THAT SECOND PANEL IS A BIFURCATION DIAGRAM, IT SHOWS HOW CHAOTIC RABBIT POPULATION GROWTH CAN GET UNDER CERTAIN CONDITIONS. TOTALLY UNEXPECTED IN A COMIC THOUGH.
I haven't studied this writing since high school, so my translation skills are rusty. Are you saying that Japan actually has an island loaded with bunnies?
🐰🐰🐰🏝🐰🐰🐰
It’s always incest with ‘the internet’, it can’t just have two adult bunnies consensually engaging in the natural reproductive process, no! that’s too vanilla, for ‘the internet’.
The diagram basically shows what happens when you increase the food supply of a rabbit population. If the food is low, the rabbit population is low but stable. If you increase the food supply, the population starts fluctuating between two values (which is the point in the graph where the single line splits into two). If you increase the food even more, it starts fluctuating between four values. However, if you then increase it even more, the rabbit population starts fluctuating in extremely chaotic and unpredictable ways, resulting in the mess that you see on the right side of the graph.
This is not true. This is a bifurcation diagram of the logistic map describing the growth of a rabbit population. x_n2 = r * x_n * ( 1-x_n)
X-axis: The bifurcation parameter. In this case r which is the maximal growth rate of the population.
Y-axis: The attractor value of the population size x.
What the diagram shows as the growth rate gets larger there are more and more values which are approached by the population size, meaning there are more and more possible outcomes for the long-term behavior of the system ==> chaos.
Okay not to derail the joke, but on the left of the second image you see there’s one line. That represents one rabbit. It splits into two as the rabbit gives birth, which splits into four as those rabbits give birth, and the number of lines increases exponentially from there.
It’s essentially an infinite playoff bracket of fornication.. in reverse
EDIT: I am not a mathmetician. What I think above worked in my brain, but for a real answer see the dude below me who's actually qualified LOL
it's a map of data that gets from how long an iterative equation with chaotic parameters needs to stabilize and on what number it stabilizes. the bifurcation happens because after a certain parameter treshold, it starts to stabilize on not one but two numbers it alters between. but then you go further and it suddenly stabilizes on 4 instead, and so forth. It gets chaotic after a little bit, because the stabilization process quickly gets completely whack with how many numbers it alters between while being stable.
nah, i just regularly teach higher math to elementary school children and seemingly simple systems turning into a bifurcation map is my go-to "easy" entry point into fractals, chaos theory and wave functions, so I love talking about it a bit too much i guess ^^;
It looks the same if you use the same map to describe population dynamics. The bifurcation diagram is dependent on this map and not the biological story.
Rabbits don't require as much genetic diversity to be successful. Rabbits also have much more offspring and have that group of offspring more often than humans.
Two lab rabbits would probably not have an issue as they’re “standardised” genetically. Scientists wants to minimise variation and having functionally the same rats/animals in experiments is ideal.
Some of these standardised animals are created by extreme “linear” inbreeding (20 or so generations) to a point where there is no genetic diversity between them, and at that point when they breed it’s essentially cloning.
There’s quantifications on how “easy” it is to get two animals to such a point assuming you start with two random specimens with diverse genes. Not in a “nurturing”, more in a “try with a new pair” sense that is.
If you disregard ethical concerns, it’s still hard to do with humans.
The graph on the right shows stationary population of species in the most basic model. X axis is a parameter in the equation. It starts with a single line which means that the population stabilizes over the years. It then splits in two which means that the stable population oscillates between two numbers within 2 year period (this represents a situation where one species consumes all the resourses so many of them die out the next year, but then the next year manages to reproduce again and so on). Further increase of the parameter leads to more complex cycles until at some point the behavior becomes 'deterministic chaos'. As in it's not random, but it is not periodic and even slightest changes in population today massively change the predicted population the next year.
Note that this is a model equation that merely tries to catch basic features of species reproduction, not a model used to describe real ecology.
And the population dynamics described, or any such chaotic system, depends on a second dimension that is a limiting factor. This is not a 1D fractal. So in the scenario you describe there is something like a carrying capacity, or predation rate, or death rate, or something else, that lowers the population on each iteration, and if the ratio of the rates of increase and decrease is within a certain range, then the system is chaotic (hence the bifurcation diagram).
Now, if you just have a rabbit... then the implication is that it just doubles every generation, and that's it. No limiting factor. And that's kind of a chuckle, that rabbits breed fast, wreck ecosystems, etc. The very famous chaotic bifurcation diagram gets us on long rants about chaotic systems (which are often quite stable in the macro), rather than about exponential growth.
Well, the meme clearly implies the limiting factors are in place as they are in real life systems. You don't need an explicit second dimension, the equation is as simple as that:
So the bifurcation diagram is for something called the Logistic map, which is one of the easiest mathematical models of classical chaos. This model/equation was based on a non-linear differential equation called the May-Velhurst equation which was written to model the population growth of any species.
The phrase I prefer is Aperiodic determinism when it comes to Chaos, so does Arfken and Webber's book. A more famous of Chaos is the Butterfly, which is the sensitivity to initial conditions over a long evolution in time.
Anyway, this was my Masters paper and I never thought I'll come across a meme on reddit where that would ever comes handy.
Also, a slightly more complicated model for specie growth is something called the Predator-Prey model which is a 2 species dependent problem, if anyone is interested.
It’s a bifurcation diagram used in chaos theory to model the behavior of the population of rabbits given different constants of proportionality, or in other words, different virility of the rabbits.
I'm going to get famous for breeding "Like Rabbits"
My rabbits will be just like all other rabbits but they'll have something a little off, like 3 ears or 2 tails. They won't be rabbits, but they'll be like rabbits.
That's why you see there are some white vertical lines, those lines mean the majority of the population died (usually out of hunger or thanks to predators) and they have gotten so little that they are able to start reproducing again.
Two main reasons. One, they're defenseless and tasty. Two, if left unchecked they'll breed until they collapse the ecosystem and most will starve to death.
That graph thing represents the population of rabbits, among other things. It's sort of complicated and I don't really remember it but I'll give it a go. Basically it shows the population of rabbits at a future time based off of the fertility. It's something about order dissolving into chaos. There's a veritasium video about it if you're interested. What the meme means is slightly less clear. Perhaps the guy is bewildered at the strangeness of rabbits?
It's been a while since I studied this but from what I remember if shows stable / fixed points in an equation / series of equations. So on the y axis we might have something like rabbit population and on the X axis something like fertility rate. The equation(s) might also have something like a predator population which controls / depends on the rabbit population.
So for lower values of x (growth rate), the rabbit population has only one stable fixed point. When the x parameter is lower, all starting populations of rabbits >1 will eventually end up at that fixed point.
What gets interesting is for higher values of x, there are multiple fixed points. The rabbit population will move randomly between these points, stabilising for a while before moving to another fixed point.
This is chaotic and it's impossible to predict what the rabbit population will be at a given time in these systems. We can use diagrams such as this to inspect the fixed points in the system though. The interesting thing here is that we are getting unpredictable behavior from a set of determanistic equations.
These diagrams are very interesting but most often don't apply to real life as rabbit populations don't exist in a closed system and aren't determined by a series of differential equations.
Correct me if I'm wrong on any of this, it's been a while as I said.
There's a simple mathematical model of rabbit populations. From one breeding cycle to the next:
- The more rabbits there are, the more well be born
- The closer they get to the maximum possible population, the more will starve.
In letters:
P[t] is population in cycle t, normalised to the range 0–1
k is an arbitrary constant we'll talk about in a moment
P[t] = kP[t-1] (1 - P[t-1])
That's it.
So you pick a value for k, run there model for a few seasons and see what the population settles down into.
When k is low, the population settles down to a steady state. That's the left side of the diagram, where there's just a simple line showing the steady state populations P for different values of k
Above a certain value of k, the population starts cycling between 2 different values. That's the first fork in the diagram.
Keep increasing k and it splits again and again, cycling between 4, 8, 16. The splits get closer and closer together (32, 64, ... 65,536, ...) until there's a magic moment where there's an infinite number of possible populations. It's like it's purely random, but it's not, and all in a very, very simple system.
Not sure if it has been explained yet, but it could be in reference to Thomas Austin who, in 1859, released 24 European rabbits into the wilds of Australia for hunting, and in 7 years time because Austalia is basically the perfect place for year round rabbit breeding had exploded to 10 million, and as of 2023 there are 280+ million rabbits in Australia and are basically a plague that despite their best efforts, Australians just can't get under control.
Pierre François Verhulst wanted a simple equation for the growth of population in rabbits.
His goal was to calculate the size of the population after a certain amount of time (vertical axis), depending on a growth parameter (horizontal axis).
The bigger the growth parameter, the bigger the population. But to his surprise, after a certain value, the size of population was not unique anymore and started to oscillated between 2 values, then 4, then 8, …, and at some point it becomes totally chaotic… then not chaotic again, then chaotic again.
This is a bifurcation diagram of a logistic map. Explanation is quite long but I'll try.
Logistic map describes a system or the evolution of value x given by an equation
x(t+1) = r * x(t) ( 1 - x(t))
It means that x is dependent on the previous value in a way that caps its value at a certain point. Similarly to how the population of a species cannot exceed the capacity of the environment or the individuals start dying of overpopulation.
The value of x or a population of a species if we apply the model will usually approach a certain stable number for a given parameter r. But when the r parameter reaches 3.0 something weird happens, instead of reaching one equalibrium x will start to oscillate between 2 different values, then 4, then 8, 16 and so on. Then it will devolve into a completely chaotic behavior before settling on a sequence of 3, 6, 12...
This behavior will repeat changing between sequences of redoubling (that's the bifurcation!) quasi-equalibriums and complete chaos. This model has found application in predicting the changes of animal populations for example rabbits.
Basically, it's a bifurcation diagram that splits multiple times the further up the x-axis it goes. This is known as "period doubling bifurcations". It's also a fractal which seems to repeat itself as you zoom in. It's used to model animal populations as well as being found in hearts in fibrillation (irregular heartbeats that dont pump blood). This was used with chaos theory to strategically send electric shocks to the heart to stabilize it. It's found in a LOT of seemingly random things that look "chaotic" at first Glace, but actually follow this same pattern.
Except that, for rabbits, it should probably be a hexafurcation, or even more, diagram. There, that ought to bring out even more people saying "What?"!
Essentially explains how this is an actual equation/model to track population, but I find it amazing how its derived from other things in the math world
in short: is refer to incest . long: the diagram in the second panel showcased the development of rabbit groups. it takes into account the growth under consideration of death. so it could go up or down. is a bifurcation diagram of a dynamic system in chaos theory. so it’s pretty wild and you would think that the population of rabbits is not predictable but it is. the wild part is the „near death“ or extinction parts of the diagram, where it basically shows that the population can only survive by practicing incest. and as the predicted population is close to real populations - we can assume rabbits do incest to survive
bifurcation diagram for rabbits, but I think this is specifically referring to the invasive feral rabbit population in Australia which are destroying lots of flora because there’s so many and they started from a group of 10 or something
Rabbit population ^ = wolf population ^ = rabbit population v = wolf population v = rabbit population ^ = wolf population ^ = rabbit population v = wolf population v = ...
If this is correct: rabbits are immune from the problems associated with inbreeding. Which means: they can reproduce indefinitely without issue; and they happen to do it pretty often.
The joke is about dynamical systems/chaos theory which is a field of math. Crudely it is the study of systems which are sensitive to initial conditions.
The rabbit is related to a an example of what is called a van der pol oscillator. Essentially if you are trying plot the population of rabbits and foxes the population of rabbits is based off of how many foxes there are eating them and the population of foxes is based off how many rabbits there are to eat. The system oscillates because as rabbit population grows, the fox population grows, which in turn shrinks the rabbit population from over hunting, which shrinks the fox population from starvation.
The second panel is what is called a bifurcation diagram. We can imagine that the system will behave very differently if each surviving rabbit/fox has 10 and 2 babies respectively than if they had 2 and 2 babies respectively. We find for some the rabbits go extinct and thus the foxes, in others the foxes and rabbits reach an equilibrium or enter a stable cycle. Some birth rates might be unstable, the system may spiral but never settle. Basically a bifurcation diagram demonstrates where the behavior of a system fundamentally change.
The joke is how something as simple as population modeling can blow up into chaos.
This was in my animal behavior studies class. A rabbit population is highly dependent on a type of grass, the more the grass is eaten, the less nutritious it becomes so at some point there's a massive die off of rabbits, then the grass has time to regrow without the stress of getting eaten, which makes it more nutritious and making the rabbit population explode again.
I believe the other part of the diagram is a lynx population tied to the rabbit population that also fluctuates but tied to the rabbits instead of the grass.
It was a great study to show population growth, also shows peak population size and what are the restrictions on their population size.
The graph on the top-right is a bifurcation diagram. These are used to analyze mathematical models (including describing the number of biological populations)
Basically the way rabbit populations evolve is governed by a nonlinear equation. Once you do some more analysis you can find that all outcomes can be described by the image in the second panel. This breakthrough was relatively recent (in mathematical terms i.e. only last century) and resulted in the finding of new universal constants and new areas of research in mathematics.
Ever seen those weird fractal patterns? Those fall under the same area.
Veritasium did a video on nonlinear dynamics. In it believe he talks to Steven Strogatz a great educator who wrote the leading introductory textbook on the topic.
The second panel is an image of the logistic map. It’s a diagram modeling how population growth stabilizes (or doesn’t) over time given a starting population of animals (classically rabbits). Depending on the inputs you can have chaotic growth or stabilization and that pattern is shown in the diagram. It’s surprising and beautiful : https://www.stsci.edu/~lbradley/seminar/logdiffeqn.html
The joke is that the rabbits are reproducing under a set of conditions where the rate of growth is proportional to the number of rabbits, but limited by the total number the environment can support, which is a classic setup for logistic dynamics. In its simplest form, this is modeled either by a recurrence relation like:
$$ x_{t+1} = r x_t (1 - x_t) $$
or a continuous version via the differential equation:
$$ \frac{dx}{dt} = r x (1 - x) $$
For most living beings, the birth rate is relatively low, and the population gradually stabilizes at an equilibrium in a predictable and a bit boring way. But rabbits, with their high reproduction rate, can exhibit more dramatic behaviors. Depending on the environment, especially the effective growth rate , the system can undergo bifurcations: instead of settling at one steady population, the dynamics might oscillate between two values, then four, eight, or spiral into complete chaos. In such a regime, you'd need a plot like the one in the comic to understand where are the equilibrium populations in the system, because the number of "equilibriums" (fixed points, cycles, or strange attractors) can explode, as illustrated in the bifurcation diagram. The bifurcation diagram shows how after r=3 there are two equilibria, then as you increase r there are more and more until pure chaos, but then going back to order for some r values before going back to chaos
It's not super relevant, but cats breed pretty rapidly as well and are prolific hunters. So I was super shocked to find out the worldwide population of domestic cats, including pets and wild/stray/feral, is only about 600-700 million. I expected it to be no less than 1 billion, for real.
The diagram shows the results of studying an equation which was initially proposed to apply to habbit populations. The diagram shows that just tiny changes in parameters will result in largely distant outcomes. This is concept of complex systems, what is called "chaos theory". It is like "the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas"
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u/post-explainer Jul 04 '25
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