Local species richness frequently is linearly related to the richness of the regional species pool from which the local community was presumably assembled. What, if anything, does this pattern imply about the relative importance of species interactions and dispersal as determinants of local species richness? Two recent papers by Hugueny and Cornell and He et al. propose that the classical island biogeography model of MacArthur and Wilson can help answer this question, by serving as a null model of the relationship between local (island) and regional (mainland) species richness in the absence of local species interactions. The two models make very different predictions, despite being derived from apparently-similar assumptions. Here we reinterpret these two models and show that their contrasting predictions can be regarded as arising from different, implicit assumptions about how species abundances vary with species richness on the mainland. We derive a more general island biogeography model of local–regional richness relationships that explicitly incorporates mainland species abundance and subsumes the two previous models as limiting cases. The new model predicts that the local–regional richness relationship can range from nearly linear to strongly curvilinear, depending on how species abundances on the mainland vary with mainland richness, as well as on rates of immigration to and extinction from islands. Local species interactions are not necessary for producing curvilinear local–regional richness relationships. We discuss the implications of our new model for the interpretation of local–regional richness relationships.