A basic rate equation model of a quantum-well semiconductor ring laser is reduced to two equations by using asymptotic methods. The reduced model allows for analytical expressions of the bifurcation points, which will simplify future model parameter estimations. The new equations allow for a two-dimensional phase-space description of the time-periodic solutions. An analysis of the bifurcation scenarios in different parameter regimes is pursued. Physical conditions for the emergence of the operating regimes are assessed quantitatively in terms of saturation processes and backscattering mechanisms.