The spectral bound and basic reproduction ratio for nonlocal dispersal cooperative problems

This paper is devoted to investigating asymptotic behavior of the spectral bound and basic reproduction ratio for nonlocal dispersal problems in the cases of small and large dispersal rates. First, we generalize the Krein-Rutman theorem to obtain a version for eventually essentially strongly positive operators. Then, we establish necessary and sufficient conditions for the existence of principal eigenvalues for nonlocal dispersal operators. Finally, we investigate the limiting profile of the spectral bound and the basic reproduction ratio for the nonlocal dispersal problems as the dispersal rates go to zero and infinity, respectively. The presented results generalize and improve some existing ones.