ASYMPTOTIC BEHAVIOR OF THE BASIC REPRODUCTION RATIO FOR PERIODIC REACTION-DIFFUSION SYSTEMS

This paper is devoted to the research on asymptotic behavior of the basic reproduction ratio for periodic reaction-diffusion systems in the case of small and large diffusion coefficients. We first study the continuity of the basic reproduction ratio with respect to parameters and the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem for large diffusion coefficients. Then we 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 a 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.