We show that in a degenerate optical parametric oscillator with saturable losses for the frequency down-converted field, the steady state can be destabilized via either a Hopf or a Turing instability. The relative order between the two bifurcations is controlled by the linear loss of the saturable absorber. If the Turing bifurcation is subcritical and the Hopf bifurcation occurs in the hysteresis domain involving the homogeneous and inhomogeneous states, steady localized structures are generated below the Hopf bifurcation and time-periodic localized structures are generated above the Hopf bifurcation.

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