Professor Bazett, thank you for an amazing video/lecture on The Logistic Growth Differential Equation and its overall impact on everyday life. One important application is population growth on planet earth.
I found your channel a year ago when I was studying the SIR model. Feel like it’s come circle with this vid. Logistic equation is a neat equation and it illustrates the fact that an equation doesn’t have to be complex to have good explanatory power. Are gonna discuss the lotka volterra equations at some point? I know that’s getting ahead a little bit but another great topic.
Haha quite true actually, sir gives a behavior much like logistic! I will do lotka-volterra properly when I do systems a little later in my ode playlist
@@DrTrefor really looking fwd to that segment. systems of ODEs are really neat. I unwittingly studied them by learning to do simple physical models based on Newton's second law. I later realized that what was going on was the idea that one can break a single higher order ODE into a series of 1st order ODEs. That struck me a quite remarkable. I'm fascinated by ODEs, and am keeping "in touch" with them through your series while I improve in other areas. Really looking forward to the rest of your ODE playlist.
Instead of using the partial fractions method to integrate consider the ODE as a second-order Bernoulli equation and solve it accordingly. The solution derived is the same one presented here.
Question In the end we see that the limit of y as x goes to infinity is 1, as we were expecting from the stable equilibrium being 1. However we see nowhere the unstable equilibrium as a limit. Shouldn't there also be some form of y so that the limit of y as x tends to infinity and equal -infinity?
3 ปีที่แล้ว +2
Thanks for the amazing video. Can you tell which software you are using for dif eq slope field
I have a question ... In PREVIOUS CALCULUS COURSE , WE LEARNED ABOUT CONSERVATIVE FIELDS , .... SO are exact differential equations also conservative ? If yes then , is the integral curve Independent of Path then ?
Thank you. Nice. Are you going to talk about chaos.? Can chaos and bifurcation diagram exists here if the function is continuous, not discrete? It seems not. What are the implications?
it doesn't seem so special to me, compared to predator pray or even the heat equation, but maybe it was because i got this in an exam question and got a little carried away with partial fractions and trying to solve for x and nothing seemed to give, it was beautiful that i could use Bernoulli's method to solve it at the brink of time tough, other interesting thing is that you can solve for 1/n the max population not that different to other DE like the RC circuit
Hi Trevor, thank you so much for this content. Question: in the video you said that when y > 1, we'll get "exponential decay" towards y = 1. I see of course why the derivative is negative when y > 1. However, how is the decay "exponential"? My understanding is that exponential decay happens when the y' = ky (where k < 0). How is this the case here? Thanks
I had a small doubt. We saw that when we tend towards y = infinity, we will eventually hit a equilibrium value. This makes sense when seen using the example of population increase that the population will increase exponentially up to a certain point and then will come to a standstill at a certain point. This shows the asymptotic stability. So, what does the asymptotically unstable point represent in a population growth example? Or is that we cannot study the asymptotically unstable point as population cannot be negative? If this is the case, is there any other scenario which can be used to understand asymptotically unstable points?
Good question. The asymptotically unstable point in this DiffEQ is the case that the dependent variable is zero. This is caused by the way we reconcile the absolute value inside the natural log, which creates the plus/minus dual sign on the other side. In the application to population growth models, it is simply a non-physical solution to consider the negative side, so EVERYTHING will occur on the positive side of this graph. One possible interpretation is if that as soon as two individuals of the species originate (for a species that requires a partner to reproduce), or just one individual for a species like bacteria, it is going to quickly grow away from zero, since those two individuals will breed and grow the population with nearly no constraints of the carrying capacity of their environment. This assumes that other factors don't get in the way, and it is possible for a population bottleneck to even start propagating a species in the first place.
If there were a hypothetical growth model, where population could be negative, then a starting point of zero would be an unstable equilibrium. Either two negative individuals propagate their trajectory of the species population, or two positive individuals propagate their trajectory of the species population. Maybe a hypothetical zombie version of the species is considered negative, and the healthy version of the species is considered positive.
There's a slightly different equation that if you plot in a way wich the x axis is the value of a and the y axis is the value in which the x got stable, you get a fractal! The equation is X(n+1) = X(n)•a•(1-X(n)) You can do what I said using the programing language C. I have the codes if someone is interested.
this playlist has really saved me, for reviewing this material again. thank you for making all these amazing videos
Again, the teaching is unbelievably great as ever but Dr. Trefor's shining hair is to a whole new level hahaha. I love it !!
Great video!
These are equations that we can apply in life
Thank you!
Professor Bazett, thank you for an amazing video/lecture on The Logistic Growth Differential Equation and its overall impact on everyday life. One important application is population growth on planet earth.
Thanks Dr.. Your videos are great for understanding the essence of the topic.
i was just reading about this in a differential equations book today and then you posted this
Cool! Love when those coincidences happen
Thank you 😊💯💯💯
This nature of Growth, is the "behind the scene" truth of every growth aspect in life, I guess !!
Thanks!
Thank you so much!
Your class is the best clearance
"You can't find a solution to that ODE."
"Why not (y-naught)?"
"..."
I found your channel a year ago when I was studying the SIR model. Feel like it’s come circle with this vid. Logistic equation is a neat equation and it illustrates the fact that an equation doesn’t have to be complex to have good explanatory power. Are gonna discuss the lotka volterra equations at some point? I know that’s getting ahead a little bit but another great topic.
Haha quite true actually, sir gives a behavior much like logistic! I will do lotka-volterra properly when I do systems a little later in my ode playlist
@@DrTrefor really looking fwd to that segment. systems of ODEs are really neat. I unwittingly studied them by learning to do simple physical models based on Newton's second law. I later realized that what was going on was the idea that one can break a single higher order ODE into a series of 1st order ODEs. That struck me a quite remarkable. I'm fascinated by ODEs, and am keeping "in touch" with them through your series while I improve in other areas. Really looking forward to the rest of your ODE playlist.
Love from India
I really liked your videos
I just learned this in ecology!
Nice! It’s definitely common in ecology for sure
Thanks ! Doctor for such an amazing series , what about getting some engineering math's to channel ?
What is the difference between growth rate and relative growth rate for a logistic equation
7:34 bruh can anyone show me the steps its driving me insane
Instead of using the partial fractions method to integrate consider the ODE as a second-order Bernoulli equation and solve it accordingly. The solution derived is the same one presented here.
Question
In the end we see that the limit of y as x goes to infinity is 1, as we were expecting from the stable equilibrium being 1. However we see nowhere the unstable equilibrium as a limit. Shouldn't there also be some form of y so that the limit of y as x tends to infinity and equal -infinity?
Thanks for the amazing video. Can you tell which software you are using for dif eq slope field
This is DESMOS
can someone give the solution for 7:48
i didnt get it
Rearrange the equation as Bernoulli Equation and solve it
Where are you getting the slope field generator from? I have been looking for a website.
Sir, why is the rate (y') proportional to (1-y/K) and not to the difference between the Total capacity and the current population (K-y)
I have a question ... In PREVIOUS CALCULUS COURSE , WE LEARNED ABOUT CONSERVATIVE FIELDS , ....
SO are exact differential equations also conservative ?
If yes then , is the integral curve Independent of Path then ?
Thank you. Nice. Are you going to talk about chaos.? Can chaos and bifurcation diagram exists here if the function is continuous, not discrete?
It seems not. What are the implications?
I do indeed plan to, and yes chaos can arise from continuous systems
@@DrTrefor Cool. Looking forward to it.!
Thank You.
I think it would be more intuitive if the Y in parenthesis is written as a proportion Y/C, where C is carrying capacity.
Can you teach us the logistic map?
Mathematicians: unchecked exponential growth is impossible in nature
Universe: hold my dark energy 😉
it doesn't seem so special to me, compared to predator pray or even the heat equation, but maybe it was because i got this in an exam question and got a little carried away with partial fractions and trying to solve for x and nothing seemed to give, it was beautiful that i could use Bernoulli's method to solve it at the brink of time tough, other interesting thing is that you can solve for 1/n the max population not that different to other DE like the RC circuit
Happy pi day ❤️❤️🙏 proffesor
Hi Trevor, thank you so much for this content. Question: in the video you said that when y > 1, we'll get "exponential decay" towards y = 1.
I see of course why the derivative is negative when y > 1. However, how is the decay "exponential"? My understanding is that exponential decay happens when the y' = ky (where k < 0). How is this the case here? Thanks
I had a small doubt. We saw that when we tend towards y = infinity, we will eventually hit a equilibrium value.
This makes sense when seen using the example of population increase that the population will increase exponentially up to a certain point and then will come to a standstill at a certain point. This shows the asymptotic stability.
So, what does the asymptotically unstable point represent in a population growth example? Or is that we cannot study the asymptotically unstable point as population cannot be negative?
If this is the case, is there any other scenario which can be used to understand asymptotically unstable points?
Good question. The asymptotically unstable point in this DiffEQ is the case that the dependent variable is zero. This is caused by the way we reconcile the absolute value inside the natural log, which creates the plus/minus dual sign on the other side. In the application to population growth models, it is simply a non-physical solution to consider the negative side, so EVERYTHING will occur on the positive side of this graph.
One possible interpretation is if that as soon as two individuals of the species originate (for a species that requires a partner to reproduce), or just one individual for a species like bacteria, it is going to quickly grow away from zero, since those two individuals will breed and grow the population with nearly no constraints of the carrying capacity of their environment. This assumes that other factors don't get in the way, and it is possible for a population bottleneck to even start propagating a species in the first place.
If there were a hypothetical growth model, where population could be negative, then a starting point of zero would be an unstable equilibrium. Either two negative individuals propagate their trajectory of the species population, or two positive individuals propagate their trajectory of the species population. Maybe a hypothetical zombie version of the species is considered negative, and the healthy version of the species is considered positive.
There's a slightly different equation that if you plot in a way wich the x axis is the value of a and the y axis is the value in which the x got stable, you get a fractal!
The equation is X(n+1) = X(n)•a•(1-X(n))
You can do what I said using the programing language C.
I have the codes if someone is interested.
Ah yes, that's a cool one!!
I must say you are not running a channel. You are running a University.
@Trefor Bazett What did you used to graph the slope field?
Its a software called GeoGebra!
Very simple equation, yet we still have things like Wright's conjecture if we add a delay.
🔥🔥🔥
Happy pi day!!! 🥳😊🥳
Happpppy pi daaaay!!!!
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