  We consider a stationary continuous model of random size population with
non-neutral mutations using a continuous state branching process with
non-homogeneous immigration. We assume the type (or mutation) of the immigrants
is random given by a constant mutation rate measure. We determine some
genealogical properties of this process such as: distribution of the time to
the most recent common ancestor (MRCA), bottleneck effect at the time to the
MRCA (which might be drastic for some mutation rate measures), favorable type
for the MRCA, asymptotics of the number of ancestors.
