Protoplanet’s growth and migration timescales (τ_growth = M/ | Ṁ |  and τ_mig = a/ | ȧ |  respectively) as a function of the protoplanet’s mass for the four different sizes of the accreted planetesimals of Fig. 6 (corresponding to the dashed line in Fig. 6). The peaks in the migration timescale correspond to the changes in the sense of migration of the protoplanet. From the two bottom panels, it can be seen that when the protoplanet’s mass is  ≳10 M_⊕, both timescales become comparable and eventually the migration timescale becomes shorter than the accretion timescale, leading to a fast migration of the protoplanet with very little accretion. For 1 km planetesimals the protoplanet growth is too slow to gain enough mass to change the type of migration before they are lost in the star (when its mass is  ~30 M_⊕). For 0.1 km planetesimals, the protoplanet can grow more massive because its accretion rate is higher than in the previous case, then it is able to open a gap in the disc and migration switches to type II, preventing it from falling in the star.