You didn't need that absolute value in P/(M-P) because P is +ve and M is the carrying capacity, so P is always less than M. Therefore the whole term becomes +ve. Anyway, nice video as always.
The starting population can be more than the carrying capacity, and the population will decrease to meet the carrying capacity. You can see it if you make a slope field.
1:53 I should point out, that I use your trick here to avoid partial fractions: factor out another P in the denominator, so you have M/(P^2 * (MP^(-1) - 1)) and then bring the P^2 to the numerator as P^(-2), then let u= the denominator.
Wow, this is so much nicer than the method I was first shown, my original lecturer showed me the solution by integrating dP/dt=aP-bP^2 Writting the DE as dP/dt=kP(1-P/M) makes it a lot neater
man, isn't this just beautiful? and such a clean procedure. although I kind of wanted to see that substitution of c (in terms of the initial population) made
that was indeed legitimate, because we're talking about a completely arbitrary constant, eg. literally anything that is not dependent on x. You can just think of C2 as absolute value of C3 and then it makes complete sense. We can do many weird things with constants in diff equations, because in the end you can propagate all the steps backwards and you will see it's all legitimate, because every step in between was also legitimate. (eg. if one of the steps involved a logarithm, then the constant will cause the expression it equals to still be positive, even if you plugged a negative number at the start)
hello! can you please do also the Logistic Differential Equation With Allee Effect? We badly need it for our special problem set in differential equation but we are having a hard time to derive the equation. Thank you in advance!
He talked through it, and they were simple enough that both constants ended up becoming one. He uses Heaviside cover-up whenever possible, and on this one, it was possible for both.
@@victoria673 P=0 is one of the stationary points of the logistic equation, in addition to P=M. Any solution should work and make mathematical sense at both the points. If it is logic that you root for, then the term ln(|p|) also has no meaning different from ln(p) as p cannot be negative. My point was the value of the constant 'c' in the final solution. c = (M-p0)/p0. What if p0=0? Is it a positive or a negative quantity??
@@elliottmanley5182 This considers the pretense that the species already exists, and that the parameters that govern its growth are constant, such as its fertility rate, death rate, and the capacity of its environment. This is only an approximation to reality, since it doesn't consider interaction with other species, changes in its environment, or evolution. At one point, obviously the species had to originate, when the population technically was zero, but that's a topic for another subject.
@@tcwan0501 What did you say to him? I could translate his response and your second comment, but I can't find anything on what that character means in your original post.
Just realised that "P naught" sounds like "Peanut."😂
Great video!
You are not alone
I learnt this before differentiation in biology. Didn't know a flying crap. Now I do.
Me: do you prefer almonds or peanuts?
BPRP: 9:00
Almonds are way better
Black pen red pen...BLUE PEN!
You didn't need that absolute value in P/(M-P) because P is +ve and M is the carrying capacity, so P is always less than M. Therefore the whole term becomes +ve. Anyway, nice video as always.
The starting population can be more than the carrying capacity, and the population will decrease to meet the carrying capacity. You can see it if you make a slope field.
Interesting that the solution is a Fermi Dirac distribution. I’ve never noticed that before.
Any reference that I could learn more about this fact?
When I learned that heaviside cover up was the residue of a simple pole, I was so happy.
Are you polish by any chance?
1:53 I should point out, that I use your trick here to avoid partial fractions: factor out another P in the denominator, so you have M/(P^2 * (MP^(-1) - 1)) and then bring the P^2 to the numerator as P^(-2), then let u= the denominator.
Wow, this is so much nicer than the method I was first shown, my original lecturer showed me the solution by integrating dP/dt=aP-bP^2
Writting the DE as dP/dt=kP(1-P/M) makes it a lot neater
man, isn't this just beautiful? and such a clean procedure. although I kind of wanted to see that substitution of c (in terms of the initial population) made
Thank you sir,,
This is awesome. I was stuck on this but your video helped me a lot!
when you said "P nought" it sounded like "peanut"
I don't think you can turn +/- C2 into C3? at the end you solve for C, but actually it could be the negative version of that too no?
that was indeed legitimate, because we're talking about a completely arbitrary constant, eg. literally anything that is not dependent on x. You can just think of C2 as absolute value of C3 and then it makes complete sense. We can do many weird things with constants in diff equations, because in the end you can propagate all the steps backwards and you will see it's all legitimate, because every step in between was also legitimate. (eg. if one of the steps involved a logarithm, then the constant will cause the expression it equals to still be positive, even if you plugged a negative number at the start)
didn't understand this lesson in my class but you helped a lot especially with the partial fraction decomposition trick
What about the discrete-time logistic equation? Many animals have mating seasons.
hello! can you please do also the Logistic Differential Equation With Allee Effect? We badly need it for our special problem set in differential equation but we are having a hard time to derive the equation. Thank you in advance!
I like your way of solving partial fractions!
Omg I love how my calc 2 class and your videos are lining up! This is amazing
So much Algebra, little calculus lol
Welcome to Diff Eq!
Where did M on the top [M/p(M-p)] in the third step go before separation
Anyone know what SONG plays quietly AT THE BEGINNING? BlackPenRedPen, you are SO AWESOME! My brain grows with every video. THANK YOU ALWAYS!!!
too late.... but doraemon op
Man I just love the way you smile...
i know some random asian guy will save my math hw thank you
Allen is watching this and can’t understand jack
The dude's playing Doremon in the background as he puts on a suit while teaching mathematics, nice.
Everyone mint ur chubbies!
how did we get from x+2=4 to this
Nice beginning doraemon music🙂
Pls solve this bprp:
Integral 0 to π sinx × d2/dx2(e^(3x) × sin 4x) dx
Why you did not cover the partial fractions steps.
He talked through it, and they were simple enough that both constants ended up becoming one.
He uses Heaviside cover-up whenever possible, and on this one, it was possible for both.
Post a video about Gödel's logic magic.
Black Pen, Red Pen, Green Pen
: )
Best derivation... It is helpful for me.. thank you
Maybe it's time to utilize this equation and start modelling the corona virus as an example of logistic growth?
Yes you can do this.
You are king
Masterful. Thank you very much.
Cosmos kids!!!!
YaY traditional blackpenredpen
: )
Fresh shirt Yacht party arms in the blazer !!
Thanks man
5:28 The nature of the problem prevents that input from ever being negative, so the absolute value wasn't needed to begin with.
It's still the principle of preparing to find all possible solutions, even the non-physical ones.
Hot in suit
Liked it, well explained derivation.
you are a grade saviour 🔥🔥🔥🔥🔥🔥
I love u
loved the solution of the fraction
1
But your explanation was excellent
Absolutely beautiful. I love your style.
People call me "'M'', never P/M.
That was so helpful tysm
What if the K value is increasing over time?
We'd need to know the details of how K varies with time, or with population, and it would make a significantly more difficult problem to solve.
Are you from Singapore? You are so cute!
He's from Taiwan.
At 2'13" and in dozens of other vids, bprp says something like "addingwads". I have never been able to work out what it is. Anyone? Isn't it?
Adding with
Plz use this form 2:13
@@hamsterdam1942 why?
@@blazedinfernape886 ah!
because it is built-in feature and more convenient
you are amazing!
No mention of a mortality rate I see; will that be the topic of the next video?
What if the initial population was zero?
@@victoria673 P=0 is one of the stationary points of the logistic equation, in addition to P=M. Any solution should work and make mathematical sense at both the points.
If it is logic that you root for, then the term ln(|p|) also has no meaning different from ln(p) as p cannot be negative.
My point was the value of the constant 'c' in the final solution.
c = (M-p0)/p0. What if p0=0?
Is it a positive or a negative quantity??
You can't have a population without any population logically speaking.
Mathematics disproves creationism! Or have we just proved it? I'm confused.
@@elliottmanley5182 This considers the pretense that the species already exists, and that the parameters that govern its growth are constant, such as its fertility rate, death rate, and the capacity of its environment. This is only an approximation to reality, since it doesn't consider interaction with other species, changes in its environment, or evolution.
At one point, obviously the species had to originate, when the population technically was zero, but that's a topic for another subject.
Nice sir
That one was fun.
What a naughty P
Refreshing
ㄤㄤㄤ
小叮噹幫你實現... 所有的願望~!!!
很喜歡你的解說，會持續收看你的影片！
謝謝~~
@@tcwan0501 What did you say to him? I could translate his response and your second comment, but I can't find anything on what that character means in your original post.
I was thinking that if I put the m on the RHS then why my answer comes wrong
Can you explain
It should still be the same.