  We study the minimal ``bathtub" toy model as an analytic tool for capturing
key processes of galaxy evolution and identifying robust successes and
challenges in reproducing observations at high redshift. The source and sink
terms of the continuity equations for gas and stars are expressed in simple
terms from first principles. The assumed dependence of star-formation rate
(SFR) on gas mass self-regulates the system into a unique asymptotic behavior,
which is approximated by an analytic quasi-steady-state solution (QSS). We
address the validity of the QSS at different epochs independent of earlier
conditions. At high z, where the accretion is assumed to consist of gas only,
the specific SFR is robustly predicted to be sSFR = [(1+z)/3]^{5/2} Gyr^{-1},
slightly higher than the cosmological specific accretion rate, in agreement
with observations at z=3-8. The gas fraction is expected to decline slowly, and
the observations constrain the SFR efficiency per dynamical time to
epsilon=0.02. The stellar-to-virial mass ratio f_sv is predicted to be constant
in time, and the observed value requires an outflow mass-lading factor of
eta=1-3, depending on the penetration efficiency of gas into the galaxy.
However, at z=2, where stars are also accreted through mergers, the simplest
model has an apparent difficulty in matching observations. The model that
maximizes the sSFR, with the outflows fully recycled, falls short by a factor 3
in sSFR, and overestimates f_sv. With strong outflows, the model can reproduce
the observed f_sv but at the expense of underestimating the sSFR by an order of
magnitude. We discuss potential remedies including a bias due to the exclusion
of quenched galaxies.
