Asymptotic behavior of the principal eigenvalue and basic reproduction ratio for time-periodic reaction-diffusion systems with time delay

This paper is devoted to asymptotic behavior of the basic reproduction ratio for time-periodic and delay reaction-diffusion systems in the cases of small and large diffusion coefficients. First, we employ the generalized Kerin-Rutman theorem to establish the existence of the principal eigenvalue for periodic and cooperative reaction-diffusion systems with delay. Then, motivated by Kerin-Rutman theorem, we provide a method to estimate the upper and lower bounds of principal eigenvalues so as to obtain the asymptotic behavior of the principal eigenvalue. In view of the relationship between the basic reproduction ratio and the principal eigenvalue, we obtain the limiting profile of the basic reproduction ratio according to the asymptotic behavior of the principal eigenvalue.