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We investigate theoretically the impact of optical feedback on quantum dot two-color laser dynamics when emitting from both the excited and the ground states. Detailed analysis of the laser dynamics is provided by numerical integrations and continuation techniques, as well as by analytical methods. As the feedback strength is increased the quantum dot laser undergoes a sequence of bifurcations involving steady-states, external cavity modes, self-pulsations and chaos. Furthermore, we report on two interesting mode competition results. First, the optical feedback favors the ground state emission; hence an increase of the feedback strength will generally lead to an increase of the ground state emission output power. Second, the optical feedback can select one lasing state or induce bistable switchings between different steady-states depending on the feedback strength and the injection current.