Fraction of time spent by the BHs above a given AGN energy injection rate Ė for the ‘EddLim’ (dotted blue), ‘sEdd’ (solid orange) and ‘sEddThm’ (densely dash-dotted violet) simulations. A red patch Ṁ_Edd shows the AGN feedback range (∼10⁴³ − 10⁴⁴ erg s⁻¹) of a BH between M_BH = 10⁶ − 10⁷ M_⊙ at the Eddington limit. The ‘EddLim’ simulation has a BH constantly accreting at the limit, thanks to the ineffectiveness of the feedback. The ‘sEdd’ case has stronger feedback events but this results in many low luminosities feedback episodes. This regime therefore reduces the overall required AGN feedback to self-regulate the BH growth. The ‘sEddThm’ simulation is discussed in Sect. 4.2.