Towers and fibered products of model categories

Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For spectral model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization.