  Perturbations of the Dirac field in the Schwarzschild spacetime are
characterized by two wave equations (for two chiralities) with effective
potentials which are iso-spectral and one of which is positive definite.
Therefore, the stability of a Dirac field in the Schwarzschild background is
straightforward. This is not so for the Schwarzschild-de Sitter case, because
potentials for both chiralities have negative gaps and the fact that the
stability is not automatically guaranteed in the asymptotically de Sitter case
was apparently omitted in the literature. Performing the time-domain
integration of the wave equations and, thereby, taking into consideration all
the quasinormal modes of the spectrum, we demonstrate stability of the Dirac
field in the Schwarzschild-de Sitter spacetime. The analysis of stability is
extended also to the Reissner-Nordstrom-de Sitter solution.
