Convergence in lotka-volterra-type delay systems without instantaneous feedbacks

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10 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)45-58
Number of pages14
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume123
Issue number1
DOIs
StatePublished - 1993

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Lotka-Volterra
Delay Systems
Type Systems
Instantaneous
Negative Feedback
Lotka-Volterra System
Delayed Feedback
Convergence Criteria
Convergence Results
Requirements
Interaction

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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abstract = "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.",
author = "Yang Kuang and Hal Smith",
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AB - 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.

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