Helioseismology has detected that solar photospheric active regions are surrounded byrnspatially extended, converging ow with typical ow velocities of 20-30 m/s. They extendrnup to 30 degrees from the center of the active region. It has been proposed that thisrnconverging ow may act as a saturation mechanism for the solar dynamo, and as such,rnmay determine the strength of solar cycles. In this work we explore questions such as:rnAre the converging ows towards active regions an e_ective saturation mechanism forrnBabcock-Leighton solar dynamo models?rnThese inows can potentially play an important role in ux-transport models of thernsolar cycle. The model of inow towards the active regions is developed within thernframework of the Surface ux Transport And Babcock-LEighton (STABLE) solar dynamornmodel. STABLE is designed to capture both the 11-year solar cycle and the evolution ofrnphotospheric magnetic ux with high _delity. The STABLE model solves the kinematicrnmagnetohydrodynamic (MHD) induction equations in a 3D, rotating, spherical shell. Therninduction equation is solved by means of the Anelastic Spherical Harmonic (ASH) code,rnwhich currently serves as the dynamical core for the STABLE model. STABLE uses thernSpotMaker spot deposition algorithm to place bipolar magnetic regions (BMRs) on thernsolar surface in response to the dynamo-generated toroidal magnetic _eld. In this way, thernmodel can be regarded as a uni_cation of BL dynamo models (2.5D in radius/latitude)rnand surface ux transport models (2.5D in latitude/longitude) into a more self-consistentrnframework that builds on the successes of each while capturing the full 3D structure ofrnthe evolving magnetic _eld. The subsequent evolution of these BMRs due to di_erentialrnrotation, meridional circulation, inow and turbulent di_usion naturally generates a meanrnpoloidal _eld as originally described by Babcock (1961) and Leighton (1964).rnWe veri_ed the STABLE model by reproducing a 2D mean-_eld benchmark and thisrnmodel, and reproduces some basic features of the solar cycle including an 11 yr periodicity,rnixrnequatorward migration of toroidal ux in the deep convection zone, and poleward propagationrnof poloidal ux at the surface. The poleward-propagating surface ux originatesrnas trailing ux in BMRs, migrates poleward in multiple non-axisymmetric streams, andrneventually reverses the polar _eld, thus sustaining the dynamo. We also present somernrepresentative dynamo simulations, focusing on the special case of kinematic magneticrninduction and axisymmetric ow _elds. Not all solutions are supercritical; it can be arnchallenge for the BL mechanism to sustain the dynamo when the turbulent di_usion nearrnthe surface is _ 1012 cm2 sÃƒÂ€Â€Â€1. However, if BMRs are su_ciently large, deep, and numerous,rnthen sustained, cyclic, dynamo solutions can be found that exhibit solar-like features.rnFurthermore, we _nd that the shearing of radial magnetic ux by the surface di_erentialrnrotation can account for most of the net toroidal ux generation in each hemisphere, asrnhas been recently argued for the Sun by [Cameron and Sch ussler(2015)].rnWe _nd that inows into active regions can indeed enhance ux cancellation andrnregulate the strength of magnetic cycles. We _nd that inows decrease the amplitude ofrnthe polar _eld, relative to a no-inows scenario (that is the tilt angle saturation [Karakrnand Miesch(2017)]). Stronger (weaker) inows lead to larger (smaller) reductions of thernpolar _eld. Our STABLE simulations show that converging ows toward the BMRsrnsigni_cantly inhibits the build-up of the polar _eld and provide a non-linear feedbackrnmechanism capable of saturating the global dynamo of the solar cycle within the Babcock-rnLeighton paradigm. To our knowledge this is the _rst 3D Babcock-Leighton model withrnexplicit BMRs to demonstrate that converging ows can serve as a viable saturationrnmechanism for the solar dynamo. We also discuss how the converging ows play a keyrnrole in determining the strength of magnetic cycles.