@@NoisqueVoaProduction Now I'm taking actuarial exams where lower-case P, capital P, and Rho will be in the same equation representing different things.
@@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.
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)
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...
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.
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!
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.
“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
(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.)
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.
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.
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?
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.
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 ...
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.
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.
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!
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
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.
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.
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 :)
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
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.
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.
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
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.
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..)
Tom Crawford, the apparently punk rock mathematician here is an amazing artist for a mathematician, lol. everyone else i've seen try to draw something more complex than a square on this channel has botched it, but that rabbit and fox actually looked like a rabbit and a fox.
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
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)
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.
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.
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.
Math uses U and V to remind me how bad my handwriting is.
The reason I stopped using cursive...
@@archivist17 bruh but in cursive you have the little curve on the v they don't look the same
My physics teacher always had us write U like ⨿ to make the difference clear.
Wow,remembered my geometry class where the u,v and w were indistinguishable... By the teacher
@@NoisqueVoaProduction Now I'm taking actuarial exams where lower-case P, capital P, and Rho will be in the same equation representing different things.
u = 0
v = 0
“This is a happy place for the population”
I mean, you can’t be unhappy if you don’t exist...
Extremely hard to be content, too.
@@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.
i can call this a trivial fact
R/im14andthisisdeep
For the grass, this would be a very happy place.
all animals were harmed during the production of this video
Only rabbits and foxes were harmed.
Not the humans, I hope
Call PETA. People Eating Tasty Animals.
@@magran17 I prefer EEDP - Everyone Eating Defenseless Plants
false.
As a man who’s taking a differential equations course, I gotta say the timing of this video so very convenient 👍
Check 3blue1brown series on differential equations
I'm doing differentiation and integration rn
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)
Summr school go brrrr
You brave soul. This is the very maths problem that made me drop out of engineering.
I’m surprised how he explains differential calculus very well and is knowledgeable, but when it comes to rabbits everything falls to pieces
I love how bad he is with words!
They're going to.. erm.. populate like rabbits 😆🤣
As great as it is, a maths degree does have it's limits...
@@TomRocksMaths how has no one picked up on this great pun yet
@@TomRocksMaths Unless it is indeterminate...
@@TomRocksMaths tongue and cheek? “Limits”?🤪
Ohh! The fluid dynamics guy.
That's me!
@@TomRocksMaths I know right?
Yass
Tom Rocks Maths Got a new sub! 😊
More than one, I guess the "change in the population of subscribers" is positive ;-)
0:26
"u are going to represent the prey"
WHAT DOES THE FOXXX SAYYY?
@@Albimar17 VVVVVVVVVVVVVVVVVVVVVVV
I misheard Brady's reply as "why me?"
??
5:18
"The change in u is equal to value of u."
Peak wisdom.
???? What's this supposed to mean
@@edwinvlasics4047 They are pretending "u" is the word, "you."
Mathematician: "Lets look at a steady state where EVERYBODY's happy..."
Rabbits getting eaten to maintain status quo: "Revolution!"
Revolution... around the steady state. :D
Basically, the steady state here is the train in Snowpiercer
There is always a communist.... well he did not factor in the lettuce now, did he?
@@kimon114 theres always a communist hater
@@jenm1 There are no "communist haters" just people who understand communism.
_u_ as in _usagi_ (rabbit in Japanese)
_v_ as in _vulpes_ (fox in Latin)
Because settling on one language is for the plebs.
Exept japanese do not use latin alphabet
@@nicolascavadini3570 Japanese uses romaji characters for many purposes
u and v are the same letter in latin though
@@nicolascavadini3570 Really Nicolas? So for example, find me 'JVCKenwood' in Japanese characters. That's a famous enough Japanese company.
It's kind of shocking how badly my professor bungled this explanation, and by contrast, how clearly this concept is laid out here.
Yeah this is where I got lost in calculus, same issue, bad prof explanation.
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...
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.
Sorry to hear that Chuck, but glad I could help out :)
I love he can stumble over words and still sound smart lol very relatable
i dont know about relatable lol, if i stumble over my words i very quickly stop sounding smart xD
The "everyone's happy when there's nothing" case (v=0, u=0) just feels existential 😂
Can confirm I'm still looking for a carnivorous rabbit... any help much appreciated.
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!
Thanks Connor!
OMG he's so cute :)
Isn't he!
@@TomRocksMaths Oh wow, subscribed. :) I'm a little surprised you noticed my comment lol.
One of the beauties of maths.. Making sense of the quality and the quantity of complexity with ease.
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.
The intro gave me Monty Python vibes
Well, that's no ordinary rabbit. That's the most foul, cruel, and bad-tempered rodent you ever set eyes on.
That is an ex-rabbit. That rabbit is no more. It's pushing up daisies.
It *is* the rabbit!
“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
BuT bUrDeN oF pRoOf
this is one of the coolest looking maths nerd I ever saw on this channel
I think this single video has killed more rabbits and foxes than the rest of Numberphile content combined
Which poses the final question: Are foxes red because of the carotin in the carrots the rabbits eat?
I thought the ultimate question was :
What does the fox say?
Woof or grrr?
@@NoisqueVoaProduction *human screaming*
(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.)
Thomas is quickly becoming one of my faves on numberphile, he's so precious
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.
He looks too "fabulous" doesnt he?....
I'm an oldie. It's the piercings I don't like. But I know I'm from a different time.
@@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.
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.
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?
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.
The man with the Navier Stokes Equation tattoo!
Yah he finally came with a new video I've been waiting for him 😂
Sure is :)
“Rabbits eating foxes”??? RUNAWAY!! RUNAWAY
"My degree is in math, not zoology."
Rabbit: Let's switch roles for a while.
Fox: Okay, I'm game.
It’s just a rabbit...... I soiled my armor!......
@Nick -- 2020 in nutshell
@@murrfeeling but you have to know that "Rabbits eating foxes" is false
When I saw “population of rabbits”, I immediately thought about the Feigenbaum-constant😅
Me too!!!!
same
yeah me as well
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 ...
Yeah. I'm surprised he never mentioned the Logistic Function.
What i like about you most dr Rocks !!! Is your simple beautiful english and your smooth explanation . You are my favorite maths . Thanks prof .
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.
Expected to hear lotka-volterra being mentioned 🙈
Me too :o
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.
Yup (also, one stable point)
@@Some.username.idk.0 To be fair that's just a stable loop of a size 0.
@@RomanQrr yeah, thats why "yup" was in the front of the sentence
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!
his smile made this whole thing about eating rabbits and brutal law of nature, seem like casual goodnight story for kids
That introduction was hysterical! The whole video was entertaining, as always. So glad to see him on your channel again!
Thanks Gio - more to come soon!
Thanks for summarizing one of the first exercises I faced in Control Engineering
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
ThatAwkwardGuy I think that’s intentional
Well because thats another video! If you want to see what hes talking about click on said end card:D
On mobile you just drag down on the video a bit and the cards disappear
You can use the developer tools (F12 in firefox) to delete the overlay.
@@onecommunistboi bless u
It’s not rabbits eating foxes, it’s rabbit-eating-foxes
As someone LITERALLY making a game using this as the core mechanic, the timing of this video cannot be any better 😂
Yeah, killer rabbits have started being a real problem recently...
The solution would be to send for Brother Maynard and the Holy Hand Grenade of Antioch.
Not sure if this is a Re:Zero reference or a Monty Python reference.
Run away!!!
@@GiuocoPianissimo thinking the same thing lol
2020 keeps getting weirder and of course, worse.
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.
Wow, what a beautiful connection between rabbit-fox population modeling and sine-cosine functions!
More Tom please!
I'll be back don't worry!
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.
Thanks - glad you enjoyed it :)
Rabbits eating foxes mathematically would be much simpler, but a much shorter video.
Why ?
0:05 That's no ordinary rabbit! It's got huge--he can leap--LOOK AT THE BONES, MAN
Wow, I seriously wish I had the chance to watch this video back in highschool, this makes so much more sense.
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 :)
Thanks Tyler - glad you enjoyed it!
I like this primate's colorful display pattern. It signals to predators that he is not a tasty meal. He is a clever primate.
"You are going to be the prey >:D"
*Brady's breathing intensifies*
hello handsome mathematician 😳😳
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
6:$5- "And everybody's happy!" Well, except the rabbits that are being eaten.
0:09 who caught the Wilhelm scream? 😂😂
I saw you on Mike Boyd’s channel when he did the mathematics entrance exam to Oxford
@00:48, intriguing!
He starts a drawing by first drawing the back of the object!!
"Rabbits are eating foxes"
Re:zero s2 watchers:
Understandable have a great day
I remember doing this in uni! Such a nice way to remember what I used to do
Lotka and Volterra, my heroes
this man is what i aspire to be
Tom is so pure
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.
This is an even better explanation of differencial equations than 3B1B's videos.
The thumbnail at the end covered up the tattoo!!
I remember learning about it in biology class in primary school with the same graph in my textbook
Love the animation at du/dt = u 😂😂😂
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.
This video is worth watching! More of this, please!
Ohhh first time I saw mathematician with tattoo 👍👍😁😁😁
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
I learned about "daisy planet" yesterday, and then I get this video 😂
Wow I own two of the books behind him.
This is also Computation and statistics
The most stylish mathematician
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.
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..)
"So we have a rabbit and a fox."
0:52
"So we have a rabbit and an anteater."
There's a really nice book for stability analysis: Introduction to Dynamics by Ian Percival and Derek Richards
Did you just get through the whole video without mentioning the names "Lotka" and "Volterra" - or was I just not paying enough attention? :D
Gorgeous, polite, funny and awesome at maths... my mother will definitely approve 😍😍😍
he can teach me math any day of the week 😎
The predator/prey cycle where pop(PRED) lags pop(PREY) is a classic model. I first learned it from the Dewdney book _Armchair Universe._
Tom Crawford, the apparently punk rock mathematician here is an amazing artist for a mathematician, lol. everyone else i've seen try to draw something more complex than a square on this channel has botched it, but that rabbit and fox actually looked like a rabbit and a fox.
You're too kind!
That animator is fire. Keep him
Thankyou so much !
2:58 I think I've seen this meme before...
AhAh!!!
oh no
Piper perrie meme
I like how I didn't even have to click the timestamp to figure out exactly what you meant
that looks like someone threw all those foxes at the rabbit while stooping time
3b1b: I think I’ve seen this before
Veritasium : me too !
I’ve seen this one! This is a classic.
>what you mean you’ve seen this? It’s brand new!
He is redefining how scientists should like.
in the foxes rabbits only situation you get litteraly an oscilating situation. Circles that go round and round, nothing more
Finally somebody is making Dynamical system theory and Fluidynamics entertaining
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
Great video!!!!
Yooo nice vid I understood everything 👍👍👍
@Prabath Hemachandra Hey do you mind checking out my channel 😀
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)
This guy just gave a talk on how to get a million dollars with maths in my school
Sure did! (and hello again Martin)
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.
He mentioned it in passing, but using u,v,w is very common for anything relating to differential equations
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.
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.
This video is basically about using algebra like a programming language