  Trapped ultra-cold atom experiments provide a unique opportunity to
understand Bose-Fermi superfluid mixtures occurring in contrasting areas of
physics. At present there are several atom-trap experiments that could
potentially explore this superfluid-mixture regime, thus warranting a detailed
understanding of the occurrence and stability of various possible thermodynamic
phases in the mixture. In the present work, we therefore construct the finite
temperature phase diagram of an interacting atomic mixture of Bose and Fermi
superfluids. Our study reveals a unique region of phase space, where the BCS
instability of the Fermi surface coincides with dynamical instability of the
homogeneous mixture towards phase separation through a first-order transition.
We illustrate how this intriguing interplay manifests in a trapped
configuration, thereby providing important constraints for observing
superfluidity in experiments.
