Principal spectral theory and asymptotic analysis for time-periodic cooperative systems with temporally nonlocal dispersal

This paper investigates the principal spectral theory and the asymptotic behavior of the principal spectrum point for a class of time-periodic cooperative systems with nonlocal dispersal operators, incorporating both coupled and uncoupled nonlocal terms. By applying the theory of resolvent positive operators and their perturbations, we first establish criteria for the existence of the principal eigenvalue. We then construct sequences of smooth upper and lower approximating matrix-valued functions, each of whose corresponding operators satisfies the principal eigenvalue existence condition. This approximation framework allows the principal spectrum point to effectively substitute for the principal eigenvalue in characterizing the global dynamics of the nonlinear system. Moreover, it facilitates the study of the asymptotic behavior of the principal spectrum point with respect to parameters under fairly general assumptions. Subsequently, for systems with both coupled and uncoupled nonlocal terms, we analyze the asymptotic behavior of the principal spectrum point in terms of the dispersal rate, dispersal range, and frequency. Finally, we illustrate the applicability of our theoretical results through a Zika virus model and a stem cell model.