Strong Competition versus Fractional Diffusion: The Case of Lotka-Volterra Interaction

Abstract

We consider a system of differential equations with nonlinear Steklov boundary conditions, related to the fractional problem where u = (u 1,…, u k ), s ∈ (0, 1), p > 0, a ij > 0 and β > 0. When k = 2 we develop a quasi-optimal regularity theory in 𝒞0, α, uniformly w.r.t. β, for every α < αopt = min (1, 2s); moreover we show that the traces of the limiting profiles as β → + ∞ are Lipschitz continuous and segregated. Such results are extended to the case of k ≥ 3 densities, with some restrictions on s, p and a ij . Since for competition of variational type the optimal regularity is known to be , these results mark a substantial difference with the case of standard diffusion s = 1, where the two competitions cannot be distinguished from each other in the limit.