Coexistence of three species in a strongly coupled elliptic system

A strongly coupled elliptic system which describes three interacting species, with homogeneous Dirichlet boundary conditions is considered. It is shown that there is no coexistence state if diffusion rates are strong, or if the intrinsic growth rates are slow. Making use of the Schauder fixed point theory, we derive some sufficient conditions to have a semi-coexistence or a coexistence state for the strongly coupled elliptic problem. Moreover, our results reveal that this problem possesses at least one coexistence state if cross-diffusions and intra-specific competitions are weak. (C) 2003 Elsevier Ltd. All rights reserved.