Spatially heterogeneous Lotka-Volterra competition

This paper studies the dynamics of the spatially-heterogeneous diffusive Lotka-Volterra competing species model. It focuses special attention in ascertaining the linear stability and multiplicity of the coexistence steady states. One of our main findings establishes that, as soon as any steady-state solution of the non-spatial model is linearly unstable somewhere in the inhabiting territory, Ω, any steady state of the spatial counterpart perturbing from it therein (as the diffusion rates, d1,d2, move away from 0) must be linearly unstable. From this general principle one can derive a number of rather astonishing consequences, as the multiplicity of the coexistence steady states when the non-spatial model exhibits founder control competition somewhere in Ω, say Ωbi, even if Ωbi is negligible empirically. Actually, this is the first available multiplicity result for small diffusion rates. Finally, based on a celebrated identity by M. Picone (1910), we are able to establish a new, rather striking, uniqueness result valid for general spatially heterogeneous models.