Of course, we were not the first to look at the connections between economics and the undead. Here’s a paper from 1982 that examines the optimal management of humans from the vampires’ point of view. Warning – the paper is very mathematical (much more than we are, but this is what happens when people who like math get bored).
Leaving the math aside, the basic idea is that vampires value humans for their blood, but taking blood turns the humans into vampires which both increases the number of predators and reduces the number of prey. The paper develops an approach for determining the optimal number of humans relative to vampires – a “steady state”. It then raises an interesting question. Say that there are currently 7 billion humans but the goal is 15 billion humans. What is the best way to get there? Do you temporary stop killing humans completely to grow the population as fast as possible? Or do you keep killing a few humans now to enjoy the blood even though it will take longer to get to the desired stock?
The answer turns out to depend on marginal utility, that is, the amount of enjoyment you get from each additional pint of blood. If the 10th pint of blood is just as good as the first, then you’ll want to get to the steady state as fast as possible (the “bang-bang” solution). On the other hand, if a pint is worth much more when you haven’t had much to drink, a more gradual path to the steady state is better.
It’s never this simple, of course. As Glen explains in Chapter 15, vampires have an incentive to behave individually in a way that undermines the optimal solution for the group.
Also, if you haven’t had enough of dynamic systems — and who has? — Bishop, Tufte and Tufte take a non-mathematical look at the how the population of zombies might rise or fall over time in Chapter 6.