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Positive solutions of a class of biological models in a heterogeneous environment. (English) Zbl 0735.35051
Authors’ summary: 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 characterize our results by the sign of the first eigenvalues of certain Schrödinger type operators.

MSC:
35J60 Nonlinear elliptic equations
92D25 Population dynamics (general)
47H11 Degree theory for nonlinear operators
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