Modelling warfare between vampires and humans: who wins?

Abstract

We adapt the Lanchester combat model to represent conflict between vampires and humans. It is assumed that vampires attack humans during the hours of darkness whilst humans attack vampires during the hours of light. The right-hand side of the differential equation model therefore depends upon the hour of the day. A key insight is that to answer the question ‘who wins’ it is not required to ‘stitch together’ the solutions to the differential equation over many days. Rather, the number of combatants surviving at the end of each day can be cast as a difference equation in terms of the numbers surviving at the end of the previous day. Two models are investigated and from the solutions the boundary delimiting the parameter regions of victory and defeat is found. Where appropriate we introduce discussion points, both mathematical and modelling. The mathematical techniques used include finding the eigenvalues and eigenvectors of a matrix, solving linear differential and discrete systems, and considering the physical meaning of the model and questioning its assumptions. Thus a variety of problem-solving skills are developed within the context of a model where students can question the underlying assumptions and propose, and investigate, models of their own.