Carnivorous dinosaurs reached larger body sizes than the predatory mammals that replaced them as the dominant big terrestrial meat-eaters. Although some workers have suggested that tyrannosaurs and other large theropods were able to achieve huge size only by becoming sluggish carrion-eaters, an alternative explanation is presented here. The maximum body size attained by a predator species reflects a balance between two conflicting demands. Population densities of huge predators must be very low to prevent their over-exploiting their food base of prey species. At the same time, however, total population sizes must be large enough to prevent chance extinction of the predator species. Calculations based on predator food requirements, prey population turnover rates, and prey:predator population density ratios, and on observed natural population densities of mammalian carnivores, suggest that a dinosaur-sized predatory mammal might find it difficult or even impossible simultaneously to maintain low enough population densities and large enough population sizes. Flesh-eating dinosaurs were able to get around this size constraint due to one, or more likely some combination, of the following factors: larger population densities and/or faster population turnover rates of their herbivorous dinosaur prey than expected for dinosaur-sized herbivorous mammals, oviparity in dinosaurs as opposed to viviparity in mammals, differences in diet between juvenile and adult theropods, and lower mass-specific food consumption rates of carnivorous dinosaurs than expected for equally large meat-eating mammals.