  We consider two systems A and B that share Einstein-Podolsky-Rosen (EPR)
steering correlations and study how these correlations will decay, when each of
the systems are independently coupled to a reservoir. EPR steering is a
directional form of entanglement, and the measure of steering can change
depending on whether the system A is steered by B, or vice versa. First, we
examine the decay of the steering correlations of the two-mode squeezed state.
We find that if the system B is coupled to a reservoir, then the decoherence of
the steering of A by B is particularly marked, to the extent that there is a
sudden death of steering after a finite time. We find a different directional
effect, if the reservoirs are thermally excited. Second, we study the
decoherence of the steering of a Schr\"odinger cat state, modeled as the
entangled state of a spin and harmonic oscillator, when the macroscopic system
(the cat) is coupled to a reservoir.
