The Transylvanian problem

Consider the Lotka-Volterra system:

where

stock of humans in an isolated Transylvanian community
number of vampires
death rate of vampires due to contact with sunlight, crucifixes, garlic, and vampire hunters
growth rate of the human population
contact coefficient.

While and are time functions, , and are constant positive parameters. The term in (1) and (2) means that each time a vampire meets a human being the former extracts blood from the latter and, by doing so, turns him into a vampire. Clearly this predator-prey model leads to cyclical time paths of and .

In Ref. 3, the optimal bloodsucking strategy for the vampire population has been determined. For this it has been assumed that the vampire community decides about the control variable , where:

bloodsucking rate (number of consumed humans) per vampire
utility derived by the average vampire from blood consumption at rate
vampires' discount rate.

This implies that the objective function (total utility per vampire) is

which has to be maximized w.r.t. the controlled state equations

By defining the stock of humans per vampire the system (4), (5) can also be written as

In Ref. 3, this control problem was solved in the three different cases of

(a)

asymptotically satiated vampires i.e. ,

(b)

blood maximizing vampires i.e. , and

(c)

unsatiable vampires i.e. .

Since the problem consists of only one control variable the optimal bloodsucking strategy is to approach a long run equilibrium of the humans to vampires ratio . This equilibrium is approached asymptotically in case (a) and as quickly as possible in cases (b) and (c). In the latter case the equilibrium is maintained by a chattering control.

In Ref. 4, the analysis of the one state variable vampirism problem was completed by solving the case of a convex-concave utility function and by showing, that a suitable state transformation can be used to establish the optimality of the candidate solution obtained by the maximum principle. Note that in the original formulation of the Transylvanian problem the usual sufficiency conditions are not satisfied.

In all these formulations, however, the resulting monotonic state trajectories and bloodsucking rates are not in accordance with empirical evidence. It is well known that the appearances of vampires follow a typical cyclical behavior. Therefore this paper's purpose is to extend the model in such a way that more realistic cyclical bloodsucking patterns are optimal.

One way of doing so would be to consider case (c) where chattering was optimal. By introducing some inertia by imposing adjustment costs on the consumption rate this cyclical behavior could be obtained as in Ref. 7 or Ref. 8. However-strictly speaking-the cyclical solutions obtained can then only be called candidates for optimal solutions since the sufficiency conditions are not satisfied.

Thus we choose to follow a path motivated by Refs. 9 and 10, in which we consider a purely concave model where the sufficiency conditions are satisfied and where the appearance of cycles can be proven analytically. In particular, we assume that the change of the consumption rate induces costs and that the vampire community also derives some utility from possessing humans and not only from consuming them.

Let us thus define:

rate of change of the consumption (bloodsucking) rate
adjustment costs caused by changes of the consumption rate
utility derived by the average vampire from having available a resource of x humans per vampires.

The extended optimal control problem becomes:

subject to

Unfortunately, the Hamiltonian is not concave in the state vector because of the term in (8). This problem can, however, be overcomed by applying the state transformation used in Ref. 4. Define the transformed resource stock by

then the problem (7) - (9) is equivalent to

subject to

where and .

Let us assume that the utility functions and are concave in and , respectively, and that the adjustment cost function is convex. Then this is a purely concave model and all cross partials are zero. The investigation of the canonical system is therefore comparably simple. In particular, the stationary points can be computed and using the Hopf bifurcation theorem it can be shown that cyclical solutions exist. These and their stability properties can then be computed numerically for some parameter values.