  This paper is devoted to the study of asymptotic behavior of the basic
reproduction ratio for periodic reaction-diffusion systems in the case of small
and large diffusion coefficients. We first establish the continuity of the
basic reproduction ratio with respect to parameters by developing the theory of
resolvent positive operators. Then we investigate the limiting profile of the
principal eigenvalue of an associated periodic eigenvalue problem for large
diffusion coefficients. We then obtain the asymptotic behavior of the basic
reproduction ratio as the diffusion coefficients go to zero and infinity,
respectively. We also investigate the limiting behavior of positive periodic
solution for periodic and cooperative reaction-diffusion systems with the
Neumann boundary condition when the diffusion coefficients are large enough.
Finally, we apply these results to a reaction-diffusion model of Zika virus
transmission.
