Qualitative Properties of the Principal Floquet Bundle and Their Applications

Abstract: Floquet bundle theory serves as a natural extension of the eigenvalue theory for elliptic and periodic parabolic operators to the spectral theory of non-autonomous, non-periodic parabolic operators, which provides a useful spectral tool for studying threshold dynamics in more general non-autonomous systems. However, current understanding and applications of the normalized principal Floquet bundle remain rather limited compared with the eigenvalue theory for elliptic or periodic parabolic problems. In this work, we first extend several fundamental properties of the principal eigenvalue to the normalized principal Floquet bundle of non-autonomous, non-periodic parabolic operators, and obtain a series of parallel results. Furthermore, we focus on the asymptotic behavior of the principal Floquet exponent with respect to the generalized frequency and diffusion rate under various typical parameter limits (including both independent and coupled parameter regimes) for the operator under homogeneous Neumann boundary conditions. The main tools are a parabolic comparison principle developed in this work for long-time limit processes, which is used to derive bounds for the principal Floquet exponent via suitably constructed sub- and super-solutions, supplemented by Harnack's inequality and long-time averaging techniques. Finally, we employ the normalized principal Floquet bundle to define two critical numbers (as extensions of the basic reproduction number) for a non-autonomous spatially diffusive SIS epidemic model, examine the extinction and weak persistence of the disease, and investigate the limiting behavior of the basic critical numbers with respect to the generalized frequency and diffusion rate.