Uniform persistence and permanence for non-autonomous semiflows in population biology

Conditions are presented for uniform strong persistence of non-autonomous semiflows, taking uniform weak persistence for granted. Turning the idea of persistence upside down, conditions are derived for non-autonomous semiflows to be point-dissipative. These results are applied to time-heterogeneous models of S-I-R-S type for the spread of infectious childhood diseases. If some of the parameter functions are asymptotically almost periodic, an almost sharp threshold result is obtained for uniform strong endemicity versus extinction in terms of asymptotic time averages. Applications are also presented to scalar retarded functional differential equations modeling one species population growth.