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And, we're going to make a major
shift.  You're going to feel like

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this is a whole different class
compared to what we were talking

3
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about last time,
because were jumping from the

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biogeochemical cycles,
or looking at the biosphere as

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00:00:19,000 --> 00:00:23,000
essentially a large biochemical
machine, to studying individual

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00:00:23,000 --> 00:00:28,000
populations of organisms,
and the communities that they make

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up when they come together.
So, before we were really talking

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about organisms as they function in
the biosphere.

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Mentally, we're grinding them all
up and thinking of them as a

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collective biochemistry basically.
And now we are going to stop

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grinding them up,
mentally, and think of them as

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individual organisms.
So, the next series of lectures,

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we're going to talk about population
ecology.

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If you remember the first lecture I
gave we talked about the hierarchy

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of organization within ecological
systems, and then we are going to

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talk about competition between
organisms with a population,

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and between organisms of different
species, and were going to talk

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about predation, and mutualism.
These are all interactions between

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organisms that affect the fitness of
organisms.  And then we'll,

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at the end, talk about community
structure.  So this is sort of the

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outline for the rest of my lectures,
not for this lecture.

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So, today we are going to talk about
properties of populations.

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We're going to analyze how we
measure growth rate,

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growth and death in populations,
and this will include populations

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that have an age structure,
and populations that don't.

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And this is all in preparation for
the next lecture where we will talk

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about human population growth.
So, in this field of population

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ecology, which is as I told you in
the first lecture,

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and some universities you could take
three courses in population ecology,

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00:03:01,000 --> 00:03:06,000
and you could get a Ph.D. in
population ecology.

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I mean, this is a whole field that
we're going to cover in two lectures.

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But what population ecologists
worry about fundamentally,

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well, they don't worry about it.
This is what they study, is what

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regulates the density of populations?
Obviously, it's a function of how

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fast they're growing,
the birth rate, and how fast they're

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dying, the death rate.
But what are the factors that

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actually influence those rates?
Is it competition with other

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organisms?  Is it the entire
structure of the community?

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Is it the availability of food?
Is it the various abiotic

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properties of the environment:
temperature, etc.

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So, they analyze these and
basically try to model the

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population growth as a function of
these various parameters.

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The other questions they ask,
is how are populations distributed

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in the environment?
Are they clustered?

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00:04:10,000 --> 00:04:14,000
Are they evenly distributed?
This has specific meanings about

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00:04:14,000 --> 00:04:19,000
their ecology.
And, the other thing that people

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are really fascinated by,
which is a really tough question,

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is why are some species' populations
extremely abundant, while

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others are rare?
And one of the discussions we always

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00:04:32,000 --> 00:04:36,000
have in my lab,
we work on an organism that's

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extremely abundant,
this prochlorococcus,

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which I told you briefly about,
is the most abundant photosynthetic

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cell on the planet.
So, my students tend to keep saying

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00:04:49,000 --> 00:04:53,000
why is it so successful?
And I keep saying, it's successful

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00:04:53,000 --> 00:04:57,000
but there are thousands of other
species who are also successful.

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00:04:57,000 --> 00:05:02,000
Abundance does not equal success.
Endurance equals success.

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If you're here in the next
generation, you're successful.

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If you're not, if your species is
disappearing, then you're not

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successful.  So,
speaking of abundance,

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let's talk about how we measure
abundance, population ecologists.

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And this is just one example.
Obviously, for microorganisms,

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or some microorganisms it's really
easy because they're tiny relative

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to their habitats.
So for the prochlorococcus that we

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work on, there are 10^5 cells per
milliliter.  So,

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we can go take a milliliter of water
and measure how many cells there.

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But for some organisms, larger ones,
that are widely distributed,

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it's not that easy.  So, one method
is mark and recapture.

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That's used a lot for things like
birds and butterflies.

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00:06:04,000 --> 00:06:09,000
For a bird, the mark would be
putting a band on the bird.

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00:06:09,000 --> 00:06:14,000
For a butterfly, they often take a
magic marker and put a mark on the

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wing.  Well, that's largely what
they do.  You try to mark

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individuals in some way that would
not influence their survivorship

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00:06:24,000 --> 00:06:29,000
rate.
So, if N equals the population size,

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that is, that's our unknown, what
we're going to do is capture,

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say, for butterflies or moths,
you use a butterfly net, or moths

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you can use a light to track them;
for birds, you put up these big mist

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nets.  They fly into them; they get
tangled up a little bit but they

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don't get hurt.
Then you band them,

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and that we let them go.
That's the way you mark them.

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00:06:59,000 --> 00:07:07,000
So, we're going to say n1 equals the
total number of marked individuals

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released.  So you capture them,
you mark them, you release them.

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00:07:15,000 --> 00:07:24,000
n2 is equal to, and then you go out
sometime later and you recapture as

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00:07:24,000 --> 00:07:32,000
many individuals as you can find,
and this would be the total number

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00:07:32,000 --> 00:07:41,000
[SIREN] that doesn't sound like a
fire drill, does it?

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00:07:41,000 --> 00:07:50,000
I assume we're good to go here.
So, n2 is the total number of

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recaptured.  And we're going to say
m2 is equal to the numbers

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recaptured that are marked.
OK, and then we assume that the

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00:08:09,000 --> 00:08:18,000
fraction of the recaptured that are
marked represent the fraction in the

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total population that was marked.
So, we say m2 over  n2 is equal to

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00:08:27,000 --> 00:08:34,000
n1 over N.
And the number that we're looking

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00:08:34,000 --> 00:08:40,000
for, population size,
is equal to n1, n2 divided by m2.

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So, of course, this assumes that
there's no effect of the marking of

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the individuals.
It assumes that there's no bias in

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the trapping for the marked or not
marked individuals.

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00:08:56,000 --> 00:09:02,000
There's all kinds of assumptions
that underlie this.

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00:09:02,000 --> 00:09:07,000
It's a start for assessing the
population size.

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00:09:07,000 --> 00:09:13,000
OK, so how do we measure population
growth?  We're going to first start

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with looking at populations that
have age structure.

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Now, I hope you printed out the
slides that were on the Web,

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because I'm depending on these
overheads a lot for this lecture

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00:09:31,000 --> 00:09:37,000
because we wouldn't get through any
of it if I wrote all this

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00:09:37,000 --> 00:09:44,000
stuff on the board.
So, we're going to talk about

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populations that have an age
structure.  And the data I'm going

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00:09:53,000 --> 00:10:03,000
to show you here is for
human populations.

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But this applies to any population
that has differential birth and

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death rates as a function of the age
of the organism,

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OK?  So, in these populations if
birth rate and death rate are high,

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the population is dominated by young
people.

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And, we'll look at this in a minute.
And, if B and D are low,

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dominated by old people,
or older I should say, since I now

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fit into the old category.
OK, so here's a typical population

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00:11:12,000 --> 00:11:19,000
age distribution for developed
countries, where each slice here,

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00:11:19,000 --> 00:11:25,000
these are females on the right,
males on the left,

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and each slice is an age category:
zero to 10 years, 10 to 20.

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And you can see that in these kinds
of populations,

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you have a fairly even age
distribution.  Long periods of no

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net growth in a population lead to
this.  In these developed countries,

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and we're going to examine why this
is, there's basically an even

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replacement rate of children for
adults.  And one of the things we

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worry about when you see this kind
of age distribution,

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although it's good in terms of
population growth,

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is when you have few young people
and a lot of older people,

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who's going to take care of them,
which is what's behind the Social

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Security crisis.
But we won't get into that.

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Since you're the young people and
I'm the old people,

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I don't want to dwell on that.
OK, so what demographers do for

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human populations is project what
the population will look like in the

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future based on the reproductive
rates of the present.

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And you can see for the US here,
it's reasonably stable if you look

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at these three snapshots.
We're going to go backwards

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starting with 1950,
and show you what the population has

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been doing since 1950.
And I'm just going to walk through

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this.  You only have one in your
handouts, but I'll show you how it's

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moving along.
Moving along, you can think of this

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as generations moving through the
population.  And this is the date up

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00:13:13,000 --> 00:13:19,000
here.  So, this is 1950,
1955, you can see this red cohort.

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A cohort is a group of individuals
that were born at roughly the same

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time.  So, you can see that red
cohort there.  And we are going

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along, 1965.  This lip here,
that we can now see, is the postwar

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baby boom.
That's what I'm a member of.

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If you can see it in this bulge in
this population.

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And now were marching along.
Here's my cohort, and I just put

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these lines on to keep you oriented.
And here comes you guys.  I think

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those are you guys,
1985.  That's roughly right,

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00:14:03,000 --> 00:14:09,000
because I never know when I've last
updated these slides.

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00:14:09,000 --> 00:14:14,000
So, and here you go.
See, here's the big bulge of all of

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these baby boomers that you guys are
going to have to take care of.

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And now, we can actually see an
echo.  This is what's called the

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00:14:24,000 --> 00:14:29,000
baby boom echo.
These are the kids of the baby

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boomers, which is you guys.
But you can only see that as we

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00:14:34,000 --> 00:14:40,000
march through it.
So, here we are at 2020.

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But you get the impression that
it's a fairly stable,

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now, even age distribution in the US
and these developed countries.

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Oops, here we go a little but more.
Sorry.  2035, 2045, OK.

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Now, in less developed countries,
the birth rate's high and the death

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rate's low.  We see a much different
age distribution.

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And here's Uganda,
with a very high reproductive rate

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showing the projections to 2050.
And here, we can march through from

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1970.  You can see that this huge
expansion, do you know what that

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noise is?  OK.
Does anybody have a hypothesis for

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what that noise is that
we could test?

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Oh, OK, I guess we can't do anything
about that.  OK,

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so here's Uganda.
And you can see the dramatic

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difference in a population where
there is large birthrates,

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and reducing death rates.  And we're
going to get into analyzing that in

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the next lecture.
I just want to show you this here

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so you have a feeling for what we
are talking about in age

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structured populations.
So, let's now look at how are going

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to analyze these populations to try
to quantify growth rates or

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00:16:09,000 --> 00:16:14,000
replacement rates.
And to do this, we set up life

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tables.  And this is basically what
insurance agencies do for human

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populations.  But we do the same
thing for populations of ecological

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interests.  We use the
same techniques.

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In this lecture,
going to use a unicorn is my example,

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because I can make up the numbers
because they don't exist.

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But in a textbook there are
examples for real organisms like

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lizards and things like that.
OK, so we need to define an age

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interval, X, and then this is the
number of intervals in the original

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cohort.  Again,
a cohort is a group of individuals

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that are born within a
defined age interval.

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I mean, I think of you guys as a
cohort.  DX is the number dying

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during that interval.
All of this is on the Web.

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These slides are on the Web.
So, you don't need to write it down,

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but you can.  And,
NX is that number of individuals

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surviving to age X.
LX is the portion of individuals

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00:17:28,000 --> 00:17:33,000
surviving to age X.
So, that's just equal to NX divided

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00:17:33,000 --> 00:17:39,000
by N0. And, we're going to look at a
table that shows this in a minute.

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And MX is something that's
measured.  It's the per capita

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births during age interval X to X
plus one.  And this is also called

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age-specific fecundity.
And you can think of it as the

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00:17:57,000 --> 00:18:03,000
number of female offspring produced
per female in a particular

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00:18:03,000 --> 00:18:11,000
age category.
OK, is everybody comfortable with

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00:18:11,000 --> 00:18:21,000
that?  So, with these definitions,
we're going to build a life table

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00:18:21,000 --> 00:18:31,000
that will allow us to actually
calculate some things of interest.

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00:18:31,000 --> 00:18:37,000
And, what do we want to calculate?
We want to calculate the

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00:18:37,000 --> 00:18:50,000
survivorship probability,

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00:18:50,000 --> 00:19:02,000
LX. We want to calculate the net
replacement rate.

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00:19:02,000 --> 00:19:10,000
No it's not really a rate,
net replacement of population per

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00:19:10,000 --> 00:19:19,000
generation, which we are calling R0.
It's basically the number of

200
00:19:19,000 --> 00:19:27,000
children people have to replace
who's there per generation.

201
00:19:27,000 --> 00:19:36,000
And then, for now, this is what we
are going to look at.

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00:19:36,000 --> 00:19:48,000
And to do that,
we are going to generate what's

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00:19:48,000 --> 00:20:00,000
called a cohort life table.
And to do this, we follow a cohort

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00:20:00,000 --> 00:20:11,000
of individuals throughout lifetime.
Or, we can also generate a static

205
00:20:11,000 --> 00:20:19,000
life table because it's not that
easy sometimes to have a group of

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00:20:19,000 --> 00:20:27,000
organisms that are born at the same
time to follow them throughout their

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00:20:27,000 --> 00:20:34,000
entire lifetime.
So there is a static life table of

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00:20:34,000 --> 00:20:40,000
taking a snapshot at one time of the
population, and calculating the age

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00:20:40,000 --> 00:20:47,000
structure.  So,
you take a snapshot,

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00:20:47,000 --> 00:20:53,000
and we look at the age structure.
And, we are going to do this in a

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00:20:53,000 --> 00:21:00,000
second so it will make more sense.
OK, so we've defined our terms.

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00:21:00,000 --> 00:21:06,000
And now, we are going to start by
calculating LX.

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00:21:06,000 --> 00:21:13,000
So, this is a cohort life table for
unicorns.  We're going to start out

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00:21:13,000 --> 00:21:19,000
with a hundred baby unicorns that we
have in our imaginary unicorn pen.

215
00:21:19,000 --> 00:21:26,000
So, this is a cohort size of 100.
And, we find that after a year there

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00:21:26,000 --> 00:21:34,000
are 50 of them left.
50 of them die in the first year.

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00:21:34,000 --> 00:21:42,000
So, the probability here, the
proportion surviving is 0.

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00:21:42,000 --> 00:21:50,000
, NX over N0,  and then a year later,
.4, .3, and then by four years older,

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00:21:50,000 --> 00:21:58,000
no unicorns left.
They don't live very long.

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00:21:58,000 --> 00:22:06,000
All right, so this is what's called
the survivorship probability,

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00:22:06,000 --> 00:22:12,000
and what we can do is look at there.
Different types of organisms have

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00:22:12,000 --> 00:22:16,000
different, what we call,
survivorship curves.  And this is

223
00:22:16,000 --> 00:22:20,000
discussed in your textbook.
We'll just describe the extremes.

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00:22:20,000 --> 00:22:25,000
These are just theoretical
survivorship curves.

225
00:22:25,000 --> 00:22:29,000
But some organisms have a very high
probability of survival as a

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00:22:29,000 --> 00:22:34,000
function of age until they
reach an old age.

227
00:22:34,000 --> 00:22:40,000
And then, they have a very low
probability of survival.

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00:22:40,000 --> 00:22:46,000
There are other organisms whose
survivorship probability drops very

229
00:22:46,000 --> 00:22:53,000
fast, right after they're born.
But if they make it through that

230
00:22:53,000 --> 00:22:59,000
interval, they're pretty good to go.
And then there are some that have a

231
00:22:59,000 --> 00:23:08,000
steady probability of dying.
So, where are humans,

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00:23:08,000 --> 00:23:20,000
do you think, on this?
Two?  No, but that's OK.

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00:23:20,000 --> 00:23:32,000
Let me ask you the other way; where
our frogs, do you think?

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00:23:32,000 --> 00:23:37,000
Yeah, OK, so you got that image.
Tons of frogs' eggs: everybody eats

235
00:23:37,000 --> 00:23:42,000
them.  Or for that matter,
the video I showed towards the end

236
00:23:42,000 --> 00:23:47,000
of the last class where there were
all those eggs of,

237
00:23:47,000 --> 00:23:53,000
what was that?  Remember all those
eggs that everybody was eating?

238
00:23:53,000 --> 00:23:58,000
Herring, thank you.  So, any
organism that puts out just tons of

239
00:23:58,000 --> 00:24:03,000
fertilized eggs,
and knowing that most of them will

240
00:24:03,000 --> 00:24:09,000
be eaten, but some of them will
survive, falls here.

241
00:24:09,000 --> 00:24:14,000
And, humans actually fall here.
Any organism that has a high

242
00:24:14,000 --> 00:24:19,000
investment in the care of offspring,
they have few offspring but they

243
00:24:19,000 --> 00:24:24,000
invest a lot into the care of those
offspring, would fall here.

244
00:24:24,000 --> 00:24:30,000
And then this, actually birds and
things fall here.

245
00:24:30,000 --> 00:24:34,000
So, here's some real but idealized
survivorship curves.

246
00:24:34,000 --> 00:24:38,000
These are humans.  And males and
females are different.

247
00:24:38,000 --> 00:24:42,000
I'm not sure whether we understand
that completely yet.

248
00:24:42,000 --> 00:24:46,000
Does anybody know whether that's
socially constructed?

249
00:24:46,000 --> 00:24:51,000
Now that there's more women
experiencing equal stress in the

250
00:24:51,000 --> 00:24:55,000
workplace as there are men that will
probably even out.

251
00:24:55,000 --> 00:24:59,000
But, I think there are more women
born, or girl babies.

252
00:24:59,000 --> 00:25:03,000
Anyway, there's some interesting
biology behind this,

253
00:25:03,000 --> 00:25:07,000
but I don't know.
I don't remember.

254
00:25:07,000 --> 00:25:11,000
And, here's grass,
of course grass spew out all these

255
00:25:11,000 --> 00:25:14,000
seeds everywhere,
and very few of them survive,

256
00:25:14,000 --> 00:25:18,000
also these frogs, etc. and birds are
commonly like this,

257
00:25:18,000 --> 00:25:22,000
where they're somewhere in between.
Why do we care so much about

258
00:25:22,000 --> 00:25:25,000
survivorship curves?
Who cares?  Well, I mean they're

259
00:25:25,000 --> 00:25:29,000
inherently interesting to population
ecologists, but there are

260
00:25:29,000 --> 00:25:33,000
also uses for them.
For example, if you want to conserve

261
00:25:33,000 --> 00:25:38,000
a species, if you're worried about a
species going extinct,

262
00:25:38,000 --> 00:25:44,000
you want to figure out whether it's
better to conserve the young ones or

263
00:25:44,000 --> 00:25:49,000
the old ones.  For example,
turtle species, you would pick a

264
00:25:49,000 --> 00:25:54,000
certain age group where the
probability of survival is high,

265
00:25:54,000 --> 00:26:00,000
and decide to target the
conservation of that age group.

266
00:26:00,000 --> 00:26:05,000
So, let's continue with,
we are building our life table here.

267
00:26:05,000 --> 00:26:11,000
So, we have the survivorship
probability, but what we really want

268
00:26:11,000 --> 00:26:17,000
to get at is understanding whether
or not the population that we are

269
00:26:17,000 --> 00:26:23,000
describing is replacing itself with
each generation.

270
00:26:23,000 --> 00:26:29,000
So, maybe we should define,
when R0 is equal to one, that meets

271
00:26:29,000 --> 00:26:35,000
the population is exactly
replacing itself.

272
00:26:35,000 --> 00:26:45,000
So, this is replacing,
so the actual growth rate of the

273
00:26:45,000 --> 00:26:55,000
population would be steady.
If R0 is less than one, the number

274
00:26:55,000 --> 00:27:02,000
of individuals is declining.
And R0 of greater than one,

275
00:27:02,000 --> 00:27:08,000
it's increasing.  So, we want to
know for our unicorns what that is.

276
00:27:08,000 --> 00:27:14,000
And to get to that, we have to know
something about the birth rates.

277
00:27:14,000 --> 00:27:20,000
So, MX is the average offspring per
female of age X.

278
00:27:20,000 --> 00:27:26,000
So, this is called the age-specific
fecundity.  And that's something

279
00:27:26,000 --> 00:27:32,000
that's a known property
of the population.

280
00:27:32,000 --> 00:27:41,000
Whoops, oh, my,
my, my, my, I'm missing a slide.

281
00:27:41,000 --> 00:27:50,000
Oh, there we go.  They're out of
order.  OK, so we have MX.

282
00:27:50,000 --> 00:28:00,000
So, how do we calculate R0?
Well, R0 is the sum of LX MX.

283
00:28:00,000 --> 00:28:17,000
With the sum of the survivorship

284
00:28:17,000 --> 00:28:25,000
times the age-specific fecundity,
and in this case, it sums up to

285
00:28:25,000 --> 00:28:33,000
three.  So, what's happening to our
unicorn population?  It's growing.

286
00:28:33,000 --> 00:28:39,000
Yeah, we are getting three unicorns
in each generation for every one

287
00:28:39,000 --> 00:28:46,000
that existed before.
So, in our imaginary unit of our

288
00:28:46,000 --> 00:28:53,000
population, we're going to be knee
deep in unicorns pretty fast.

289
00:28:53,000 --> 00:29:00,000
OK, so I forgot my watch, so I have
to look at my computer.

290
00:29:00,000 --> 00:29:06,000
What if we can't follow cohort?
Oh, thank you.

291
00:29:06,000 --> 00:29:11,000
How do we create the same kind of
analysis for a population that we

292
00:29:11,000 --> 00:29:16,000
can't follow through time,
but can only look at as a snapshot?

293
00:29:16,000 --> 00:29:22,000
OK, this is where we go to the
slide.  If you don't have it in your

294
00:29:22,000 --> 00:29:27,000
handout, it doesn't matter.
I just got off the web this morning.

295
00:29:27,000 --> 00:29:32,000
I couldn't find a skeleton of the

296
00:29:32,000 --> 00:29:37,000
unicorn because,
of course, that's totally imaginary,

297
00:29:37,000 --> 00:29:42,000
but I found a mastodon.  So, just
imagine that this is a unicorn,

298
00:29:42,000 --> 00:29:47,000
and I couldn't find a unicorn horn,
so this is a sheep's.  But, all

299
00:29:47,000 --> 00:29:52,000
these principles apply.
I just discovered Images in Google,

300
00:29:52,000 --> 00:29:57,000
which is really exciting.  So,
you're going to get subjected

301
00:29:57,000 --> 00:30:02,000
to this for awhile.
So, OK, so what you can do,

302
00:30:02,000 --> 00:30:06,000
and this has actually been done with
mountain sheep,

303
00:30:06,000 --> 00:30:11,000
is you go out you find dead sheep,
you find skeletons of sheep that

304
00:30:11,000 --> 00:30:15,000
have died for whatever causes.
And you go out, and you sample

305
00:30:15,000 --> 00:30:19,000
until you have,
say, 100 skeletons.

306
00:30:19,000 --> 00:30:24,000
And that's your cohort that you're
looking at, at one point in time.

307
00:30:24,000 --> 00:30:28,000
And from their horn, you can
actually tell how old they

308
00:30:28,000 --> 00:30:33,000
were when they died.
You can count the number of rings,

309
00:30:33,000 --> 00:30:38,000
so that's what's here, annual horn
rings.  This is for a dall mountain

310
00:30:38,000 --> 00:30:43,000
sheep.  So, you can say well now it
died when it was two.

311
00:30:43,000 --> 00:30:47,000
That one died when it was 10.
That one died when it was whatever

312
00:30:47,000 --> 00:30:52,000
age.  And then you can create the
same kind of life table,

313
00:30:52,000 --> 00:30:57,000
a static life table, where you have
a hundred skeletons.

314
00:30:57,000 --> 00:31:02,000
That is your cohort.
You look at the number dying of age

315
00:31:02,000 --> 00:31:06,000
zero to one, the number of one year
olds, the number that died when they

316
00:31:06,000 --> 00:31:11,000
were one year old,
the number that died when they were

317
00:31:11,000 --> 00:31:16,000
a two-year-old etc.
And so, from these data,

318
00:31:16,000 --> 00:31:20,000
these are the data that you
collected, you can calculate this

319
00:31:20,000 --> 00:31:25,000
column, NX, so NX is DX,
or NX minus DX equals NX plus one.

320
00:31:25,000 --> 00:31:30,000
Does that make sense?  I can never
tell whether.

321
00:31:30,000 --> 00:31:35,000
I know if I write this on the board
it might be easier,

322
00:31:35,000 --> 00:31:40,000
but it's so obvious isn't it?
We are just saying that this is the

323
00:31:40,000 --> 00:31:45,000
number that died at the age.
This is the number you started with,

324
00:31:45,000 --> 00:31:50,000
so that's how many are going to have
that age, that age,

325
00:31:50,000 --> 00:31:55,000
and that age.  And then,
once you have this column,

326
00:31:55,000 --> 00:32:00,000
your proportion surviving LX,
you can calculate LX.  LX equals NX

327
00:32:00,000 --> 00:32:05,000
divided by N0, OK?
So, we are doing exactly the same

328
00:32:05,000 --> 00:32:10,000
thing as we did before.
It's just that we're getting the NX

329
00:32:10,000 --> 00:32:15,000
column instead of getting it by
following the cohort.

330
00:32:15,000 --> 00:32:20,000
We're getting it by calculating it
based on how old dead organisms were

331
00:32:20,000 --> 00:32:25,000
when they died.
And in my ecology class that I

332
00:32:25,000 --> 00:32:30,000
teach, some years we actually go out
to the Mount Auburn Cemetery.

333
00:32:30,000 --> 00:32:37,000
And you can do this from human
gravestones.  You can go to the

334
00:32:37,000 --> 00:32:44,000
cemetery, and pick out a number of
gravestones, and see the age at

335
00:32:44,000 --> 00:32:51,000
which humans died.
You create yourself a cohort,

336
00:32:51,000 --> 00:32:58,000
and you can create a life table.
And you can do that for different

337
00:32:58,000 --> 00:33:06,000
eras, and see how replacements
have changed.

338
00:33:06,000 --> 00:33:13,000
OK, now so that's the analysis for
populations that have an age

339
00:33:13,000 --> 00:33:21,000
structure.  Now we are going to go
more into simpler type of population,

340
00:33:21,000 --> 00:33:29,000
and that is a population with a
stable age distribution.

341
00:33:29,000 --> 00:33:46,000
And to do this,

342
00:33:46,000 --> 00:33:51,000
you're going to help me,
and we're going to use your calculus

343
00:33:51,000 --> 00:33:55,000
that you've all been studying.
So, instead of the unicorn now,

344
00:33:55,000 --> 00:34:00,000
have your imaginary population be a
population of microbes

345
00:34:00,000 --> 00:34:06,000
that divide in half.
They multiplied by dividing in half.

346
00:34:06,000 --> 00:34:12,000
So, each one of these is a microbe
that's dividing in half.

347
00:34:12,000 --> 00:34:18,000
This is your mental image.
This is what's called exponential

348
00:34:18,000 --> 00:34:25,000
growth.  It's obvious how that
happens.  And we're going to model

349
00:34:25,000 --> 00:34:31,000
this population,
we're going to first assume

350
00:34:31,000 --> 00:34:42,000
unlimited resources.
OK, so we're going to say that the

351
00:34:42,000 --> 00:34:56,000
rate of population increase is equal
to the average birth rate minus the

352
00:34:56,000 --> 00:35:11,000
average death rate times
the number of cells.

353
00:35:11,000 --> 00:35:19,000
OK, so we are going to now turn this
into math, and that is to say the

354
00:35:19,000 --> 00:35:28,000
dN/dt, the increase in population
where N is the population number is

355
00:35:28,000 --> 00:35:36,000
equal to the birth rate minus the
death rate times N which is the

356
00:35:36,000 --> 00:35:46,000
number of cells, OK?
And then, we're going to let B minus

357
00:35:46,000 --> 00:35:58,000
D, the birth rate minus the death
rate, be what we call r.

358
00:35:58,000 --> 00:36:08,000
And, this is what's called the
intrinsic rate of increase of a

359
00:36:08,000 --> 00:36:19,000
population.  OK,
what are the units of r?

360
00:36:19,000 --> 00:36:30,000
One over time, exactly, time to the
minus one.

361
00:36:30,000 --> 00:36:36,000
So, let's look at that more
carefully.  And also,

362
00:36:36,000 --> 00:36:43,000
it's a little misleading to say it's
the rate of increase because r can

363
00:36:43,000 --> 00:36:49,000
be positive or negative,
however it turns out.  It can be

364
00:36:49,000 --> 00:36:56,000
positive or negative,
but that's what it's called.

365
00:36:56,000 --> 00:37:02,000
So, we have the dN/dt equals rN.
We're substituting r in this

366
00:37:02,000 --> 00:37:09,000
equation for one over N
times dN/dt equals r.

367
00:37:09,000 --> 00:37:15,000
OK, so ours has the unit time to the
minus one.  And so,

368
00:37:15,000 --> 00:37:21,000
let's ask a question.
Given N0  I give you the population

369
00:37:21,000 --> 00:37:27,000
density at some time which we're
going to call T equals zero.

370
00:37:27,000 --> 00:37:33,000
Given a population growing
according to this,

371
00:37:33,000 --> 00:37:39,000
which is exponential growth,
what if we want to know the

372
00:37:39,000 --> 00:37:45,000
population, what N
is at any time T?

373
00:37:45,000 --> 00:37:51,000
We want an equation that will give
us, given N0 what would the

374
00:37:51,000 --> 00:37:57,000
population density be at some time,
T?  What do you have to do to this

375
00:37:57,000 --> 00:38:03,000
to get that?  Yeah,
so who wants to do that for me?

376
00:38:03,000 --> 00:38:10,000
Come on.  You guys did this freshman
year.  It's the easiest thing there

377
00:38:10,000 --> 00:38:17,000
is, right?  Every class I've had has
had somebody who was willing to come

378
00:38:17,000 --> 00:38:24,000
up and do this.
OK, so we'll just add a T there.

379
00:38:24,000 --> 00:38:31,000
So, N at sometime T is equally to
N0 e to the rT.

380
00:38:31,000 --> 00:38:38,000
And so,   We could say,
then, r equals natural log of NT

381
00:38:38,000 --> 00:38:46,000
minus natural log of N0 divided by T.
And I like to write it that way

382
00:38:46,000 --> 00:38:53,000
because then, we know what this
looks like, right?

383
00:38:53,000 --> 00:39:01,000
Let's plot that.  This is N and
this is T.  What does

384
00:39:01,000 --> 00:39:08,000
that look like?
I know this is really rudimentary

385
00:39:08,000 --> 00:39:14,000
but remember we're modeling
population growth.

386
00:39:14,000 --> 00:39:20,000
So here, if we plot the log of N,
and this is what we do with cultures

387
00:39:20,000 --> 00:39:26,000
of microorganisms.
That's a flask.  Those are a lot of

388
00:39:26,000 --> 00:39:33,000
microbes in there.
And what we do is we sample it at

389
00:39:33,000 --> 00:39:39,000
various points in time,
and if you take the log we get a

390
00:39:39,000 --> 00:39:46,000
nice straight line that we can draw
a regression through.

391
00:39:46,000 --> 00:39:53,000
And what's the slope of that line
equal to?  r.  Exactly.

392
00:39:53,000 --> 00:40:00,000
The growth rate in the units: N to
the minus one.

393
00:40:00,000 --> 00:40:12,000
OK, what's the Y intercept?
N0.  OK, now suppose we want to

394
00:40:12,000 --> 00:40:25,000
calculate the doubling time of the
population, the time it

395
00:40:25,000 --> 00:40:37,000
takes to double.
How would we do that?

396
00:40:37,000 --> 00:40:48,000
Let's first define it.
It's the time, T, that it takes for

397
00:40:48,000 --> 00:40:59,000
NT to equal to N0,
right?  If we start with N0 the

398
00:40:59,000 --> 00:41:10,000
population doubles.
Then, that's the time at NT.

399
00:41:10,000 --> 00:41:22,000
So, we want to solve for that T for
the time it takes for the

400
00:41:22,000 --> 00:41:33,000
population to double.
Since natural log of NT over N0

401
00:41:33,000 --> 00:41:45,000
equals rT, then the natural log of,
sorry, 2N0 over N0 equals rT, and T

402
00:41:45,000 --> 00:41:57,000
equals the natural log of two
divided by r equals our

403
00:41:57,000 --> 00:42:05,000
doubling time.
Does that make sense?

404
00:42:05,000 --> 00:42:09,000
I'll put this out there so you can
see it better.

405
00:42:09,000 --> 00:42:14,000
What's the natural log of two?
0.69, thank you, always a handy

406
00:42:14,000 --> 00:42:18,000
thing to have in our repertoire.
So, that's just the way, it's

407
00:42:18,000 --> 00:42:23,000
easier to think about the time it
takes for a population to double

408
00:42:23,000 --> 00:42:26,000
often, then the instantaneous
growth rate.