High-redshift submillimetre galaxies ( SMGs ) are some of the most rapidly star-forming galaxies in the Universe .
Historically , galaxy formation models have had difficulty explaining the observed number counts of SMGs .
We combine a semi-empirical model with 3-D hydrodynamical simulations and 3-D dust radiative transfer to predict the number counts of unlensed SMGs .
Because the stellar mass functions , gas and dust masses , and sizes of our galaxies are constrained to match observations , we can isolate uncertainties related to the dynamical evolution of galaxy mergers and the dust radiative transfer .
The number counts and redshift distributions predicted by our model agree well with observations .
Isolated disc galaxies dominate the faint ( S _ { 1.1 } \la 1 mJy , or S _ { 850 } \la 2 mJy ) population .
The brighter sources are a mix of merger-induced starbursts and galaxy-pair SMGs ; the latter subpopulation accounts for \sim 30 - 50 per cent of all SMGs at all S _ { 1.1 } \ga 0.5 mJy ( S _ { 850 } \ga 1 mJy ) .
The mean redshifts are \sim 3.0 - 3.5 , depending on the flux cut , and the brightest sources tend to be at higher redshifts .
Because the galaxy-pair SMGs will be resolved into multiple fainter sources by ALMA , the bright ALMA counts should be as much as 2 times less than those observed using single-dish telescopes .
The agreement between our model , which uses a Kroupa IMF , and observations suggests that the IMF in high-redshifts starbursts need not be top-heavy ; if the IMF were top-heavy , our model would over-predict the number counts .
We conclude that the difficulty some models have reproducing the observed SMG counts is likely indicative of more general problems – such as an under-prediction of the abundance of massive galaxies or a star formation rate–stellar mass relation normalisation lower than that observed – rather than a problem specific to the SMG population .