Uniform persistence and flows near a closed positively invariant set

H. I. Freedman, Shigui Ruan, Moxun Tang

Research output: Contribution to journalArticlepeer-review

256 Scopus citations

Abstract

In this paper, the behavior of a continuous flow in the vicinity of a closed positively invariant subset in a metric space is investigated. The main theorem in this part in some sense generalizes previous results concerning classification of the flow near a compact invariant set in a locally compact metric space which was described by Ura-Kimura (1960) and Bhatia (1969). By applying the obtained main theorem, we are able to prove two persistence theorems. In the first one, several equivalent statements are established, which unify and generalize earlier results based on Liapunov-like functions and those about the equivalence of weak uniform persistence and uniform persistence. The second theorem generalizes the classical uniform persistence theorems based on analysis of the flow on the boundary by relaxing point dissipativity and invariance of the boundary. Several examples are given which show that our theorems will apply to a wider varity of ecological models.

Original languageEnglish (US)
Pages (from-to)583-600
Number of pages18
JournalJournal of Dynamics and Differential Equations
Volume6
Issue number4
DOIs
StatePublished - Oct 1994

Keywords

  • AMS Subject Classifications: 34C35, 58F25, 92A15
  • Positively invariant set
  • dissipativeness
  • limit set
  • persistence
  • prolongational set

ASJC Scopus subject areas

  • Analysis

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