We examine the spheroid growth and star formation quenching experienced by galaxies from z~3 to the present by studying the evolution with redshift of the quiescent and spheroid-dominated fractions of galaxies from the CANDELS and GAMA surveys. We compare the observed fractions with predictions from a semi-analytic model which includes prescriptions for bulge growth and AGN feedback due to mergers and disk instabilities. We facilitate direct morphological comparison by converting our model bulge-to-total stellar mass ratios to Sersic indices. We then subdivide our population into the four quadrants of the sSFR-Sersic index plane and study the buildup of each of these subpopulations. We find that the fraction of star forming disks declines steadily, while the fraction of quiescent spheroids builds up over cosmic time. The fractions of star forming spheroids and quiescent disks are both non-negligible, and stay nearly constant over the period we have studied, at about 10% and 15-20% respectively. Our model is qualitatively successful at reproducing the evolution of the two main populations (star forming disk-dominated galaxies and quiescent spheroid-dominated galaxies), and approximately reproduces the relative fractions of all four types, but predicts a stronger decline in star forming spheroids, and increase in quiescent disks, than seen in the observations. A model with an additional channel for bulge growth via disk instabilities agrees better overall with the observations than a model in which bulges may grow only through mergers. We study evolutionary tracks of some individual galaxies as they experience morphological transformation and quenching, and examine the importance of different physical drivers of this transformation (major and minor mergers and disk instabilities). We find that complex histories with multiple transformative events are the norm.
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