Predators and Prey - Numberphile

[@Skip_Sandwich_DX]

Math uses U and V to remind me how bad my handwriting is.

[@archivist17]

The reason I stopped using cursive...

[@____-pb1lg]

@@archivist17 bruh but in cursive you have the little curve on the v they don't look the same

[@jrdnajh]

My physics teacher always had us write U like ⨿ to make the difference clear.

[@NoisqueVoaProduction]

Wow,remembered my geometry class where the u,v and w were indistinguishable... By the teacher

[@Skip_Sandwich_DX]

@@NoisqueVoaProduction Now I'm taking actuarial exams where lower-case P, capital P, and Rho will be in the same equation representing different things.

[@dougthedonkey1805]

u = 0 v = 0 “This is a happy place for the population” I mean, you can’t be unhappy if you don’t exist...

[@archivist17]

Extremely hard to be content, too.

[@lubieplacki2772]

@@archivist17 Satisfaction is made out of lack of deprivation, it only exists through a negative, like cleaniness is lack of filth. We need first to feel the thorn before we can enjoy its removal. Asymmetry.

[@oldcowbb]

i can call this a trivial fact

[@anthonygrow1482]

R/im14andthisisdeep

[@wtspman]

For the grass, this would be a very happy place.

[@b.a.r.c.l.a.y9701]

all animals were harmed during the production of this video

[@anuppokhrel2357]

Only rabbits and foxes were harmed.

[@jasper3706]

Not the humans, I hope

[@magran17]

Call PETA. People Eating Tasty Animals.

[@KuraIthys]

@@magran17 I prefer EEDP - Everyone Eating Defenseless Plants

[@Triantalex]

false.

[@DiracComb.7585]

As a man who’s taking a differential equations course, I gotta say the timing of this video so very convenient 👍

[@williammedeiros2699]

Check 3blue1brown series on differential equations

[@messinberver4683]

I'm doing differentiation and integration rn

[@demenion3521]

i'd say you rather find this in some course on nonlinear dynamics and not so often in a differential equations course. but maybe in this case, the equations might actually have a nice solution (didn't try to solve it)

[@jpalacios117]

Summr school go brrrr

[@YourCrazyOverlord]

You brave soul. This is the very maths problem that made me drop out of engineering.

[@gametheory6948]

I’m surprised how he explains differential calculus very well and is knowledgeable, but when it comes to rabbits everything falls to pieces

[@turdferguson3400]

I love how bad he is with words! They're going to.. erm.. populate like rabbits 😆🤣

[@TomRocksMaths]

As great as it is, a maths degree does have it's limits...

[@abijo5052]

@@TomRocksMaths how has no one picked up on this great pun yet

[@rogerjohan9621]

@@TomRocksMaths Unless it is indeterminate...

[@rafaelsoto4899]

@@TomRocksMaths tongue and cheek? “Limits”?🤪

[@azhar07464]

Ohh! The fluid dynamics guy.

[@TomRocksMaths]

That's me!

[@Redditard]

@@TomRocksMaths I know right?

[@Redditard]

Yass

[@5hirtandtieler]

Tom Rocks Maths Got a new sub! 😊

[@brunoc.3669]

More than one, I guess the "change in the population of subscribers" is positive ;-)

[@AlbiinoBlacksheep]

5:18 "The change in u is equal to value of u." Peak wisdom.

[@edwinvlasics4047]

???? What's this supposed to mean

[@alexandermcclure6185]

@@edwinvlasics4047 They are pretending "u" is the word, "you."

[@TomRocksMaths]

Can confirm I'm still looking for a carnivorous rabbit... any help much appreciated.

[@hotdogskid]

I love he can stumble over words and still sound smart lol very relatable

[@samayahone3497]

i dont know about relatable lol, if i stumble over my words i very quickly stop sounding smart xD

[@hamiltonianpathondodecahed5236]

0:26 "u are going to represent the prey"

[@Albimar17]

WHAT DOES THE FOXXX SAYYY?

[@drenzine]

@@Albimar17 VVVVVVVVVVVVVVVVVVVVVVV

[@sohee7597]

I misheard Brady's reply as "why me?"

[@Triantalex]

??

[@chuckvanderbildt]

It's kind of shocking how badly my professor bungled this explanation, and by contrast, how clearly this concept is laid out here.

[@Mortiis558]

Yeah this is where I got lost in calculus, same issue, bad prof explanation.

[@cmuller1441]

The difference between a video and a real course in Math is proofs. Here he just says: it goes arround the happy (1,1) point without even proving that the curves are closed! Couldn't those be spirals converging to the happy point or growing bigger and bigger? He draws things that looks like circles. He didn't even prove that the loops are convex... And of course those loops can't look like circles near the axes. In particular near 0,0 the curve cannot go down like a circle would do or you would cross the x axis...

[@Mortiis558]

c muller I am in particular referring to his explanation of dv/dt. I got integrating just fine but when dv/dt was introduced and included in integration I just lost the path and couldnt follow anymore.

[@TomRocksMaths]

Sorry to hear that Chuck, but glad I could help out :)

[@connoryoung1993]

I did a couple mathematical biology units for my undergrad. Tom, you explained this much better in 17 minutes and conveyed so much more than my lecturers did in two semesters!

[@TomRocksMaths]

Thanks Connor!

[@iTeerRex]

One of the beauties of maths.. Making sense of the quality and the quantity of complexity with ease.

[@wobaguk]

Mathematician: "Lets look at a steady state where EVERYBODY's happy..." Rabbits getting eaten to maintain status quo: "Revolution!"

[@xevira]

Revolution... around the steady state. :D

[@aksela6912]

Basically, the steady state here is the train in Snowpiercer

[@kimon114]

There is always a communist.... well he did not factor in the lettuce now, did he?

[@jenm1]

@@kimon114 theres always a communist hater

[@kimon114]

@@jenm1 There are no "communist haters" just people who understand communism.

[@stevemarethyu3003]

I used to do a version of this with my fifth graders. It was for science/ecology, so we didn't get into the math, but it was a great way to show how connected everything in an ecosystem is.

[@hcblue]

The "everyone's happy when there's nothing" case (v=0, u=0) just feels existential 😂

[@QuestionEverythingButWHY]

“A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. Science advances one funeral at a time.” ― Max Planck

[@revimfadli4666]

BuT bUrDeN oF pRoOf

[@andywright8803]

I very clearly remember doing a version of this when I was 15, writing endless results on a piece of paper and graphing the results as I went with lined paper and a bit of imagination. I was fascinated and had to tear myself away to revise my O levels.

[@zh84]

I first came across this in 1983, on a free tape of sample BASIC programs supplied with my Sinclair ZX Spectrum. This was called "Evolution, or Foxes and Rabbits", and you could type in the starting values for the two populations and then watch as they evolved in time.

[@bachirblackers7299]

What i like about you most dr Rocks !!! Is your simple beautiful english and your smooth explanation . You are my favorite maths . Thanks prof .

[@TomRocksMaths]

[@jackjensen422]

I recall a computer game that I played at school in about 7th grade, where you set up how many of each sort of animal you want in a population and then run a simulation, and it displays a 15x15 or so grid of animal sprites appearing and disappearing as predator-prey interactions play out over generations... Does anybody recall that game?

[@amauryleblanc7979]

this is one of the coolest looking maths nerd I ever saw on this channel

[@giosanchez2714]

That introduction was hysterical! The whole video was entertaining, as always. So glad to see him on your channel again!

[@TomRocksMaths]

Thanks Gio - more to come soon!

[@souptime8635]

The intro gave me Monty Python vibes

[@roderickwhitehead]

Well, that's no ordinary rabbit. That's the most foul, cruel, and bad-tempered rodent you ever set eyes on.

[@laurihei]

That is an ex-rabbit. That rabbit is no more. It's pushing up daisies.

[@pdutube]

It *is* the rabbit!

[@nadya-catherineismail7804]

Thomas is quickly becoming one of my faves on numberphile, he's so precious

[@R_V_]

Call me reactionary or anything, but his tattoos and piercings make me feel quite uncomfortable. I prefer hearing to his videos than watching them. Of course, it's just a personal distaste.

[@laharl2k]

He looks too "fabulous" doesnt he?....

[@Toobula]

I'm an oldie. It's the piercings I don't like. But I know I'm from a different time.

[@R_V_]

@@Toobula I may not be an oldie, but in a way I feel from a different time too. I'd say "from better times", but of course your mileage may vary.

[@hawkmoths]

I struggle to understand what his appearance, specifically a few small holes and a few tattoos, have to do with his ability to teach a concept effectively and make it enjoyable. But that's just me.

[@EuphoricIntentions]

I wonder if there is anything to the notion that a fox isn’t being drawn well, but a rabbit is. Also, the graph I think in regards to time, should be a wave with undefined portions in a peak and trough that extend to Infinity so that the flow of time isn’t interrupted.

[@OlafDoschke]

Which poses the final question: Are foxes red because of the carotin in the carrots the rabbits eat?

[@NoisqueVoaProduction]

I thought the ultimate question was : What does the fox say?

[@yomanxy]

Woof or grrr?

[@jasper3706]

@@NoisqueVoaProduction *human screaming*

[@Tfin]

(For anyone taking this seriously, the answer is "no." Well, it starts with that, anyway. It then goes on about Bugs Bunny and actual rabbit diets for a bit.)

[@RomanQrr]

I just realized that from this explanation there are 3 types of starting points: points that are in the stable loop. points that lead to extinction, and points that lead to extinction of the foxes, while rabbits are booming.

[@Some.username.idk.0]

Yup (also, one stable point)

[@RomanQrr]

@@Some.username.idk.0 To be fair that's just a stable loop of a size 0.

[@Some.username.idk.0]

@@RomanQrr yeah, thats why "yup" was in the front of the sentence

[@fatsteve3120]

It might be worth looking at the Lotka-Volterra equations and how they precisely behave (as opposed to the somewhat simplistic drawing of the graphs presented in this video). As it turns out, unless there is an artificial, external extinction of either the predator population or the prey population, their populations can get infinitesimally close to zero and still recover!

[@Zolzulia]

his smile made this whole thing about eating rabbits and brutal law of nature, seem like casual goodnight story for kids

[@tsuchan]

Tom was amazing! The fox/rabbit field animation at 01:01 was confusing because it didn't make sense it until I realised that it was misleading. It didn't include the 'lag' Tom talked about.

[@TomRocksMaths]

Thanks - glad you enjoyed it :)

[@gigamungus5764]

16:33 literally can't see what he's talking about because of the end screen cards lol, it pops up at the exact time and place to completely obscure the video

[@audreka13]

ThatAwkwardGuy I think that’s intentional

[@onecommunistboi]

Well because thats another video! If you want to see what hes talking about click on said end card:D

[@aakifaslam6098]

On mobile you just drag down on the video a bit and the cards disappear

[@jon_j__]

You can use the developer tools (F12 in firefox) to delete the overlay.

[@cheeseboy8241]

@@onecommunistboi bless u

[@nickcarter4006]

“Rabbits eating foxes”??? RUNAWAY!! RUNAWAY

[@murrfeeling]

"My degree is in math, not zoology."

[@pierreabbat6157]

Rabbit: Let's switch roles for a while. Fox: Okay, I'm game.

[@robertwynkoop7112]

It’s just a rabbit...... I soiled my armor!......

[@fewwiggle]

@Nick -- 2020 in nutshell

[@DendrocnideMoroides]

@@murrfeeling but you have to know that "Rabbits eating foxes" is false

[@Kaesekuchen002]

Thanks for summarizing one of the first exercises I faced in Control Engineering

[@toopteeps6653]

The man with the Navier Stokes Equation tattoo!

[@definitelydan2711]

Yah he finally came with a new video I've been waiting for him 😂

[@TomRocksMaths]

Sure is :)

[@GenericAnimeBoy]

0:05 That's no ordinary rabbit! It's got huge--he can leap--LOOK AT THE BONES, MAN

[@SgtSupaman]

I'm glad they clarified at the end, because I was definitely wondering how they expected any amount of rabbits less than 2 to keep reproducing and also how populations dropping below 1 didn't result in extinction.

[@carlborgen]

Expected to hear lotka-volterra being mentioned 🙈

[@medore13]

Me too :o

[@xiaomarou9890]

When I saw “population of rabbits”, I immediately thought about the Feigenbaum-constant😅

[@himanshutahiliani1235]

Me too!!!!

[@sachiel197]

same

[@stulebackery1363]

yeah me as well

[@bachirblackers7299]

In fact still there is something to do with !!! Since it's about growth of two populations simultaneously in the same area ! Also we need to consider the nature is not only rabbit n fox ... There must be man n other creatures ... Plants .... Water ... Illness ... Death ... weakness ...

[@PhilBagels]

Yeah. I'm surprised he never mentioned the Logistic Function.

[@dougthedonkey1805]

It’s not rabbits eating foxes, it’s rabbit-eating-foxes

[@kevinfontanari]

Yeah, killer rabbits have started being a real problem recently...

[@nottmjas]

The solution would be to send for Brother Maynard and the Holy Hand Grenade of Antioch.

[@GiuocoPianissimo]

Not sure if this is a Re:Zero reference or a Monty Python reference.

[@gregvandenberg2859]

Run away!!!

[@praneet6230]

@@GiuocoPianissimo thinking the same thing lol

[@amauryleblanc7979]

2020 keeps getting weirder and of course, worse.

[@roseofthetree]

As someone LITERALLY making a game using this as the core mechanic, the timing of this video cannot be any better 😂

[@My-Say]

OMG he's so cute :)

[@archivist17]

Isn't he!

[@TomRocksMaths]

[@My-Say]

@@TomRocksMaths Oh wow, subscribed. :) I'm a little surprised you noticed my comment lol.

[@HasekuraIsuna]

_u_ as in _usagi_ (rabbit in Japanese) _v_ as in _vulpes_ (fox in Latin)

[@olmostgudinaf8100]

Because settling on one language is for the plebs.

[@nicolascavadini3570]

Exept japanese do not use latin alphabet

[@alexv3357]

@@nicolascavadini3570 Japanese uses romaji characters for many purposes

[@EastBurningRed]

u and v are the same letter in latin though

[@tsuchan]

@@nicolascavadini3570 Really Nicolas? So for example, find me 'JVCKenwood' in Japanese characters. That's a famous enough Japanese company.

[@AeroCraftAviation]

0:09 who caught the Wilhelm scream? 😂😂

[@avi12]

Wow, what a beautiful connection between rabbit-fox population modeling and sine-cosine functions!

[@skankhunt-zw6gg]

@00:48, intriguing! He starts a drawing by first drawing the back of the object!!

[@5hirtandtieler]

Really great, digestible video. I missed the nondimensionality at first, so was confused how a steady state was at [supposedly] (1 fox, 1 rabbit) but glad it was touched on and explained at the end :)

[@TomRocksMaths]

Thanks Tyler - glad you enjoyed it!

[@Johnnyb3g00d]

Wow, I seriously wish I had the chance to watch this video back in highschool, this makes so much more sense.

[@demenion3521]

when being confronted with such equations, i would start by looking at the nullclines (all the curves where one of the derivatives vanishes). this would give you the lines u=0, u=1, v=0 and v=1. and then you notice that the flow can only be perpendicular to the nullclines, so that you can kinda construct the circular behaviour. you still need to determine the direction of the flow at as least one point though (unless there are limiting cycles that separate the phase space). the only thing, you cannot determine that easily is the fact that you actually get circles here instead of spirals. that really requires linear stability analysis around the fixed points

[@greyed]

6:$5- "And everybody's happy!" Well, except the rabbits that are being eaten.

[@itisALWAYSR.A.]

Tom is so pure

[@darekgala4667]

Very elegant model which describes real life phenomena :) I wonder if this graph should look more like whirpool rather than circles? Populations will oscilate instead going towards equilibrium? :D

[@Quarky_]

There's a really nice book for stability analysis: Introduction to Dynamics by Ian Percival and Derek Richards

[@SlimThrull]

"So we have a rabbit and a fox." 0:52 "So we have a rabbit and an anteater."

[@jzieba0204]

I remember learning about it in biology class in primary school with the same graph in my textbook

[@boboblacksheep5003]

He is redefining how scientists should like.

[@gmurphy1028]

"You are going to be the prey >:D"

[@NoisqueVoaProduction]

*Brady's breathing intensifies*

[@felicityc]

being a fox i had a heart attack in the first five seconds...

[@rogerdearman]

On the family ZX spectrum 48k there was an included cassette with some test software included (including a breakout\arkenoid clone amongst others), One of the included programs was a fox abbits simulation. Memories

[@Cosmic910]

I saw you on Mike Boyd’s channel when he did the mathematics entrance exam to Oxford

[@5445tashi]

I remember doing this in uni! Such a nice way to remember what I used to do

[@businessguide6219]

This video is worth watching! More of this, please!

[@user-vn7ce5ig1z]

The predator/prey cycle where pop(PRED) lags pop(PREY) is a classic model. I first learned it from the Dewdney book _Armchair Universe._

[@George12String]

PhD student in Ecology here. The statement "you look at actual data of any predator prey situation and you see exactly this" is an incredible overstatement.

[@whatskraken3886]

this man is what i aspire to be

[@ProbablyTheBestUkuleleDadEver]

One the free bits of software that came with 16K Sinclair Spectrum was exactly this, you could set the initial state and got a graph produced on screen to show the populations, it was called Foxes and Rabbits.

[@xxMrPHDxx]

8:52 Now I'm curious about that circular thing. I want a video on that

[@TomRocksMaths]

I'm planning to cover the extra detail on my channel soon - so keep your eyes peeled :)

[@xxMrPHDxx]

@@TomRocksMaths Cool. I'm so excited

[@ryandean3162]

It's simply because any stable, repeating system of two variables in this way is either going to be constant, which would be the center point, which would make a straight line over time if you plotted it out, or it's going to vary in a regular fashion, which would be a sine wave, which means circular or at least ellipsoid. If you add in more variables, like the food source for the rabbits since they can easily devastate it if there's too many, or diseases like myxomatosis, or the fact that foxes have more prey than just rabbits, or a whole host of other real world considerations, you can get all sorts of weird lines and shapes that include things like a complete collapse of the feedback loop.

[@upsydaysy3042]

@@TomRocksMaths aaaaaaand subscribed! First time ever I get any maths whatsoever. You are on another level.

[@TomRocksMaths]

@@upsydaysy3042 Amazing - you're very welcome :)

[@archivist17]

More Tom please!

[@TomRocksMaths]

I'll be back don't worry!

[@rupen42]

This is an even better explanation of differencial equations than 3B1B's videos.

[@anthonylaviale3021]

They glossed over it, but it's not obvious the trajectories loop back on themselves. With slightly different equations you can get trajectories that spiral down towards the equilibrium point, which would mean the oscillations get smaller and smaller. You can also get trajectories that spiral out, with stronger and stronger oscillations. You can even get areas where you spiral in and others where you spiral out. Dynamic systems are really interesting and get many cool behaviours.

[@timseguine2]

He mentioned it in passing, but using u,v,w is very common for anything relating to differential equations

[@vancetilley7942]

The host/parasite relationship between honey bees and varroa mites follow this same pattern. As the bees build up their Spring numbers the varroa mites lag behind, only peaking after the bee’s numbers begin to decline in autumn/winter. It is this lag that creates high mite numbers as the bee numbers seasonally decrease leading to colony morbidity - often in winter. 🐝

[@YawnGod]

I like this primate's colorful display pattern. It signals to predators that he is not a tasty meal. He is a clever primate.

[@CptFUBAR]

The thumbnail at the end covered up the tattoo!!

[@azlandpilotcar4450]

Rabbits eating foxes mathematically would be much simpler, but a much shorter video.

[@DendrocnideMoroides]

Why ?

[@TheHereticAnthem20]

5:18 "The change in you is equal to the value of you" Philosophical

[@ajaisingh9129]

Love the animation at du/dt = u 😂😂😂

[@LaVieDeReine86]

Gorgeous, polite, funny and awesome at maths... my mother will definitely approve 😍😍😍

[@TomRocksMaths]

[@vladimir520]

I'm entering 11th grade, already kitted out with quite a lot of knowledge on mathematical analysis. Excited to learn it! (despite the pandemic, because of which I hope I'm taught analysis from home..)

[@romanlist2643]

What ARE baby foxes called though? 14:00

[@jpalacios117]

That animator is fire. Keep him

[@suvamranjit7491]

"Rabbits are eating foxes" Re:zero s2 watchers: Understandable have a great day

[@mrmojorisingii]

2:58 I think I've seen this meme before...

[@DeivisAteivis]

AhAh!!!

[@minewarz]

oh no

[@hwiatslgeord2887]

Piper perrie meme

[@thatoneguy9582]

I like how I didn't even have to click the timestamp to figure out exactly what you meant

[@englishmotherfucker1058]

that looks like someone threw all those foxes at the rabbit while stooping time

[@qualia765]

This gave me a question. What is the difference between a vector field and a phase plane? My guess is that a phase plane is always is a circle that could be stretched like that, while a vector field can be anything, but I'm not sure.

[@ultrozy]

Why there are exactly closed trajectories in the phase plane? There could be spirals, which arrows lead towards steady point (meaning it's asymptotically stable) or away from it (meaning its unstable)

[@mirceapopa1485]

Finally somebody is making Dynamical system theory and Fluidynamics entertaining

[@roseofthetree]

I learned about "daisy planet" yesterday, and then I get this video 😂

[@bachirblackers7299]

This is a better illustration of whats happening in the Mendelbrot Set . What if we add more elements in the chain ... With multiple predators n preys . Always a number looks like he's chasing his squared Nr same as squaroot ... Thanks Dr

[@rubengomes8258]

The term alpha*u*v in dv/dt represents more food available for the foxes. I don't understand why this term has the factor v. I thought it should be alpha*u. Explanation?

[@Theraot]

What does the fox say? 13:50

[@koraptd6085]

14:00 the irony is the fact that modern meaning of word "kids" derives from term for the goat offspring that is usually killed by foxes ;p

[@princetyagii]

The most stylish mathematician

[@TomRocksMaths]

[@patmull1]

I guess this is the simplest model, but I've seen also some predator-prey simulations when the prey population cannot grow forever because logically, according to how things work in nature, overpopulation will lead to fewer resources, thus some of the prey may die out of hunger. That was the only slightly confusing moment, other than that, fantastic explanation.

[@MusicFanatical1]

Wow I own two of the books behind him.

[@VivaanTheCraft3r]

Yooo nice vid I understood everything 👍👍👍

[@VivaanTheCraft3r]

@Prabath Hemachandra Hey do you mind checking out my channel 😀

[@davidgillies620]

Alexander Dewdney in _Scientific American_ did several of these population dynamics explorations in the 1980s. Often they were simulations which were well within the capabilities of home computers of the era. I remember one, set on a toroidal world (like Asteroids) called Wa-Tor, which modelled sharks and fish.

[@CorrectHorseBatteryStaple472]

I'm a bit unconvinced by the equations. Might have been helpful to see the step where they take out the constants. I'm guessing the equations are "correct" but a lot got skipped... For example, you can't just say that more foxes = more competition = foxes die... There is only "competition" if there aren't enough rabbits to keep all the foxes fed. I'm guessing the foxes and rabbits were just the motivating example for this kind of analysis. Definitely a simpler approach than I was ever given at school.

[@thewaytruthandlife]

in the foxes rabbits only situation you get litteraly an oscilating situation. Circles that go round and round, nothing more

[@biktor75]

Lotka and Volterra, my heroes

[@PositronQ]

This is also Computation and statistics