Context: The earlier introduced compartmental description, well-known from the description of infection diseases and epidemics, was applied here to describe the nonlinear temporal evolution of the baryonic matter in interstellar gas and stars in the presence of triggered star formation. Aims: The competition of triggered star formation, spontaneous star formation, stellar feedback and stellar evolution was theoretically investigated to understand the baryonic matter cycle including luminous baryonic matter in main-sequence stars and weakly luminous matter in white dwarfs, neutron stars and black holes. Of particular interest was the understanding of the cosmic star formation history and the redshift dependence of the gas and stellar fractions using compartmental models. Methods: For stationary rates of spontaneous and triggered star formation, continuous stellar feedback and stellar evolution exact and approximate analytical solutions of the time evolution of the fractions of stellar and locked-in stellar matter were derived involving the time dependence of the gaseous fraction G(t). The high accuracy of the analytical solutions is proven by comparison with the exact numerical solutions of the GSL equations. Results: The inclusion of the triggered star formation process explains the observed cosmological star formation rate, the integrated stellar density at redshifts below z=8, and the present-day gas and stellar fractions very well. The generalized GSL-model provides excellent fits to the observed redshift dependencies of the star formation rate and the integrated stellar density. Moreover, it explains the observed present-day gas and stellar fractions, and it makes predictions on the future evolution of these fractions in the universe.