  We consider the time evolution of systems in which a spatially homogeneous
scalar field is coupled to fermions. The quantum back-reaction is taken into
account in one-loop approximation. We set up the basic equations and their
renormalization in a form suitable for numerical computations. The initial
singularities appearing in the renormalized equations are removed by a
Bogoliubov transformation. The equations are then generalized to those in a
spatially flat Friedmann-Robertson-Walker universe.
  We have implemented the Minkowski space equations numerically and present
results for the time evolution with various parameter sets. We find that
fermion fluctuations are not in general as ineffective as assumed previously
but show interesting features which should be studied further. In an especially
interesting example we find that fermionic fluctuations can ``catalyze'' the
evolution of bosonic fluctuations.
