The consequences are explored of an observationally established relationship of the star formation rate (SFR) of star-forming galaxies with their stellar mass (M) and cosmic time (t), such that SFR ∝M t−2.5. It is shown that small systematic differences in SFR dramatically amplify in the course of time: galaxies with above-average SFR run into quasi-exponential mass and SFR growth, while galaxies with below-average SFR avoid such exponential growth and evolve with moderate mass increase. It is argued that galaxies following the first path would enormously overgrow if they keep forming stars all the way to the present, hence should quench star formation and turn passive. By the same token, those instead avoiding the quasi-exponential growth may keep forming stars up to the present. Thus, it is conjectured that this divergent behaviour can help us to understand the origin of the dichotomy between passive, spheroidal galaxies and star-forming, disc galaxies.