Spectral and Nonlinear Effects in Certain Elliptic Systems of Three Variables

In this paper, the authors use the method of decomposing operator to prove the existence of positive solutions to certain elliptic biological interacting systems of three species in all possible patterns of interactions in terms of the combinations of competition, symbiosis, and predation under the homogeneous Robin–Dirichlet boundary conditions. The existence of positive solutions on large domains and the asymptotic stability of positive steady states for some cases are also proved. The main idea is that the existence of positive solutions and their properties can be characterized by the spectral properties of certain operators of Schrodinger type and by the equilibria of the system.