M_infolw − M_Res(t₀)/M_BH(t₀) plane in three different redshifts bins according to L-Galaxies run on the Millennium merger trees. While the y-axis depicts how gas-rich is the environment around the MBH, the x-axis illustrates how powerful are the gas inflows towards the MBH. The blue horizontal and vertical lines correspond to the chosen ℛ^th, and $M_{\mathrm{inflow}}^{\mathrm{th}}$. Each row corresponds to each of the events that can trigger cold gas inflows toward the MBH: major mergers (top), minor mergers (middle) and disc instabilities (bottom). The plot has been computed by using the free parameters of the Boosted model reported in Table 1. To guide the reader, major (minor) mergers correspond to galaxy interactions which involve galaxies with baryonic masses differing by less (more) than a factor of 2.