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The purpose of this paper is to investigate the principal spectral theory andasymptotic behavior of the spectral bound for cooperative nonlocal dispersalsystems, specifically focusing on the case where partial diffusion coefficientsare zero, referred to as the partially degenerate case. We propose twosufficient conditions that ensure the existence of the principal eigenvalue inthese partially degenerate systems. Additionally, we study the asymptoticbehavior of the spectral bound for nonlocal dispersal operators with small andlarge diffusion coefficients, considering both non-degenerate and partiallydegenerate cases. Notably, we find a threshold-type result as the diffusioncoefficients tend towards infinity in the partially degenerate case. Finally,we apply these findings to discuss the asymptotic behavior of the basicreproduction ratio in a viral diffusion model.

Principal spectral theory and asymptotic behavior of the spectral bound for partially degenerate nonlocal dispersal systems