  This paper is devoted to the study of the asymptotic behavior of the
principal eigenvalue and basic reproduction ratio associated with periodic
population models in a patchy environment for small and large dispersal rates.
We first deal with the eigenspace corresponding to the zero eigenvalue of the
connectivity matrix. Then we investigate the limiting profile of the principal
eigenvalue of an associated periodic eigenvalue problem as the dispersal rate
goes to zero and infinity, respectively. We further establish the asymptotic
behavior of the basic reproduction ratio in the case of small and large
dispersal rates. Finally, we apply these results to a periodic Ross-Macdonald
patch model.
