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Uniform persistence and flows near a closed positively invariant set. (English) Zbl 0811.34033
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.

37C10 Dynamics induced by flows and semiflows
34D05 Asymptotic properties of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
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