Positive solutions for large elliptic systems of interacting species groups by cone index methods

This article considers the existence of positive solutions for various systems of four nonlinear coupled elliptic partial differential equations subjected to zero Dirichlet boundary conditions. In applications, the components can be interpreted as concentrations of four interacting populations or chemicals. The species are divided into two groups which interact in prey-predator or cooperating Volterra–Lotka type of relations. Within each group, the species can interact in other possibilities. Topological cone index method is used for proving the existence of positive coexistence solutions. Sufficient conditions are found in terms of the signs of the principal eigenvalues of various simple related operators.  2003 Elsevier Inc. All rights reserved.