  Synchronization is one of the most striking instances of collective behavior,
occurring in many natural phenomena. For example, in some ant species, ants are
inactive within the nest most of the time, but their bursts of activity are
highly synchronized and involve the entire nest population. Here we revisit a
simulation model that generates this synchronized rhythmic activity through
autocatalytic behavior, i.e., active ants can activate inactive ants, followed
by a period of rest. We derive a set of delay differential equations that
provide an accurate description of the short- and medium-time simulations for
large ant colonies. Analysis of the fixed-point solutions, complemented by
numerical integration of the equations, indicates the existence of stable
limit-cycle solutions when the rest period is greater than a threshold and the
event of spontaneous activation of inactive ants is very unlikely, so that most
of the arousal of ants is done by active ants. Furthermore, we argue that the
persistent oscillations observed in the simulations for colonies of finite size
are due to resonant amplification of demographic noise.
