| Abstract: |
| We investigate biological concepts such as persistence, permanence, and extinction within the framework of random dynamical systems in infinite-dimensional settings. Focusing on asymptotically random cocycles, we establish structural properties of the associated random attractors, including existence, compactness, invariance, minimality, and connectedness under suitable assumptions. These results provide a rigorous foundation for describing the long-term behavior of randomly evolving populations. In particular, we show how the qualitative features of random attractors can be used to characterize persistence and extinction phenomena, thereby linking dynamical properties with biologically meaningful outcomes. This approach offers a unified perspective for analyzing population dynamics under randomness in infinite-dimensional spaces. |
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