Positive solutions of a class of biological models in a heterogeneous environment

In this paper we discuss existence of positive solutions to a general nonlinear elliptic system of reaction-diffusion equations representing a predator-prey or competition model of interaction between two species, in a heterogeneous environment. We also consider homogeneous Dirichlet and/or Robin boundary conditions. In the predator-prey case we give necessary and sufficient conditions for the existence of positive solutions, while in the competition case we give sufficient conditions. We use index theory in a positive cone to attack our problem and characterise our results by the sign of the first eigenvalues of certain Schrodinger type operators.