  Feedback delay has been identified as a key ingredient in the quorum sensing
synchronization of synthetic gene oscillators. While this influence has been
evidenced at the theoretical level in a simplified system of degrade-and-fire
oscillators coupled via a common activator protein, full mathematical
certifications remained to be provided. Here, we prove from a rigorous
mathematical viewpoint that, for the very same model, the synchronized
degrade-and-fire oscillations are 1/ unstable with respect to out-of-sync
perturbations in absence of delay, and 2/ are otherwise asymptotically stable
in presence of delay, no matter how small is its amplitude. To that goal, we
proceed to an extensive study of the population dynamics in this system, which
in particular identifies the mechanisms of, and related criteria for, the
delay-dependent stability of periodic orbits with respect to out-of-sync
perturbations. As an additional outcome, the analysis also reveals that,
depending on the parameters, multiple stable partially synchronized periodic
orbits can coexist with the fully synchronized one.
