If a vampire must feed on human blood, how long would it take a certain number of vampires to wipe out the human population? And if those human victims then turned to vampires, how soon would vampires outnumber humans? Mathematical modeling turned to the world of the supernatural as a surprising number of researchers have tackled the complexities of human and vampire coexistence. A 1982 paper by Richard Hartl and Alexander Mehlmann titled "The Transylvanian Problem of Renewable Resources" took a look at how vampires would affect the human population.
In doing so, they divided vampires into three categories: the "asymptotically satiated vampire," the "blood maximizing vampire," and the "unsatiable vampire." Regardless of the type of vampire, though, they found that bloodsuckers can't help but face diminishing resources:
"[W]e are facing a typical consumption-resource trade off. The vampire society derives utility from consumption of blood but in sucking the blood of a human being and in turning him to a vampire the resource of human beings is reduced whereas the number of vampires is increased. Both of these effects diminish the resource of humans per vampire curtailing future possibilities of consumption."
However, a 2007 article modeled a vampire apocalypse and concluded that “vampires would eliminate humans within three years.” Which provoked a refutation in 2008. More research has been done, but the results vary according to which vampire mythology one ascribes to. Are the vampires in Bram Stoker’s universe more dangerous than those in Anne Rice’s universe? What about the vampires in the worlds of Twilight or The Historian? Read an overview of the available research at Atlas Obscura.
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