And, we're going to make a
major shift.
You're going to feel like this
is a whole different class
compared to what we were talking
about last time,
because were jumping from the
biogeochemical cycles,
or looking at the biosphere as
essentially a large biochemical
machine, to studying individual
populations of organisms,
and the communities that they
make up when they come together.
So, before we were really
talking about organisms as they
function in the biosphere.
Mentally, we're grinding them
all up and thinking of them as a
collective biochemistry
basically.
And now we are going to stop
grinding them up,
mentally, and think of them as
individual organisms.
So, the next series of
lectures, we're going to talk
about population ecology.
If you remember the first
lecture I gave we talked about
the hierarchy of organization
within ecological systems,
and then we are going to talk
about competition between
organisms with a population,
and between organisms of
different species,
and were going to talk about
predation, and mutualism.
These are all interactions
between organisms that affect
the fitness of organisms.
And then we'll,
at the end, talk about
community structure.
So this is sort of the outline
for the rest of my lectures,
not for this lecture.
So, today we are going to talk
about properties of populations.
We're going to analyze how we
measure growth rate,
growth and death in
populations, and this will
include populations that have an
age structure,
and populations that don't.
And this is all in preparation
for the next lecture where we
will talk about human population
growth.
So, in this field of population
ecology, which is as I told you
in the first lecture,
and some universities you could
take three courses in population
ecology, and you could get a
Ph.D.
in population ecology.
I mean, this is a whole field
that we're going to cover in two
lectures.
But what population ecologists
worry about fundamentally,
well, they don't worry about
it.
This is what they study,
is what regulates the density
of populations?
Obviously, it's a function of
how fast they're growing,
the birth rate,
and how fast they're dying,
the death rate.
But what are the factors that
actually influence those rates?
Is it competition with other
organisms?
Is it the entire structure of
the community?
Is it the availability of food?
Is it the various abiotic
properties of the environment:
temperature,
etc.?
So, they analyze these and
basically try to model the
population growth as a function
of these various parameters.
The other questions they ask,
is how are populations
distributed in the environment?
Are they clustered?
Are they evenly distributed?
This has specific meanings
about their ecology.
And, the other thing that
people are really fascinated by,
which is a really tough
question, is why are some
species' populations extremely
abundant, while others are rare?
And one of the discussions we
always have in my lab,
we work on an organism that's
extremely abundant,
this prochlorococcus,
which I told you briefly about,
is the most abundant
photosynthetic cell on the
planet.
So, my students tend to keep
saying why is it so successful?
And I keep saying,
it's successful but there are
thousands of other species who
are also successful.
Abundance does not equal
success.
Endurance equals success.
If you're here in the next
generation, you're successful.
If you're not,
if your species is
disappearing,
then you're not successful.
So, speaking of abundance,
let's talk about how we measure
abundance, population
ecologists.
And this is just one example.
Obviously, for microorganisms,
or some microorganisms it's
really easy because they're tiny
relative to their habitats.
So for the prochlorococcus that
we work on, there are 10^5 cells
per milliliter.
So, we can go take a milliliter
of water and measure how many
cells there.
But for some organisms,
larger ones,
that are widely distributed,
it's not that easy.
So, one method is mark and
recapture.
That's used a lot for things
like birds and butterflies.
For a bird, the mark would be
putting a band on the bird.
For a butterfly,
they often take a magic marker
and put a mark on the wing.
Well, that's largely what they
do.
You try to mark individuals in
some way that would not
influence their survivorship
rate.
So, if N equals the population
size, that is,
that's our unknown,
what we're going to do is
capture, say,
for butterflies or moths,
you use a butterfly net,
or moths you can use a light to
track them; for birds,
you put up these big mist nets.
They fly into them;
they get tangled up a little
bit but they don't get hurt.
Then you band them,
and that we let them go.
That's the way you mark them.
So, we're going to say n1
equals the total number of
marked individuals released.
So you capture them,
you mark them,
you release them.
n2 is equal to,
and then you go out sometime
later and you recapture as many
individuals as you can find,
and this would be the total
number [SIREN]
that doesn't sound like a fire
drill, does it?
I assume we're good to go here.
So, n2 is the total number of
recaptured.
And we're going to say m2 is
equal to the numbers recaptured
that are marked.
OK, and then we assume that the
fraction of the recaptured that
are marked represent the
fraction in the total population
that was marked.
So, we say m2 over n2 is equal
to n1 over N.
And the number that we're
looking for, population size,
is equal to n1,
n2 divided by m2.
So, of course,
this assumes that there's no
effect of the marking of the
individuals.
It assumes that there's no bias
in the trapping for the marked
or not marked individuals.
There's all kinds of
assumptions that underlie this.
It's a start for assessing the
population size.
OK, so how do we measure
population growth?
We're going to first start with
looking at populations that have
age structure.
Now, I hope you printed out the
slides that were on the Web,
because I'm depending on these
overheads a lot for this lecture
because we wouldn't get through
any of it if I wrote all this
stuff on the board.
So, we're going to talk about
populations that have an age
structure.
And the data I'm going to show
you here is for human
populations.
But this applies to any
population that has differential
birth and death rates as a
function of the age of the
organism, OK?
So, in these populations if
birth rate and death rate are
high, the population is
dominated by young people.
And, we'll look at this in a
minute.
And, if B and D are low,
dominated by old people,
or older I should say,
since I now fit into the old
category.
OK, so here's a typical
population age distribution for
developed countries,
where each slice here,
these are females on the right,
males on the left,
and each slice is an age
category: zero to 10 years,
10 to 20.
And you can see that in these
kinds of populations,
you have a fairly even age
distribution.
Long periods of no net growth
in a population lead to this.
In these developed countries,
and we're going to examine why
this is, there's basically an
even replacement rate of
children for adults.
And one of the things we worry
about when you see this kind of
age distribution,
although it's good in terms of
population growth,
is when you have few young
people and a lot of older
people, who's going to take care
of them, which is what's behind
the Social Security crisis.
But we won't get into that.
Since you're the young people
and I'm the old people,
I don't want to dwell on that.
OK, so what demographers do for
human populations is project
what the population will look
like in the future based on the
reproductive rates of the
present.
And you can see for the US
here, it's reasonably stable if
you look at these three
snapshots.
We're going to go backwards
starting with 1950,
and show you what the
population has been doing since
1950.
And I'm just going to walk
through this.
You only have one in your
handouts, but I'll show you how
it's moving along.
Moving along,
you can think of this as
generations moving through the
population.
And this is the date up here.
So, this is 1950,
1955, you can see this red
cohort.
A cohort is a group of
individuals that were born at
roughly the same time.
So, you can see that red cohort
there.
And we are going along,
1965.
This lip here,
that we can now see,
is the postwar baby boom.
That's what I'm a member of.
If you can see it in this bulge
in this population.
And now were marching along.
Here's my cohort,
and I just put these lines on
to keep you oriented.
And here comes you guys.
I think those are you guys,
1985.
That's roughly right,
because I never know when I've
last updated these slides.
So, and here you go.
See, here's the big bulge of
all of these baby boomers that
you guys are going to have to
take care of.
And now, we can actually see an
echo.
This is what's called the baby
boom echo.
These are the kids of the baby
boomers, which is you guys.
But you can only see that as we
march through it.
So, here we are at 2020.
But you get the impression that
it's a fairly stable,
now, even age distribution in
the US and these developed
countries.
Oops, here we go a little but
more.
Sorry.
2035, 2045, OK.
Now, in less developed
countries, the birth rate's high
and the death rate's low.
We see a much different age
distribution.
And here's Uganda,
with a very high reproductive
rate showing the projections to
2050.
And here, we can march through
from 1970.
You can see that this huge
expansion, do you know what that
noise is?
OK.
Does anybody have a hypothesis
for what that noise is that we
could test?
Oh, OK, I guess we can't do
anything about that.
OK, so here's Uganda.
And you can see the dramatic
difference in a population where
there is large birthrates,
and reducing death rates.
And we're going to get into
analyzing that in the next
lecture.
I just want to show you this
here so you have a feeling for
what we are talking about in age
structured populations.
So, let's now look at how are
going to analyze these
populations to try to quantify
growth rates or replacement
rates.
And to do this,
we set up life tables.
And this is basically what
insurance agencies do for human
populations.
But we do the same thing for
populations of ecological
interests.
We use the same techniques.
In this lecture,
going to use a unicorn is my
example, because I can make up
the numbers because they don't
exist.
But in a textbook there are
examples for real organisms like
lizards and things like that.
OK, so we need to define an age
interval, X, and then this is
the number of intervals in the
original cohort.
Again, a cohort is a group of
individuals that are born within
a defined age interval.
I mean, I think of you guys as
a cohort.
DX is the number dying during
that interval.
All of this is on the Web.
These slides are on the Web.
So, you don't need to write it
down, but you can.
And, NX is that number of
individuals surviving to age X.
LX is the portion of
individuals surviving to age X.
So, that's just equal to NX
divided by N0.
And, we're going to look at a
table that shows this in a
minute.
And MX is something that's
measured.
It's the per capita births
during age interval X to X plus
one.
And this is also called
age-specific fecundity.
And you can think of it as the
number of female offspring
produced per female in a
particular age category.
OK, is everybody comfortable
with that?
So, with these definitions,
we're going to build a life
table that will allow us to
actually calculate some things
of interest.
And, what do we want to
calculate?
We want to calculate the
survivorship probability,
00:18:39
 
LX.
We want to calculate the net
replacement rate.
No it's not really a rate,
net replacement of population
per generation,
which we are calling R0.
It's basically the number of
children people have to replace
who's there per generation.
And then, for now,
this is what we are going to
look at.
And to do that,
we are going to generate what's
called a cohort life table.
And to do this,
we follow a cohort of
individuals throughout lifetime.
Or, we can also generate a
static life table because it's
not that easy sometimes to have
a group of organisms that are
born at the same time to follow
them throughout their entire
lifetime.
So there is a static life table
of taking a snapshot at one time
of the population,
and calculating the age
structure.
So, you take a snapshot,
and we look at the age
structure.
And, we are going to do this in
a second so it will make more
sense.
OK, so we've defined our terms.
And now, we are going to start
by calculating LX.
So, this is a cohort life table
for unicorns.
We're going to start out with a
hundred baby unicorns that we
have in our imaginary unicorn
pen.
So, this is a cohort size of
100.
And, we find that after a year
there are 50 of them left.
50 of them die in the first
year.
So, the probability here,
the proportion surviving is
0.5, NX over N0,
and then a year later,
.4, .3, and then by four years
older, no unicorns left.
They don't live very long.
All right, so this is what's
called the survivorship
probability, and what we can do
is look at there.
Different types of organisms
have different,
what we call,
survivorship curves.
And this is discussed in your
textbook.
We'll just describe the
extremes.
These are just theoretical
survivorship curves.
But some organisms have a very
high probability of survival as
a function of age until they
reach an old age.
And then, they have a very low
probability of survival.
There are other organisms whose
survivorship probability drops
very fast, right after they're
born.
But if they make it through
that interval,
they're pretty good to go.
And then there are some that
have a steady probability of
dying.
So, where are humans,
do you think,
on this?
Two?
No, but that's OK.
Let me ask you the other way;
where our frogs,
do you think?
Yeah, OK, so you got that
image.
Tons of frogs' eggs:
everybody eats them.
Or for that matter,
the video I showed towards the
end of the last class where
there were all those eggs of,
what was that?
Remember all those eggs that
everybody was eating?
Herring, thank you.
So, any organism that puts out
just tons of fertilized eggs,
and knowing that most of them
will be eaten,
but some of them will survive,
falls here.
And, humans actually fall here.
Any organism that has a high
investment in the care of
offspring, they have few
offspring but they invest a lot
into the care of those
offspring, would fall here.
And then this,
actually birds and things fall
here.
So, here's some real but
idealized survivorship curves.
These are humans.
And males and females are
different.
I'm not sure whether we
understand that completely yet.
Does anybody know whether
that's socially constructed?
Now that there's more women
experiencing equal stress in the
workplace as there are men that
will probably even out.
But, I think there are more
women born, or girl babies.
Anyway, there's some
interesting biology behind this,
but I don't know.
I don't remember.
And, here's grass,
of course grass spew out all
these seeds everywhere,
and very few of them survive,
also these frogs,
etc.
and birds are commonly like
this, where they're somewhere in
between.
Why do we care so much about
survivorship curves?
Who cares?
Well, I mean they're inherently
interesting to population
ecologists, but there are also
uses for them.
For example,
if you want to conserve a
species, if you're worried about
a species going extinct,
you want to figure out whether
it's better to conserve the
young ones or the old ones.
For example,
turtle species,
you would pick a certain age
group where the probability of
survival is high,
and decide to target the
conservation of that age group.
So, let's continue with,
we are building our life table
here.
So, we have the survivorship
probability, but what we really
want to get at is understanding
whether or not the population
that we are describing is
replacing itself with each
generation.
So, maybe we should define,
when R0 is equal to one,
that meets the population is
exactly replacing itself.
So, this is replacing,
so the actual growth rate of
the population would be steady.
If R0 is less than one,
the number of individuals is
declining.
And R0 of greater than one,
it's increasing.
So, we want to know for our
unicorns what that is.
And to get to that,
we have to know something about
the birth rates.
So, MX is the average offspring
per female of age X.
So, this is called the
age-specific fecundity.
And that's something that's a
known property of the
population.
Whoops, oh, my,
my, my, my, I'm missing a
slide.
Oh, there we go.
They're out of order.
OK, so we have MX.
So, how do we calculate R0?
Well, R0 is the sum of LX MX.
00:28:00
 
With the sum of the
survivorship times the
age-specific fecundity,
and in this case,
it sums up to three.
So, what's happening to our
unicorn population?
It's growing.
Yeah, we are getting three
unicorns in each generation for
every one that existed before.
So, in our imaginary unit of
our population,
we're going to be knee deep in
unicorns pretty fast.
OK, so I forgot my watch,
so I have to look at my
computer.
What if we can't follow cohort?
Oh, thank you.
How do we create the same kind
of analysis for a population
that we can't follow through
time, but can only look at as a
snapshot?
OK, this is where we go to the
slide.
If you don't have it in your
handout, it doesn't matter.
I just got off the web this
morning.
I couldn't find a skeleton of
the unicorn because,
of course, that's totally
imaginary, but I found a
mastodon.
So, just imagine that this is a
unicorn, and I couldn't find a
unicorn horn,
so this is a sheep's.
But, all these principles
apply.
I just discovered Images in
Google, which is really
exciting.
So, you're going to get
subjected to this for awhile.
So, OK, so what you can do,
and this has actually been done
with mountain sheep,
is you go out you find dead
sheep, you find skeletons of
sheep that have died for
whatever causes.
And you go out,
and you sample until you have,
say, 100 skeletons.
And that's your cohort that
you're looking at,
at one point in time.
And from their horn,
you can actually tell how old
they were when they died.
You can count the number of
rings, so that's what's here,
annual horn rings.
This is for a dall mountain
sheep.
So, you can say well now it
died when it was two.
That one died when it was 10.
That one died when it was
whatever age.
And then you can create the
same kind of life table,
a static life table,
where you have a hundred
skeletons.
That is your cohort.
You look at the number dying of
age zero to one,
the number of one year olds,
the number that died when they
were one year old,
the number that died when they
were a two-year-old etc.
And so, from these data,
these are the data that you
collected, you can calculate
this column, NX,
so NX is DX,
or NX minus DX equals NX plus
one.
Does that make sense?
I can never tell whether.
I know if I write this on the
board it might be easier,
but it's so obvious isn't it?
We are just saying that this is
the number that died at the age.
This is the number you started
with, so that's how many are
going to have that age,
that age, and that age.
And then, once you have this
column, your proportion
surviving LX,
you can calculate LX.
LX equals NX divided by N0,
OK?
So, we are doing exactly the
same thing as we did before.
It's just that we're getting
the NX column instead of getting
it by following the cohort.
We're getting it by calculating
it based on how old dead
organisms were when they died.
And in my ecology class that I
teach, some years we actually go
out to the Mount Auburn
Cemetery.
And you can do this from human
gravestones.
You can go to the cemetery,
and pick out a number of
gravestones, and see the age at
which humans died.
You create yourself a cohort,
and you can create a life
table.
And you can do that for
different eras,
and see how replacements have
changed.
OK, now so that's the analysis
for populations that have an age
structure.
Now we are going to go more
into simpler type of population,
and that is a population with a
stable age distribution.
00:33:29
 
And to do this,
you're going to help me,
and we're going to use your
calculus that you've all been
studying.
So, instead of the unicorn now,
have your imaginary population
be a population of microbes that
divide in half.
They multiplied by dividing in
half.
So, each one of these is a
microbe that's dividing in half.
This is your mental image.
This is what's called
exponential growth.
It's obvious how that happens.
And we're going to model this
population, we're going to first
assume unlimited resources.
OK, so we're going to say that
the rate of population increase
is equal to the average birth
rate minus the average death
rate times the number of cells.
OK, so we are going to now turn
this into math,
and that is to say the dN/dt,
the increase in population
where N is the population number
is equal to the birth rate minus
the death rate times N which is
the number of cells,
OK?
And then, we're going to let B
minus D, the birth rate minus
the death rate,
be what we call r.
And, this is what's called the
intrinsic rate of increase of a
population.
OK, what are the units of r?
One over time,
exactly, time to the minus one.
So, let's look at that more
carefully.
And also, it's a little
misleading to say it's the rate
of increase because r can be
positive or negative,
however it turns out.
It can be positive or negative,
but that's what it's called.
So, we have the dN/dt equals
rN.
We're substituting r in this
equation for one over N times
dN/dt equals r.
OK, so ours has the unit time
to the minus one.
And so, let's ask a question.
Given N0 I give you the
population density at some time
which we're going to call T
equals zero.
Given a population growing
according to this,
which is exponential growth,
what if we want to know the
population, what N is at any
time T?
We want an equation that will
give us, given N0 what would the
population density be at some
time, T?
What do you have to do to this
to get that?
Yeah, so who wants to do that
for me?
Come on.
You guys did this freshman
year.
It's the easiest thing there
is, right?
Every class I've had has had
somebody who was willing to come
up and do this.
OK, so we'll just add a T
there.
So, N at sometime T is equally
to N0 e to the rT.
And so, We could say,
then, r equals natural log of
NT minus natural log of N0
divided by T.
And I like to write it that way
because then,
we know what this looks like,
right?
Let's plot that.
This is N and this is T.
What does that look like?
I know this is really
rudimentary but remember we're
modeling population growth.
So here, if we plot the log of
N, and this is what we do with
cultures of microorganisms.
That's a flask.
Those are a lot of microbes in
there.
And what we do is we sample it
at various points in time,
and if you take the log we get
a nice straight line that we can
draw a regression through.
And what's the slope of that
line equal to?
r.
Exactly.
The growth rate in the units:
N to the minus one.
OK, what's the Y intercept?
N0.
OK, now suppose we want to
calculate the doubling time of
the population,
the time it takes to double.
How would we do that?
Let's first define it.
It's the time,
T, that it takes for NT to
equal to N0, right?
If we start with N0 the
population doubles.
Then, that's the time at NT.
So, we want to solve for that T
for the time it takes for the
population to double.
Since natural log of NT over N0
equals rT, then the natural log
of, sorry, 2N0 over N0 equals
rT, and T equals the natural log
of two divided by r equals our
doubling time.
Does that make sense?
I'll put this out there so you
can see it better.
What's the natural log of two?
0.69, thank you,
always a handy thing to have in
our repertoire.
So, that's just the way,
it's easier to think about the
time it takes for a population
to double often,
then the instantaneous growth
rate.
