- [Instructor] Let's now
think a little bit more
about how we might
model population growth,
and as we do so, we're gonna become
a little bit more familiar
with the types of formulas
that you might see in an
AP Biology formula sheet.
So in a previous video,
we introduced the idea
of per capita growth rate of a population,
and we used the letter r for that.
And so let's say that the
per capita growth rate
for a population is 0.2.
That means that on average,
for every one individual
in that population,
a year later, it would have
grown by 20%, by two-tenths.
So for every one, you would
now have 1.2 of that population
a year later.
Now, as we mentioned, many populations,
or the ones that reproduced sexually,
you'd need at least two,
a male and a female,
but there's populations of certain things
that can just reproduce on their own.
They can just bud, or they can divide,
if we're talking about
especially unicellular organisms.
And from this notion, we
can get a related notion,
which is our maximum per capita
growth rate of a population
and let me just write that there.
You could view this as
your per capita growth rate
if the population is
not limited in any way,
if there's ample resources.
Water, food, land, territory,
whatever that population needs to grow,
but that still is talking
about per capita growth rate
of population, we're just viewing as,
this is the unfettered
one, this is the maximum.
And from that, we can set up
an exponential growth equation,
and we've seen this in other videos
where the rate of change of our population
with respect to time, N is our population,
so dN dt is our rate
of change of population
with respect to time.
Or our population growth
rate right over here,
let me write this down.
Population growth rate.
If we're dealing with a population
that in no way is being
limited by its ecosystem,
which in reality is not realistic,
at some point you would be.
Well then, the rate of
growth of population
is going to be your maximum per capita
growth rate of population,
times your population itself.
And we could see it set
up a little table here
to see how these would
relate to each other,
so let me do that, let me set up a table.
And so let's think about
what the rate of change
of population will be, our
population growth rate,
for certain populations.
So let's think about
what it's going to be,
when our population is 100.
When our population is 500,
and when our population is 900.
So, given these populations,
what would be your population
growth rate for each of them?
Pause this video and try to answer that.
Well, when our population is 100,
our population growth rate
is just going to be 0.2 times that.
So 0.2, let me write this down,
this is just going to be, dN
dt is just going to be 0.2,
our maximum per capita
growth rate of population
times our population, times 100,
which is equal to 20.
So we're going to grow per year by 20
when our population is 100.
Now what about when our population is 500?
What is going to be our
population growth rate?
Pause the video again
and try to answer that.
Well, once again, we just take our maximum
per capita growth rate and
multiply it times our population,
so .2 times 500, our population
growth rate is now 100.
If we're talking about bunnies,
and if our time is in years,
this would be 100 bunnies per year,
or 100 individuals per year.
And let's think about it
when our population is 900.
What's our population growth rate then?
Pause the video again.
Alright, well we're just
gonna take 0.2 times 900,
so it is going to be 180
individuals per year.
Now, as I just mentioned,
this is talking about a
somewhat unrealistic situation
where a population can
just grow and grow and grow
and never be limited in any way.
We know that land is
limited, food is limited,
water is limited.
And so, there's this notion
of a natural carrying capacity
of a given population
in a given environment.
And to describe that,
we'll use the letter K.
And so let's say, for the organisms
that we're studying here,
let's say they're bunnies,
and they're bunnies on a
relatively small island,
let's say that the
natural carrying capacity
for that island is 1000.
The the island really can't
support more than 1000 bunnies.
So how would we change this
exponential growth equation
right over here, exponential,
to reflect that?
Well, what mathematicians
and biologists have done
is they've modified this,
they multiplied this times the factor,
to get us what's known as logistic growth.
Logistic growth.
So this is exponential growth,
and what we're gonna now talk
about is logistic growth.
And what they do is they start
with the exponential growth,
so my population growth rate
you could view as your maximum
per capita growth rate,
times your population, so
that's exactly what we had
right over here, but then they
multiplied that by a factor
so that this thing slows down
the closer and closer we get
to the carrying capacity.
And the factor that they add is,
your carrying capacity
minus your population,
over your carrying capacity.
Now let's see if this
makes intuitive sense.
So let's set up another table here.
And I'll do it with the same values.
So let's say we have N,
so our population, what's going
to be our population growth
when our population's 100, when it's 500,
and when it's 900?
So I encourage you, pause this video,
and figure out what dN dt
is at these various times.
Well, at 100, it's going to be,
I'll do this one, I'll write it out,
it's going to be 0.2 times 100,
times the carrying capacity is 1000,
so it's gonna be 1000 minus 100,
all of that over 1000.
So this is 900 over 1000,
this is going to be 0.9.
And then 0.2 times 100
is 20, so 20 times 0.9,
this is going to be equal to 18.
So it's a little bit lower,
it's being slowed down a little
bit, but it's pretty close.
Now let's see what happens
when we get to N equals 500.
Pause this video and figure what dN dt,
our population growth rate,
would be at that time.
So in this case, it's
going to be 0.2 times 500,
times 500, times this factor here,
which is now going to be 1000
minus 500, that's our population now,
minus 500, all of that over 1000.
Now what's this going to be?
This is 100, which we had there,
but it's going to be
multiplied by 500 over 1000,
which is .5.
So we're only going to grow half as fast
as we were in this situation.
'Cause once again, we don't have
an infinite amount of resources here.
So this is going to be 100 times .5,
which is equal to 50.
And then if you look at
this scenario over here,
when our population is 900, what is dN dt?
Pause the video again.
Well, it is going to be 0.2 times 900,
which is 180, times this factor,
which is going to be 1000 minus 900,
all of that over 1000.
So now this factor's going to be,
100 over 1000, which is .1, 0.1.
This part right over here is,
this part right over here is 180.
180 times 1/10th is
going to be equal to 18.
So now our population
growth has slowed down.
Why is that happening?
Here, your population rate,
the rate of growth is growing
and growing and growing,
because the more bunnies or
whatever types of individuals
you have, there's just more to reproduce,
and they're just gonna
keep growing exponentially.
But here, they're
getting closer and closer
to the carrying capacity of
whatever environment they're at.
And so at 900, they're awfully close,
so now you're gonna have some bunnies
that are going hungry,
and maybe they're not in the
mood to reproduce as much,
or maybe they're getting killed,
or they're dying of, this
is very unpleasant thinking,
they're dying of starvation,
or they're not able to
get water, dehydration.
Who knows what might be happening?
And we could also think
about this visually.
If we were to make a quick
graph right over here,
where if this is time,
and if this is population,
our exponential growth right here
would describe something
that looks like this.
So, for exponential growth, our population
will grow like this.
The more our population
is, the faster it grows,
the more it is, the faster it grows,
the more it is, the faster it grows,
and it'll just keep
going forever until it,
just, there's no limit in theory.
And obviously, we know
that's not realistic.
Now with logistic growth,
I'll do this in red,
in logistic growth, in
the beginning it looks
a lot like exponential growth,
it's just a little bit slower.
But then as the population
gets higher and higher,
it gets a good bit slower,
and it's limited by the
natural carrying capacity
of the environment for that population.
So K would be right over there.
It would asymptote up to it,
but not quite approach it.
And if you wanted to get the
limit, what would happen?
Well, what happens at a population
of 1000 in this circumstance?
Well then, this factor right
over here just becomes zero,
so your population at that point
just wouldn't grow anymore if
you were to even get there.
