Humans vs. Vampires Apocalypse Calculator
This calculator simulates a vampire apocalypse using a predator-prey model. Adjust the initial conditions and interaction rates to see who would win in a battle for survival: humans, vampires, or the slayers who hunt them.
Simulation Parameters
The Mathematics of Monsters: Modeling the End of the World
While often relegated to horror movies and Gothic novels, the concept of a vampire apocalypse poses a fascinating mathematical problem. It serves as a perfect introduction to epidemiological modeling—the same science used to track real-world outbreaks like influenza or COVID-19. By treating vampirism as an infectious disease where the "infected" (vampires) consume the "susceptible" (humans), we can use differential equations to predict the fate of humanity.
The Lotka-Volterra Model
This calculator relies on a variation of the Lotka-Volterra equations, also known as the predator-prey equations. Developed independently by Alfred J. Lotka (1925) and Vito Volterra (1926), these equations describe the dynamics of biological systems in which two species interact. In our case:
- The Prey (Humans): Their population grows naturally but decreases when "eaten" (bitten/turned) by predators.
- The Predators (Vampires): Their population only grows by consuming prey. If they eat all the humans, they starve and die out.
Why Do Vampires usually Win in Movies?
In fiction, vampires often overwhelm humanity quickly. Mathematically, this is due to the reproductive ratio ($R_0$). If a single vampire creates just one new vampire per month, the population grows exponentially ($2, 4, 8, 16...$). Within a few years, the vampire population would exceed the human population. This calculator allows you to test the "Buffy Factor"—the introduction of Slayers. By removing vampires from the equation at a rate faster than they can reproduce, slayers act as an "antiviral" agent, potentially stabilizing the system.
Real-World Applications
Believe it or not, scientists have published actual papers on this topic. In 2009, researchers from the University of Oxford and the University of Ottawa published "Mathematical Models of Vampire Influence," exploring scenarios based on different vampire mythologies (e.g., Dracula vs. Twilight). Their conclusion? Without a highly efficient staking rate (slayers), humans are wiped out incredibly fast.
Frequently Asked Questions (FAQ)
We use a logarithmic scale because the difference between the human population (Billions) and the starting vampire population (1) is massive. On a standard linear graph, the vampire line would be invisible at the bottom until it was too late. The log scale allows you to see the growth trends of both populations simultaneously.
This represents the efficiency of vampire hunters. It is the percentage of the vampire population killed by slayers each month. In ecological terms, this is the "predator death rate" caused by external forces. High staking rates are the only way to prevent a total human collapse in this model.
Mathematically, it is very difficult. If vampires are immortal (do not die naturally) and must feed on humans to create new vampires, the human population will eventually crash unless the vampire birth rate is zero. Slayers provide the necessary "mortality" variable to the vampire population.
In this simplified model, vampires do not die of starvation; they only die if staked. However, if the human population drops to zero, the vampire growth rate halts. Advanced models might include a "decay" rate for vampires when $H$ approaches zero.