On the evolution of slow dispersal in multi-species communities

For any $N geq 2$, we show that there are choices of diffusion rates ${d_i}_{i=1}^N$ such that for $N$ competing species which are ecologically identical and having distinct diffusion rates, the slowest diffuser is able to competitive exclude the remainder of the species. In fact, the choices of such diffusion rates is open in the Hausdorff topology. Our result provides some evidence in the affirmative direction regarding the conjecture by Dockery et al. in cite{Dockery1998}. The main tools include Morse decomposition of the semiflow, as well as the theory of normalized principal bundle for linear parabolic equations.