Theoretical studies of ecosystem models have generally concluded that large numbers of species will not stably co-exist if the species are all competing for the same limited set of resources. Here we describe a simple multi-trait model of competition where the presence of $N$ resources will lead to the stable co-existence of up to $2^N$ species. Our model also predicts that the long-term dynamics of the population will lie on a neutral attractor hyperplane. When the population shifts within the hyperplane, its dynamics will behave neutrally, while shifts which occur perpendicular to the hyperplane will be subject to restoring forces. This provides a potential explanation of why complex ecosystems might exhibit both niche-like and neutral responses to perturbations. Like the neutral theory of biodiversity, our model generates good fits to species abundance distributions in several datasets but does so without needing to evoke inter-generational stochastic effects, continuous species creation or immigration dynamics. Additionally, our model is able to explain species abundance correlations between independent but similar ecosystems separated by more than 1400 km inside the Amazonian forests.