Compact almost automorphic dynamics of linear non-autonomous differential equationswith exponential dichotomy and of delayed biological models

In this work, we study the dynamics of linear non-autonomous differential equations with exponential dichotomy and compact almost automorphic perturbations. First, we prove that if the homogeneous system is exponentially dichotomous and the coefficient matrix is compact almost automorphic, then the associated Green's function is compact bi-almost automorphic and uniformly continuous relative to the principal diagonal of the two-dimensional Euclidean space. Next, we demonstrate the invariance of the compact almost automorphic function space under convolution products with Green's function as the kernel. These results ensures that the unique bounded solution of a linear non-autonomous differential equation, under exponential dichotomy and with compact almost automorphic perturbation, is itself compact almost automorphic. Finally, we investigate the existence and the global exponential stability of a unique positive compact almost automorphic solution for a nonlinear non-autonomous delayed biological model with nonlinear harvesting or immigration terms and mixed delays. For more information and the Latex file see https://ejde.math.txstate.edu/Volumes/2025/91/abstr.html