Introduction to Population Models and Logistic Equation (Differential Equations 31)

[@xCaleb]

I honestly cannot believe that there aren’t hundreds of colleges and universities having a bidding war on Prof. Leonard’s salary in order to secure him as a mathematics professor for their institution right now. This man is an invaluable gift to all learners of mathematics.

[@mgmartin51]

Most colleges and universities don’t care about the quality of the teaching

[@datascienceds7965]

If universities had him, we the public would miss a brilliant Prof. I know it's bit of selfish thought Lol

[@zb5775]

As someone else has said, almost all universities/colleges don't give a flying fk about the quality of teaching. They don't. It's not even in their awareness. I did a degree from 18-22, then went back 20yrs later, so I had more maturity and perspective. And I realized how indifferent the "system" really is.

[@funmaster5249]

You are singlehandedly the greatest champion of explaining difficult math to the average human being. Thank you so much for existing and doing the things you do.

[@Frostasy]

I miss you. My class has gone past your videos and I feel like I need you.

[@doggiegirls]

he needs to come back ;-;

[@sharkleberryfin2987]

it sounds like prof leonard is your ex lol

[@xantimiki]

Did you find videos fit for your class?

[@funssisfun]

@@xantimiki let’s hope, after all that was 4 years ago

[@naimon6615]

Because of parts of your videos lessons,My GED Math score is 172 which is college ready. Thank you very much

[@lissetdeleon9442]

Nai Mon that awesome! Congratulations, your hard work paid off.

[@ProfessorLeonard]

Great Job!!!

[@RoundArthur]

I liked the video because you went through the nitty' gritty' of the math. I am the kind of person who needs to know who, what, where, when, and why when it comes to math. Thank YOU!

[@g-abeshawel9603]

thank you my teacher from Ethiopia

[@wisamdoghoz3471]

You guys I have noticed someting off about the explosion condition. It does explode when p0 > m but only at a specific point in the futrue when mtk=ln(P0/m) which gives us zero in the denominator and in return either plus or minus infinity depending if you are approaching the point from the left which gives us plus infinity i.e. the explosion we are talking about. However, if you move a tiny bit in the future you get minus infinity and after a little while it reaches zero and stays there. It was pretty confusing to me at the beginning and needed to spend some time to get it right.

[@hexhackbangla8368]

thats what i was looking for , i have a great respect for him

[@nathantraylor6173]

Thank you for everything. It's funny how things never make sense until they do.

[@usernameisamyth]

Thank you very much, Sir I (almost) finished them all. They were really helpful. I really appreciate your time and effort to make this possible. Again, thanks.

[@TimothyShevgun]

Great video! Great job! Logistic equation is a hardcore thing indeed. Much more difficult than one may guess, especially when it comes to the general case. This should be the best explanation on the topic I have ever seen so far. However, I must notice there is a sort of EXPLANATORY MISTAKE where it comes to 'explosion/extinct' version of the model at 59:30. No formal mistakes are though. Part 1. Dynamics analysis. Let's look at the equation dP/dt = k*P*(P-M), where k>0 and M>0. We can identify 3 cases. 1) If current value P>M (for instance, as P0>M), then factor (P-M)>0 and the right-hand side (RHS) of the equation gives us dP/dt>0. It means that P(t)->+inf. 2) If current value 0M, but tau0 is region of interest, this issue stays in the shade.

[@ayushishrivastava150]

Sir your videos are amazingly helpful, after watching your videos I feel more confident with topics related to differential equations.. Kindly consider partial differential equations next. Thank-you.

[@əliabdulkərimova]

May God be pleased with him,Amen🤲

[@davekes856]

Please donate to this man. Even $1 will help keep these videos coming!

[@joshescobar3065]

Professor Leonard treats math like a language, teaches you how to speak it, and how to use it. An attribute that is paramount for any professor, but sadly an attribute that most lack.

[@hussantamimi9918]

All the way up professor! I don't know how many times i should say thank u.

[@onemanenclave]

You can support him financially.

[@hussantamimi9918]

@@onemanenclave which i did

[@xdxdxdxdxd48]

ur the best teacher

[@selousscout9664]

Very hard. Brutal video

[@hitm43]

1:00:00 if the second term in the denominator is getting larger and larger and we subtract it from Po, wouldn't that mean the denominator approaches negative infinity and therefore P(t) -> 0? The conclusion of P(t) -> infinity makes more sense if the second term in the denominator approaches Po such that the denominator gets smaller and smaller and therefore P(t) larger and larger. Not sure if that is what was meant here.

[@carolinemitchell4322]

Holy cow. Fantastic. Thank you so much!

[@doomerman965]

Hey professor please do Laplace transforms soon!

[@agrotizer476]

The glasses are a disguise. He takes them off to fly around and fight Super Villains. His only weakness is kryptonite.

[@pipertripp]

Great presentation, BTW. I'm using Boyce and Diprima and learning on my own. It's a tonne of fun, but your presentation is really useful. I'm definitely going to check out your ODE play list. It will be really good reinforcement of what I've read in the text. Thanks for all of this, it's clearly a lot of work.

[@pipertripp]

@Allan 112358 I'm using windows 10 with python. Specifically the anaconda python distribution and I'm doing my coding in Jupyter labs/notebooks. I'm doing a lot of the coding by hand to learn in better detail how to numerically solves systems of ODEs and the various numerical methods that can be used to do that (Fwd/bkwd Euler, Runge/Kutta, Adams-Bashforth, etc). That said, there are some really excellent python packages that will do this sort of thing for you: scipy, sympy, gekko have solves, and numpy is a great support library that is generally useful. For plotting/visualizing results I just use mathplotlib, the python standard. All of this is free, btw, which is an added bonus.

[@pipertripp]

@Allan 112358 best of luck with it!

[@pipertripp]

@Allan 112358 a cheap windoze laptop would be fine, it money is an issue that's probably the best option in the short term.

[@pipertripp]

@Allan 112358 all of those techs work great on mac, windoze, or linux so you should be fine.

[@benradick1489]

Sir I think there is a mistake around 58:43. Isn't the limit of P(t) in this case going to be 0, no matter the starting condition? For M>Po, the equation is a scaled version of: P=1/(1-e^t) And for M infinity is 0. Perhaps I have made a mistake?

[@robertoberidojr.435]

I notice this also. Even though case 2 where Po>M is suppose to be called explosion but the limit of the resulting function as t approaches infinity would go to zero from the negative P value.

[@robertoberidojr.435]

The one in case 2 where Po< M is just alright since it's called extinction and the limit should go to zero as t approaches infinity. The function is decreasing to zero

[@n.trushaev5132]

Yes, I'm confused about this, as well. For an initial population P0 > M, we have (P0 - M)e^kMt = P0 - M < P0, but eventually (P0 - M)e^kMt reaches and surpasses the initial value P0. Furthermore, as (P0 - M)e^kMt < P0 approaches the initial value P0, the denominator tends to zero, and we should therefore have population explosion. We then approach a _bifurcation_ point at time t* such that (P0 - M)e^kMt* = P0, and then for all subsequent points in time t' > t*, we should have a negative population size.

[@csliew22]

For the explosion model, I see my book as dP/dt=rP, in here, it is dP/dt=kP(P-M), can any tell me how these two equations are interchangeable?

[@csliew22]

Or can anyone tell me, in the section birth rate proportional to population, why (δ/k) could be equal to M, δ is initial death rate?

[@xnorgate5894]

Man I remember everyone hated solving these things. They were just nasty, especially during the test ughhh.

[@josephalan9067]

At the very end when youre explaining how when P > M, P(t) → infinity.... I dont understand how the fraction is shrinking but not going negative. What we have essentially in the denominator is P - (P - M)e^t - so as t increases, e^t is sharply increasing. And P - M is positive since P > M. And no matter how small a positive a value P - M is, since it is being multiplied by an exponential function increasing without bound, its overall value is quickly rising. Then we are subtracting that value from P....so would that not result in a negative number? Some value minus (some value minus a smaller value) times a huge value. For example 10 - (10 - 3)100000 = 10 - (7)100000 = negative. Im sure I am missing something but I was struggling to understand what

[@aysezulalyilmaz729]

Can a threshold value of a population be a rational number, or do I need to approximate the number to the nearest integer while making calculations?

[@Festus2022]

At 20:10 How can B-sub1 be considered a constant, if it inversely varies with the population in a linear fashion? Dr. L doesn't explain this. Then he makes the great leap to the Logistic Equation. A little explanation here would go a long way!!!

[@nikolozperadze4887]

Amazing Stuff!!

[@MrCuteguylol]

Hi! Can you upload some videos on DEs solvable for p, x, y Envelopes, singular solution Clairauts Equation Ricatti Equation Thank you :D

[@nicholi8933]

If you had been my professor I may have learned something in dif eq. Though I seem to know more than I thought, this was still good to go over. I know in chemistry reactions can reach an equilibria, is this not a term used for your first case?

[@isobar5857]

This man is priceless.

[@ameerhamzashahi1006]

hello sir birth rate per thousand of people per change in time, is it in fraction?

[@pipertripp]

So in the very last bit about the threshold, you basically get an infinitely large population in a finite time... if the denominator were to go negative (after t hits the value at which Po - (Po - M)*e^(mkt) < 0) then the population would suddenly switch to being negative, which makes no physical sense whatsoever. Am I reading that right?

[@robertoberidojr.435]

I notice this also. Even though case 2 where Po>M is suppose to be called explosion but the limit of the resulting function as t approaches infinity would go to zero from the negative P value.

[@robertoberidojr.435]

The one in case 2 where Po< M is just alright since it's called extinction and the limit should go to zero as t approaches infinity. The function is decreasing to zero

[@n.trushaev5132]

Yes, I'm confused about this, as well. For an initial population P0 > M, we have (P0 - M)e^kMt = P0 - M < P0, but eventually (P0 - M)e^kMt reaches and surpasses the initial value P0. Furthermore, as (P0 - M)e^kMt < P0 approaches the initial value P0, the denominator tends to zero, and we should therefore have population explosion. We then approach a _bifurcation_ point at time t* such that (P0 - M)e^kMt* = P0, and then for all subsequent points in time t' > t*, we should have a negative population size.

[@robertoberidojr.435]

Very great

[@alnavski98]

Don't remember if you've already mentioned this but will you also make videos on PDE's as well? That would be nice

[@milano97]

can someone help me with comparing to another formula found. textbooks and even blackpenredpen, patrickjmt use the formula: dP/dt = kP(1-P/M). Is this the same logistics equation to prof Leonard or not? I dont get where he is getting kP(P-M) where M is So/K

[@Simran-ru7wc]

Same doubt

[@dildobaggins2759]

Why is B1 a constant with the logistic equation?

[@chanakyasinha8046]

He has taught, its linear decrement of birth rate with the size of present population... If it happens to be variable, the differential equation will be more tough.

[@dildobaggins2759]

@@chanakyasinha8046 theres easier ways to see the logistic equation in my opinion...

[@chanakyasinha8046]

@@dildobaggins2759 how?

[@sumjabali]

The king 👑

[@dhaneshamirpuri7059]

Anyone know of a Number Theory lecture series!

[@ahmadrashid971]

You're great

[@DutchNorthAtlanticAlliance]

Good mathematics, especially since we can force out any beta sub 1's without any repercussions and even keep factoring these out without any moral compass needed or without violating any youtube inclusivity policy 😂 the more alpha mathematics we do, the stronger and smarter we get my fellow Mathematicians 💪😎🤟

[@potatoepicnessfilms]

what do you mean people die

[@waelbassam6002]

Q/solve this question Let P=population of the fish k=carrying capacity of fish α=growth rate of fish Now answer Using the mathematical modeling of fishes (1) If population of fish is small ,p≈0 ? *

[@donnamoore9514]

the video image is too poor, you need to fix it more

[@horizonbrave1533]

Professor, I'm not sure where Diff. Equatiosn factors (no pun intended) into overall Calculus. If Derivatives are usually taught in Calc I, and Integrels in Calc II, I assume DIfferential Equations are in Calc III? Also....Star Wars Episode IX looks awesome.

[@onemanenclave]

D.E. follows Calc 3.

[@briskioO]

DE uses many techniques from calc I, calc II, and some calc III. It is simply impossible to solve differential equations without knowing the techniques used in calculus

[@RayJackson88]

Episode 9 looks like the kind of mess you're forced to pick up while walking your dog...

[@rockfordlines3547]

AND THEY'RE DOING WHAT BUNNIES DO!

[@muyuanliu3175]

P goes to zero not in this way

[@yashrajjaiswal6051]

thanks to him we can now calculate the population of earth that was swiped out after thanos snap 😂😂😂😂🤣🤣😅

[@jayakumarvenkatesan8688]

35:00 sc

[@imnotasher4892]

🥵