Hi. It's Mr. Andersen and in this video I'm going to talk about exponential
growth which is how populations can explode. Most students understand exponential growth,
but the math sometimes gets a little tricky. And so I am going to step you through that
in a couple of ways. And so let's start with this rabbit right here. Let's say it's part
of a population. We don't only have one rabbit, but we have a number of rabbits in our population.
We refer to that in all of the equations as N. N is going to be the population size. Now
that's going to change as we go through time. But this is going to be our original population
which is going to be N. Let's stack those rabbits up so we can count them. So our N
to start is going to be 10. So we have 10 rabbits at time 0. Now that population is
either going to increase, it's going to decrease or it's going to stay the same. And what things
are determining that? It's going to be our growth rate. And so this is the second letter
you should remember. And that's r. r is going to refer to how much it's changing over time.
And there's really only two things that are going to change that population. We're going
to have new rabbits, that's going to be births. And then we're going to have dead rabbits
and that's going to be deaths. And so those two things are going to contribute to the
change in the population. But it's per capita. In other words we have to divide by the N.
Which is going to be the original population size. And so let me make some baby rabbits.
So if I click here we've got 5 baby rabbits. And so our births would be five. And then
let's say I want to kill a couple of rabbits. Let's kill that guy. Don't worry, they're
okay. They're just virtual rabbits. And so I kill that one as well. And so we've got
births of 5. We've now got deaths of 2. And what was our N to begin with? It was 10. And
so we figure out our r value. That's going to be 5 minus 2 divided by 10, which is 3
over 10, which is going to be 0.3. And so our r value is 0.3 or our growth rate is 0.3.
What does that really mean? It's the factor at which our population is increasing. And
so if I take 10 times 0.3 I'm going to get 3. And that's how much our population increased.
And one thing you should know about that growth rate is that if the ecosystem is stable the
growth rate is essentially going to stay the same. It's not going to change over time.
And so you might think, well, if the growth rate stays the same, isn't the population
just going to increase along a consistent amount? Not really. And so let's watch what
happens. Now we're going to take 0.3 growth rate for the next generation and instead of
multiplying it times 10 we now have to multiply it times 13. And if we do that and we'll use
this equation right here, this is the change in N or the change in t. We're going to take
our growth rate with is 0.3 and now multiply it times 13. Well we don't get three anymore.
We get 3.9. And so I'm going to round. That sounds a lot like 4 rabbits. So I'm going
to add 4 rabbits. And now our population is up to 17. So even though r stayed the same
since we multiplied it times a larger value, we're going to get more rabbits in the next
generation. So let's do generation 3. We're now taking 0.3 times 17. And I get 5.1, which
is a lot like 5 rabbits. And I'm going to add those two rabbits. Or if we now multiply
that growth rate times 22, I get 6.6 which is pretty close to 7 rabbits. So we're going
to add those 7 rabbits. And so we now have got a population of 29. And so you can see
that the population is increasing. But if I were to ask you a question, I could ask
you some hard questions. The first one is not so hard. What's the population going to
be in year 5? Well to do that you take 29 times 0.3 and then we'd add that to 29. But
what if I asked you 10. Or even 30? Well this problem get's pretty hard. And so you're quickly
going to want a little bit of help. And for me when I want help the first place I go to
is to a spreadsheet. And let's go to the spreadsheet. So we're going to go to Excel. Kind of remember
those numbers there. And so let's kind of rebuild that chart. So on the left side we're
going to have 0 as our first time and 1 as our second time. If you didn't know to do
this in Excel, I can select both of those, grab this little corner here and I can increase
and it will do the counting for me. And so let's go up to population in time 30. Okay.
So we've filled that in. Now what's our original population? That's going to be 10. So I'm
got to just put that in to start with. Now what's the next population? So I'm going to
put a formula in this box. And to do that I'm just going to put an = sign. So I'm going
to put an = sign, you can see the formula right here. And so what did we do? Remember
we're going to take the original population, so let me click on that. So that's going to
be 10. And then we're going to add that, to that, we're going to add our growth rate which
was 0.3 times then we're going to click on that again. And so let's see what we get.
So we get 13. And so what we did is again, we took what was in this cell and we added
it to what was in this cell times 0.3. And so again through the magic of a spreadsheet
I can simply grab this and it's just going to use that same thing over and over. It's
going to iterate on that. And so what we're going to get is it's going to do all of the
math for us. And so what did we have in the first one? 13, 17, 22. This sounds familiar.
Now it's obviously not rounded off and so this isn't the correct number of rabbits,
but we see now this exponential curve here. Or this J shaped curve. Or sometimes we call
it like a hockey stick curve, because it's quickly turning up like that. And so now we
could quickly answer those questions. At time 5 we should have around 37 rabbits. What was
the next one? I think 10. We should have around 138 rabbits. And if we go all the way down
here to time 30 we're going to have 200 and, 26199 rabbits. So that's a lot of rabbits
really really quickly. And so you can see how exponential growth takes off. But what's
fun about a spreadsheet is I could play around with it a little bit. Let's say instead of
0.3, if I change my growth rate to 0.1. So if I do that, what are we going to get? Well
if I move this all the way down again were going to get another J shaped curve. Now it's
not going to be as steep as that one was. And we didn't have as many rabbits at the
end. But we're still going to be exponential growth. Or if I were to go edit that variable
again. Let's make it 0. So let's say we take it times 0. What would we get then? Well it's
10. In other words if I go all the way down here, what are we going to get for a value?
Well, we're not increasing at all. And so it's going to be 10. It's going to stay 10
the whole time. A really hard question that I ask the students is this. Let's say we get
a negative growth rate. So let's make it -0.3. What are we going to get for a value there?
Well we get 7 here. But if I go all the way down, what do we get? Oh that's weird. We're
going to approach kind of a limit. We're going to approach 0. And that's because we're going
to take off larger amounts to begin with. And then we're going to take off less amounts
as the population gets smaller and smaller and smaller. So again that's spreadsheets.
But you don't always have a spreadsheet with you. Sometimes you just need a calculator
and a little bit of algebra. So let's go to the algebra. This is going to be the algebraic
solution to this. And so we've got an equation for exponential growth. And so change in N
over t is going to be equal to N, which is our population size, times 1 plus r, where
r is going to be the growth rate and then we're going to raise that to t, where t is
going to be equal to time. And so let's make sure that this works. Let's start right here
with 0. And so let's plug in our numbers. So we're going to put 10 in here for n. That's
that original population. 1 and then r is going to go right here. It's going to be 0.3.
And then we raise it to the 0 power. Because our time is going to be 0. Well if we simplify
that a little bit, anything raised to the 0 power is always 1. And so that's going to
be 10. And so that works out so far. But don't trust me. Let's keep going. Let's go to the
next one. Let's go to 1. So if we now put in 1 for time. It's going to be, N is still
10. This is 1 plus r again. And the r is not changing. But we're raising it to the one
power. Anything raised to the 1st power is going to be itself. So it's going to be 1.3.
And we get 13. Let's try that again with 2. So if we go to the second power again, we
plug in 2 here. It's the only thing that we're changing. So we get 1.3 to the second power.
So we're going to have to square 1.3 which is 1.69 and we get 16.9 which is a lot like
17. Or I could just throw out another time. So let's say we go time 30. So then we're
going to raise it to the 30th power. And so I get 26,199 which is going to match up exactly
with our spreadsheet. We're going to have a lot of rabbits really really quickly. And
so that's kind of an algebraic solution. It's a quick way to be given a time and then figure
out how much it's going to grow. And so a good question I could ask you is this. Let's
look at bacteria rather than rabbits. Okay? And so let's say we have one bacteria. E.
coli can reproduce in about 20 minutes. In other words 1 can make an exact copy of itself
in about 20 minutes. And so let's say that none of them die. Let's say we get rid of
the death rate. So we've got our births over N. And so how many births would we have? If
we're just dividing in half we're going to have one new E. coli. What was our original
population? It was going to be 1 as well. So now we're going to get an r of 1. And so
instead of increasing by 0.3, we're now increasing by 1 which is really increasing by 100%. So
we had 1 bacteria and now we have 100% as many bacteria or twice as many. Or 100% in
addition to that original. And so now we have 2. And so what are we going to have on the
next round? We're going to have 4. And 8. And 16. And then 32. And you can see how exponential
growth gives us a huge amount of bacteria really, really quickly. And so the question
I might ask is, is the sky the limit? And so if we're looking at exponential growth
you know, after 20 rounds like this we're going to be way up in here. I can't even read
this, 5 million, something like that. And so it does it just keep going and going and
going? No. Because what I told you is kind of a lie. r is not going to stay the same
forever. As it starts to grow they're going to run out of food, resources, space. And
so our r is going to start to change. And so then we're going to start to move into
what's logistic growth. And that's going to be a totally different set of equations and
I'm going to include that in another video. And so that's exponential growth. And I hope
that was helpful.