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| Uniform persistence and other related notions have been extensively studied for autonomous and non-autonomous dynamical systems. We introduce a concept of uniform persistence above and below a minimal set of an abstract monotone skew-product semiflow. When the minimal set has a continuous separation we characterize the uniform persistence in terms of the principal spectrum. Applications to the case in which the flow is generated by the solutions of a family of non-autonomous functional differential equations will be shown, as well as a method for the calculus of the upper Lyapunov exponent of the minimal set. Finally, we will illustrate our theory with the study of the behavior of a family of almost periodic neural networks, in a situation where persistence actually implies permanence. |
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