Global dynamics and asymptotic spreading of a diffusive age-structured model in spatially periodic media

The paper is concerned with the persistence and spatial propagation of populations with age structure in spatially periodic media. We first provide a complete characterization of the global dynamics for the problem via investigating the existence, uniqueness and global stability of the nontrivial equilibrium. This leads to a necessary and sufficient condition for populations to survive, in terms of the principal eigenvalue of the associated linearized problem with periodic boundary conditions. We next establish the spatial propagation dynamics for the problem and derive the formula for the asymptotic spreading speed. The result suggests that the propagation fronts of populations are uniform for all age groups with a common spreading speed. Our approach involves developing the theory of generalized principal eigenvalues and the homogenization method to address novel challenges arising from the nonlocal age boundary condition.