A large fraction of the luminous distant submm-wave galaxies recently detected using SCUBA on the JCMT appear to be associated with interacting optical counterparts. We investigate these systems using a simple hierarchical clustering model of galaxy evolution, in which the large luminosity of the SCUBA galaxies is assumed to be generated at the epoch of galaxy mergers in a burst of either star formation activity or the fuelling of an active galactic nucleus (AGN). The models are well constrained by the observed spectrum of the FIR/submm background radiation and the 60-micron counts of low-z IRAS galaxies. The ratio between the total amount of energy released during mergers and the mass of dark matter involved must increase sharply with z at z<1, and then decrease at higher z, independent of the fraction of the luminosity of mergers that is produced by starbursts and AGN. One additional parameter - the reciprocal of the product of the duration of the enhanced luminosity produced by the merger and the fraction of mergers that induce an enhanced luminosity, which we call the activity parameter - is introduced, to allow the relationship between merging dark matter haloes and the observed counts of distant dusty galaxies to be investigated. The observed counts can only be reproduced if the activity parameter is greater by a factor of about 5 and 100 at redshifts of 1 and 3 respectively, compared with the present epoch. Hence, if mergers account for the SCUBA galaxies, then the merger process must have been much more violent at high redshifts. We discuss the counts of galaxies and the intensity of background radiation in the optical/near-IR wavebands in the context of these hierarchical models, and thus investigate the relationship between the populations of SCUBA and Lyman-break galaxies. Comment: 18 pages, 12 figures, in press at MNRAS. Final proof version