https://www.explainxkcd.com/wiki/api.php?action=feedcontributions&feedformat=atom&user=47.151.234.78explain xkcd - User contributions [en]2026-04-18T23:31:40ZUser contributionsMediaWiki 1.30.0https://www.explainxkcd.com/wiki/index.php?title=2945:_Broken_Model&diff=4105752945: Broken Model2026-04-17T20:48:44Z<p>47.151.234.78: /* Explanation */ fixed wording</p>
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<div>{{comic<br />
| number = 2945<br />
| date = June 12, 2024<br />
| title = Broken Model<br />
| image = broken_model_2x.png<br />
| imagesize = 278x448px<br />
| noexpand = true<br />
| titletext = In addition to eating foxes, rabbits can eat grass. The grass also eats foxes. Our equations chart the contours of Fox Hell.<br />
}}<br />
<br />
==Explanation==<br />
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." Instead of fixing the model, though, she instead begins to calculate using this flawed model, and notes that this model implies that rabbits are carnivores.<br />
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The equations start with N<sub>rabbits</sub>(0)=400 and N<sub>foxes</sub>(0)=20, the number of rabbits and foxes at time 0, followed by what looks like the {{w|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.<br />
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If this was indeed the case the rabbits would likely soon render the foxes extinct. They might not, if each rabbit requires to eat a very small amount of fox, and they hunt in packs, so that a single fox feeds many rabbits, but it would need to be a very small amount, very infrequently. The reason this doesn't happen in reality for rabbits is that they outnumber the foxes (20 foxes vs 400 rabbits) and thus enough of them can survive being preyed upon to maintain the species. 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 (assuming that there is not some specific nutrition requirement that is only fulfilled by the foxes). However, this reality is terrifying for the foxes, because they are rendered as prey.<br />
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The title text extends the joke by looking at an even more flawed model that incorporates grass. In the real world, rabbits eat grass and foxes do not directly interact with grass. In this model, the fox-eating rabbits also continue to eat grass and foxes do interact with grass. Even more unfortunately for them, this is in the form of the fox eating grass, creating a "Fox Hell".<br />
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As a matter of fact, grass pulls nutrients from air and soil and synthesizes its energy through [https://soandmo.com/blogs/blog/where-does-the-grass-plant-get-its-energy-from photosynthesis], and may use foxes that have already died from other causes as fertilizer. Foxes do occasionally [https://www.wildlifeonline.me.uk/animals/article/red-fox-diet-what-foxes-eat eat grass], although not as food, but for other health reasons.<br />
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==Transcript==<br />
:[Miss Lenhart is holding a pointer to a whiteboard, indicating the last part of the last line of text.]<br />
:Miss Lenhart: Hmm, looks like I accidentally swapped the predation terms. <br />
:Miss Lenhart: If this were an ecology class, I would have to fix that. <br />
:Miss Lenhart: Unfortunately for those 20 poor foxes, this is calculus, and the math says these 400 rabbits are hungry for meat.<br />
<br />
:[There are three lines on the white board, where the '...' ellipses in the bottom line are illegible:]<br />
:N<sub>rabbits</sub>(0) = 400<br />
:N<sub>foxes</sub>(0) = 20<br />
:dN<sub>0</sub>/dt = ... dN<sub>...</sub>/... = ... <br />
<br />
:[Caption below the panel:]<br />
:Every broken mathematical model is just a glimpse into a terrifying alternate universe.<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics with lowercase text]]<br />
[[Category:Math]]<br />
[[Category:Biology]]<br />
[[Category:Animals]]</div>47.151.234.78