Central Florida University professor Costas Efthimiou has published a paper demonstrating that vampires cannot possibly exist because if they did, and fed regularly on humans (creating more of their kind in the process) the entire population would eventually become vampires and, with their food supply exhausted, die of starvation. In his model, it would only take two and a half years.
It's a neat mathematical demonstration, but I beg to point out that mathematical models rarely translate intact into the real world. Professor Efthimiou has overlooked several important variables that would affect the accuracy of his calculations.
First of all, the professor erroneously assumes that a vampire always kills his (or her) victims, which we know is not true, since one of the staples of vampire fiction is the person slinking around with a scarf to cover those tell-tale fang marks.
Second, he assumes that anyone bitten by a vampire automatically becomes one, which may or may not be true, depending on which theory and practice of vampirism holds true--in many fictional worlds, it requires rather more difficult circumstances or elaborate rituals, most commonly involving an exchange of blood rather than a mere one-way donation. In vampire fiction, once the bitten victim's plight is revealed, you often have a quest to slay the vamp responsible before the victim "turns"--why bother if the victim's death and recruitment to the undead are inevitable?
Third, he assumes that all vampires created live indefinitely to feed on the human population, when all of us know perfectly well that vampire mortality is very high, especially in those early years before they learn how to cope with their limitations and leverage their special powers. As with any species, many vampires must be created if the species is to survive. Take oysters, for example--the average oyster sheds an average of a million eggs in each spawning season--why isn't the world neck deep in oysters? Because other sea inhabitants--particularly small fish--eat the overwhelming majority before they can even begin growing a shell. (I use the oyster since, as the daughter of an oyster malacologist, I know a tad more about their reproductive cycle than I really care to know and relish the rare chance to make practical use of this knowledge.) If you consider the number of vampires who die of sunburn and garlic allergy, the number slain by rival bloodsuckers, and add in the depredations of generations of slayers like Buffy, I think you'll find the mathematical odds much more friendly to the notion that a small number of savvy, wary vampires do manage to coexist with us.
Though the professor's paper is certainly fascinating--he also has some interesting things to say about ghosts and zombies: Ghosts, Vampires and Zombies: Cinema Fiction vs Physics Reality.