Persistence and Permanence for a Class of Functional Differential Equations with Infinite Delay

Abstract

The paper deals with a class of cooperative functional differential equations (FDEs) with infinite delay, for which sufficient conditions for persistence and permanence are established. Here, the persistence refers to all solutions with initial conditions that are positive, continuous and bounded. The present method applies to a very broad class of abstract systems of FDEs with infinite delay, both autonomous and non-autonomous, which include many important models used in mathematical biology. Moreover, the hypotheses imposed are in general very easy to check. The results are illustrated with some selected examples.

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Acknowledgments

Work supported by Fundação para a Ciência e a Tecnologia, under UID/MAT/04561/2013.

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Correspondence to Teresa Faria.

Additional information

Dedicated to Professor John Mallet-Paret, on the occasion of his 60th birthday.

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Cite this article

Faria, T. Persistence and Permanence for a Class of Functional Differential Equations with Infinite Delay. J Dyn Diff Equat 28, 1163–1186 (2016). https://doi.org/10.1007/s10884-015-9462-x

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Keywords

  • Infinite delay
  • Persistence
  • Permanence
  • Quasimonotone condition
  • Lotka–Volterra model

AMS Subject Classification

  • 34K12
  • 34K25
  • 92D25