  Low frequency ionization oscillations involving plasma and neutral density
(breathing modes) are the most violent perturbations in Hall thrusters for
electric propulsion. Because of its simplicity, the zero-dimensional (0-D)
predator-prey model of two nonlinearly coupled ordinary differential equations
for plasma and neutral density has been often used for the characterization of
such oscillations and scaling estimates. We investigate the properties of its
continuum analog, the one-dimensional (1-D) system of two nonlinearly coupled
equations in partial derivatives (PDE) for plasma and neutral density. This is
a more general model, of which the standard 0-D predator-prey model is a
special limit case. We show that the 1-D model is stable and does not show any
oscillations for the boundary conditions relevant to Hall thruster and the
uniform ion velocity. We then propose a reduced 1-D model based on two coupled
PDE for plasma and neutral densities that is unstable and exhibit oscillations
if the ion velocity profile with the near the anode back-flow (toward the
anode) region is used. Comparisons of the reduced model with the predictions of
the full model that takes into account the self-consistent plasma response show
that the main properties of the breathing mode are well captured. In
particular, it is shown that the frequency of the breathing mode oscillations
is weakly dependent on the final ion velocity but shows a strong correlation
with the width of the ion back-flow region.
