  We examine the stability of feedback-regulated star formation (SF) in
galactic nuclei and contrast it to SF in extended discs. In galactic nuclei the
dynamical time becomes shorter than the time over which feedback from young
stars evolves. We argue analytically that the balance between stellar feedback
and gravity is unstable in this regime. We study this using numerical
simulations with pc-scale resolution and explicit stellar feedback taken from
stellar evolution models. The nuclear gas mass, young stellar mass, and SFR
within the central ~100 pc (the short-timescale regime) never reach
steady-state, but instead go through dramatic, oscillatory cycles. Stars form
until a critical surface density of young stars is present (such that feedback
overwhelms gravity), at which point they begin to expel gas from the nucleus.
Since the dynamical times are shorter than the stellar evolution times, the
stars do not die as the gas is expelled, but continue to push, triggering a
runaway quenching of star formation in the nucleus. However the expelled gas is
largely not unbound from the galaxy, but goes into a galactic fountain which
re-fills the nuclear region after the massive stars from the previous burst
cycle have died off (~50 Myr timescale). On large scales (>1 kpc), the
galaxy-scale gas content and SFR is more stable. We examine the consequences of
this episodic nuclear star formation for the Kennicutt-Schmidt (KS) relation:
while a tight KS relation exists on ~1 kpc scales in good agreement with
observations, the scatter increases dramatically in smaller apertures centered
on galactic nuclei.
