a linear model in a log-ratio transformed space, and a linear model in the space of relative abundances---and show that compositional LV more accurately describes community dynamics. Finally, we demonstrate the clinical utility of compositional modeling by predicting patient outcomes on a dataset of immunocompromised patients. Our results indicate that microbial dynamics in the simplex are nonlinear, and that interactions occur in the space of relative abundances.