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Permanence in ecological systems with spatial heterogeneity

Published online by Cambridge University Press:  14 November 2011

Robert Stephen Cantrell ,
Chris Cosners  and
Vivian Hutson
Robert Stephen Cantrell
Affiliation:
Department of Mathematics and Computer Science, The University of Miami, Coral Gables, FL 33124, U.S.A.
Chris Cosners
Affiliation:
Department of Mathematics and Computer Science, The University of Miami, Coral Gables, FL 33124, U.S.A.
Vivian Hutson
Affiliation:
Department of Applied Mathematics, The University of Sheffield, Sheffield S102TN, U.K.
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Synopsis

A basic problem in population dynamics is that of finding criteria for the long-term coexistence of interacting species. An important aspect of the problem is determining how coexistence is affected by spatial dispersal and environmental heterogeneity. The object of this paper is to study the problem of coexistence for two interacting species dispersing through a spatially heterogeneous region. We model the population dynamics of the species with a system of two reaction–diffusion equations which we interpret as a semi-dynamical system. We say that the system is permanent if any state with all components positive initially must ultimately enter and remain within a fixed set of positive states that are strictly bounded away from zero in each component. Our analysis produces conditions that can be interpreted in a natural way in terms of environmental conditions and parameters, by combining the dynamic idea of permanence with the static idea of studying geometric problems via eigenvalue estimation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 3 , 1993 , pp. 533 - 559
Copyright
Copyright © Royal Society of Edinburgh 1993

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