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The bismuth-based perovskites are an interesting class of materials that exhibit a variety of coupled ferroic properties. Through epitaxial growth in the (001) pseudo-cubic [(001)pc] orientation, various phases with variable ferroelectric polarization can be stabilized. Using density-functional theory calculations, we predict the phase stability of the bismuth-based perovskite oxides as a function of pseudo-cubic in-plane (IP) lattice constant, mimicking (001)pc epitaxial films. We find that the BiMnO3, BiCoO3, and BiNiO3 systems each exhibit only one stable phase over a wide range of IP lattice constants. In the BiFeO3 (BFO) and BiCrO3 (BCO) systems, by comparison, we find several phases that are energetically favorable, depending on the value of the IP strain. The BFO phases predicted to be stable, in order of increasing compressive IP strain, are monoclinic Cc, triclinic P1, monoclinic Cm, and tetragonal P4mm. In the BCO system, we find two orthorhombic Pbnm phases, respectively, under no IP strain and under compressive IP strain, and one monoclinic Cc phase to be stable under tensile IP strain. Our results serve to guide experimental efforts in terms of selecting growth substrates with the goal of achieving desired epitaxial-stabilized perovskite phases and to support future investigations of the tunability of BXO properties with epitaxial strain.