# Humans vs Vampires Calculator (Population Model)

URL: https://speedycalculator.com/calculators/other/humans-vs-vampires/
Last updated: 2026-10-30

> The humans vs vampires calculator simulates how a population of humans, vampires and vampire hunters changes over days, months or centuries, using coupled differential equations solved with the Runge–Kutta method, and tells you whether – and when – humanity falls.

## Inputs

- Scenario
- Initial humans
- Annual human population growth (%)
- Initial vampires
- Chance a bitten human becomes a vampire (%)
- Vampire feeding (aggression toward humans)
- Vampires restrain themselves (avoid killing, keep ≥ 1,000 humans per vampire)
- Initial hunters
- New hunters recruited per hunter per year (when the threat is high)
- Vampires destroyed per hunter per year (when vampires are easy to find)
- Attacks on hunters per vampire per year
- Chance a bitten hunter becomes a vampire (%)
- Hunters retiring or dying of other causes (% per year)
- Limit the hunters' organisation to 0.1% of humans
- Simulation length
- Time unit
- Chart scale

## Results

- Outcome
- Populations over time
- Time until humanity is wiped out
- Humans at the end
- Vampires at the end
- Hunters at the end
- Peak number of vampires
- Peak reached after
- Humans per vampire at the end

## Formula

```
dH/dt = r·H − bites·(c + k) − recruits · dV/dt = bites·c + q·A − K · dS/dt = recruits − A − δ·S · bites = f·V·H/(H + S + V), K = κ·S·V²/(V² + 300²), A = α·V·S/(S + V + 1), recruits = ρ·S·V/(V + 300)
```

## Explanation

Vampire stories hide a neat piece of population biology. Vampires are predators, humans are their prey, and – unlike wolves and rabbits – every successful attack can create a new predator. Without anything to stop them, vampires grow exponentially until the prey runs out. Add vampire hunters who prey on the vampires and you get a three-level food chain, the same family of models (Lotka–Volterra) that ecologists use for lynx and hares.

This calculator lets you play with that system. Pick a ready-made scenario – a single vampire with nobody to stop it, a discreet coven that keeps its numbers low, hunters and vampires circling each other in slowly fading cycles, hunters who quickly reach a stalemate, or the dusk of humanity when hunters are overwhelmed – or build your own world with custom numbers.

The model tracks humans, vampires and hunters together and solves the equations with the fourth-order Runge–Kutta method. The chart (linear or logarithmic) shows all three populations, and the results report the outcome, the peak number of vampires and when humanity disappears, if it does.

### How to use the Humans vs Vampires Calculator

- **Scenario**
- **Initial humans**
- **Annual human population growth**
- **Initial vampires**
- **Chance a bitten human becomes a vampire**
- **Vampire feeding (aggression toward humans)**
- **Vampires restrain themselves (avoid killing, keep ≥ 1,000 humans per vampire)**
- **Initial hunters**
- **New hunters recruited per hunter per year (when the threat is high)**
- **Vampires destroyed per hunter per year (when vampires are easy to find)**
- **Attacks on hunters per vampire per year**
- **Chance a bitten hunter becomes a vampire**
- **Hunters retiring or dying of other causes**
- **Limit the hunters' organisation to 0.1% of humans**
- **Simulation length**
- **Time unit**
- **Chart scale**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Each day vampires feed f times a year per vampire, spread over humans, hunters and fellow vampires. A bitten human turns with probability c, otherwise dies (or, for a restrained coven, usually survives). Hunters destroy vampires but find them hard to track when vampires are rare, recruit new hunters while the threat is high, retire at a steady rate and can themselves be bitten and turned.

### Worked example

In the lone-predator scenario one vampire feeds 12 times a year and half of the victims turn. Each vampire then creates about 6 new vampires a year, so the number grows roughly like e6t while humans are plentiful. Reaching one million takes about ln(1,000,000) ÷ 6 ≈ 2.3 years, and with the feeding slowing as humans become scarce the simulation shows humanity wiped out after about 3 years and 4 months – with roughly 510,000 vampires left and nothing to eat.

### What the scenarios show

ScenarioTypical outcome over 40 yearsLone predatorHumans gone after about 3 years 4 monthsDiscreet covenStable: about 1,000 humans per vampireFading cyclesVampires peak near 440 after about 4 years, then swing in fading cycles towards about 290Quick stalemateHunters hold vampires near 110 for decadesDusk of humanityHumans gone after about 1 year 3 months

## Worked example

Inputs — Scenario: Lone predator – one vampire, nobody to stop it; Simulation length: 5; Time unit: Years; Chart scale: Linear.

Results — **Outcome = Humanity is wiped out after 3 years and 4.1 months.**; Time until humanity is wiped out = 3 years and 4.1 months; Humans at the end = 0; Vampires at the end = 509,572; Hunters at the end = 0; Peak number of vampires = 509,572.

## Key terms

- **Lotka–Volterra model**: Coupled equations for predator and prey populations that can cycle.
- **Runge–Kutta method**: A numerical method that advances differential equations in small, accurate steps.
- **Refuge effect**: Predators struggle to find prey when prey is scarce, which stabilises the system.
- **Carrying capacity**: The largest population an environment (here, the human food supply) can sustain.

## FAQ

**How fast would vampires take over the world?**
With one vampire who feeds once a month and turns half of its victims, the vampire population doubles roughly every six weeks; a city of one million is gone in about 3 years and 4 months in this model.

**Why can't vampires exist mathematically?**
The classic argument (Efthimiou and Gandhi, 2007) shows that unchecked vampires grow exponentially and would have run out of humans within a few years – stories need restraint or hunters to stay plausible.

**What makes the populations oscillate?**
When hunters are many, vampires decline; with fewer vampires, recruitment of hunters drops and they retire, letting the vampires recover. That feedback loop produces predator–prey cycles.

**Why do the cycles fade in some scenarios?**
Hunters find rare vampires hard to track (a "refuge" effect), which damps the cycles and leads to a stable balance.

**What does the logarithmic scale show?**
It spreads small and large numbers evenly, so 20 vampires and 1,000,000 humans fit on one chart.

**Is this a real epidemic model?**
It uses the same mathematics as predator–prey and SIR epidemic models, but the numbers are playful assumptions, not data.


## Sources

- [Lotka–Volterra equations – Wikipedia](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations)
- [Runge–Kutta methods – Wikipedia](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods)
- [Compartmental models in epidemiology – Wikipedia](https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology)