Spatial versus non-spatial dynamics for diffusive Lotka–Volterra competing species models

Abstract

This paper studies the dynamics of the diffusive Lotka–Volterra competition model under small dispersion rates and Dirichlet boundary conditions. Its main goal is ascertaining the connections between the qualitative behavior of the positive solutions of the parabolic model for small diffusions and the dynamics of its associated non-spatial model given by switching them off. After sharpening very substantially some previous results of Furter and Lopez-Gomez (Proc R Soc Edinb 127A:281–336, 1997), we characterizes the singular limit, as diffusions go to zero, of any sequence of coexistence steady-state solutions. It turns out that they must approximate, point-wise in the inhabiting territory, the global attractor of the non-spatial model, uniformly on compact subsets of the habitat zones where a global hyperbolic attractor exist. As a very special consequence of our general theorem, the Dirichlet counterpart of the singular perturbation theorem of Hutson et al. (World Sci Ser Appl Anal 4:501–533, 1995) holds. Further, a multiplicity result is given when the underlying non-spatial model exhibits a founder control competition somewhere in the territory. This is the first multiplicity theorem available in the literature for small diffusivities.