  We introduce a differential equation for star formation in galaxies that
incorporates negative feedback with a delay. When the feedback is
instantaneous, solutions approach a self-limiting equilibrium state. When there
is a delay, even though the feedback is negative, the solutions can exhibit
cyclic and episodic solutions. We find that periodic or episodic star formation
only occurs when two conditions are satisfied. Firstly the delay timescale must
exceed a cloud consumption timescale. Secondly the feedback must be strong.
This statement is quantitatively equivalent to requiring that the timescale to
approach equilibrium be greater than approximately twice the cloud consumption
timescale. The period of oscillations predicted is approximately 4 times the
delay timescale. The amplitude of the oscillations increases with both feedback
strength and delay time.
  We discuss applications of the delay differential equation (DDE) model to
star formation in galaxies using the cloud density as a variable. The DDE model
is most applicable to systems that recycle gas and only slowly remove gas from
the system. We propose likely delay mechanisms based on the requirement that
the delay time is related to the observationally estimated time between
episodic events. The proposed delay timescale accounting for episodic star
formation in galaxy centers on periods similar to P 10 Myrs, irregular galaxies
with P 100 Myrs, and the Milky Way disk with P~ 2Gyr, could be that for
exciting turbulence following creation of massive stars, that for gas pushed
into the halo to return and interact with the disk and that for spiral density
wave evolution, respectively.
