Dynamo models relying on the Babcock-Leighton mechanism are successful in reproducing most of the solar magnetic field dynamical characteristics. However, considering that such models operate only above a lower magnetic field threshold, they do not provide an appropriate magnetic field regeneration process characterizing a self-sustainable dynamo. In this work we consider the existence of an additional alpha-effect to the Babcock-Leighton scenario in a mean-field axisymmetric kinematic numerical model. Both poloidal field regeneration mechanisms are treated with two different strength-limiting factors. Apart from the solar antisymmetric parity behavior, the main solar features are reproduced: cyclic polarity reversals, midlatitudinal equatorward migration of strong toroidal field, poleward migration of polar surface radial fields, and the quadrature phase shift between both. Long-term variability of the solutions exhibits lenghty periods of minimum activity followed by posterior recovery, akin to the observed Maunder Minimum. Based on the analysis of the residual activity during periods of minimum activity, we suggest that these are caused by a predominance of the alpha-effect over the Babcock-Leighton mechanism in regenerating the poloidal field.