  Nonlinear numerical simulations are performed to investigate the density
evolution in the dust layer of a protoplanetary disk due to the gravitational
instability and dust settling toward the midplane. We assume the region where
the radial pressure gradient at equilibrium is negligible so that the
shear-induced instability is avoided, and also restrict to an axisymmetric
perturbation as a first step of nonlinear numerical simulations of the
gravitational instability. We find that there are two different evolutional
paths of the gravitational instability depending on the nondimensional gas
friction time, which is defined as the product of the gas friction time and the
Keplerian angular velocity. If the nondimensional gas friction time is equal to
0.01, the gravitational instability grows faster than dust settling. On the
other hand, if the nondimensional gas friction time is equal to 0.1, dust
aggregates settle sufficiently before the gravitational instability grows. In
the latter case, an approximate analytical calculation reveals that dust
settling is faster than the growth of the gravitational instability regardless
of the dust density at the midplane. Thus, the dust layer becomes extremely
thin and may reach a few tenth of the material density of the dust before the
gravitational instability grows.
