Simulating Competition and Logistic Growth
ฝัง
- เผยแพร่เมื่อ 25 ส.ค. 2018
- Resource limits bend exponential curves into S-shaped logistic curves. The fourth in a series on evolution.
Made with Blender and python.
Github: github.com/Helpsypoo/primer
Special thanks to supporters on Patreon, especially:
Jordan Scales
世珉 匡
Eric Helps
Ben Kamens
Christy Serbus
Sean Barker
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Streaming myself working on these monstrosities:
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More on logistic growth on Khan Academy:
www.khanacademy.org/math/ap-c...
www.khanacademy.org/science/h...
License information:
Creative Commons 4.0 (CC-BY-NC). More info at creativecommons.org/licenses/...
Speaking of attribution:
The 3D bunny model comes from the Stanford Computer Graphics Laboratory: graphics.stanford.edu/data/3Ds...
The music is "Summer Days" by Kai Engel, distributed under a CC-BY-NC license via the Free Music Archive. freemusicarchive.org/music/Kai...
Several other inputs into the graphics are from public domain contributions to www.blendswap.com.
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A number of people have commented that at 5:22, the green population should be expected to beat out the blue population because the blue population suffers from mutations while the green population benefits from them. That effect is real, but it's too small to tip the scales, and I determined it was too complex for the main message of the video. But because some people are interested, I thought I'd lay it out in more detail here.
Only 1% of replications by blue creatures produce a green creature (4:52), and another 1% produce orange creatures. This means blue's replication rate is effectively only 98% of the stated value. 9.8% instead of 10%. The green population gets an expected influx of creatures each time step equal to 0.001 (a tenth of a percent) times the number of blue creatures.
Here's a desmos graph of the expected changes per time step for blue and green, from the equation in the video. (The green equation has an added term for bonus creatures from blue's mutations.)
www.desmos.com/calculator/zhig0dcftg
x is the number of blue creatures, and there are sliders for the other parameters if you want to play with them. N is the total, and N-x is the number of green creatures.
We can see that when there are 49 or more blue creatures we expect to gain green creatures and lose blue creatures. And if there are 44 or more blues, we expect to lose more blues than greens. But for any number of blues less than this, we expect green to do worse. Green is never expected to outnumber blue.
This all assumes no oranges. The fact that the functions add to less than zero for any value reflects the fact that orange is gaining in any of these scenarios.
So it's true that in this setup blue suffers losses from mutations to different colors while green benefits, but the magnitude of this effect with the given parameters is too small to make up the difference in base replication chance between green and blue. Additional sims I ran reflect this fact, with green regularly losing, but I chose to animate the first sim I ran because it was a good reminder of how chaotic this system is.
I appreciate folks commenting with a critical eye, and this will help me know what to explain more clearly vs what to gloss over in future videos. It was also kind of fun to make the desmos graph. If you like it and want to see more desmos graphs when relevant to other videos, please reply to this comment and tell me so!
Thanks!
new subscriber here, whatever u put out i think im going to watch either way :)
More of this ! 🤤
But dosn't this also influence the "death rate" because blue lose every mutation one who can not replicate you could maybe give both yellow and green a 1% mutation rate to get blue idk it is just an idea
Primer wow
sometimes youtube algorithm really gives you the good stuff like this
You are correct
Yeah!
if it is sometimes,then it is more likely a mistake
Then we get actually happened videos only
Yeah your right
The way the blobs wiggle their hands.
The way the blobs win our hearts.
'hands'
The derp.... I love derpy stuff, like Quagsire and these blobs
They vibin
@@henryg.8762 I need to watch his video now “5 ways NOT to think.”
I'm a fan for the excited orange blob
Nick Louie bandwagon fan
I'd like this but it's at orange blob's favourite number, 69.
@@Ziess 87, let's get it to 420
Me too
You mean scp 999
My God, In collage I was unable to figure out from where numbers comes out and memorize all stuff related to logistic growth without understanding it at all just to pass my matlab laboratories. It's so hard to explain this so simple. Thank you, channels like this deserve gold medal and monetization factor of x100
Hahah your last sentence is what all of us think but couldnt quite express
That's the problem with most universities. They produce graduates that are good at remembering stuff, not necessarily understanding stuff.
Collage btw
*College :\
College*
And I’m 10
This is one of the only channels that I will drop everything to watch when a new video comes out. Love it!
nigga
rude
@@bestbread3446 not really, since it's not directed at anyone or anything. CHECK YO' PRIVILEGE, SON
would you drop your child?
LillyCookies DDAENG no, but im sure you were dropped as a child.
"Sorry, buddy."
Orange blob: *blank face*
If it's too crowded, you could also just snap your fingers
At which point that whole "logistics curve" thing would quickly bring the population back up to carrying capacity. No matter how powerful the snap, its effects are ultimately temporary. :P
@@eclipserepeater2466 I know, Thanos was not a mathematician haha
+DEMON XD sorry boys we took over the planet and have gone to the moon and invented the internet
Love that sub
*Orange blobs appear*
Blue blobs: _I don’t feel so good_
You should simulate the predator-prey model I think it would be really fun
Blue blobs being sore losers
I'm pretty sure he has done this in a later video now
Kiritsu yes
He did
@MsBizzyGurl na its more like gis first natural selection simulation. There he made it that when a blob was 20% bigger than the others it could simply eat them.
Can't believe I've missed out on this superb content until now ! Keep up the good work man !
Yeah same. Simulating natural selection just showed up in my recommended.
Same just found this channel yesterday
I love how when you run the simulations it's 3d models just sliding along a wall
I believe it's a flat plane viewed from above, and they are just looking up because they know how to smile at the camera
Clear and very well animated!
For the mathematically inclined, you can read delta as the derivative of N with respect to time, since it is the variation of N. Thus, before adding the crowding term, you get dN/dt = aN, a differential equation whose solution is indeed exponential, N = exp(at) + c. By adding the crowding term, you get dN/dt = aN^2 + bN, and the solution to that differential equation is the logistic function, as stated.
++ Thanks for spelling it out. :)
Help me out here, if dN/dt = aN, then the integral of dN/dt with respect to t is N. Thus, integral of aNdt = aNt = N, right? What am I missing?
Thank you!
@@hubierti No, since N actually is a function of t itself (you could write our differential equation as N'(t) = aN(t)) you can't just leave it be when you integrate with respect to t - what we're looking for is a function whose derivative has the original function as a factor. And in fact, the only function f(t) whose derivative f'(t) = some constant * f(t) is the exponential function, and the derivative of e^ax = a*e^ax so the solution must be, like Vincent stated, N(t) = e^at + C since this gives N'(t) = a*e^at = a*N(t).
I am smol brain and i have no ide wat haf of dis is
This reminds me of some of my favorite teachers over the years. You take the time to explain a concept in detail, and show every step of your work, while explaining WHY it's changing and how it's affecting everything else. You also recap things we've gone over before, and recap when you finish a concept. It's wonderful, and honestly, has pushed me to add physics classes to the bio classes i was planning on when I return to school. You've rekindled my love of math, and I applaud you for it. Thank you.
so thanos should've just modified creature's stats to reach the ideal equilibrium at some point in time instead of killing the half of everything which is only a temporary solution. not so smart huh
Yes
He'd have to increase birth rates for almost all creatures (except humans, since they're the problem-)
And increase how quickly plants grow (to sustain current human populations, and the soon to be increased animal populations from increasing their birthrates)
And since he can't do both of those in one snap (those are seperate actions after all), he'd have to snap twice.
Which would kill him more than likely.
Also those variables would have to be very precisely changed
So.. no, he really couldn't fix it that way.
I do agree that decreasing population size was the dumbest possible solution though
Even if he didnt add , there was still a n equilibrium point , the thing is humans modified earth to increase ita carrying capacity to their own good and thus they influenced their own stats and caps
but don't forget that in real life the "stats" are determined by the environment of the creatures, and aren't fixed unconditionally.
This guy literally just debunked Thanos.
"We've seen that creatures can grow exponentially-" (flip flop)
"But in the real world-" (stop)
"A realistic growth curve would look something like this." (HOW CAN YOU DO THIS TO ME)
Thanks - I've been looking forward to this video for a while - you have a style that makes it very easy to listen to and understand. I'm looking forward to what comes next.
Thanks, I really appreciate the kind words.
Thanos’ plan layout when trying to *save the universe* .
ellathon22 this actually shows how futile Thanos strat is. Population would grow back in no time and even promote a higher replication ratio
@@yzraeloa3787 Plus you have an initial universal famine, since most logistics will be severly damaged due to the loss of half the officials, half the technicians, half the farmers etc.
Thanos never really thought this through at all^^
@@TheRedMooncorp It took me about five seconds of thought when I heard his plan to realize he'd spent about zero seconds on it.
@@IamCoalfoot I feel you.
I mean even choosing specifically half is so stupid.
Even if Thanos would be right, the optimal number of deaths would differ from world to world, but sure let's treat a well organized society with population controlls, a new colony with 500 people and a sprawling over populated world of slums the same 😂😅
TheRedMooncorp population controls are a meme
Sooo... you spent the first 3/4 of this video explaining the very basics of calculus? I'm not complaining. In fact I'm very glad you are doing this because it is very useful in explaining these kinds of situations where the rate changes over time. So good job, plus you made it more interesting than any lecture could be.
It's been 50 days! We miss your videos!
Haha, thanks for the enthusiasm. Rest assured that more are coming. I'm getting impatient too. :P
:D
@@PrimerBlobs :D
Plz do a video on simulation of life and why these ameobas are pear shaped
They ameobas cuz they look squidgy and there hands stretch from their bodies so there pseudopods
Oh how my AP Calculus teacher ruined variable specification for me, I know that if I didn't specify N' = Delta when doing a project like this I'd have gotten tossed out a window or something for not defining my variables sufficiently.
Other than that petty observation which most people would just be able to figure out if they're not baboon-tier, this video is genuinely an excellently animated, excellently voiced, and phenomenally descriptive example of the fact that there truly are still amazing content creators on this platform. This will be the first patreon that I support, and I wholeheartedly look forward to your next videos in the future!
I love how you condescendingly call people who are less mathematically gifted or educated "baboon-tier". It says a lot about you that you consider people to be tiered in that way, and consider yourself in a superior "tier".
I absolutely adore this series of videos, I can't watch enough of them! please keep making them
I'm horrible at maths and science so I don't always understand all the numbers in these videos but all the concepts are really well explained and easy to grasp with a second watch and well-worded google searches. The simulations are also really interesting, easy to follow, and incredibly entertaining, and all of this combines to make your videos some of the best I've seen on TH-cam- thank you so much!
Thanos should watch your videos.
I just stumbled upon these videos and I really /really/ enjoy how simplified they are. I enjoy watching these kind of things. Keep up the good work
Wowwww I wished my calc teacher taught it like this when we learnt logistical functions. Never realized why dN/dt=N is exponential and dN/dt=-N^2 logistical.
I knew that it was exponential only from integration, cool to see the visuals though
B1rd1e yeah I know how to integrate it but it didn’t make sense conceptually.
@@h_3795 can you maybe explain? I really wanna know how to get the function of N from the Delta function
dvir komlosh shoot idk man. I can give you the proof with integration but idk conceptually. But here’s what I think. The reason why tho that dN/dt=N is exponential is that it doesn’t have to move to the right in order to increase faster. Simply increasing makes it increase faster. With dN/dt=t, which results in a quadratic, much slower, you have to increase t in order to increase faster, which lowers the slope.
dvir komlosh oh I’ve just figured it out. You know that the derivative of y=e^x is e^x, so dy/dx e^x=e^x. Since y=e^x, dy/dx=y
Those videos are amazing, very high quality. Thank you and great work!
I should be studying... but as a former biology student, and as I'm fascinated by genetics, I just can't stop watching this series
Man your videos are fantastic, I'm sure many people would kill for more frequently released episodes
Ooh differential equations in disguise
Shh
I will reply to not shh
And use my home account tomorrow
@@buygeorgeivegotit7203 F
Please don't stop making videos, your videos are the beauty of TH-cam and are really satisfying, enjoyable and interesting.
Just found this channel, fantastic videos! I’ve been looking for a good educational economics channel for a while
This is an awesome way to grasp derivatives too wow this is an insanely clear explanation too thank you 🙏🏽❤️
I have my BA in biology I wish a few of my professors would have used something like this, a lot of people have a hard time understanding population growth. Very informative
I love your videos! This is going to be useful in for future for sure :) It's nothing to complicated, but you always go over things that are easy to overlook.
I wish there were more of these videos! They’re very interesting and informative.
I would like to say that I absolutely LOVE this. I also love the fact that so many people are on board with this series, this topic, and this channel! It actually surprises me how many people like this stuff besides me. Thanks Primer!!! Also since you're going this high so fast you might want to put that face reveal at a higher sub level. Just saying.
I like how you explain these differential equations in such an intuitive way.
Never forget that he retconned where the orange blobs came from. Personally, I prefer the original lore, and I'm not calling it 'Legends' or some dumb name like that.
TELL US
Excellent quality for the visuals and commentary - great stuff!
*Edit* : read response by channel before liking my comment ;)
I doubt that green won because of luck in the simulation at 5:19.
Blue loses offsprings to mutation (both orange and green) which makes blue worse than the stats suggest. Additionally green has a higher birthrate as blue mutates into green. Although I have to admit that you didn't show the mutation rates, so they could be small enough to be negligible.
I think for a better comparison both orange and green could mutate back to blue with the same likelyhood.
This white haired guy is right
I didn't add a label for the mutation rate, but I did say that orange and green each come out of 1% of blue's mutations. So blue is still 98% as good as the written stats suggest.
@@PrimerBlobs yeah 98% is only for one generation though. Blue is destined to extinction
@@PrimerBlobs oh your videos are lovely and almost unique btw. Can you make a video about yourself, how do you make these videos and how long does it take? Thanks a lot
@@PrimerBlobs You are right. I missed that! And yes, 1% is pretty negligible. The mutation rate does really "feel" a lot higher though. But I'll take your word!
These videos are so chill and fun to watch. I'm glad I clicked on one when it was randomly recommended. I love watching the adventures of these little blobs!
This is one of the best channels on TH-cam. Huge fan.
I love how this TH-camr just sprouted up with confidence and good quality. Most people start nervous and slow with no audio or something like that. I'm not saying that's a bad thing, I'm just saying that this is way better. It's awesome.
0:12
*_"Sorry buddy."_*
I never understood graphs and stuff, like parabels (i don't know if that's the right word, i'm german, so i don't know the special terms for that) in school, but you are really outstanding when it comes to explaining something. Great job man!
This is so entertaining, calming and educating. Great job!
I can say that this is helping me with my trapping course because you talk about the carrying capacity. Keep it up, your vids are epic.
The video, the music and the presentation of the information are excellent
Thanks for giving me a solid base to my next simulation game.
These videos are so simple, yet SO entertaining
I love this stuff, please keep doing it, I think your style is awesome
I am so glad I found this channel
It’s funny and I’m learning about something that is extremely complex easily
Really enjoying this series. Great work
So calming and easy to enjoy. This is so satisfying
Would love to see your take on something like the rabbit and fox problem, or some other version of populations that have growth rates variant of eachother. Love the videos!
This guy could write codes for old school RuneScape. Lots of equations like this. Really helped me get good at math as a kid. Great vids
Excellent material. Very clear and informative. Great work!
I always love coming and watching your videos because I always learn something from it :3
TH-cam recommended me your videos, and I've been binging since. I've been reading a lot about world leaders, especially the suffering going on in North Korea. I've always wondered how one person can end up on top like that, and it would be really interesting to see a simulation based on these structures. I'm still getting through these videos, so I might come across it, but I've just been so fascinated in the way that the world works, which has taken me 25 years to get to this level of curiosity. I saw the doves and hawks video, and I am wondering if it's a derivative of that, but at a way larger scale. Thank you for these videos, so much food for thought. I look forward to any videos you might make in the future :~)
Wow! I just love watching the blobs. Great animation by the way!
0:28 could of been a really nice city landscape ya know
0:06 even though we have limits in the real world it’s still nice to see that the human population looks similar to that graph over the past 150-200 years
Finally TH-cam recommended something stimulating... Keep up the great content
Nice too see all the math about probability etc. in action^^
I’m in a calc 2 class rn and just had a test on differential equations (of which logistic growth is one) and it just occurred to me that logistic growth is just the next polynomial step up from direct growth (adding one term)
This is like 3Blue1Brown standard quality! Good stuff man!
Ur channel is awesome. Keep up the interesting lovely stuff!
these are really interesting videos, little tibits like this can make a morning so much better :D
So far I am loving this series.
shoutout to my guy for putting so much work for 7 minutes of goodness
Fantastic series!
5 seconds and I subscribed. God bless youtube recommendations for this good stuff!
You are the only TH-camr of quality
Why are these videos so fun to watch
Love your videos!
i'm really mad you didn't simplify the bell curve into a quadratic equation. im so invested in how beautiful functions look
I have multiple questions:
Which programming language do you use ?
How do you visualize all these stats ?
Where can I learn/read more about genetic algorithms, which is related to this topic?
Thank you so much for the videos
I like each and every one of them
Damn, now I want to tweak the code you're using to create random mutation, multiple environments, hunting, competing for mates... Sounds like a really fun hands off full "game of life" project ;)
This quality videos are the reason why TH-cam is an amazing place
this is the best channel to procrastinate to.
This channel is amazing
A good lesson about logistic growth without a single mention of calculus.... Impressive. Clearly calculus is still evident, but the idea was presented flawlessly without it. Good job!
I love your channel! awesome content. Thank you!
I learned about this channel from 3blue1brown's website, in his FAQ, the first question he answers is what does he use to animate his videos? He mentions you as an example of not only someone who took inspiration from him, but did the best in coming up with your own unique style, so I thought you were worth checking out based on that. I'm glad I did, and what's more, I liked this video so much that I've just subscribed :)
OMG these animations are soo neat
Fantastic channel. This will grow, a lot.
Your content is great! Please keep on doing videos❤❤❤
i love this channel bc it sceince/maths but i understand it and its fun bc its about things i will click on XD keep it up!!!!
Very well presented. Thank you
i love this channel so much
I really hope you're a teacher somewhere. I learn more in these short videos than I did sitting in a high school class lol. The simulations and animations are top notch
I love your videos! They are pretty lovely!
You are amazing, truly.
Just a hypothesis: the reason that green did better then blue was because blue had no way to "get founded" out of another species. When blue turned yellow or green, and eventually died out, it had no way of coming back. If green dies out though, it gan be brought back with blue.
These are so good
Great video!
Please more videos like these, from Uruguay
I came back to watch it again since now I understand this math!
I'm binge watching all primers videos
Excellent material!!!!