  We address the cosmological evolution of violent gravitational instability in
high-redshift, massive, star-forming galactic discs. To this aim, we integrate
in time the equations of mass and energy conservation under self-regulated
instability of a two-component disc of gas and stars. The disc is assumed to be
continuously fed by cold gas at the average cosmological rate. The gas forms
stars and is partly driven away by stellar feedback. The gas and stars flow
inward through the disc to a central bulge due to torques that drive angular
momentum outwards. The gravitational energy released by the mass inflow down
the gravitational potential gradient drives the disc turbulence that maintains
the disc unstable with a Toomre instability parameter Q~1, compensating for the
dissipative losses of the gas turbulence and raising the stellar velocity
dispersion. We follow the velocity dispersion of stars and gas as they heat and
cool respectively and search for disc `stabilization', to be marked by a low
gas velocity dispersion comparable to the speed of sound ~10 km/s. We vary the
model parameters that characterize the accreted gas fraction, turbulence
dissipation rate, star-formation rate, and stellar feedback. We find that as
long as the gas input roughly follows the average cosmological rate, the disc
instability is a robust phenomenon at high redshift till z~1, driven by the
high surface density and high gas fraction due to the intense cosmological
accretion. For a broad range of model parameter values, the discs tend to
`stabilize' at z~0-0.5 as they become dominated by hot stars. When the model
parameters are pushed to extreme values, the discs may stabilize as early as
z~2, with the gas loss by strong outflows serving as the dominant stabilizing
factor.
