Summary: This paper studies population models with time delays that represent the dispersal time for individuals to move from one patch to other patches. Two kinds of threshold between the globally stable coexistence and extinction of the populations in all patches are obtained for an autonomous models and a periodic model. The autonomous case extends results of a model without any delays of Y. Takeuchi [Acta. Appl. Math. 14, 49–57 (1989; Zbl 0665.92017)] to a model with dispersing time.
It is shown that time delays decrease the threshold for the permanence of the populations in two ways. First, the populations go extinct for any length of time delays if they go extinct in the absence of the delays. Second, the time delays can drive the populations from coexistence to extinction even if the populations are permanent when there are no time delays. When the environment in all patches is periodic in time, sufficient conditions for the existence and uniqueness of a globally stable positive periodic solution and sufficient conditions for the extinction of the populations are obtained. The results indicate that the “super”-rich food conditions can be removed.