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Broken Model

Title text: In addition to eating foxes, rabbits can eat grass. The grass also eats foxes. Our equations chart the contours of Fox Hell.

Explanation

This explanation may be incomplete or incorrect: Created by a KILLER RABBIT. Do NOT delete this tag too soon. Otherwise, the KILLER RABBIT may come for you.

This comic shows Miss Lenhart explaining a mathematical model of a predator-prey relationship. The model has the terms swapped, showing that 400 rabbits are preying on 20 foxes. The teacher realizes this mistake and says "If this were an ecology class, I would have to fix that." The caption notes that mathematical models can provide a glimpse into a terrifying alternate universe.

The equations start with N_rabbits(0)=400 and N_foxes(0)=20, the number of rabbits and foxes at time 0, followed by what looks like the Lotka–Volterra equations, a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact. One of the pair of equations describes the number of prey over time, the other the number of predators over time, differing only by a negative sign (and coefficients). It is easy to mix up which equation describes which species, leading to the inverted predator-prey relation described.

If this was indeed the case the rabbits would soon render the foxes extinct. The reason this doesn't happen in reality for rabbits is that they outnumber the foxes (20 foxes vs 400 rabbits) and thus can survive being preyed upon. Often the predator takes the old and sick animals first, thus keeping the rest of the animals more healthy. But following the math of the wrong formula would soon lead to zero foxes. As per the title text, the rabbits could survive without the foxes to prey on, since they still eat grass.

The comic plays on the idea that even seemingly simple mathematical models can have unexpected and potentially horrifying consequences, especially if they are flawed or inaccurate. This is a common theme in xkcd, which often uses humor to explore complex scientific and philosophical concepts.

The title text is a humorous addition to the comic, playing on the absurd implications of the flawed mathematical model. It suggests that the model is so wrong that it creates a bizarre, terrifying "Fox Hell" where rabbits and grass both prey on foxes, leading to a chaotic and ultimately fatal situation for the foxes. This is an example of xkcd's signature dark humor, where even seemingly harmless concepts like mathematical models can have darkly comical consequences. The title text extends the absurdity of the comic's premise, taking the flawed model to its most extreme and unsettling conclusion.

The title text also plays on the real world situation where it seems like foxes actually do eat grass, although maybe not as food, but for other health reasons. So in our world the grass mainly gets eaten by rabbits but also some of it by foxes. In Miss Lenhart's model the foxes is both eaten by grass, and by rabbits, who then also eats grass, meaning the rabbits are like foxes and the foxes are like grass in our world. The grass would then be more like rabbits. But not quite like our world since it seems the rabbits actually hunt the foxes (grass) for food. Not like foxes eat grass in our world.

Transcript

[Miss Lenhart is holding a pointer to a whiteboard, indicating the last part of the last line of text.]

Miss Lenhart: Hmm, looks like I accidentally swapped the predation terms.

Miss Lenhart: If this were an ecology class, I would have to fix that.

Miss Lenhart: Unfortunately for those 20 poor foxes, this is calculus, and the math says these 400 rabbits are hungry for meat.

[There are three lines on the white board, where the '...' parts in the bottom line is illegible:]

N_rabbits(0) = 400

N_foxes(0) = 20

dN₀/dt = ... dN_.../... = ...

[Caption below the panel:]

Every broken mathematical model is just a glimpse into a terrifying alternate universe.

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