The paper studies the growth of inhomogeneities of the inflaton field at the post-inflationary stage of the evolution of the Universe in realistic inflationary models and the dynamics of inhomogeneities at the nonlinear stage of evolution. At the beginning of this stage, the inflaton field can be considered almost homogeneous, and the expansion law of the Universe coincides with that at the dust-like stage. Over time, inhomogeneities of the inflaton field grow due to gravitational instability. If the stage of reheating is long enough, these inhomogeneities can give rise to the first gravitationally bound structures. For the study, a numerical code was written to simulate the evolution of inhomogeneities at the post-inflationary stage of expansion of the Universe.