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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.

MSC:
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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