Niche and fitness differences control the outcome of competition, but determining their relative importance in invaded communities -- which may be far from equilibrium -- remains a pressing concern. Moreover, it is unclear whether classic approaches for studying competition, which were developed predominantly for pairs of interacting species, will fully capture dynamics in complex species assemblages. We parameterized a population dynamic model using competition experiments of two native and three exotic species from a grassland community. We found evidence for minimal fitness differences or niche differences between the native species, leading to slow replacement dynamics and priority effects, but large fitness advantages allowed exotics to unconditionally invade natives. Priority effects driven by strong interspecific competition between exotic species drove single-species dominance by one of two exotic species in 80% of model outcomes, while a complex mixture of non-hierarchical competition and coexistence between native and exotic species occurred in the remaining 20%. Fungal infection, a commonly hypothesized coexistence mechanism, had weak fitness effects, and is unlikely to substantially affect coexistence. In contrast to previous work on pairwise outcomes in largely native-dominated communities, our work supports a role for nearly-neutral dynamics and priority effects as drivers of species composition in invaded communities. [link to publication]