  The quasiparticle excitations and dynamical stability of an atomic
Bose-Einstein condensate coupled to a quantum degenerate Fermi gas of atoms at
zero temperature is studied. The Fermi gas is assumed to be either in the
normal state or to have undergone a phase transition to a superfluid state by
forming Cooper pairs. The quasiparticle excitations of the Bose-Einstein
condensate exhibit a dynamical instability due to a resonant exchange of energy
and momentum with quasiparticle excitations of the Fermi gas. The stability
regime for the bosons depends on whether the Fermi gas is in the normal state
or in the superfluid state. We show that the energy gap in the quasiparticle
spectrum for the superfluid state stabilizes the low energy energy excitations
of the condensate. In the stable regime, we calculate the boson quasiparticle
spectrum, which is modified by the fluctuations in the density of the Fermi
gas.
