Combining different theoretical approaches, curvature modulated sorting in lipid bilayers fixed on non-planar surfaces is investigated. First, we present a continuous model of lateral membrane dynamics, described by a nonlinear PDE of fourth order. We then prove the existence and uniqueness of solutions of the presented model and simulate membrane dynamics using a finite element approach. Adopting a truly multiscale approach, we use dissipative particle dynamics (DPD) to parameterize the continuous model, i.e. to derive a corresponding macroscopic model.Our model predicts that curvature modulated sorting can occur if lipids or proteins differ in at least one of their macroscopic elastic moduli. Gradients in the spontaneous curvature, the bending rigidity or the Gaussian rigidity create characteristic (metastable) curvature dependent patterns. The structure and dynamics of these membrane patterns are investigated qualitatively and quantitatively using simulations. These show that the decomposition time decreases and the stability of patterns increases with enlarging moduli differences or curvature gradients. Presented phase diagrams allow to estimate if and how stable curvature modulated sorting will occur for a given geometry and set of elastic parameters. In addition, we find that the use of upscaled models is imperative studying membrane dynamics. Compared with common linear approximations the system can evolve to different (meta)stable patterns. This emphasizes the importance of parameters and realistic dynamics in mathematical modeling of biological membranes.