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Conditions for permanence in well-known biological competition models

Published online by Cambridge University Press:  17 February 2009

Jan H. van Vuuren  and
John Norbury
Jan H. van Vuuren
Affiliation:
Department of Applied Mathematics, Stellenbosch University, Private Bag XI, 7602 Matieland, South Africa; email: [email protected]
John Norbury
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles', OX1 3LB, United Kingdom; email: [email protected]
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Abstract

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Reaction-diffusion systems are widely used to model the population densities of biological species competing for natural resources in their common habitat. It is often not too difficult to establish positive uniform upper bounds on solution components of such systems, but the task of establishing strictly positive uniform lower bounds (when they exist) can be quite troublesome. Two previously established criteria for the permanence (non-extinction and non-explosion) of solutions of general weakly-coupled competition-diffusion systems with diagonally convex reaction terms are used here as background to develop more easily verifiable and concrete conditions for permanence in various well-known competition diffusion models. These models include multi-component reaction-diffusion systems with (i) the by now classical Lotka-Volterra (logistic) reaction terms, (ii) higher order “logistic” interaction between the species, (iii) logistic-logarithmic reaction terms, (iv) Ayala-Gilpin-Ehrenfeld θ-interaction terms (which are used to model Drosophila competition), (v) logistic-exponential interaction between the species, (vi) Schoener-exploitation and (vii) modified Schoener-interference between the species. In (i) a known condition for permanence (for the ODE-system) is recovered, while in (ii)–(vii) new criteria for permanence are established.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 2 , October 2000 , pp. 195 - 223
Copyright
Copyright © Australian Mathematical Society 2000

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