Stability and limiting properties of generalized principal eigenvalue for inhomogeneous nonlocal cooperative system

The principal eigenvalue for linear elliptic operator has been known to be one of very useful tools to investigate many important partial differential equations. Due to the pioneering works of Berestycki et al. \cite{BCV1,BCV2}, the study of qualitative properties for the principal eigenvalue of nonlocal operators has attracted a lot of attention of the community from theory to application (For examples \cite{LL22-1,LLS22,LZ1,LZ2,SLLY,DDF,XLR}). In this paper, motivated from the study of mathematical modeling the dynamics of infectious diseases in \cite{NV1, ZZLD, WD}, we analyze the asymptotic properties of the principal eigenvalue of nonlocal inhomogeneous cooperative system with respect to the dispersal rate and dispersal range. This can be done thanks to the deep results of Rainer \cite{Ra13}, Kriegl and Michor \cite{KM03} on the stability of eigenvalue of the variable matrices of zero-order coefficients and extends many results from \cite{BCV1,LL,NV1}. Our work provides a fundamental step to investigate the nonlinear system modeling the spreading of the transmitted diseases as mentioned.