The amplitudes of the magnetic field generated by various α2-dynamo models are investigated for the case when the magnetic Reynolds number,R, slightly exceeds the critical value,R(0), at which dynamo action is marginally possible. The basic assumption is that the α-effect is modified by the presence of the magnetic field by an amount proportional to the local magnetic energy density, according to the quenching law α = α0− α1|B|2, whereBis the magnetic field and α0and α1are positive constants. Solutions were obtained by expanding the field in powers of the amplitude, a, ofB, and by solving the resulting perturbation equations up to third order in a with the help of the adjoint kinematic dynamo system. An expression for the growth rate, γ1, of an infinitesimal disturbance whenR — R(0)is small was obtained, and the amplitude, a(0), of the final (stable) state was derived. Numerical integrations were performed for five models originally due to Roberts (1972) as well as for a model studied by Rädler and Wiedemann (1989). In all cases, the eigenvalues, the eigenfunctions, and the parameters needed to determine the final steady state (i.e., γ1and the Landau constant) show satisfactory numerical evidence of convergence. It was found that the bifurcation is supercritical for all six models. The critical Reynolds number for the dipole family of solutions is quite close to that of the quadrupole family of solutions in each case, as are the final steady-state amplitudes. The six models behave differently at finite amplitudes, and in particular there appears to be no correlation between the relative amplitudes of the steady dipole and quadrupole solutions and the relative sizes ofR(0)for the two families.