  As an atomic Bose Einstein condensate (BEC) is coupled to a source of
uncondensed atoms at the same temperature and to a sink (extraction towards an
atom laser) the idealized description in terms of a Gross-Pitaevsky equation
(GP) no longer holds. Under suitable physical assumptions we show that the
dissipative BEC obeys a Complex Ginzburg Landau equation (CGL) and for some
parameter range it undergoes a space time patterning. As a consequence, the
density of BEC atoms within the trap displays non trivial space time
correlations, which can be detected by monitoring the density profile of the
outgoing atom laser. The patterning condition requires a negative scattering
length, as e.g. in $^7$Li. In such a case we expect a many domain collapsed
regime, rather than a single one as reported for a closed BEC.
