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Hi. It's Mr. Andersen and in
this video I'm going to talk about exponential
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growth which is how populations can explode.
Most students understand exponential growth,
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but the math sometimes gets a little tricky.
And so I am going to step you through that
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in a couple of ways. And so let's start with
this rabbit right here. Let's say it's part
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of a population. We don't only have one rabbit,
but we have a number of rabbits in our population.
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We refer to that in all of the equations as
N. N is going to be the population size. Now
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that's going to change as we go through time.
But this is going to be our original population
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which is going to be N. Let's stack those
rabbits up so we can count them. So our N
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to start is going to be 10. So we have 10
rabbits at time 0. Now that population is
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either going to increase, it's going to decrease
or it's going to stay the same. And what things
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are determining that? It's going to be our
growth rate. And so this is the second letter
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you should remember. And that's r. r is going
to refer to how much it's changing over time.
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And there's really only two things that are
going to change that population. We're going
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to have new rabbits, that's going to be births.
And then we're going to have dead rabbits
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and that's going to be deaths. And so those
two things are going to contribute to the
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change in the population. But it's per capita.
In other words we have to divide by the N.
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Which is going to be the original population
size. And so let me make some baby rabbits.
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So if I click here we've got 5 baby rabbits.
And so our births would be five. And then
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let's say I want to kill a couple of rabbits.
Let's kill that guy. Don't worry, they're
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okay. They're just virtual rabbits. And so
I kill that one as well. And so we've got
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births of 5. We've now got deaths of 2. And
what was our N to begin with? It was 10. And
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so we figure out our r value. That's going
to be 5 minus 2 divided by 10, which is 3
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over 10, which is going to be 0.3. And so
our r value is 0.3 or our growth rate is 0.3.
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What does that really mean? It's the factor
at which our population is increasing. And
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so if I take 10 times 0.3 I'm going to get
3. And that's how much our population increased.
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And one thing you should know about that growth
rate is that if the ecosystem is stable the
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growth rate is essentially going to stay the
same. It's not going to change over time.
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And so you might think, well, if the growth
rate stays the same, isn't the population
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just going to increase along a consistent
amount? Not really. And so let's watch what
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happens. Now we're going to take 0.3 growth
rate for the next generation and instead of
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multiplying it times 10 we now have to multiply
it times 13. And if we do that and we'll use
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this equation right here, this is the change
in N or the change in t. We're going to take
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our growth rate with is 0.3 and now multiply
it times 13. Well we don't get three anymore.
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We get 3.9. And so I'm going to round. That
sounds a lot like 4 rabbits. So I'm going
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to add 4 rabbits. And now our population is
up to 17. So even though r stayed the same
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since we multiplied it times a larger value,
we're going to get more rabbits in the next
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generation. So let's do generation 3. We're
now taking 0.3 times 17. And I get 5.1, which
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is a lot like 5 rabbits. And I'm going to
add those two rabbits. Or if we now multiply
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that growth rate times 22, I get 6.6 which
is pretty close to 7 rabbits. So we're going
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to add those 7 rabbits. And so we now have
got a population of 29. And so you can see
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that the population is increasing. But if
I were to ask you a question, I could ask
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you some hard questions. The first one is
not so hard. What's the population going to
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be in year 5? Well to do that you take 29
times 0.3 and then we'd add that to 29. But
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what if I asked you 10. Or even 30? Well this
problem get's pretty hard. And so you're quickly
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going to want a little bit of help. And for
me when I want help the first place I go to
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is to a spreadsheet. And let's go to the spreadsheet.
So we're going to go to Excel. Kind of remember
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those numbers there. And so let's kind of
rebuild that chart. So on the left side we're
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going to have 0 as our first time and 1 as
our second time. If you didn't know to do
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this in Excel, I can select both of those,
grab this little corner here and I can increase
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and it will do the counting for me. And so
let's go up to population in time 30. Okay.
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