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

Published online by Cambridge University Press:  17 April 2009

Afshin Ghoreishi  and
Roger Logan
Afshin Ghoreishi
Affiliation:
Department of Mathematics, Bowdoin College Brunswick, ME 04011, United States of America
Roger Logan
Affiliation:
Department of Mathematics, The College of Charleston, Charleston SC 29424, United States of America
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Abstract

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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 1 , August 1991 , pp. 79 - 94
Copyright
Copyright © Australian Mathematical Society 1991

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