Summary: Single species nonautonomous dispersal models with delays are considered. An interesting result on the effect of dispersal for persistence and extinction is obtained. That is, if the species is persistent in a patch then it is also persistent in all other patches; if the species is permanent in a patch then it is also permanent in all other patches; if the species is extinct in a patch then it is also extinct in all other patches.
Furthermore, some new sufficient conditions for the permanence and extinction of the species in a patch are established. The existence of positive periodic solutions is obtained in the periodic case by employing Z. Teng and L. Chen’s [Comput. Math. Appl. 37, No. 7, 61-71 (1999; Zbl 0942.34061)] results on the existence of positive periodic solutions for functional differential equations.