Persistence in Infinite-Dimensional Systems

The concept of persistence reflects the survival of all components of a model ecosystem. Most of the results to date are restricted to ordinary differential equations or to dynamics on locally compact spaces. The concept is investigated here in the setting of a $C^0 $-semigroup which is asymptotically smooth. Since the equations of population dynamics often involve delays or diffusion this seems the appropriate setting. Conditions are placed on the flow on the boundary which, given the presence of a global attractor provided by the assumption of dissipativeness and asymptotic smoothness, are necessary and sufficient for persistence.