  We study theoretically the superfluidity and stability of a Bose-Einstein
condensate (BEC) whose interatomic scattering length is periodically modulated
with optical Feshbach resonance. Our numerical study finds that the properties
of this periodic BEC are strongly influenced by the modulation strength. When
the modulation strength is small, only the Bloch waves close to the Brillouin
zone edge suffer both Landau and dynamical instabilities. When the modulation
strength is strong enough, all Bloch waves become dynamically unstable. In
other words, the periodic BEC loses its superfluidity completely.
