Modern Economy, 2012, 3, 826-831 http://dx.doi.org/10.4236/me.2012.37105 Published Online November 2012 (http://www.SciRP.org/journal/me) Intertemporal Model of Co-Existence of Two Rival Species: A Case of Vampires and Humans Co-Habitation Wadim Strielkowski1, Evgeny Lisin2, Emily Welkins3 1Institute of Economic Studies, Faculty of Social Sciences, Charles University in Prague, Prague, Czech Republic 2Moscow Power Engineering Institute (Technical University), Moscow, Russia 3University of Strasbourg, Strasbourg, France Email: [email protected], [email protected], [email protected] Received October 9, 2012; revised November 11, 2012; accepted November 20, 2012 ABSTRACT This article presents a serious analysis of a ridiculous subject. It provides a new approach to modeling intertemporal interactions between vampires and humans based popular fiction literature, comic books, films and TV series. It is mainly concerned with the following question: would peaceful co-existence of humans and vampires be possible from a scientific point of view? And if so, why do not humans encounter vampires more often. Mathematical modeling enables us to conclude that several popular culture sources outline the models describing plausible and peaceful vampire and human co-existence. Keywords: Supernatural Economics; Mathematical Modeling; Popular Culture; Predator-Prey System; Vampires 1. Introduction Carver [6] noted that Marx used the vampire metaphor at least three times in “Capital”. For example, in one of the Vampires as well as the economic significance of vam- cases Marx describes British industry as “vampire-like” pirism and optimal bloodsucking strategies (e.g. pre- which “could but live by sucking blood, and children’s venting the depletion of renewable human resources) blood too” [7]. made their way into the mainstream research literature Marx’s colleague and long-time sponsor Frederick becoming an inspiration for several academic papers (e.g. Engels also used the vampire metaphor in his writing. In [1-4]). one of his works entitled “The Condition of the Working The description of vampires, the man’s natural preda- Class in England”, Engels identified and blamed the tors, can be found in legends and folklore. The word “vampire property-holding class” as the source of all “vampire” itself is considered to originate in Hungarian social troubles [8]. language where it is spelled as “vampir”. The first myths Marx described vampires’ habits, their greediness and and legends about vampires probably existed at the dawn their lounging for blood with such vividness that it often of human history. In the 19th century, ancient Mesopota- crossed the boundaries of the mere metaphor. Although mian texts dating back to 4000 B.C. were translated into many researchers perceive Marx’s vampires as a meta- English revealing some mentioning of “seven spirits” phoric abstract of bourgeois bloodsuckers feeding on that are very much like the description of vampires as we working people (see [9,10]) Marx’s knowledge of vam- think of them today [5]. pires was very detailed. Frayling [11] states that in one The fact that vampires constituted a threat to humans particular case, when describing Wallachian peasants throughout the history of mankind (whether this threat performing forced labour for their boyars, Marx refered was real or imaginary) can be shown on the examples of to one specific “boyar” who was “drunk with victory” recent archaeological findings at ancient burial sites and who might had been no one but Wallachian prince where some human remains showed signs of being Vlad (called “The Impaler”)—or well-known Count Drac- staked, strapped or gagged with a stone, a typical way to ula from the eponymous book. This is very peculiar be- slay the vampire according to the superstitious beliefs. cause Bram Stoker’s “Dracula”, did not see the light of The vampire metaphor in economics can be found in the day until 1897, some 14 years after Marx’s death. the works of Karl Marx and his followers. For instance, This brings us to the point of posing a question: would Copyright © 2012 SciRes. ME W. STRIELKOWSKI ET AL. 827 it be possible for vampires to peacefully co-exist with coefficient of a lethal outcome for vampire-human inter- humans? Starting from Bram Stoker’s “Dracula”, the action for humans). theme of vampirism has been widely exploited by many Let us calibrate the parameters of this specific (vam- authors: Stephen King, Anne Rice, Stephenie Meyer, piric) case of predator-prey model. The calculation pe- Elizabeth Kostova or Charlaine Harris, just to name a riod is set at 1 year with a step of 5 days (t = 0 ··· 73). few. In addition to that, vampires often appear in comic The coefficient of human population growth k for the books and films and TV series based on these books (e.g. given period is very small and can be neglected, there- “Buffy the Vampire Slayer”). We reviewed some of the fore k = 0. The coefficient of lethal outcome for humans popular literature sources, comic books, and films on interacting with vampires can be calculated according to vampires and identified four types of scenarios describe- the scenario presented in the Stoker-King mode y0 t t ing vampire-human interactions. These scenarios were y0q , where y0 = 1, q = 2. The probability of a human used to draw models of vampire-human confrontation being turned into a vampire is very high, thence b = 1. using the predator-prey model specified above. Jonathan Harker and Abraham van Helsing could not be, by all means, considered efficient vampire slayers, there- 2. The Stoker Model fore we can put c = 0. The resulting model can be presented in a form of the Bram Stoker’s “Dracula” describe interactions between following Cauchy problem: vampires and humans in the following way: a vampire dx selects a human victim and gets into its proximity (it axy dt typically happens after dark and the vampire needs the victim to invite her/him in). Often the vampire does not dy axy require permission to enter the victim’s premises and dt (1) 9 attacks the sleeping victim [12,13]. The vampire bites the x01.6510 victim and drinks the victim’s blood, then returns to feed y 01 for 4 - 5 consecutive days, whereupon the victim dies, is buried and rises to become another vampire (unless a Due to the fact that the total sum of humans and vam- wooden stake is put through its heart). Vampires usually pires does not change in time (human population does need to feed every day, so more and more human beings not grow and humans gradually become vampires), the are constantly turned into vampires [14]. predator-prey model is diminished to a simple problem Assume the events described in “Dracula” were real. of an epidemic outbreak [16]. How would things evolve given the Stoker model dy- It can be assumed that for any moment t there holds an 9 namics described in both sources? Let us take 1897 as equality xtytxy 00, where x0 1.65 10 . The the starting point (i.e. the year Stoker’s novel was first system of differential equations can be presented in a form published). In 1897, the world population was about of a single differential equation: 1650 million people [15]. dy axt yt ayt y x yt , (2) The initial conditions of the Stoker model are the fol- dt 00 lowing: 1 vampire, 1650 million people, there are no with the initial condition yy01 0 . organized groups of vampire slayers. The date we choose This equation belongs to the class of logistic equations for the first vampire has little bearing on our argument (e.g. the Verhulst equation that describes the growth of and therefore can be set arbitrarily. The model can be population). Solving the Cauchy problem for this equa- presented in a form of a diagram (Diagram 1). tion yields: Where H denotes humans and V denotes vampires, H0 dy is the initial state of human population, v0 is the initial atd yx y y state of vampire population and the aHV describes an 00 interaction between a human and a vampire (with a as the ddyy yxat00 d yyxy00 lnyyxy ln 00 (3) lnyyxyayxtt0000000 ln y y0 ln ln ay00 x t t 0 yxy00 x 0 y y 0 eay00 x t t 0 Diagram 1. The stoker model. yxyx00 0 Copyright © 2012 SciRes. ME 828 W. STRIELKOWSKI ET AL. By solving the problem above we get the following growth of vampire population. Within the two months of equation: Dracula’s arrival to England, there would have been 4 thousand vampires in operation. The model analyzed in xy00 y0 ytt0 (4) this scenario is very similar to an epidemic outbreak yx e yxtt00 0 00 caused by a deadly virus (e.g. Ebola or SARS). Accord- The solution clearly shows that with passing time the ing to the Stoker model, vampires need just half a year to number of vampires grows and very soon there are no take up man’s place in nature. Therefore, under this sce- humans left: limt ytxy00. nario the co-existence of humans and vampires seems The solution to this problem is presented below (Fig- highly unrealistic. ure 1). It is clearly visible that the human population is drastically reduced by 80% by the 165th day from the 3. The Harris-Meyer-Kostova Model moment when the first vampire arrives. This means that Stephenie Meyer’s “Twilight series”, Charlaine Harris’s the human population reaches its critical value and prac- “Sookie Stockhouse (Southern Vampire) series”, “True tically becomes extinct (following the definitions of Blood” (TV series) and Elizabeth Kostova’s “The Histo- “Critically Endangered species” by the International Un- rian” show the world where vampires peacefully co-exist ion for Conservation of Nature) [17].