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Zbl 0811.34033
Freedman, H.I.; Ruan, Shigui; Tang, Moxun
Uniform persistence and flows near a closed positively invariant set. (English)
[J] J. Dyn. Differ. Equations 6, No.4, 583-600 (1994). ISSN 1040-7294; ISSN 1572-9222/e

The behavior of a continuous flow in the vicinity of a closed positively invariant set in a metric space is studied. The obtained results generalize results of Ura-Kimura and Bhatia on classification of a flow near a compact invariant set in a locally compact metric space. Applying the obtained results, the authors prove two persistence theorems. One of the theorems unifies and generalizes earlier persistence results based on Lyapunov-like functions and those about the equivalence of weak uniform persistence and uniform persistence. The second theorem generalizes persistence results based on analysis of a flow on the boundary by relaxing point dissipativity and invariance of the boundary. The obtained results are illustrated by considering several ecological systems.
[E.Ershov (St.Peterburg)]

MSC 2000:: *37-99 Dynamic systems and ergodic theory
37C10 Vector fields, flows, ordinary differential equations
34D05 Asymptotic stability of ODE
34G20 Nonlinear ODE in abstract spaces

Keywords: persistence theorems; ecological systems

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