Area under rate function gives the net change

WEBVTT
Kind: captions
Language: en

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- [Voiceover] Let's say
that something is traveling

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at a constant rate of
five meters per second.

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That's its velocity in one dimension.

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If it was negative, we'd
be moving to the left.

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If it's positive, it's
moving to the right.

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Let's say that we care
about what is our change

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in distance over, the delta
symbol represents change,

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over a change in time of four seconds.

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Over four seconds.

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I could say from t equal
zero to t is equal to four.

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That's our change in time.

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That's our four-second
interval that we care about.

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Well, one way to think
about it is what a rate

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by definition is nothing but
a change in some quantity.

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In this case, it's distance over

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a change in some other quantity.

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In this case, we're thinking about time.

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Or another way to think
about it, if we multiply

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both sides times change in
time, you get your change

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in distance is equal to your
rate times change in time.

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This is very close to
you, you might remember

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from pre-algebra, distance
is equal to rate times time.

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Time, that just comes from the
definition of what a rate is.

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It's a change in one quantity

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with respect to another quantity.

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If you just apply this, you can say,

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"Okay, my rate is a constant five meters

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"per second and my delta
t is four seconds."

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So times four seconds.

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Well, that's just going to give you 20.

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That's just going to get you 20.

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Let me do that in same color

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that I had for the change in distance.

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That's going to be 20.

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Then the seconds, cancel
the seconds, 20 meters.

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So my total change in distance over those

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four seconds is going to be 20 meters.

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Nothing new here.

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Nothing too fancy.

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But I want to do now is
connect this to the area

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under the rate function
over this time period.

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So let's graph that.

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Let's graph it.

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That's my rate axis.

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This is my time axis.

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This is going to be in seconds.

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This is going to be in meters per second.

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Let's see, one, two, three, four, five.

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Let's see if it's about enough,

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and then I go one, two, three, four, five.

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Our rate, at least in this
example, is a constant, is

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a constant five meters per second.

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It's a constant five meters per second.

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That is my r of t in this example.

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What did we just do here?

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We just multiplied our change in time

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times our constant rate.

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We just multiplied our change in time.

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So from time equal zero
seconds to four seconds.

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It's this length here,

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if we think on that axis.

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Now we multiply it
times our constant rate.

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We multiply it times this right over here.

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If I multiply this base times this height,

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what am I going to get?

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I'm going to get this area
under the rate function.

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Now that area is going to be 20.

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If we went with the units of
them, obviously you're used

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to things of area being
something units squared

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because it's usually meters times meters

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or miles times miles or inches
times inches so it'll be

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inches squared, meters
squared, or miles squared.

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But here if we go at the units
of the axis of the meters

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per second times seconds which
is going to get you meters.

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But the important thing here is

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that the units here or
the area here is 20.

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So this for that very simple
example, it looks like the area

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under the rate curve is
equal to the net change

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over that time period where the rate is

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something with respect to time.

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Well let's test the little burrow.

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It's just going to be more intuition here.

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Let's say that we have a
different rate function.

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Let's say that we will
make with the different,

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let's say instead we had a rate function.

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I'll use on this yellow.

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Let's say on a rate function, that is...

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Let me make it a little bit interesting.

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Let's say it's one meter per second,

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for our time is zero is
less than or equal to time

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which is less than or
equal to two seconds.

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Honestly, this is all in
seconds where we're of time.

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There's two meters per second,

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for t is greater than two seconds.

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What's that going to look like?

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Actually, try to graph it
yourself, and just say,

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"Well, what is the
total change in distance

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"over the first, let's say five seconds?"

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We want to do delta t over, not the first

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four seconds but the first five seconds.

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Well, let's graph it.

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Let's graph it.

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So this is one meter per second.

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One meter per second.

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That is two meters per second.

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That's in meters per second.

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That's my rate axis.

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This right over there is
going to be my time axis.

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One, they're obviously
not of the same scale,

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three, four, five.

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What is this rate function look like?

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Well, my rate is one meter per second

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between time is zero and
two, including two seconds,

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and then the rate jumps.

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Nothing can accelerate
instantly like this.

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You'd need an infinite
force or an infinitely

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small mass I guess to, or maybe there's

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something that's thinking about...

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Anyway, I only get two complex here

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but this is unrealistic mode.

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It's not typical for something
to just have spontaneous

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velocity increase like that
but let's just go with it.

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Then after the two seconds, we are

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at a constant rate of
two meters per second.

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Now, what is our total change in distance

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over the first five seconds here?

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So here, we care about
the first five seconds,

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or we can break up the problem.

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We could say, "Well, over
the first two seconds."

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Change in time is two seconds,

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times our constant rate
over those two seconds.

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It's going to be two seconds
times one meter per second.

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Well, that's going to give us two meters.

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So this here is going to be,

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shall we do that in orange color,

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that's going to give us two meters there,

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and then we look at the next section.

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Our change in time here is three seconds,

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and then we multiply
that, times our constant

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two meters per second.

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That's going to give us an area of six.

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If we look at the units, in both cases,

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we're multiplying seconds times meters

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per second which is
going to give us meters.

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This is going to be two plus
six meters or eight meters.

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So hopefully this is giving
you the intuition that the area

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under the rate curve
or the rate function is

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going to give you our total
net change in whatever

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that rate thing was finding the rate of.

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In this case, it is
distance per unit time.

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If you take the area
under the rate function,

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that kind of distance for
the speed or this velocity

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function over some period
of time, that area is

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going to be our total
net change in distance.


Human brain and growing intelligence

WEBVTT
Kind: captions
Language: en

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This right over here is a, I guess you could say it's a

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Side view of a human brain and I probably don't have to tell you that the brain is a

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fascinating organ

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It's what makes you you and me me everything that you are I

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Think about, that

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we're doing, we're seeing it's really all occurring in the brain. Even this

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reality that you see around you, that you think is external to your brain. It's really a

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simulation that your brain constructs for itself based on sensory input from the outside world, and so you could imagine is this incredibly

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complex and you could even say

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mysterious organ. And even though we don't understand everything about how the human brain works and I could imagine we will be exploring

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how the brain works and learning more about it for decades if not centuries

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to come there are things that we do know about the human brain.

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We do know for example that it's made of it's made up of the estimates

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I've seen of anywhere, or is roughly 80 to 100

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billion -- let me write that in caps

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billion

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billion neurons and neurons are

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the cells that

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essentially do all of the

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processing of our brain so this right over here is a is a if you were to kind of really

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zoom in on the brain on a very, very small level. This right over here is a

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blow-up of what a neuron looks like. There are other Khan Academy videos that go in much more depth on the anatomy of a neuron

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and the functions of a neuron, but you see here that you have a you you have

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it's a cell with a nucleus

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just like other cells and what it does is essentially it receives input it receives signals from other neurons

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and then when it's stimulated accordingly

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it'll send signals to other neurons and this

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stimulation and inhibition and sending signals from one neuron to another and through kind of chemic

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chemoelectrical signals this is the stuff of our thoughts. Our thoughts are these these continuous

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stimulations and inhibitions from one neuron to

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another from one neuron to another neuron

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It's not you know when when you're thinking a thought or you're looking at a dog or you're trying to solve an algebra problem,

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it's not just one or two neurons that are firing; we're talking on the order of of

00:02:22.180 --> 00:02:27.660
thousands or millions or maybe even billions of neurons firing and you have kind of these

00:02:28.330 --> 00:02:34.739
impulses that are traveling through your brain depending on the connections that you're forming, the associations that you're

00:02:35.140 --> 00:02:40.259
forming in the concept that you're looking at and these connections usually I'd say hey 80 to 100 billion

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that's a huge number that's on the order of the number of stars in a Milky Way sized galaxy. That's an

00:02:47.980 --> 00:02:55.049
unimaginably large number but to kind of blow your mind even more in the human brain in the human brain

00:02:55.299 --> 00:02:57.160
there are

00:02:57.160 --> 00:02:59.160
approximately 100

00:02:59.560 --> 00:03:01.450
trillion

00:03:01.450 --> 00:03:03.450
trillion synapses

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synapses, and another way to think about them is -- let me write synapses

00:03:08.560 --> 00:03:11.069
synapses or another way to think about these are kind of the connections between

00:03:11.440 --> 00:03:14.940
these hundred billion neurons or these 80 to 100 billion neurons

00:03:14.940 --> 00:03:22.199
So you're thinking each neuron roughly has a thousand connections to other neurons so it's an incredibly incredibly complex thing

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So one question that you might say it's okay. Well, how do neurons relate to something

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like

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intelligence? Your ability to solve a problem or to

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reason through things and what we do know if you look at the animal kingdom

00:03:40.950 --> 00:03:45.569
is that the amount of neurons and the level of connections between the neurons is a

00:03:46.239 --> 00:03:52.319
reasonable indicator if you go from species to species for intelligence, so for example?

00:03:53.019 --> 00:03:54.370
jellyfish

00:03:54.370 --> 00:03:55.630
jellyfish

00:03:55.630 --> 00:03:57.519
Have eight hundred

00:03:57.519 --> 00:03:59.519
eight hundred neurons

00:03:59.799 --> 00:04:02.399
while human beings have eighty to a hundred

00:04:02.920 --> 00:04:03.760
billion neurons

00:04:03.760 --> 00:04:10.560
That human beings are kind of at the at the top of the food chain at least when it comes to how many neurons there

00:04:10.560 --> 00:04:15.359
Are and the number of synapses, etcetera, kind of go with that.

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But what about within a species, is intelligence something

00:04:21.280 --> 00:04:26.099
intelligence that is something that is fixed are you just kind of born with a certain number of neurons?

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And a certain number of connections between them

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Or is it or is it something that can somehow change over time because of the environment that you're in? And this is the interesting

00:04:35.320 --> 00:04:39.790
thing and this is something that I am I'm always thinking about it, and it's fascinating and I always think about it whenever I'm

00:04:40.400 --> 00:04:44.169
Alone I'm thinking about things around what I'm trying to tackle a new subject

00:04:44.170 --> 00:04:49.120
Is that there's actually a lot of research here on the fact that you can change?

00:04:49.610 --> 00:04:57.159
you can change the structures in your brain through training the structures in your brain, so you can change it, so

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This is this is changeable

00:05:01.520 --> 00:05:03.520
Changeable or maybe ourselves

00:05:03.530 --> 00:05:05.559
You can change it can

00:05:06.350 --> 00:05:07.760
you can

00:05:07.760 --> 00:05:13.330
change your intelligence based on essentially training your brain. And

00:05:13.670 --> 00:05:16.840
the analogy that you can have here and you know you could say it's a metaphor

00:05:16.840 --> 00:05:19.959
But in some ways it's a very direct metaphor

00:05:20.060 --> 00:05:25.300
We all know that you can train your muscles; that if you lift weights if you strain your muscles

00:05:26.300 --> 00:05:28.130
that then your muscle kind of

00:05:28.130 --> 00:05:31.809
Your body recognizes that it needs to be stronger in that part of your body

00:05:31.810 --> 00:05:34.929
And it will build that part of bodies as more you strain it the more

00:05:34.930 --> 00:05:37.330
You stress it you give it a little time for repair

00:05:37.330 --> 00:05:39.330
And then your muscle will come back stronger

00:05:39.590 --> 00:05:45.189
And what a lot of the researchers have found over the last few decades is this is also true of the brain. You

00:05:45.500 --> 00:05:52.929
see it as a child develops. This is some imagery that I got from Mindset Works

00:05:52.930 --> 00:05:58.030
this is a site run by Carol Dweck was one of the researchers who really thinks about how the

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How when you struggle, and when you try to tackle things more how that affects your ability to actually process it

00:06:06.290 --> 00:06:12.939
actually affects your intelligence and what this imagery right over here is this is a an image of a

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Child's I guess you could say neuron density or the synapsel

00:06:21.840 --> 00:06:25.090
and the word density at birth, and we see that as a child learns languages,

00:06:25.090 --> 00:06:32.769
they explore the world around them, as they struggle as a persevere as they try to emulate those around them, their brain actually gets

00:06:32.990 --> 00:06:38.319
denser the more that a child uses tries to kind of form connections

00:06:38.570 --> 00:06:45.109
Struggle stresses their brain the connections between the neurons become more numerous you see that here and they can also become

00:06:45.539 --> 00:06:50.718
stronger so you might you actually might have a a once again more connections form and

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They can become more active and when you think about the brain as you know when you have a thought

00:06:56.960 --> 00:06:59.449
It's all of these associations between these different neurons

00:06:59.449 --> 00:07:04.429
and they're all firing and stimulating each other what when you build those new connections is those new

00:07:04.619 --> 00:07:11.299
Associations and a lot or those those new synapses it allows you to form new communication in your brain you new associations

00:07:11.300 --> 00:07:16.639
It'll essentially allow you to think faster and better and to reason in ways that you couldn't have done before

00:07:16.889 --> 00:07:18.199
And you might say okay?

00:07:18.199 --> 00:07:20.359
Well that just happens from you know child to birth

00:07:20.360 --> 00:07:23.719
but what about later on in life is this something that I can change now if I'm

00:07:23.849 --> 00:07:28.878
10 years old or 20 years old or 50 years old and all the research says. Yes

00:07:28.879 --> 00:07:32.058
You absolutely can and the key is how do you stimulate?

00:07:32.699 --> 00:07:37.188
your brain just as someone of any age can stimulate their body and

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Allow their muscles to grow by lifting weights by stressing them

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There's been there been many studies that have shown it for example in animals that if you have an animal living in an unstated

00:07:47.580 --> 00:07:51.529
Environment you actually don't have that many connections form you don't have that much much

00:07:53.219 --> 00:08:00.799
Density of the Brain especially the synapse density but but then if you have an animal that's in a stimulating environment an environment that's constantly

00:08:00.959 --> 00:08:08.509
challenging an environment where it's interacting with each others, and it's solving problems, then you actually have a higher density of of

00:08:09.839 --> 00:08:17.419
Neural connections you have you essentially the animal has become smarter so what I find really interesting about this and I always

00:08:17.789 --> 00:08:23.239
Remembered in my brain is it's very easy. You know a lot of us it's it's tempting to say okay?

00:08:23.240 --> 00:08:27.169
I got what I got that's what my brain is and you know I'll just have to deal with it

00:08:27.319 --> 00:08:30.588
But what all the research tells us is that is not the case the more you?

00:08:30.749 --> 00:08:36.888
Stimulate the brain the more you struggle the more you kind of try to tackle new things you can actually train

00:08:37.709 --> 00:08:41.719
Train yourself. Let me see if you can actually train yourself to

00:08:42.269 --> 00:08:49.159
Be to be smart or whether you're smart or not smart is not something that you were just born with you can actually train yourself

00:08:49.160 --> 00:08:52.670
And the best way to train yourself and this it's it's very analogous

00:08:52.670 --> 00:08:57.830
and there's a lot of research is very similar to the way that you would train if you're lifting weights if you're lifting weights if

00:08:58.110 --> 00:08:59.339
You can lift five

00:08:59.339 --> 00:09:04.969
Pounds a lot you're not just gonna keep lifting five pounds you want to push yourself you want to try lifting with without you know

00:09:04.970 --> 00:09:06.649
Consult your doctor or your trainer

00:09:06.649 --> 00:09:08.240
You don't want to blow out your back or anything

00:09:08.240 --> 00:09:14.029
But you want to keep lifting heavier and heavier weights and essentially making your muscle fatigue making your muscles struggle

00:09:14.029 --> 00:09:18.259
It's not always an easy thing, but it pays off and the when you train your brain

00:09:18.260 --> 00:09:21.289
It's essentially the same thing you want to tackle new problems

00:09:21.290 --> 00:09:23.209
You don't want to tackle problems that are easy

00:09:23.209 --> 00:09:24.959
But that's just like lifting the two-pound weight

00:09:24.959 --> 00:09:29.659
over and over again you want to lift the 10-pound weight the 20-pound weight to eventually get to the

00:09:29.790 --> 00:09:36.440
100-pound weight and so that means tackling problems doing problems that are going to be difficult. Where you you might make a mistake

00:09:36.440 --> 00:09:40.640
But making that mistake is just like you know this picture of arnold where maybe he's not able to lift this weight

00:09:40.830 --> 00:09:43.429
But then as his mom as his body kind of adjusts

00:09:43.500 --> 00:09:47.299
It won't take too much time before he is able to lift that

00:09:47.700 --> 00:09:52.729
to lift that weight so the important Takeaway is you you can completely change your intelligence by

00:09:53.130 --> 00:09:58.309
Challenging yourself by putting yourself in a stimulating environment and your brain really is like a muscle

00:09:58.410 --> 00:10:02.930
The more you train it the more you struggle. The more you stimulate it the smarter you'll get


Logistic function application

WEBVTT
Kind: captions
Language: en

00:00:00.303 --> 00:00:02.630
- So we've seen in the
last few videos if we start

00:00:02.063 --> 00:00:04.682
with a logistic differential equation

00:00:05.787 --> 00:00:10.566
where we have r which is
essentially is a constant

00:00:10.566 --> 00:00:12.582
that says how fast our we growing

00:00:12.582 --> 00:00:15.673
when we're unconstrained
by environmental limits.

00:00:15.779 --> 00:00:17.713
Then we have K which we can view

00:00:17.713 --> 00:00:21.685
as the maximum population
given our constraints.

00:00:21.079 --> 00:00:23.168
We saw that if we wanted to solve this,

00:00:23.879 --> 00:00:25.488
and we didn't want one
of the constant solutions

00:00:25.488 --> 00:00:26.640
of N of T is equal to zero,

00:00:26.064 --> 00:00:27.863
or N of T is equal to K,

00:00:28.439 --> 00:00:29.867
and we did this in the last few videos.

00:00:30.062 --> 00:00:35.062
We got the solution that N of T is equal

00:00:39.219 --> 00:00:44.219
to our initial N naught
times our maximum population.

00:00:46.516 --> 00:00:51.480
All of that over our initial population

00:00:53.844 --> 00:00:58.193
plus the difference between
our maximum population,

00:00:58.193 --> 00:01:00.340
and the initial population.

00:01:00.684 --> 00:01:05.684
So K minus N naught times
E to the negative rt.

00:01:18.782 --> 00:01:19.796
That this right over here.

00:01:19.922 --> 00:01:21.524
This logistic function.

00:01:21.524 --> 00:01:26.524
This logistic function is
a nonconstant solution,

00:01:28.256 --> 00:01:29.831
and it's the interesting one we care about

00:01:29.831 --> 00:01:31.639
if we're going to model population

00:01:31.639 --> 00:01:36.258
to the logistic differential equation.

00:01:36.731 --> 00:01:39.495
So now that we've done all
that work to come up with this,

00:01:39.495 --> 00:01:40.343
let's actually apply it.

00:01:40.343 --> 00:01:43.620
That was the whole goal, was
to model population growth.

00:01:43.096 --> 00:01:46.024
So let's come up with some assumptions.

00:01:47.046 --> 00:01:48.942
Let's first think about,

00:01:48.942 --> 00:01:50.954
well let's say that I have an island.

00:01:52.178 --> 00:01:54.296
So let's say that this is my island,

00:01:54.627 --> 00:01:59.627
and I start settling it with a 100 people.

00:02:00.027 --> 00:02:02.604
So I'm essentially saying N naught

00:02:02.847 --> 00:02:04.789
Let me do that in the N naught color.

00:02:04.789 --> 00:02:09.297
So I'm saying N naught is equal to 100.

00:02:09.297 --> 00:02:10.342
Let's say that this environment,

00:02:10.747 --> 00:02:15.747
given current technology
of farming and agriculture,

00:02:15.921 --> 00:02:18.522
and the availability of
water and whatever else,

00:02:18.891 --> 00:02:23.891
let's say it can only
support 1,000 people max.

00:02:27.591 --> 00:02:32.591
So you get the idea, so
we get K is equal to 1000.

00:02:33.934 --> 00:02:36.591
That's the limit to the population.

00:02:36.851 --> 00:02:37.820
So now what we have to think about

00:02:37.082 --> 00:02:38.827
is what is r going to be?

00:02:39.565 --> 00:02:40.971
So we have to come up
with some assumptions.

00:02:40.971 --> 00:02:45.971
So, let's say in a generation
which is about 20 years,

00:02:46.926 --> 00:02:48.677
well I'll just assume in 20 years,

00:02:48.677 --> 00:02:51.205
yeah I think it's reasonable
that the population grows

00:02:51.205 --> 00:02:55.741
by, let's say that the
population grows by 50%.

00:02:55.741 --> 00:02:59.292
In 20 years you have 50% growth.

00:03:00.088 --> 00:03:05.088
50% increase, increase
in the actual population.

00:03:06.415 --> 00:03:08.510
So what would you have to
have your annual increase

00:03:09.365 --> 00:03:13.114
in order to after 20 years to grow by 50%.

00:03:13.529 --> 00:03:15.684
Well to think about that
I'll get out my calculator.

00:03:16.471 --> 00:03:19.063
One way to think about it, growing by 50%,

00:03:19.063 --> 00:03:23.385
that means that you are at
1.5 your original population,

00:03:23.526 --> 00:03:25.994
and if I take that to the 120th power,

00:03:26.189 --> 00:03:29.837
and we'll just do 1 divided by 20th.

00:03:29.978 --> 00:03:33.073
This essentially says how
much am I going to grow by

00:03:33.073 --> 00:03:36.185
or what is going to, this is telling me

00:03:36.185 --> 00:03:41.185
I'm going to grow by a factor
of 1.02 every year, 1.02048.

00:03:44.242 --> 00:03:46.327
So one way to think
about it is if every year

00:03:47.092 --> 00:03:52.092
I grow by 0.020 I'll just
round five then over 20 years

00:04:08.987 --> 00:04:13.987
as this compounds I
will have grown by 50%.

00:04:16.301 --> 00:04:17.810
So that would be our r.

00:04:17.081 --> 00:04:20.252
This is essentially how much
we're going to grow each year.

00:04:20.987 --> 00:04:25.987
Let me write that, growth each year.

00:04:26.906 --> 00:04:29.915
We're going to assume
our t here is in years.

00:04:30.244 --> 00:04:34.350
So we're going to assume
that our t is in years.

00:04:34.915 --> 00:04:37.231
So t is in years.

00:04:37.813 --> 00:04:40.381
So what would our logistic
function look like,

00:04:40.674 --> 00:04:42.622
given all these assumptions?

00:04:42.635 --> 00:04:47.635
We would have N of T, let me N of T,

00:04:48.096 --> 00:04:53.096
is equal to, is equal to N naught times K.

00:04:54.447 --> 00:04:56.605
That's going to be 100 times 1000.

00:04:57.374 --> 00:05:02.374
So it's going to be 100 times 1000.

00:05:03.411 --> 00:05:06.608
My initial population
times my maximum population

00:05:06.077 --> 00:05:11.070
divided by my initial population,

00:05:15.388 --> 00:05:19.733
plus the difference between
my final and initial.

00:05:19.733 --> 00:05:22.065
So that's 1000 minus 100,
so that's going to be,

00:05:22.603 --> 00:05:24.701
this right over here is going to be 900.

00:05:25.004 --> 00:05:29.248
900 times e to the negative r.

00:05:29.731 --> 00:05:34.731
So the negative 0.0205 times t.

00:05:40.551 --> 00:05:43.158
So it will be equal to that,

00:05:43.407 --> 00:05:47.881
and to verify that this actually is,

00:05:47.881 --> 00:05:50.038
this actually does describe population

00:05:50.038 --> 00:05:50.964
in the that way we thought

00:05:50.964 --> 00:05:53.695
the logistic differential equation would

00:05:53.957 --> 00:05:55.423
let's actually plot it.

00:05:55.423 --> 00:05:58.048
So let me just pause this
video and then plot it.

00:05:58.663 --> 00:05:59.781
So there you go, I made a plot

00:05:59.781 --> 00:06:00.844
and I copy and pasted it here,

00:06:01.411 --> 00:06:06.398
and we see the behavior
that we wanted to see.

00:06:06.398 --> 00:06:10.599
We see the population right
over here at year zero,

00:06:10.678 --> 00:06:12.702
it's starting at 100.

00:06:12.918 --> 00:06:14.999
Let me do this in a color that
you're more likely to see.

00:06:15.988 --> 00:06:18.992
Population starts at
100, and we can see that,

00:06:18.992 --> 00:06:22.087
let's see after 20 years our population

00:06:22.087 --> 00:06:25.681
looks like it's almost grown to 150.

00:06:25.775 --> 00:06:27.962
So it looks like, at
least in the beginning,

00:06:27.962 --> 00:06:32.407
this term, this right
over here is dominating.

00:06:32.407 --> 00:06:37.407
We are growing by this .0205,
that would be 2.05% per year,

00:06:40.211 --> 00:06:43.129
which gets us close to 50% growth,

00:06:43.017 --> 00:06:44.933
and we see that's what
happening initially.

00:06:45.086 --> 00:06:48.120
So we go from 100 to 150
in the first 20 years,

00:06:48.012 --> 00:06:49.063
in the first generation.

00:06:49.063 --> 00:06:52.620
Then in the next generation
we should add another 75

00:06:53.187 --> 00:06:56.210
if we weren't being kind of
constrained by the environment.

00:06:56.417 --> 00:06:59.163
So 150 plus 75 would be 225,

00:06:59.467 --> 00:07:01.513
and it looks like we got
after 20 years to about 200.

00:07:01.927 --> 00:07:03.391
So we're a little bit slower.

00:07:03.759 --> 00:07:05.058
We're a little bit slower than kind

00:07:05.058 --> 00:07:07.253
of the pure exponential growth.

00:07:07.253 --> 00:07:09.980
But the pure exponential
growth would probably

00:07:09.098 --> 00:07:13.249
have us tracking something closer to here,

00:07:14.354 --> 00:07:15.441
but still growing pretty well.

00:07:16.224 --> 00:07:18.224
But then as our population gets larger

00:07:18.224 --> 00:07:19.490
and larger and larger as we're getting

00:07:19.049 --> 00:07:20.941
closer and closer to
the maximum population,

00:07:21.382 --> 00:07:26.382
our rate of growth is approaching zero.

00:07:26.514 --> 00:07:30.008
So we constantly approach
our maximum population,

00:07:30.069 --> 00:07:33.308
but we never quite get there.

00:07:33.308 --> 00:07:34.843
It's really an asymptote.

00:07:34.843 --> 00:07:37.620
We're just approaching it as
time goes on and on and on.

00:07:38.065 --> 00:07:40.002
But you can kind of set your own threshold

00:07:40.587 --> 00:07:42.210
and say ok, when do we get

00:07:42.021 --> 00:07:44.992
to kind of 90% of maximum population?

00:07:45.503 --> 00:07:47.097
That looks like that happens,

00:07:47.097 --> 00:07:50.163
90% of maximum population happened

00:07:50.163 --> 00:07:55.163
after 210 years on this island.

00:07:55.744 --> 00:07:57.183
So on a human scale that seems

00:07:57.183 --> 00:07:58.779
like a long time, many generations.

00:07:58.779 --> 00:08:00.857
But, I guess in a cosmic
scale it's not that long.

00:08:01.559 --> 00:08:02.575
Not even a cosmic scale,

00:08:02.575 --> 00:08:04.656
even just slightly longer
than a human scale.

00:08:05.633 --> 00:08:10.198
So, it'll happen, well this
describes what's happening.

00:08:10.198 --> 00:08:12.065
So this is a pretty interesting model,

00:08:12.065 --> 00:08:14.638
and I'd be interested
to see how it compares

00:08:14.638 --> 00:08:17.933
with actual data out there
for actual population growth.

00:08:17.933 --> 00:08:21.951
With that said, it's not this you know,

00:08:21.951 --> 00:08:24.857
everything that we've done
so far has always assumed,

00:08:24.857 --> 00:08:28.945
we're kind of assuming this
idea of a Malthusian limit.

00:08:28.979 --> 00:08:31.598
But what we've learned from human history,

00:08:31.598 --> 00:08:35.287
that this Malthusian limit
seems to keep getting pushed

00:08:35.287 --> 00:08:39.029
higher and higher based on
the improvement of technology.

00:08:39.029 --> 00:08:40.224
That we're able to grow more crops

00:08:40.224 --> 00:08:41.140
in a certain amount of area.

00:08:41.140 --> 00:08:42.142
We have better rule of law,

00:08:42.241 --> 00:08:43.928
so people don't kill each other as often.

00:08:43.928 --> 00:08:47.481
We have better control
of water and irrigation,

00:08:47.481 --> 00:08:49.208
and all of these things.

00:08:49.208 --> 00:08:51.234
So that we're able to increase the limits

00:08:51.468 --> 00:08:53.825
far beyond what we thought.

00:08:53.825 --> 00:08:57.343
I would guess if you told Thomas Malthus

00:08:57.343 --> 00:09:00.025
that in the year 2014
we have seven billion

00:09:00.025 --> 00:09:02.023
people on the earth, he would have said

00:09:02.023 --> 00:09:04.119
that's far beyond the Malthusian limit.

00:09:04.119 --> 00:09:05.072
He probably would have guessed

00:09:05.072 --> 00:09:06.970
the Malthusian limit was like a billion,

00:09:06.097 --> 00:09:08.639
or two billion given the
technology of the time,

00:09:09.512 --> 00:09:11.785
and we're already at 7 billion.

00:09:11.785 --> 00:09:13.603
As technology improves
and agriculture improves,

00:09:13.603 --> 00:09:15.401
and rule of law improves,
and everything improves,

00:09:15.401 --> 00:09:17.396
we might be able to get, who knows?

00:09:17.396 --> 00:09:18.759
There might be a time,
we might think it's crazy

00:09:18.759 --> 00:09:21.160
for their to be 20 billion
people on the planet,

00:09:21.016 --> 00:09:23.074
but given today's technology.

00:09:23.218 --> 00:09:25.219
But if technology improves
then optimistic scenarios

00:09:25.228 --> 00:09:27.883
that maybe we could keep going.

00:09:27.883 --> 00:09:29.887
That's not necessarily a good thing,

00:09:29.923 --> 00:09:33.455
but that just might be what it is.


Solving the logistic differential equation part 1

WEBVTT
Kind: captions
Language: en

00:00:00.535 --> 00:00:02.598
-Let's now attempt to find a solution for

00:00:03.165 --> 00:00:06.666
the logistic differential equation.

00:00:06.666 --> 00:00:08.790
And we already found
some constant solutions,

00:00:08.079 --> 00:00:09.424
we can think through
that a little bit just

00:00:10.135 --> 00:00:14.674
as a little bit of review
from the last few videos.

00:00:14.674 --> 00:00:17.052
So if this is the t-axis
and this is the N-axis

00:00:17.052 --> 00:00:20.067
we already saw that if N
of zero, if a time equals

00:00:20.202 --> 00:00:22.673
zero, or a population
is zero, there is no one

00:00:22.673 --> 00:00:24.888
to reproduce and this
differential equation is

00:00:24.888 --> 00:00:27.367
consistent with that, because
if N is zero, this thing

00:00:27.367 --> 00:00:29.581
is going to be zero, and
so our rate of change

00:00:29.581 --> 00:00:31.399
is going to be zero with respect to time,

00:00:31.399 --> 00:00:33.212
so our population just won't change.

00:00:33.212 --> 00:00:34.378
It'll just stay at zero.

00:00:34.378 --> 00:00:35.797
Which is nice because that's what would

00:00:35.797 --> 00:00:38.065
actually happen in a real population.

00:00:38.065 --> 00:00:40.863
That gives us the constant solution that N

00:00:40.863 --> 00:00:42.491
of t is equal to zero, that's one solution

00:00:42.491 --> 00:00:44.940
of this differential equation,
not that interesting.

00:00:44.094 --> 00:00:47.159
A zero population will
never grow or change.

00:00:48.005 --> 00:00:50.000
The other constant solution is what if our

00:00:50.000 --> 00:00:53.377
population started at the maximum of what

00:00:53.377 --> 00:00:56.635
the environment could sustain, and in that

00:00:56.635 --> 00:00:58.798
situation this term is
going to be K over K,

00:00:58.798 --> 00:00:59.615
which is one.

00:00:59.615 --> 00:01:01.753
One minus one is zero,
and so in there, the

00:01:01.753 --> 00:01:03.071
population would also not change.

00:01:03.071 --> 00:01:04.949
It would just stay at K,
and so the rate of change

00:01:04.949 --> 00:01:06.048
would just stay at zero.

00:01:06.939 --> 00:01:09.046
So that's another constant solution, that

00:01:09.046 --> 00:01:14.046
we start at the maximum
population and then

00:01:14.046 --> 00:01:16.217
this differential equation tells us

00:01:16.631 --> 00:01:18.554
a scenario that never changes.

00:01:18.554 --> 00:01:19.630
But we said, hey look,
there could be something

00:01:20.314 --> 00:01:23.718
interesting that happens
if our initial condition,

00:01:23.718 --> 00:01:27.396
which we called N sub-not,
that's our N of zero.

00:01:27.396 --> 00:01:29.391
Time equals zero if we
started someplace in between,

00:01:29.391 --> 00:01:33.290
maybe closer, much below
our eventual ceiling

00:01:33.029 --> 00:01:34.925
I guess, because if we're a lot below our

00:01:35.186 --> 00:01:37.112
eventual ceiling this thing is going to be

00:01:37.112 --> 00:01:38.800
a small fraction, this
thing is going to be

00:01:38.008 --> 00:01:41.059
close to one, and so the
rate of change is going

00:01:41.851 --> 00:01:44.189
to be pretty close to
being proportional to N.

00:01:45.085 --> 00:01:47.481
When we're thinking that,
it might look something

00:01:48.246 --> 00:01:51.767
like that, as N increases,
our rate of change

00:01:51.767 --> 00:01:54.811
increases, but then as N
approaches K this thing

00:01:55.207 --> 00:01:56.621
is going to approach
one, this thing is going

00:01:56.621 --> 00:01:58.936
to approach zero, and so
it's going to overwhelm

00:01:58.936 --> 00:02:01.349
this and our rate of change is going

00:02:01.349 --> 00:02:02.653
to approach zero.

00:02:02.653 --> 00:02:04.307
So we can imagine a scenario where we

00:02:04.307 --> 00:02:08.562
asymptote towards K.

00:02:09.648 --> 00:02:13.447
Let's see if we can
solve this to find this,

00:02:13.447 --> 00:02:18.203
this N of t to find the actual analytic

00:02:18.203 --> 00:02:20.115
expression for this N
of t right over here.

00:02:20.115 --> 00:02:22.258
Because this is interesting,
this is what could

00:02:22.258 --> 00:02:25.223
be used to model populations that would be

00:02:25.223 --> 00:02:28.297
more consistent with a Malthusian mindset.

00:02:28.963 --> 00:02:30.229
So let's see if we can do that.

00:02:30.229 --> 00:02:31.573
To do that we just have
to realize this is a

00:02:31.573 --> 00:02:35.545
separable differential equation, and we're

00:02:35.545 --> 00:02:37.170
assuming is a function
of d, we're going to

00:02:37.017 --> 00:02:40.394
solve for an N of t that satisfies this.

00:02:40.547 --> 00:02:42.531
So we just have to separate
it from the explicit

00:02:42.531 --> 00:02:45.173
ts, but there are no explicit ts here, so

00:02:45.173 --> 00:02:46.468
it's quite easy to do.

00:02:46.468 --> 00:02:48.670
So what I'm going to
do is I'm going to take

00:02:48.067 --> 00:02:50.968
this part right over here,
and I'm going to divide

00:02:51.571 --> 00:02:52.800
both sides by that.

00:02:52.008 --> 00:02:53.540
I'm going to leave the r
on the right hand side,

00:02:54.332 --> 00:02:56.744
it'll make things, I
think, a little bit easier

00:02:56.744 --> 00:02:58.997
as we try to solve for N of t.

00:02:58.997 --> 00:02:59.931
So let me just do that.

00:02:59.931 --> 00:03:04.931
So this is going to be equal to one over N

00:03:05.443 --> 00:03:10.250
times one minus N over K.

00:03:10.025 --> 00:03:15.025
One minus N over K
times dN dT, times dN dT

00:03:20.925 --> 00:03:23.162
is equal to r.

00:03:26.205 --> 00:03:29.044
Another way we could think about it, well

00:03:29.044 --> 00:03:31.891
actually, let me just continue
to tackle it this way.

00:03:31.891 --> 00:03:34.465
So we get that, and now
what I want to do is

00:03:34.465 --> 00:03:36.208
take the anti-derivative
of both sides with

00:03:36.208 --> 00:03:38.283
respect to t.

00:03:38.958 --> 00:03:40.515
Well, this is pretty straightforward.

00:03:40.515 --> 00:03:42.752
That's just going to be
rt times some constant,

00:03:42.752 --> 00:03:43.758
but what's this going to be?

00:03:43.812 --> 00:03:45.305
This is kind of this messy thing.

00:03:45.305 --> 00:03:47.090
And maybe if we could
break this out, if we

00:03:47.009 --> 00:03:49.502
can expand this into two
fractions, do a little

00:03:49.583 --> 00:03:51.641
bit of partial fraction expansion, maybe

00:03:51.641 --> 00:03:53.126
we can come up with an expression that's

00:03:53.126 --> 00:03:57.875
a little bit easier to find
the anti-derivative of.

00:03:57.875 --> 00:03:59.599
I'm hoping that I can find an A and a B

00:03:59.599 --> 00:04:04.599
where A over N plus B
over one minus N over K

00:04:09.439 --> 00:04:11.124
is equal to this business.

00:04:11.124 --> 00:04:16.124
Is equal to one over N
times one minus N over k.

00:04:17.226 --> 00:04:18.545
Let's see if we can find an A and a B.

00:04:18.545 --> 00:04:20.122
This is just partial
fraction expansion, if this

00:04:20.122 --> 00:04:21.997
looks unfamiliar to you I encourage you to

00:04:21.997 --> 00:04:24.004
review that part on Khan Academy.

00:04:24.004 --> 00:04:25.737
Do a search for partial fraction expansion

00:04:25.737 --> 00:04:27.628
on Khan Academy.

00:04:27.628 --> 00:04:28.682
So how do we do it?

00:04:28.682 --> 00:04:30.584
We'll just add these two right over here.

00:04:31.508 --> 00:04:33.419
If we take the sum,
right over here, this is

00:04:33.419 --> 00:04:38.419
going to be A times this, which is A minus

00:04:38.647 --> 00:04:43.647
A over K N plus B times
this, plus BN, over

00:04:46.289 --> 00:04:47.803
the product of these two.

00:04:47.803 --> 00:04:50.261
So it's just going to be N times one

00:04:50.261 --> 00:04:52.404
minus N over k.

00:04:52.404 --> 00:04:53.930
One way to think about
it, I just multiplied

00:04:53.093 --> 00:04:55.371
the numerator and the
denominator of this one

00:04:56.208 --> 00:05:00.666
by one minus N over k,
so A times one minus

00:05:00.666 --> 00:05:02.893
N over k is that, and times one minus N

00:05:02.893 --> 00:05:04.834
over k is that, and I
multiplied the numerator

00:05:04.834 --> 00:05:06.902
and the denominator of this both by N, BN

00:05:07.514 --> 00:05:09.551
and N times that, and then I added the two

00:05:09.551 --> 00:05:11.333
now that I had the same denominator.

00:05:11.333 --> 00:05:13.137
So that's just adding fractions

00:05:13.137 --> 00:05:14.553
with unlike denominators.

00:05:14.553 --> 00:05:16.776
It's going to be equal to this, one over

00:05:16.776 --> 00:05:21.485
N times one minus N over K.

00:05:21.485 --> 00:05:23.808
Now we can try to think about what is

00:05:23.808 --> 00:05:25.014
A and B going to be equal to?

00:05:25.014 --> 00:05:26.790
What can they be equal to?

00:05:26.079 --> 00:05:27.407
Well I have a constant term here.

00:05:28.118 --> 00:05:29.249
I don't have any N term here.

00:05:29.249 --> 00:05:31.361
I could say that maybe I have a zero times

00:05:31.361 --> 00:05:34.092
an N, and that actually
helps things a little bit,

00:05:34.092 --> 00:05:37.301
because maybe we could
say that this thing,

00:05:37.301 --> 00:05:40.535
that this plus this,
which are the coefficients

00:05:40.553 --> 00:05:44.889
on N, are going to sum
up to zero, and that

00:05:44.889 --> 00:05:48.951
this, which is rA, is
going to be equal to one.

00:05:49.509 --> 00:05:50.356
That's pretty nice.

00:05:50.356 --> 00:05:53.818
So we can say that A is equal to one, A is

00:05:53.818 --> 00:05:56.514
equal to one, and that
if A is equal to one

00:05:56.514 --> 00:06:01.514
we have negative one over
K, negative one over K,

00:06:03.045 --> 00:06:06.966
plus B is equal to zero.

00:06:07.371 --> 00:06:09.280
Well what's B going to be?

00:06:09.028 --> 00:06:14.028
Well B would be equal to one over K.

00:06:15.251 --> 00:06:19.020
So we can re-write this
as, we can re-write it

00:06:19.002 --> 00:06:24.002
as one over N plus one over K,

00:06:32.944 --> 00:06:37.373
plus one over K over,

00:06:40.818 --> 00:06:42.523
let me just do this,

00:06:42.531 --> 00:06:46.432
over one minus N over K,

00:06:50.957 --> 00:06:55.957
and then times dN dt, dn dt

00:07:00.164 --> 00:07:04.156
is equal to r.

00:07:04.165 --> 00:07:06.658
So I just did a little
partial fraction expansion,

00:07:06.658 --> 00:07:08.024
and I like giving myself a little bit of

00:07:08.024 --> 00:07:09.102
real estate, let me clear this out.

00:07:09.804 --> 00:07:12.175
You can rewind the video and review that,

00:07:12.175 --> 00:07:14.213
if you find it necessary,

00:07:14.213 --> 00:07:16.156
so let's do that.

00:07:16.156 --> 00:07:17.455
So how does this help us?

00:07:17.455 --> 00:07:20.447
Well, hey, you probably
might be recognizing

00:07:20.447 --> 00:07:21.855
the anti-derivative of
one over N, and you might

00:07:21.855 --> 00:07:22.868
even see this.

00:07:22.868 --> 00:07:25.045
So let's just think
through this a little bit.

00:07:25.045 --> 00:07:26.871
We know the anti-derivative of one over N

00:07:26.871 --> 00:07:29.280
is a natural log, just did a new color,

00:07:29.028 --> 00:07:32.297
is a natural log of the
absolute value of N.

00:07:32.549 --> 00:07:34.374
We can see the derivative
of that with respect

00:07:34.374 --> 00:07:39.374
to N is equal to one
over N, and if we were

00:07:40.827 --> 00:07:42.183
to find the derivative
of this with respect

00:07:42.183 --> 00:07:45.490
to t, derivative with
respect to t of the natural

00:07:45.049 --> 00:07:49.341
log of the absolute
value of N is equal to,

00:07:49.782 --> 00:07:50.826
that's going to be the derivative of this

00:07:50.826 --> 00:07:53.607
with respect to N times
the derivative of N

00:07:53.607 --> 00:07:57.975
with respect to t, times dN dt.

00:07:57.975 --> 00:07:59.294
We could do the same thing over here.

00:07:59.294 --> 00:08:03.405
Notice, I have an expression.

00:08:03.405 --> 00:08:05.214
What would be the derivative
of this expression

00:08:05.214 --> 00:08:07.105
right over here?

00:08:07.105 --> 00:08:08.753
It would be negative one over K, if I'm

00:08:08.753 --> 00:08:10.788
talking the derivative with respect to N,

00:08:10.788 --> 00:08:12.484
it would be negative one over K.

00:08:12.484 --> 00:08:13.881
I have a positive one
over K here, and I could

00:08:13.881 --> 00:08:15.378
even make it negative.

00:08:15.378 --> 00:08:19.219
I can turn this, I can
clear this out of the

00:08:19.219 --> 00:08:22.444
way, instead of having a positive there I

00:08:22.444 --> 00:08:25.572
could have, I guess I could
say a double negative.

00:08:26.218 --> 00:08:28.270
So a double negative,
I haven't changed the

00:08:28.027 --> 00:08:30.581
value of if, and notice,
now this is the derivative

00:08:30.824 --> 00:08:34.077
of this, which makes it very
nice for u-substitution,

00:08:34.077 --> 00:08:36.837
or you might be already
used to doing this thing

00:08:36.837 --> 00:08:39.659
in your head, and so we
know that the derivative

00:08:39.066 --> 00:08:40.813
with respect to N.

00:08:41.408 --> 00:08:43.203
Let me write this in a new color.

00:08:43.203 --> 00:08:47.553
We know that the derivative
with respect to N of the

00:08:47.553 --> 00:08:52.553
natural log of one minus
N over K, once again this

00:08:53.881 --> 00:08:55.005
comes straight out of the chain rule, it's

00:08:55.005 --> 00:08:57.057
going to be the derivative
of this with respect

00:08:57.057 --> 00:09:00.532
to N, which is negative one over K times

00:09:00.532 --> 00:09:03.677
the derivative of this
whole thing with respect

00:09:03.677 --> 00:09:07.435
to this thing, which is times one over

00:09:07.435 --> 00:09:10.758
one minus N over K, which is exactly what

00:09:10.758 --> 00:09:12.273
we have over here.

00:09:12.273 --> 00:09:13.404
But if I were to take
the derivative of this

00:09:13.404 --> 00:09:15.979
with respect to t, so the derivative with

00:09:15.979 --> 00:09:19.981
respect to t of the
natural log of one minus

00:09:19.981 --> 00:09:22.033
N over K, it's going to be the derivative

00:09:22.501 --> 00:09:24.442
of this with respect to N, which is that,

00:09:24.442 --> 00:09:25.858
which is what we just found.

00:09:25.858 --> 00:09:27.863
It's going to be this.

00:09:27.927 --> 00:09:30.496
So copy and paste.

00:09:30.496 --> 00:09:33.679
It's going to be that,
it's going to be that.

00:09:33.679 --> 00:09:35.723
Times the derivative
of N with respect to t,

00:09:36.119 --> 00:09:40.135
just straight out of
the chain rule, dN dt.

00:09:40.279 --> 00:09:43.392
Well notice, we have a
dN dt, we can multiply

00:09:43.392 --> 00:09:45.005
it times each of these things.

00:09:45.005 --> 00:09:46.376
Actually, why not, let's just do that

00:09:46.376 --> 00:09:49.431
just to make it clear, cause this really

00:09:49.431 --> 00:09:51.064
isn't so much differential equations,

00:09:51.064 --> 00:09:53.520
but sometimes some of the calculus that we

00:09:53.052 --> 00:09:55.981
learned not too long ago, and even frankly

00:09:56.449 --> 00:09:59.184
some of the algebra, it's
nice to not skip steps.

00:09:59.184 --> 00:10:02.388
So if I distribute this, edit, paste.

00:10:02.388 --> 00:10:05.704
Woops, I wanted to copy and paste.

00:10:06.413 --> 00:10:11.413
So, copy and paste, so
I have that, and then

00:10:12.632 --> 00:10:17.632
I have, I can copy and
paste, so that I'm just

00:10:22.085 --> 00:10:27.085
distributing this
business right over here.

00:10:28.794 --> 00:10:31.989
So I get that, now let me
clean it up a little bit.

00:10:31.989 --> 00:10:34.084
So I have that right
over here, and of course

00:10:34.939 --> 00:10:38.215
we have this being equal to r.

00:10:38.215 --> 00:10:40.327
If I take the anti-derivative
of this with respect

00:10:40.327 --> 00:10:45.260
to t, well I'm just going to get this

00:10:55.519 --> 00:11:00.519
minus this right over that.

00:11:00.687 --> 00:11:01.421
So let's do that.

00:11:01.421 --> 00:11:03.530
Let's take the anti-derivative
with respect to t,

00:11:03.545 --> 00:11:05.458
so the left hand side I'm going to get the

00:11:05.458 --> 00:11:09.244
natural log, the anti-derivative of that

00:11:09.244 --> 00:11:13.178
with respect to t is the natural log of

00:11:13.178 --> 00:11:17.936
the absolute value of N, and then minus

00:11:17.936 --> 00:11:21.153
the anti-derivative of
this with respect to t

00:11:21.153 --> 00:11:26.104
is the natural log of
the absolute value of

00:11:26.104 --> 00:11:31.104
one minus N over K is equal to,

00:11:31.863 --> 00:11:33.958
oh, and let's not forget we're going to

00:11:33.958 --> 00:11:35.216
have some constant here,
I want to find a pretty

00:11:35.216 --> 00:11:37.952
general solution, let's call it C one, is

00:11:37.952 --> 00:11:40.016
equal to the anti-derivative
of this with respect

00:11:40.592 --> 00:11:45.592
to t is r times t, maybe
plus some other constant,

00:11:46.436 --> 00:11:48.607
plus some other constant,

00:11:48.607 --> 00:11:50.311
just like that.

00:11:50.467 --> 00:11:54.033
Now I'm going to assume
that my N of t meets

00:11:54.033 --> 00:11:56.686
this assumption right over here.

00:11:56.686 --> 00:11:59.400
So I'm going to assume,

00:12:08.277 --> 00:12:13.277
assume N of t is going to be less than K

00:12:13.394 --> 00:12:15.015
and greater than zero.

00:12:15.015 --> 00:12:17.564
That means that this thing,
this N is always going

00:12:17.564 --> 00:12:19.846
to be positive, and if N is always between

00:12:19.846 --> 00:12:21.789
zero and K, that means that this thing is

00:12:21.789 --> 00:12:23.615
always going to be positive.

00:12:23.615 --> 00:12:25.884
So that helps me clear
things up a little bit.

00:12:25.884 --> 00:12:28.572
Actually let me do it in this color.

00:12:28.572 --> 00:12:30.686
That helps me get rid of that,

00:12:30.686 --> 00:12:35.300
helps me get rid of that.

00:12:35.003 --> 00:12:39.048
I can instead throw some parentheses here,

00:12:39.345 --> 00:12:43.958
and why don't I subtract
the C one from both sides.

00:12:43.958 --> 00:12:47.225
That would get rid of
it here, so edit, cut,

00:12:47.225 --> 00:12:50.611
and paste, so I have the C one there.

00:12:50.611 --> 00:12:52.642
I know I'm overwriting my own work, which

00:12:52.642 --> 00:12:56.220
is not making it look
as clean as possible.

00:12:56.022 --> 00:12:58.438
I'm just trying to clean
it up a little bit.

00:12:58.636 --> 00:13:00.447
I have an arbitrary constant
that I haven't really

00:13:00.447 --> 00:13:02.258
solved for yet minus another constant.

00:13:02.258 --> 00:13:03.906
Let me just call this some other constant,

00:13:03.906 --> 00:13:06.227
let me just call it a general C.

00:13:06.227 --> 00:13:11.227
So let me just call this C,

00:13:11.396 --> 00:13:16.396
so clear, and I'm just
going to call that C.

00:13:16.708 --> 00:13:19.786
I am just going to call that C.

00:13:19.786 --> 00:13:21.863
And I could even simplify
this a little bit,

00:13:21.863 --> 00:13:23.420
but actually I just realized I'm thirteen

00:13:23.042 --> 00:13:24.977
minutes into this video,
which is longer than

00:13:25.355 --> 00:13:25.452
I like to do.

00:13:26.325 --> 00:13:27.763
So let's continue this in the next video.

00:13:27.763 --> 00:13:30.180
I was getting excited
because I'm so close.

00:13:30.018 --> 00:13:32.920
I'm so close to solving for an N of t

00:13:33.082 --> 00:13:35.416
that satisfies our logistic
differential equation.

00:13:35.416 --> 00:13:37.690
Very, very exciting.


Solving the logistic differential equation part 2

WEBVTT
Kind: captions
Language: en

00:00:00.163 --> 00:00:02.195
-(Voice over) So we left off in part one

00:00:02.195 --> 00:00:04.728
getting pretty close to finding our N of T

00:00:04.728 --> 00:00:07.825
that satisfies the logistic
differential equation

00:00:07.825 --> 00:00:12.276
where it's initial condition
is between zero and K,

00:00:12.276 --> 00:00:13.844
and now we just have to really just

00:00:13.844 --> 00:00:16.361
do some algebra to finish things up.

00:00:16.812 --> 00:00:18.146
So we left with this.

00:00:18.146 --> 00:00:19.209
For our N of T this must be true.

00:00:19.776 --> 00:00:22.511
Now, we could use a
little bit of logarithm

00:00:22.511 --> 00:00:24.764
properties to rewrite this left hand side

00:00:24.764 --> 00:00:27.772
as the logarithm of.

00:00:28.395 --> 00:00:29.397
The logarithm of something minus

00:00:29.595 --> 00:00:30.762
the logarithm of something else,

00:00:30.762 --> 00:00:32.633
that's going to be the first something,

00:00:33.016 --> 00:00:34.034
The logarithm of the first something

00:00:34.178 --> 00:00:36.977
divided by the second something.

00:00:37.461 --> 00:00:40.139
One minus N over K,

00:00:40.188 --> 00:00:42.214
and this is of course going to be equal to

00:00:42.074 --> 00:00:43.587
all this business that we have,

00:00:44.253 --> 00:00:45.768
is going to be equal to,

00:00:45.768 --> 00:00:47.115
actually let me just write it.

00:00:47.115 --> 00:00:50.943
It's going to be equal to, equal to,

00:00:50.943 --> 00:00:55.943
R times T plus C, plus C,

00:00:57.395 --> 00:00:58.679
and now what we could do,

00:00:58.679 --> 00:01:00.062
this is the same thing as saying that

00:01:00.062 --> 00:01:03.619
E to the R T plus C is going to be

00:01:03.619 --> 00:01:06.263
equal to this thing right over here.

00:01:06.263 --> 00:01:08.479
This, the natural log of this,

00:01:08.479 --> 00:01:10.527
is equal to the exponent
that I have to raise

00:01:10.527 --> 00:01:12.592
E to to get to this right over here,

00:01:13.177 --> 00:01:14.676
so I could just write that.

00:01:14.777 --> 00:01:15.844
We could just write,

00:01:15.844 --> 00:01:16.861
or another way of thinking about it,

00:01:16.861 --> 00:01:19.293
we could take E, if this is equal to that,

00:01:19.293 --> 00:01:21.340
we could take E to this
power on this left hand side,

00:01:21.763 --> 00:01:23.781
and E to this power on
this right hand side,

00:01:23.943 --> 00:01:25.294
and they should be equal.

00:01:25.821 --> 00:01:27.724
So E to this power is just going

00:01:27.724 --> 00:01:29.292
to be what's inside the parenthesis.

00:01:29.292 --> 00:01:31.598
It's just going to be N.

00:01:31.598 --> 00:01:35.261
It's just going to be N over one minus,

00:01:35.876 --> 00:01:38.497
one minus, N over K.

00:01:38.088 --> 00:01:39.485
I'll do that in a green color,

00:01:40.277 --> 00:01:41.372
so you can keep track of
where things came from.

00:01:43.211 --> 00:01:45.222
Is equal to E to this business.

00:01:45.546 --> 00:01:50.546
Is equal to E to the R T.

00:01:52.095 --> 00:01:53.978
R, I'll do the T in white.

00:01:54.461 --> 00:01:58.594
R T plus C, plus C.

00:01:59.249 --> 00:02:02.827
Now this I could rewrite if I want to.

00:02:03.062 --> 00:02:05.194
If I have E to something
plus something else,

00:02:05.194 --> 00:02:08.221
I could rewrite this as to the R T times

00:02:08.464 --> 00:02:13.464
E times E to the C, and
just to simplify things,

00:02:14.068 --> 00:02:15.899
this is just going to be
another constant here,

00:02:16.511 --> 00:02:17.526
and I could, if this is C,

00:02:17.661 --> 00:02:18.677
I could call it C one,

00:02:18.677 --> 00:02:20.561
but I'm just going to
call that a constant.

00:02:20.066 --> 00:02:21.834
So I could just say
that that's going to be

00:02:22.428 --> 00:02:26.433
equal to some constant times E to the R T,

00:02:26.928 --> 00:02:29.727
and now we just need to solve for N.

00:02:29.727 --> 00:02:31.103
And once again, at any point while we're

00:02:31.103 --> 00:02:32.927
working on this you get inspired,

00:02:32.927 --> 00:02:34.930
feel free to solve for N.

00:02:35.761 --> 00:02:36.562
So let's see.

00:02:36.562 --> 00:02:38.027
One way to solve for N.

00:02:38.027 --> 00:02:38.675
Let's see.

00:02:38.675 --> 00:02:40.177
If we could take the reciprocal of both

00:02:40.177 --> 00:02:41.644
sides of this, we're going to get.

00:02:42.705 --> 00:02:44.777
Let me draw a line here
just so you know we're.

00:02:45.577 --> 00:02:47.376
Let me draw a line.

00:02:47.376 --> 00:02:49.927
So, if we take the reciprocal
of both sides of this,

00:02:49.927 --> 00:02:52.223
we're going to get one over N,

00:02:52.007 --> 00:02:57.007
one minus N over K over
light green N is equal to,

00:02:58.454 --> 00:03:00.126
is equal to, and so let's,

00:03:00.126 --> 00:03:02.674
we could say it's equal to,

00:03:03.262 --> 00:03:06.227
it's equal to one over C
time E to the negative R T.

00:03:06.227 --> 00:03:08.728
But one over C, that's just
going to be another constant.

00:03:09.044 --> 00:03:10.129
So I could write one over C here,

00:03:10.894 --> 00:03:11.827
and I take the reciprocal.

00:03:11.827 --> 00:03:12.595
And once again, this is just

00:03:12.595 --> 00:03:13.296
going to be another constant,

00:03:13.296 --> 00:03:15.144
so I'm going to be a
little bit of hand waving,

00:03:15.144 --> 00:03:16.262
and say well O.K we're going to

00:03:16.262 --> 00:03:17.278
get another constant here.

00:03:17.278 --> 00:03:20.161
I could have called
this C one, this C two.

00:03:20.161 --> 00:03:22.594
I could call this C three if I want

00:03:22.594 --> 00:03:24.012
to make it clear that these are not

00:03:24.012 --> 00:03:25.561
going to be the same number.

00:03:25.561 --> 00:03:27.944
This is E to the this power.

00:03:28.328 --> 00:03:30.720
This is the reciprocal of that.

00:03:30.072 --> 00:03:31.237
Actually, maybe I'll do that just to

00:03:31.885 --> 00:03:33.700
make it a little bit clearer.

00:03:33.007 --> 00:03:36.284
That would have been C
one right over there,

00:03:36.977 --> 00:03:39.178
and so this is going to be, sorry,

00:03:39.178 --> 00:03:41.126
the reciprocal of this C three

00:03:41.126 --> 00:03:42.343
and E to the negative,

00:03:42.343 --> 00:03:44.407
the reciprocal of E to the R
T is E to the negative R T.

00:03:44.983 --> 00:03:49.983
E to the negative, E to the negative R,

00:03:51.943 --> 00:03:54.461
E to the negative R T.

00:03:54.677 --> 00:03:56.511
And let's see, if we divide the numerator

00:03:56.511 --> 00:03:59.977
and the denominator by N,

00:03:59.977 --> 00:04:00.960
or if we divide E.

00:04:00.096 --> 00:04:01.329
I want to think, if we divide this

00:04:02.193 --> 00:04:04.094
term by in we're going to get.

00:04:04.094 --> 00:04:08.172
So this term by N is
going to be one over N,

00:04:08.311 --> 00:04:09.318
and then this term by N is just going

00:04:10.011 --> 00:04:12.062
to be minus one over K.

00:04:12.086 --> 00:04:13.654
So this is just going to be minus

00:04:14.428 --> 00:04:16.517
one over K is equal to this.

00:04:18.952 --> 00:04:21.130
This is equal to this business.

00:04:21.130 --> 00:04:23.357
So copy and paste.

00:04:23.711 --> 00:04:25.168
It's going to be equal to that.

00:04:25.521 --> 00:04:27.448
This is good algebra practice here.

00:04:27.448 --> 00:04:28.175
Now let's see.

00:04:28.175 --> 00:04:29.477
We could take this one over K,

00:04:29.477 --> 00:04:31.568
add it to both sides, so let's do that.

00:04:32.127 --> 00:04:36.960
So let me just cut, so
let me cut and paste it.

00:04:36.096 --> 00:04:37.748
I'm going to add it to both sides,

00:04:38.612 --> 00:04:40.794
so that should be a plus one over K.

00:04:41.594 --> 00:04:44.277
So, plus one over K.

00:04:44.277 --> 00:04:46.144
And now to solve for N,

00:04:46.144 --> 00:04:48.222
I just take the reciprocal of both sides.

00:04:48.561 --> 00:04:51.808
So I'm going to get, I'm going to get N.

00:04:52.178 --> 00:04:54.344
And I'll write it in kind
of the function notation.

00:04:54.344 --> 00:04:57.379
N of T, actually let me make my T in white

00:04:57.694 --> 00:04:58.978
since I've been taking the trouble all

00:04:58.978 --> 00:05:00.776
this time of rewriting this in white,

00:05:00.776 --> 00:05:02.527
is equal to one over.

00:05:02.527 --> 00:05:07.527
Is going to be equal to one
over all of this business.

00:05:08.511 --> 00:05:12.926
Is going to be equal to one
over all of this business.

00:05:13.793 --> 00:05:16.577
So copy and paste.

00:05:16.093 --> 00:05:18.691
Is going to be equal to that.

00:05:20.011 --> 00:05:21.235
And that by itself.

00:05:22.111 --> 00:05:24.530
That's by itself is already interesting.

00:05:25.244 --> 00:05:27.710
That by itself is already interesting.

00:05:28.861 --> 00:05:29.943
So I could write it like this.

00:05:30.928 --> 00:05:33.543
And if I want, if I don't like, let's see,

00:05:33.543 --> 00:05:34.910
well yeah I could just,

00:05:34.091 --> 00:05:36.143
if I don't like having this K kind

00:05:36.962 --> 00:05:37.895
of a fraction in a fraction,

00:05:37.895 --> 00:05:39.957
I could rewrite it as.

00:05:41.039 --> 00:05:42.563
Actually, maybe I'll do it over here.

00:05:43.048 --> 00:05:48.048
N of, N of T is equal to.

00:05:49.289 --> 00:05:51.613
I'll just multiply the numerator
and the denominator by.

00:05:51.096 --> 00:05:53.097
I'll just multiply it,

00:05:53.961 --> 00:05:56.694
actually let me just leave
it like that for now.

00:05:57.127 --> 00:06:00.478
And what I'm going do, what
I'm now going to do, is say,

00:06:00.478 --> 00:06:01.345
"Well look.

00:06:01.345 --> 00:06:04.611
"We're assuming that N
of zero is N sub knot."

00:06:04.728 --> 00:06:09.012
So we're assuming that N of zero,

00:06:09.228 --> 00:06:12.776
N of zero, is equal to N sub knot.

00:06:12.776 --> 00:06:14.228
So let's write this thing.

00:06:14.228 --> 00:06:15.253
Let's solve for the constant.

00:06:15.478 --> 00:06:17.144
Let's figure out what this could be

00:06:17.144 --> 00:06:19.362
if we know what our initial condition is.

00:06:19.677 --> 00:06:23.644
So N of zero, N of zero,

00:06:23.644 --> 00:06:26.593
is going to be equal to,

00:06:27.228 --> 00:06:31.111
is going to be equal to one, one over.

00:06:31.243 --> 00:06:32.194
When T is zero,

00:06:32.194 --> 00:06:33.978
this is just going to be equal to one,

00:06:33.978 --> 00:06:36.427
so it's just going to be our constant C

00:06:36.427 --> 00:06:41.427
three plus one over K plus one over the

00:06:41.844 --> 00:06:44.249
maximum population that our population,

00:06:44.249 --> 00:06:46.512
that our environment, can handle,

00:06:46.512 --> 00:06:49.876
and that's going to be equal to N knot,

00:06:49.876 --> 00:06:51.710
and now we can solve for our constant.

00:06:52.056 --> 00:06:53.912
So we get, let's see I'm probably going to

00:06:53.912 --> 00:06:55.326
need a lot of real estate for this,

00:06:55.326 --> 00:06:57.561
so I can take the reciprocal
of both sides again,

00:06:57.561 --> 00:06:59.889
so this is something that
we're doing a lot of.

00:07:00.162 --> 00:07:05.162
C three plus one over K is
equal to one over N knot.

00:07:06.562 --> 00:07:08.083
Is equal to one over N knot.

00:07:08.083 --> 00:07:09.894
Just took the reciprocal of both sides,

00:07:09.894 --> 00:07:12.813
and so we get our
constant C three is equal

00:07:12.813 --> 00:07:17.813
to one over N knot minus one over K.

00:07:18.262 --> 00:07:20.661
Minus one over K.

00:07:21.062 --> 00:07:23.293
So we can rewrite our solution,

00:07:23.293 --> 00:07:27.106
which we'll call the logistic
function, we get, we get,

00:07:27.106 --> 00:07:29.877
this is fun now, N of T.

00:07:30.793 --> 00:07:35.793
N of T is equal to one
over, our constant is this,

00:07:38.126 --> 00:07:41.161
so it's going to be, let
me copy and paste this.

00:07:42.076 --> 00:07:44.861
So copy and paste.

00:07:45.002 --> 00:07:47.379
It's going to be that,
that's our constant.

00:07:48.496 --> 00:07:51.499
So it's going to be that times

00:07:51.526 --> 00:07:55.127
E to the negative R T,

00:07:55.711 --> 00:07:59.228
to the negative R times T,

00:07:59.228 --> 00:08:04.228
plus one over K, plus one over K.

00:08:07.744 --> 00:08:10.428
And if we don't like having
all of these denominators in,

00:08:10.428 --> 00:08:13.144
all of these fractions in the denominator,

00:08:13.144 --> 00:08:15.477
why don't we multiply everything times the

00:08:15.477 --> 00:08:18.145
numerator and the denominator by N knot K.

00:08:18.145 --> 00:08:19.470
So I'm going to multiply the

00:08:19.047 --> 00:08:22.221
numerator times N knot K,

00:08:22.644 --> 00:08:26.378
and I'm going to multiply
the denominator by N knot K.

00:08:27.669 --> 00:08:28.745
N knot K.

00:08:28.745 --> 00:08:29.811
And so what do we get?

00:08:29.811 --> 00:08:31.972
This is all going to be equal to.

00:08:32.205 --> 00:08:37.205
In the numerator I have N
knot times K, N knot times K.

00:08:38.677 --> 00:08:42.089
And in the denominator I am going to have,

00:08:42.089 --> 00:08:43.360
let's see if I multiply this term right

00:08:43.035 --> 00:08:46.738
over here times N knot
K I'm going to have K.

00:08:48.769 --> 00:08:51.011
If I multiply this term times N knot K,

00:08:51.811 --> 00:08:53.833
I'm going to have N knot,

00:08:54.293 --> 00:08:56.277
so it's going to be minus N knot,

00:08:57.512 --> 00:09:02.512
minus N not, times E to the negative R T,

00:09:03.595 --> 00:09:08.595
Times E to the negative R T, negative R T,

00:09:10.635 --> 00:09:13.346
and then if I multiply this times N not K,

00:09:13.346 --> 00:09:15.362
I'm going to get N knot.

00:09:15.431 --> 00:09:18.026
So plus N knot.

00:09:18.146 --> 00:09:19.395
And there you have it.

00:09:19.395 --> 00:09:21.277
We have found a solution for the

00:09:21.277 --> 00:09:22.894
logistic differential equation.

00:09:22.894 --> 00:09:24.789
We will call this logistic function,

00:09:24.789 --> 00:09:26.808
and in future videos we
will explore it more,

00:09:26.979 --> 00:09:28.844
and we will see what it actually does.

00:09:28.844 --> 00:09:29.820
If you were to plot this,

00:09:29.082 --> 00:09:29.911
and I encourage you to do so,

00:09:30.649 --> 00:09:31.457
either on the internet,

00:09:31.457 --> 00:09:31.530
you could try Wolfram Alpha,

00:09:32.187 --> 00:09:33.468
or if you're on your graphing calculator,

00:09:33.468 --> 00:09:36.044
you will see that it has the exact

00:09:36.044 --> 00:09:38.110
properties that we want it to have.

00:09:38.011 --> 00:09:39.195
It starts at N knot.

00:09:39.294 --> 00:09:40.369
It starts to increase
at an increasing rate,

00:09:41.044 --> 00:09:42.395
but then it starts to slow down as

00:09:42.395 --> 00:09:44.512
we reach our maximum population.

00:09:44.512 --> 00:09:46.761
And so that is actually
a very neat function.

00:09:46.761 --> 00:09:48.777
If you model a population with this,

00:09:48.777 --> 00:09:50.679
you can kind of start to make predictions

00:09:50.679 --> 00:09:54.261
about what might the
population be at time whatever.

00:09:54.261 --> 00:09:57.542
And so, hopefully, you
found that satisfying.


Logistic differential equation intuition

WEBVTT
Kind: captions
Language: en

00:00:00.626 --> 00:00:02.781
In the last video, we took a stab at

00:00:02.781 --> 00:00:05.572
modeling population as a function of time.

00:00:05.572 --> 00:00:07.153
And we said, okay, well
maybe the rate of change

00:00:07.153 --> 00:00:09.114
of population with respect to time

00:00:09.114 --> 00:00:10.649
is going to be proportional to

00:00:10.649 --> 00:00:11.885
the population itself.

00:00:11.915 --> 00:00:13.573
That the rate will increase

00:00:13.573 --> 00:00:15.526
as the population increases.

00:00:15.743 --> 00:00:17.471
And when you actually try to solve

00:00:17.471 --> 00:00:18.704
this differential equation,

00:00:18.704 --> 00:00:20.931
you try to find an N of
T that satisfies this,

00:00:20.931 --> 00:00:22.902
we found that an exponential would work.

00:00:22.902 --> 00:00:25.858
An exponential satisfies
this differential equation.

00:00:25.913 --> 00:00:27.577
And it would look like this visually.

00:00:27.912 --> 00:00:30.503
It would look like this,

00:00:30.645 --> 00:00:32.450
it would look like this visually.

00:00:32.045 --> 00:00:33.987
Where you're starting at
a population of N not,

00:00:34.392 --> 00:00:35.981
this is the time axis,

00:00:35.981 --> 00:00:37.402
this is the population axis.

00:00:37.663 --> 00:00:38.708
And as time increases,

00:00:39.113 --> 00:00:41.122
population increases exponentially.

00:00:41.709 --> 00:00:43.182
Now we said there's an issue there.

00:00:43.359 --> 00:00:45.430
What if Thomas Malthus is right?

00:00:45.587 --> 00:00:47.542
That the environment can't support...

00:00:48.041 --> 00:00:49.305
let's say that the
environment can't support...

00:00:49.674 --> 00:00:50.815
let me do this in a new color.

00:00:50.815 --> 00:00:53.118
Let's say that the environment
really can't support

00:00:53.118 --> 00:00:58.118
more than K, more than a population of K.

00:01:02.252 --> 00:01:04.263
Then clearly the population can't just

00:01:04.372 --> 00:01:06.676
go and go right through the ceiling.

00:01:06.676 --> 00:01:07.882
They're not going to be able to have food,

00:01:07.882 --> 00:01:10.941
or water or resources
or whatever it might be.

00:01:11.472 --> 00:01:12.323
They might generate too much pollution.

00:01:12.323 --> 00:01:13.278
Who knows what it might be.

00:01:13.278 --> 00:01:18.147
And so this first stab
at modeling population

00:01:18.147 --> 00:01:19.445
doesn't quite do the trick.

00:01:19.445 --> 00:01:23.690
Especially if you are
kind of in Malthus's camp.

00:01:23.069 --> 00:01:24.135
And that's where PF, and once again

00:01:24.756 --> 00:01:26.522
I'm sure I'm mispronouncing the name,

00:01:26.638 --> 00:01:29.354
Verhulst is going to
come into the picture.

00:01:29.354 --> 00:01:30.680
Because he read Malthus's work,

00:01:30.068 --> 00:01:32.313
and said, "Well yeah, I think
I can do a pretty good job

00:01:32.925 --> 00:01:35.197
"of modeling the type of behavior

00:01:35.197 --> 00:01:37.116
"that Malthus is talking about."

00:01:37.362 --> 00:01:39.763
And he says what we really want

00:01:41.104 --> 00:01:43.288
is something...

00:01:43.361 --> 00:01:44.755
Let me write it, so the rate...

00:01:44.922 --> 00:01:45.805
let's try to model.

00:01:45.805 --> 00:01:47.444
Let's set up another
differential equation.

00:01:50.975 --> 00:01:52.800
And now let's say okay,

00:01:52.008 --> 00:01:57.008
instead of, if N is substantially smaller

00:01:58.005 --> 00:02:00.296
than what the environment
can support, then yeah,

00:02:00.791 --> 00:02:02.767
that makes sense to
have exponential growth.

00:02:06.719 --> 00:02:09.479
But maybe we can dampen this,

00:02:09.513 --> 00:02:11.576
or maybe we can bring this growth to zero

00:02:11.576 --> 00:02:15.652
as N approaches K.

00:02:16.163 --> 00:02:18.940
And so how can we actually modify this?

00:02:18.094 --> 00:02:19.589
Maybe we can multiply it by something

00:02:20.635 --> 00:02:23.636
that for when N is small,

00:02:23.665 --> 00:02:25.693
when N is much smaller than K,

00:02:25.892 --> 00:02:28.407
this term right over here
is going to be close to one

00:02:28.567 --> 00:02:30.396
and when N is close to K,

00:02:30.396 --> 00:02:32.292
this term is close to zero.

00:02:32.292 --> 00:02:33.139
So let me write that.

00:02:34.221 --> 00:02:38.304
So these are our goals for
this term, right over here.

00:02:38.349 --> 00:02:43.349
When N is much smaller than K,

00:02:51.008 --> 00:02:52.541
so now the population is
not constrained at all,

00:02:53.333 --> 00:02:54.275
people can have babies

00:02:54.275 --> 00:02:55.441
and those babies can be fed

00:02:55.441 --> 00:02:57.474
and then they can have
babies, et cetera, et cetera,

00:02:57.078 --> 00:03:02.078
then this thing should be close to one.

00:03:02.914 --> 00:03:05.220
And so then you have
essentially our old model.

00:03:05.221 --> 00:03:06.930
But then as N approaches K,

00:03:10.713 --> 00:03:14.180
then this thing should approach,

00:03:14.426 --> 00:03:17.744
then this term or this expression

00:03:17.744 --> 00:03:19.438
should approach zero.

00:03:19.487 --> 00:03:21.219
And what that does is as N approaches

00:03:21.219 --> 00:03:25.651
the natural limit, the
ceiling to population,

00:03:25.805 --> 00:03:27.546
then no matter what this is doing,

00:03:27.546 --> 00:03:28.911
if this thing is approaching zero,

00:03:28.911 --> 00:03:30.765
that's going to make the actual

00:03:30.765 --> 00:03:33.200
rate of growth approach zero.

00:03:33.002 --> 00:03:35.363
So food is going to be more scarce,

00:03:35.561 --> 00:03:37.234
it's going to be harder to find things.

00:03:37.665 --> 00:03:40.670
And so what can I construct here,

00:03:40.067 --> 00:03:41.776
dealing with N and K that
will have these properties?

00:03:42.431 --> 00:03:44.970
And for fun, you might actually want to

00:03:44.097 --> 00:03:44.986
pause the video and see
if you can construct

00:03:45.859 --> 00:03:48.194
a fairly simple algebraic statement

00:03:48.194 --> 00:03:50.706
using N and K and maybe the number one

00:03:50.706 --> 00:03:52.215
if you find the need,

00:03:52.783 --> 00:03:55.500
for an expression that
has these properties.

00:03:56.869 --> 00:03:57.715
Well, let's see.

00:03:57.715 --> 00:03:58.786
What if we start with a one

00:04:01.329 --> 00:04:05.434
and we subtract N over K?

00:04:13.066 --> 00:04:14.649
Does this have those properties?

00:04:15.945 --> 00:04:16.961
Well yeah, sure it does.

00:04:16.961 --> 00:04:19.387
When N is really small, or I should say

00:04:19.387 --> 00:04:21.897
when it's a small fraction of K,

00:04:23.543 --> 00:04:24.644
this is going to be a small fraction,

00:04:24.644 --> 00:04:25.941
then this whole thing is going to be

00:04:25.941 --> 00:04:27.400
pretty close to one.

00:04:27.004 --> 00:04:28.933
It's going to be a
little bit less than one.

00:04:29.544 --> 00:04:30.958
And when N approaches K,

00:04:30.958 --> 00:04:33.247
as N gets closer and
closer and closer to K,

00:04:33.682 --> 00:04:34.774
then this thing right over here

00:04:35.602 --> 00:04:37.274
is going to approach one,

00:04:37.274 --> 00:04:38.587
which means this whole expression

00:04:38.587 --> 00:04:39.875
is going to approach zero.

00:04:39.875 --> 00:04:42.720
Which is exactly what we wanted.

00:04:42.072 --> 00:04:44.178
And this thing right
over here is actually,

00:04:44.968 --> 00:04:47.536
and this is used in tons of applications,

00:04:47.536 --> 00:04:50.582
not just in population modeling

00:04:50.582 --> 00:04:53.641
but that's kind of one of its first

00:04:53.641 --> 00:04:55.960
reasons or motivations.

00:04:56.186 --> 00:04:57.261
This differential equation right over here

00:04:57.936 --> 00:04:59.135
is actually quite famous.

00:04:59.359 --> 00:05:02.450
It's called the logistic
differential equation.

00:05:13.346 --> 00:05:14.169
And in the next video,

00:05:14.169 --> 00:05:15.198
we're actually going to solve this.

00:05:15.639 --> 00:05:17.360
And it's actually, this is a separable

00:05:17.036 --> 00:05:17.978
differential equation.

00:05:18.302 --> 00:05:19.365
You can actually solve it just using

00:05:19.932 --> 00:05:21.540
standard techniques of integration.

00:05:21.054 --> 00:05:22.358
it's a little bit hairier than this one,

00:05:22.844 --> 00:05:24.498
so we're going to work
through it together.

00:05:24.498 --> 00:05:25.823
And we're going to look at the solution.

00:05:25.823 --> 00:05:28.768
The solution to the logistic
differential equation

00:05:28.768 --> 00:05:33.752
is the logistic function, which once again

00:05:33.752 --> 00:05:37.330
essentially models population in this way.

00:05:37.033 --> 00:05:39.209
But before we actually solve for it,

00:05:39.506 --> 00:05:41.320
let's just try to interpret this

00:05:41.032 --> 00:05:43.239
differential equation and think about

00:05:43.527 --> 00:05:46.123
what the shape of this
function might look like.

00:05:46.589 --> 00:05:47.602
And to do that,

00:05:52.723 --> 00:05:56.811
let me draw some axes here.

00:05:58.069 --> 00:05:59.323
Let me draw some axes here.

00:05:59.944 --> 00:06:01.508
So that's my time axis,

00:06:01.508 --> 00:06:03.201
that's my population axis,

00:06:03.801 --> 00:06:05.340
let me scroll up a little
bit because sometimes

00:06:05.034 --> 00:06:06.131
the subtitles show up around here

00:06:07.031 --> 00:06:08.552
and then people can't see what's going on.

00:06:09.454 --> 00:06:11.126
So let's think about a
couple of permutations,

00:06:11.126 --> 00:06:12.750
a couple situations.

00:06:12.075 --> 00:06:16.583
So if our initial, if our
N at time equals zero,

00:06:18.097 --> 00:06:18.738
remember N is a function of T.

00:06:19.648 --> 00:06:22.580
If at time equals zero N is equal to zero,

00:06:24.178 --> 00:06:25.231
then this term is going to be zero

00:06:25.708 --> 00:06:27.917
and then your rate of
change is going to be zero,

00:06:27.917 --> 00:06:29.988
and so you're not going
to add any population.

00:06:30.244 --> 00:06:31.128
And that's good.

00:06:31.554 --> 00:06:33.318
Because if your population is zero,

00:06:33.318 --> 00:06:34.703
how are you going to actually be able

00:06:34.703 --> 00:06:35.666
to add population?

00:06:35.666 --> 00:06:37.755
There's no one there to have children.

00:06:37.884 --> 00:06:40.326
So there's actually one constant solution

00:06:40.326 --> 00:06:42.487
to this differential equation,

00:06:44.648 --> 00:06:46.810
which is just N of T is equal to zero.

00:06:46.081 --> 00:06:49.057
And this satisfies the
logistic differential equation.

00:06:49.008 --> 00:06:50.903
Hey, if your population starts at zero

00:06:51.695 --> 00:06:53.843
if N sub not is zero,

00:06:53.934 --> 00:06:55.441
then you're just going
to be at zero forever.

00:06:55.441 --> 00:06:57.954
Well that's actually what
would happen in real life

00:06:57.954 --> 00:06:59.680
if there's no one there to have kids.

00:07:00.018 --> 00:07:01.097
Now let's think about another situation.

00:07:01.097 --> 00:07:02.081
What if our population,

00:07:02.954 --> 00:07:04.939
what if N not is equal to K?

00:07:05.904 --> 00:07:07.953
What happens if N not is equal to,

00:07:11.489 --> 00:07:12.685
so that's K right over there,

00:07:12.685 --> 00:07:14.788
what happens if at time equals zero,

00:07:15.182 --> 00:07:16.706
this is our population.

00:07:16.706 --> 00:07:18.640
Well if N is equal to K,

00:07:18.064 --> 00:07:19.077
then this is one minus one,

00:07:19.653 --> 00:07:20.665
then this thing is zero,

00:07:20.665 --> 00:07:21.833
and so our rate of population change

00:07:21.833 --> 00:07:22.835
is going to be zero.

00:07:23.223 --> 00:07:24.904
So essentially if my population is zero

00:07:24.904 --> 00:07:25.926
then after a little bit of time,

00:07:25.926 --> 00:07:27.342
my population is still the same K.

00:07:27.623 --> 00:07:29.867
If my rate of change
of population is zero,

00:07:29.867 --> 00:07:31.950
that means my population
is staying constant.

00:07:32.352 --> 00:07:34.390
And so my population's just going

00:07:34.039 --> 00:07:35.498
to stay there at K.

00:07:36.084 --> 00:07:36.578
And that's actually believable.

00:07:37.334 --> 00:07:38.687
Malthus would actually probably say

00:07:38.687 --> 00:07:40.311
that you're gonna have, maybe it grows

00:07:40.311 --> 00:07:43.674
a little bit beyond the
capacity of the environment

00:07:43.674 --> 00:07:46.577
and then you have some flood,

00:07:46.577 --> 00:07:48.406
or some hurricane or some famine

00:07:48.406 --> 00:07:49.408
and it goes around.

00:07:49.534 --> 00:07:51.716
But for our purposes, you can never model

00:07:51.716 --> 00:07:52.758
anything perfectly.

00:07:52.948 --> 00:07:55.116
For our purposes, that's pretty good.

00:07:55.116 --> 00:07:57.110
You're at the limit of what the

00:07:57.011 --> 00:07:59.016
environment can handle,

00:07:59.115 --> 00:08:01.650
so you just kind of stay there.

00:08:01.065 --> 00:08:02.320
That's actually another constant solution.

00:08:02.905 --> 00:08:05.147
that N of T, if it starts,

00:08:05.147 --> 00:08:06.769
and now you can kind of appreciate

00:08:06.769 --> 00:08:08.378
why initial conditions are important.

00:08:08.378 --> 00:08:09.530
If you start at zero you're
going to stay at zero,

00:08:09.053 --> 00:08:10.107
if you start at K you're
going to stay at K.

00:08:11.345 --> 00:08:13.947
So that is N of T just stays K.

00:08:14.694 --> 00:08:16.780
But now let's think of a
more interesting scenario.

00:08:17.253 --> 00:08:20.089
Let's assume an initial population

00:08:20.089 --> 00:08:22.492
that's someplace between zero and K.

00:08:22.492 --> 00:08:24.095
So this is going to be...

00:08:25.484 --> 00:08:26.575
I'm going to assume initial population

00:08:27.052 --> 00:08:30.958
that is someplace, it's greater than zero,

00:08:31.605 --> 00:08:34.030
so there are people to
actually have children,

00:08:34.025 --> 00:08:35.583
and it is less than K,

00:08:35.808 --> 00:08:38.823
so we aren't fully maxing
out the environment.

00:08:38.929 --> 00:08:40.885
Or the land or the food or the water

00:08:40.885 --> 00:08:41.874
or whatever it might be.

00:08:41.874 --> 00:08:43.138
So what's going to happen?

00:08:43.462 --> 00:08:44.535
So, and once again, I'm just going to

00:08:44.535 --> 00:08:45.636
kind of sketch it,

00:08:45.636 --> 00:08:47.595
and then we're going to
solve it in the next video.

00:08:48.215 --> 00:08:50.071
So when N is a lot less than K,

00:08:50.071 --> 00:08:51.128
it's a small fraction of K,

00:08:53.548 --> 00:08:54.589
this term is going to be the main one

00:08:54.958 --> 00:08:56.427
that's influencing it.

00:08:56.427 --> 00:08:58.425
Because this is a small fraction of K,

00:08:58.425 --> 00:08:59.363
I mean even the way I drew it,

00:08:59.363 --> 00:09:02.441
it looks like it's about
a sixth or a seventh

00:09:03.143 --> 00:09:04.186
or an eighth of K,

00:09:04.501 --> 00:09:06.279
so it's one minus 1/8.

00:09:06.279 --> 00:09:11.163
So it's gonna be 7/8 times this.

00:09:11.335 --> 00:09:14.883
So really, this is what's going to dictate

00:09:16.067 --> 00:09:17.125
what our rate of growth is.

00:09:17.417 --> 00:09:18.852
And if this is kind of dictating it,

00:09:18.852 --> 00:09:20.929
we're kind of looking more of a...

00:09:21.145 --> 00:09:22.532
well, let's just think of it this way.

00:09:22.582 --> 00:09:23.691
As the population grows,

00:09:23.691 --> 00:09:27.172
the rate of change is going to grow.

00:09:27.484 --> 00:09:32.484
So it's going to look something like this.

00:09:33.004 --> 00:09:34.076
As our population gets larger,

00:09:35.012 --> 00:09:37.101
our slope is getting higher.

00:09:37.209 --> 00:09:38.724
It's getting steeper and steeper.

00:09:39.153 --> 00:09:41.041
But then as N approaches K,

00:09:41.246 --> 00:09:42.951
then this thing is gonna become,

00:09:42.989 --> 00:09:44.698
this is gonna be close to one minus...

00:09:44.698 --> 00:09:46.099
close to one.

00:09:46.099 --> 00:09:48.118
And so this is going to
become a very small number,

00:09:48.118 --> 00:09:50.204
it's going to make this
whole thing approach zero.

00:09:50.646 --> 00:09:52.162
So as N approaches K,

00:09:52.771 --> 00:09:54.548
the whole thing, the rate of change

00:09:54.548 --> 00:09:57.029
is going to flatten out.

00:09:57.459 --> 00:09:59.440
And we're gonna asymptote towards K.

00:09:59.858 --> 00:10:03.441
And so the solution to the

00:10:03.441 --> 00:10:05.493
logistic differential equation

00:10:05.961 --> 00:10:07.874
should look something like this.

00:10:07.874 --> 00:10:09.679
Depending on what your
initial conditions are.

00:10:09.679 --> 00:10:10.756
If your initial condition is here,

00:10:10.756 --> 00:10:12.091
maybe it does something like this.

00:10:12.091 --> 00:10:13.415
If your initial condition's here,

00:10:13.415 --> 00:10:16.393
maybe it does something like this.

00:10:16.393 --> 00:10:17.559
And once again, this is what's fun

00:10:17.559 --> 00:10:18.361
about doing differential equations.

00:10:18.361 --> 00:10:19.964
Before even doing the fancy math,

00:10:19.964 --> 00:10:21.221
you can kind of get an intuition,

00:10:21.221 --> 00:10:22.987
just by thinking through
the differential equation

00:10:22.987 --> 00:10:26.325
of what it is likely to be.

00:10:26.438 --> 00:10:30.449
Down here, or when N
is much smaller than K

00:10:30.072 --> 00:10:32.056
its rate of increase is increasing

00:10:32.704 --> 00:10:34.106
as N increases,

00:10:34.148 --> 00:10:35.224
and over here as N gets close to K,

00:10:35.908 --> 00:10:38.996
its rate of increase is decreasing.

00:10:39.351 --> 00:10:40.458
Now let's actually in the next video,

00:10:40.458 --> 00:10:43.526
actually solve for what
N, a solution to this

00:10:44.138 --> 00:10:47.129
and see if that confirms
what our intuition is.