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Convergence in Lotka–Volterra-type delay systems without instantaneous feedbacks

Published online by Cambridge University Press:  14 November 2011

Yang Kuang  and
Hal L. Smith
Yang Kuang
Affiliation:
Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, U.S.A.
Hal L. Smith
Affiliation:
Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, U.S.A.
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Synopsis

Most of the convergence results appearing so far for delayed Lotka–Volterra-type systems require that undelayed negative feedback dominate both delayed feedback and interspecific interactions. Such a requirement is rarely met in real systems. In this paper we present convergence criteria for systems without instantaneous feedback. Roughly, our results suggest that in a Lotka–Volterra-type system if some of the delays are small, and initial functions are small and smooth, then the convergence of its positive steady state follows that of the undelayed system or the corresponding system whose instantaneous negative feedback dominates. In particular, we establish explicit expressions for allowable delay lengths for such convergence to sustain.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 1 , 1993 , pp. 45 - 58
Copyright
Copyright © Royal Society of Edinburgh 1993

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