On the long term spatial segregation for a competition–diffusion system

Article type: Research Article

Authors: Squassina, Marco

Affiliations: Dipartimento di Informatica, Università degli Studi di Verona, Cá Vignal 2, Strada Le Grazie 15, I-37134 Verona, Italy. E-mail: [email protected]

Note: [] The author was supported by the MIUR national research project “Variational and Topological Methods in the Study of Nonlinear Phenomena”.

Abstract: We investigate the long term behavior for a class of competition–diffusion systems of Lotka–Volterra type for two competing species in the case of low regularity assumptions on the data. Due to the coupling that we consider the system cannot be reduced to a single equation yielding uniform estimates with respect to the inter-specific competition rate parameter. Moreover, in the particular but meaningful case of initial data with disjoint support and Dirichlet boundary data which are time-independent, we prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially segregated state satisfying some variational inequalities.

Keywords: competition–diffusion systems, Lotka–Volterra model, spatial segregation, population dynamics, asymptotic behaviour, stationary solution, dissipative systems

DOI: 10.3233/ASY-2008-0868

Journal: Asymptotic Analysis, vol. 57, no. 1-2, pp. 83-103, 2008