Analytical solution given in Eqs. (B.11), (B.15) to the grains-and-boulders model introduced in Sect. 5 for different boulder–grain particle mass ratios: (a) total number of grains N_g; (b) expected boulder mass 〈m〉_B. The total mass ratio of grains and boulders is , so boulders are expected to grow to 7 times their initial mass, as indicated by the horizontal black line. The horizontal axes refer to dimensionless time . In (b), the filled area indicates the physical spread of the mass distribution given by Eq. (B.17).