paper a day : periodic bloodsucking rates for vampires

Alex forwarded me this reference:

R. F. Hartl, A. Mehlmann, and A. Novak
Journal of Optimization Theory and Applications, Vol.75 No. 3, 1992

In this paper, we present a new approach for modeling the dynamic intertemporal confrontation between vampires and humans. It is assumed that the change of the vampiristic consumption rate induces costs and that the vampire community also derives some utility from possessing humans and not only from consuming them. Using the Hopf bifurcation theorem it can be shown that cyclical bloodsucking strategies are optimal. These results are in accordance with empirical evidence.

Keywords: maximum principle; limit cycles; economics of human resources; vampire myth

To the feather-fool and lobcock, the pseudo-scientist and materialist, these deeper and obscurer things must, of course, appear a grandam’s tale.
Montague Summers. The Vampire in Europe

This paper analyzes a mathematical model of bloodsucking rates for vampires using control theory. However, as they note in the introduction,

To a traditional vampirologist, the use of optimal control theory against vampires, as exercised in Ref. 6, seems highly questionable. This is due to the fact that the application of Pontryagin’s principle requires the derivation of a shadow price for vampires. Such a price is, however, nonexistent since vampires do not have a shadow.

As a predator-prey scenario, we can model the dynamics of the population using some differential equations. The problem for the vampires is to set a bloodsucking rate (humans per vampire) so as to maximize a utility function subject to the dynamics. However, the model has to be made more sophisticated to account for the cyclical bloodsucking patterns found in real vampires. The modifications are twofold — firstly, vampires also derive some utility from posessing humans rather than just sucking blood from them, and secondly, changing the consumption rate penalizes the utility. So in this push-and-pull framework they can derive some cycles, appropriately named “cycles of fear” in which the bloodsucking rate is modulated over time to achieve a stable tradeoff and net utility.

The full version, which is not to be missed, can be found via SpringerLink.
For some earlier comments on the optimal destruction of vampires and macroeconomic policy (which involves the shadow price), see this related JSTOR article.

My research is waaaaaay too boring.