  We study the dynamical stability of the BTZ black string against fermonic and
gravitational perturbations. The BTZ black string is not always stable against
these perturbations. There exist threshold values for $m^2$ related to the
compactification of the extra dimension for fermonic perturbation, scalar part
of the gravitational perturbation and the tensor perturbation, respectively.
Above the threshold values, perturbations are stable; while below these
thresholds, perturbations can be unstable. We find that this non-trivial
stability behavior qualitatively agrees with that predicted by a
thermodynamical argument, showing that the BTZ black string phase is not the
privileged stable phase.
