On the uniqueness of the positive steady state for Lotka-Volterra models with diffusion

We give criteria for the uniqueness and stability of the componentwise positive steady state for the diffusive Lotka-Volterra model of several competing species under Dirichlet boundary conditions, thereby extending known results for cases of only two species. In the case of an underlying spatial domain which is one dimensional interval, we obtain an estimate for a quantity occurring in the hypotheses of a number of uniqueness results. The estimate sharpens those results and gives a partial negative answer to a conjecture of Korman and Leung.