分野別セミナー

概要：
This talk discusses several recent findings on the dynamics of the spatially-heterogeneous diffusive Lotka-Volterra competing species model. First, it delivers a general (optimal) singular perturbation result generalizing, very substantially, the pioneering theorem of Hutson, López- Gómez, Mischaikow and Vickers (1994) for their mutant model, later analyzed, very sharply, by W. M. Ni and his collaborators. Then, it establishes that, as soon as any steady-state solution of the non-spatial model is linearly unstable somewhere in the inhabiting territory, \Omega, any steady state of the spatial counterpart perturbing from it therein as the diffusion rates separate away from zero must be linearly unstable. From this feature 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 \Omega, say \Omega_{bi}, even if \Omega_{bi} is negligible empirically. Actually, this is the first existing multiplicity result for small diffusion rates. Finally, based on the Picone identity, we can establish a new, rather striking, uniqueness result valid for general spatially heterogeneous models. This result generalizes, very substantially, those of W. M. Ni and collaborators for the autonomous model.