A new analysis of the nearly century-old Lotka–Volterra theory allows us to link species interactions to biodiversity patterns, including: species abundance distributions, estimates of total community size, patterns of community invasibility, and predicted responses to disturbance. Based on a few restrictive assumptions about species interactions, our calculations require only that the community is sufficiently large to allow a mean-field approximation. We develop this analysis to show how an initial assemblage of species with varying interaction strengths is predicted to sort out into the final community based on the species’ predicted target densities. The sorting process yields predictions of covarying patterns of species abundance, community size, and species interaction strengths. These predictions can be tested using enrichment experiments, examination of latitudinal and productivity gradients, and features of community assembly.