Self-regulated galaxy evolution within a self-consistently varying galaxy-wide initial mass function

Semi-analytical evolution models of galaxies are a useful and computationally inexpensive tool for a fast assessment of individual properties and their evolution. In this work, specifically the influence of a metallicity- and star-formation-rate- (SFR) dependent, galaxy-wide stellar initial mass function (IGIMF) on the self-regulation of star formation in a galaxy is of interest. We aim to investigate the impact of a variable IGIMF, and especially its influence on the equilibrium SFR of the system, its gas fraction, the metallicity evolution, and the gas depletion timescale, τ_̊m gas, all in comparison with two non-varying galaxy-wide IMFs (gwIMFs). A two-phase two-component model with gas and stellar components was integrated using the Cash--Karp method with adaptive step size. For comparison, the calculations were carried out using non-varying gwIMFs and a variable IGIMF, implemented using the code, which is based on the IGIMF theory. The input parameters were the effective radius, the final age, and the accretion rate of the galaxy. Our highly simplified model does not represent a full physical galactic network; however, it is suitable for the purpose of this study. GalIMF All models---both non-varying gwIMFs and the IGIMF--reproduce reasonable gas fractions, gas-depletion timescales, and the main sequence of star-forming galaxies. However, only the IGIMF model accurately predicts the mass--metallicity relation and provides a more comprehensive description of quenched elliptical galaxies. For massive ellipticals all models suggest the need for an additional gas-heating source to reach a quenched state. Using a different stellar yield table in the IGIMF model does not significantly affect the results. In all models, the galaxies evolve in a self-regulated way, which is determined by the accretion rate. The self-regulated constancy of the SFR reflects the constant SFRs of nearby star-forming galaxies. The specific gas-accretion rate of all galaxies appears to be comparable to the Hubble constant. The inclusion of outflows improves the results for the canonical gwIMF model, but not significantly, while for the IGIMF model it has no significant impact.