  We investigate toy models for spatial and temporal instabilities in
collective neutrino oscillations induced by neutrino self-interactions, with
special emphasis on inhomogeneous systems with densities following a profile.
Simulations are based on a mathematica program that solves the Liouville
equation with or without vacuum terms, refractive terms from a background
medium, and neutrino-neutrino forward scattering, in one space dimension and in
time. A discrete number of momentum modes are characterized by the neutrino
velocity projection on the spatial direction. We also consider the effects of
charged current interaction source terms and neutral current scattering
contributions. We find that refractive effects from the medium, in particular
for density distributions with a profile, and neutral current non-forward
scattering off the background medium can strongly influence fast collective
flavor transformations. Specifically we find that if both are present, fast
flavor conversions can be strongly suppressed or at least delayed.
