Cycles of fear: Periodic bloodsucking rates for vampires

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 75, No. 3, DECEMBER 1992

Cycles of Fear: Periodic Bloodsucking Rates for Vampires

R. F. H A R T L , l A. M E H L M A N N , 2 A N D A. N O V A K 3

Communicated by G. Leitmann

Abstract. In this paper, we present a new approach for modelling 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.

Key Words. Maximum principle, limit cycles, economics of human resources, vampire myths.

"To the feather-fool and lobcock, the pseudo-scientist and materialist, these deeper and obscurer things must, of course, appear a grandma's tale."

From The Vampire in Europe, by Montague Summers.

1. Introduction

While the behavior of vampires has been studied and documented over long periods of tirae (see, e.g., Ref. 1), neither the economic significance of vampirism nor the optimality of bloodsucking strategies has been analyzed by means of rational modelling. Vampiristic activities seemed to be of inter- est only to scholars of anthropology (Ref. 2) or, much more regrettable, to the Hammer Film Productions.

~Associate Professor, Department of Operations Research, Technische Universit/~t Wien, Wien, Austria.

2Associate Professor, Department of Operations Research, Technische Universit~it Wien, Wien, Austria.

3University Assistant, Institute of Statistics, Universit/it Wien, Wien, Austria.

559 0022-3239/92/1200-0559506.50/0 © 1992 Plenum Publishing Corporation

560 JOTA: VOL. 75, NO. 3, DECEMBER 1992

This situation has changed completely with the appearance of Refs. 3 and 4. Starting from a dynamic control model of confrontation between vampire-predators and human-preys, the authors indicate how the vampires' utility from blood intake can be optimized through sophisticated depletion of renewable human resources.

However, the problem with all these results is that the derived monotonic state trajectories and bloodsucking rates are not in accordance with empirical evidence. According to the main body of literature on vampirism research, the appearances of vampires follow a typical cyclical behavior (Ref. 5). The purpose of this paper is to extend the model in such a way that more realistic cyclical bloodsucking patterns are optimal.

Another possible extension, proposed in Ref. 6, provides human beings with an intertemporal welfare trade-off by allocation of labor services between production of economic goods and of useful instruments against vampires (such as stakes, garlic, or rosary beads). To a traditional vampirol- ogist, 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.

2. Transyivanian Problem

Consider the Lotka-Volterra system

~ = - a v + dvh, (1)

l;t = nh - dvh ; (2)

here, h is the stock of humans in an isolated Transylvanian community; v is the number of vampires; a is the death rate of vampires due to contact with sunlight, crucifixes, garlic, and vampire hunters; n is the growth rate of the human population; d is the contact coefficient.

While h and v are time functions, a, n, and d are constant positive parameters. The term dvh 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 v and h.

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 c( t ) , where: c( t ) is the bloodsucking rate (number of consumed humans) per vampire; U(c) is the utility derived

JOTA: VOL. 75, NO. 3, DECEMBER 1992 561

by the average vampire from blood consumption at rate c; r is the vampires' discount rate. This implies that the objective function (total utility per vampire) is

0 ~ exp(-rt)U(c) d t , (3)

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

13 = - a v + co , (4)

I~ = n h - cv . (5)

By defining the stock of humans per vampire x ( t ) = h ( t ) / v ( t ) , the system (4), (5) can also be written as

~: = ( n + a - c ) x - c. (6)

In Ref. 3, this control problem was solved for the three different cases:

(a) asymptotically satiated vampires, i.e., U" < 0; (b) blood maximizing vampires, i.e., U"= 0; (c) unsatiable vampires, i.e., U"> 0.

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 x °~. 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 optima1.

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 the following symbols: u(t) is the rate of change of the consumption (bloodsucking) rate; A(u) is the adjustment cost caused by changes of the consumption rate; V(x) is the utility derived by the average vampire from having available a resource of x humans per vampire. The extended optimal control problem becomes

f0 ° exp(-rt)[ U(c) + V(x) - A (u)] dt ~ max, (7)

subject to

2 = ( n + a - c ) x - c , x(0)=x0, (8)

~=u, c(0)=c0[or c(0) free]. (9)

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

y=ln(1 +x) ¢~ x = e x p ( y ) - 1. (1o)

Then, the problem (7)-(9) is equivalent to

f0 ° exp(-rt)[ U(c) + W(y) - A(u)l dt ~ max, 01)

subject to

p=(n+a) [1 - e x p ( - y ) ] - c , y(0)=y0, (12)

d=u, c(O)=co [or c(0) freel, (13)

where

W(y)= V[exp(y)- 1] and y0=ln(1 +x0).

Let us assume that the utility functions U(c) and W(y) are concave in c and y, respectively, and that the adjustment cost function A(u) 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.

3. Mathematical Analysis

To find optimal solutions to our control problem (11)-(13), we make use of Pontryagin's maximum principle and formulate the current-value Hamiltonian

H= U(c)+ W(y)-A(u)+)~{b[1 - e x p ( - y ) ] - c} +¢tu, (14)

where ,~ and/1 denote the current-value adjoint variables to y and c, respectively. Note that we set b = n + a for further convenience.

As the optimal change u of the bloodsucking rate c has to maximize the Hamiltonian, the assumptions made on the function A(u) imply that u has to follow

Hu=0 ~ A'(u)=u, (15)

and hence u = u(/l). The time derivatives of the adjoint variables are given by

= r,~ - Hy = A[r- b exp(-y)] - W'(y), (16)

/i =r/ t - He= r/l + , ~ - U'(c). (17)

As the Hamiltonian (14) is concave jointly in the states and the control, the canonical system, consisting of the two state equations (12) and (13), the adjoint equations (16) and (17), and the optimality condition (15), together with the limiting transversality condition

lira exp(-r t ) {,~ (t)y(t) + u (t)c(t)} = 0, (18) t ~ o O

yield not only necessary but even sufficient conditions for optimal solutions of our problem. Our aim is to show that the canonical system possesses stable periodic solutions. One way to do this is to apply the Hopf bifurcation theorem. Among other things, this requires that the canonical system has an equilibrium and that the Jacobian evaluated at the steady state possesses a pair of purely imaginary eigenvalues and no other eigenvalue with zero real part.


First, we want to establish the existence of a steady state. A steady state (y, c, A.,/1, u) of the canonical system is a solution to the following system of nonlinear equations:

b[1 - exp( -y ) ] - c = 0, (19)

u = 0 , (20)

A.[r- b exp( -y ) ] - W'(y) = 0, (21)

W + ~ - U'(c) =0, (22)

-Iz + A'(u) = 0. (23)

From (20) and (23), the results u ~ = 0 and p~ =A'(0) follow immediately. Moreover, (19) and (22) yield the relations

c°~=b[1-exp(-y~)] and A~=U'(c~)-rA'(O). (24)

From (24) and (21), it can be seen that the steady-state value y~ has to be a solution of

f (y ) = { U'(b[ 1 - exp( -y) ] ) - rA'(0)}

x [r - b exp( -y ) ] - W'(y) = 0. (25)

Specifying U(c) as a linear function U(c)= UoC, with u0 > rA'(O), we see that f (y) is monotonically increasing and therefore at most one steady state can exist. Moreover, if

lira W'(y) = oo and lira W'(y) = 0 (26) y~O y~co

hold, one and only one equilibrium exists.

4. Stability Analysis

Next, we linearize the canonical system to analyze the stability behavior of the steady state. This yields

~:J(z®)(z ~-z), (27) where z = (y, c, ~,/J) and J(z ~) is the Jacobian evaluated at the steady state (y, e, ~, / l) . The Jacobian is given by the 4 × 4 matrix

0 0 0 l/A" J(z) = ~b e x p ( - y ~) - W" 0 r - b e x p ( - y ~) 0 '

0 - U " 1 r

where the derivatives of the functions have to be evaluated at steady state.


According to Ref. 9, the eigenvalues of Jacobian matrices of the type

~a, l a,2 a,3 a14 1

j(z)=la21 a22 a14 a24 [ , ' - a , , - a 2 ,

J [.a32 a42 -aj2 r-a22

are given by

~1,z.3., = r/2 ± ~/(r/2) 2 - K/2 • ( t / 2 ) ~ - 4 de t (J ) ,

where K is the sum of the determinants

all a13 + a22 a24 + 2 a12 a14 .

a31 r--alj a42 r--a221 a32 --a21

Moreover, such a Jacobian has a pair of purely imaginary eigenvalues if and only if the relations

K 2 + 2r2K = 4 det(J), (28)

K> 0, (29)

hold. For our model, the constant K and the determinant det(J) are given

by

K= b exp(-y~°)[r- b exp(-y~)] + U"/A", (30)

d e t ( J ) = {b exp(-y°°)[A?°+ U"[r-b exp(-y~)]] - W"}/A". (31)

In the remainder of this section, we make use of a linear utility function U(c) =UoC and assume A(u)= au2/2. Then, we obtain

K= b e x p ( - y ~ ) [ r - b exp(-y~°)], (32)

det(J) = {uob exp( -y ~) - W"}/a. (33)

The bifurcation condition (28) is equivalent to

-4{u0b exp(-y ~) - W"}/a + b exp(-y°~)[r- b exp(-y~)]

× [2r 2 + b e x p ( - y ~ ) [ r - b exp(-y*)]] = 0, (34)


Choosing a as bifurcation parameter, we can evaluate the critical value acrit from (34),

a=it =b exp(-y°~)[r-b exp(-y~)]

2r 2 + b exp(-y°°)[r- b exp(-y°~)] × (35)

4[u0b exp(-y ~) - W"]

Note that the steady-state values of (y, c, ~,/~) do not depend on the parameter a. By choosing the parameter values b, u, r and the function W(y), we can calculate the steady state and if r>b exp(-y ~) holds, then the critical value acrit can be calculated from (34). In this case, the Jacobian evaluated at the equilibrium has a pair of purely imaginary eigenvalues ~1,2 = • ico. Moreover, the crossing velocity

0 Re( ~,,2)/Oal,~=,~,, ~.0. (36)

Therefore, we conclude that isolated periodic solutions will exist either for a > a~r~t or a < acr~t. To determine the stability of the cycles and the direction of the bifurcation, further computations either analytically or numerically are necessary. As the analytical proof of the stability of cycles generated by a Hopf bifurcation is rather cumbersome, even in very simple models (see, e.g., Ref. 11), we will present a numerical example leading to a stable cycle.

Specify the following functions:

U(e) = uoc, W(y) =y~/2, A(u) = au2/2.

Furthermore, choose the parameter values r=0.53, b=6.5, and Uo=0.44. For these values, the steady state is given by

(y, c, ;t,/~) = (5.2507, 6.4659, 0.4400, 0.0000).

According to (35), the critical value ct~rit = 14.6294 can be computed. To determine the stability of cycles and the direction of the bifurcation, further numerical investigations with the code BWDO (see Ref. 12) were carried out. It turned out that stable cycles occur for a > acrlt. Using the boundary-value problem solver COLSYS (see Refs. 13 and 14), we can find a sta31e limit cycle, e.g., for a = 15.0. This cycle in the (x, c)-plane is illus- trated in Fig. 1.

5. Conclusions

The phenomenon of cyclical bloodsucking patterns has been noted by many scholars of vampirology, but as yet no attempt has been made to offer a rational interpretation of it. In this paper, we presented a theoretical

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5,50 / - " " ' ~ ~ 6.55 6.50

6.40 6 . 3 5 - I

_ I

5.30

0 200 400 600 800 lOflO 1200

Fig. t. Cycle of tear for vampires.

explanation of those cycles of fear. The model proposed has two features that distinguish it from traditional approaches to demographic oscillations. First, it is based on a controlled Lotka-Volterra type of system. Second, the fluctuations are the result of optimal choices of consumption over time.

References

1. SUMMERS, M., The Vampire in Europe, Kegan Paul, Trench, and Trubner, Lon- don, England, 1929.

2. Du BOULA¥, J., The Greek Vampire: A Study of Cyclic Symbolism in Marriage and Death, Man : The Journal of the Royal Anthropological Institute of Great Britain and Ireland, Vol. 17, pp. 219-238, 1982.

3. HARTL, R. F., and MEUt.MANN, A., The Transylvanian Problem of Renewable Resources, R~vue Frangaise d'Automatique, Informatique et de Recherche Operationetle, Vol. 16, pp. 379-390, 1982.

4. HARTL, R. F., and MEHLMANN, A., Convex-Concave Utility Function: Optimal Blood Consumption for Vampires, Applied Mathematical Modelling, Vol. 7, pp. 83-88, 1983.

5. KrNG, S., Salem's Lot, New American Library, New York, New York, 1969.


6. SNOWER, D., Macroeconomic Policy and the Optimal Destruction of Vampires, Journal of Political Economy, Vol. 90, pp. 647-655, 1982.

7. STEINDE, A., FEICHTINGER, G., HARTE, R. F., and SOROER, G., On the Opti- mality of Cyclical Employment Policies: A Numerical Investigation, Journal of Economic Dynamics and Control, Vol. t0, pp. 457-466, 1986.

8. LUHMER, A., STEINDL, A., FEICHTINGER, G., HARTL, R. F., and SORGER, G., ADPULS in Continuous Time, European Journal of Operational Research, Vol. 34, pp. 171-177, 1988.

9. DOCKNER, E. ]., and FEICHTINGER, G., On the Optimality of Limit Cycles in Dynamic Economic Systems, Journal of Economics, Vol. 53, pp. 31-50, 1991.

10. WIRE, F., Cyclical Strategies in Two-Dimensional Optimal Control Problems: Necessary Conditions and Existence, Annals of Operations Research, Vol. 37, pp. 345-356, 1992.

l l. FEICHTINGER, G., NOVAK, A., and WIRE, F., Limit Cycles in Intertemporal Adjustment Models: Theory and Applications, Research Report, Department of Operations Research, Technische Universit/it Wien, 1992.

12. HASSARD, B. D., KAZARINOFF, N. D., and WAN, Y. H., Theory and Applications of Hopf Bifurcation, Mathematical Society, Lecture Notes, London, England, 1981.

13. ASCHER, U., CHRISTIANSEN, J., and RUSELL, R. D., A Collocation Solver for Mixed-Order Systems of Boundary-Value Problems, Mathematics of Computa- tion, Vol. 33, pp. 659-679, 1978.

14. STEINDL, A., COLSYS, ein Kollokationsvelfahren zur L6sung yon Randwert- Probtemen bein Systemen Gew~hnlicher Differentialgleichungen, Master's Thesis, Technische Universit~it Wien, t981.

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Cycles of fear: Periodic bloodsucking rates for vampires