Public concern over biodiversity loss is often rationalized as a threat to ecosystem functioning, but biodiversity-ecosystem functioning (BEF) relations are hard to empirically quantify at large scales. We use a realistic marine food-web model, resolving species over five trophic levels, to study how total fish production changes with species richness. This complex model predicts that BEF relations, on average, follow simple Michaelis-Menten curves when species are randomly deleted. These are shaped mainly by release of fish from predation, rather than the release from competition expected from simpler communities. Ordering species deletions by decreasing body mass or trophic level, representing 'fishing down the food web', accentuates prey-release effects and results in unimodal relationships. In contrast, simultaneous unselective harvesting diminishes these effects and produces an almost linear BEF relation, with maximum multispecies fisheries yield at ≈40% of initial species richness. These findings have important implications for the valuation of marine biodiversity.