Hi. It's Mr. Andersen and in this video I'm going to talk about logistic
growth. You should probably watch my video on exponential growth if you haven't done
so first. But logistic growth is what happens after. In other words after a population eventually
maybe crashes or finds a natural kind of a carrying capacity. And so let's start by talking
about population growth in general. Imagine we have two rabbits. Our N value is 2. So
N is always going to refer to the population size. Now it's going to change over time,
but to start let's say N is equal to 2. r is going to tell us how fast that population
is going to change. We call that growth rate. It really is determined by two things. Births
and deaths. And let's say that these rabbits have two baby rabbits. And so we'd have two
births minus zero deaths. So that's going to be 2 minus 0 or 2. And then we divide it
by our original N, which was 2. So it's going to be 2 over 2 or 1. And if you have a growth
rate of one, things are going to go really, really quickly. But we could have had a growth
rate of 0.5. How could we get that? Let's say 2 rabbits are born. Minus 1 dies. So that
would be 2 minus 1 or 1. Over 2 and that's going to be 0.5. The take home message about
exponential growth remember is if r is ever greater than zero, we're going to show fast
exponential growth. And so let's say it's 1. What's going to happen? Well those four
rabbits are going to become eight. And then we're going to take 8, in this case, times
the growth rate of 1, and so we're now going to have 16. And now we're going to have 32.
And now we're going to have 64. And we're going to have 128. And we're going to have
256. And pretty soon my whole screen is going to be filled up with rabbits. And I heard
my fan come on on my computer because it's really taxing to put that many rabbits on
the screen. And so does that occur in nature? No. And Darwin noted that as well. He looked
at, instead of rabbits, elephants. Elephants take a long time to reproduce. In other words
they have to be 30 before they can reproduce. And they live until they are about 90. And
what he found was that even though they might produce 3 pair of offspring during that time,
if you just let them keep breeding over and over and over again, and after five centuries,
or 500 years later, you're going to have 15,000,0000 elephants. And so exponential growth goes
really really quickly. And we know that it can't last forever. Eventually we're going
to hit a limit. And so let's look at those four rabbits again. Eventually they're going
to reach what's called K or carrying capacity. It's the maximum amount in a population that
can be supported in a general kind of, in an ecosystem or in an area. And so why is
that? Well rabbits eat things. And that is grass. And when the run out of grass they
die. Now I don't have the ratio right, you have to have way more producers than you're
going to have consumers. And so let's put our rabbits on the grass. What are they going
to do? They're going to eat the grass. And as they do so, that grass is going to go away
and they're going to have to move on to another area. And so luckily there's more grass for
them to find. But eventually they'll eat their way into a corner. So this rabbit right here
has a problem. If this grass didn't grow back, it would really be stuck. And so if we add
much more than four rabbits, we would have exceeded the carrying capacity. And in class
I ask my students to build models that showed logistic growth. And a lot of them use paper
and pencil, but one creative group used Minecraft. And what they did was constructed a little
chamber here. And they put one chicken in it. And every time the chicken would step
on one of these platforms, two doors would open up. And two more chickens would come
out. So pretty soon we had three chickens. And they'd step on platforms. And pretty soon
we had 5 chickens. And it just went really, really, really quickly. Pretty soon there
were chickens everywhere. And I ask them well it's got to show logistic growth. And these
bright students Gabe, Ethan and David said, well watch what happens. Eventually as they
stand on the platforms it becomes so crowded they can't get off anymore. And so they can't
release anymore chickens into the area and so eventually it reaches kind of a carrying
capacity of chickens in this one container. Now let me show you another model. This is
a NetLogo model called Rabbits, Grass and Weeds. So let me launch that for a second.
So in this model the rabbits are going to be these little white things with ears. And
the grass are going to be these little green squares. And so we start with 5. Now the rabbits
are going to find grass if they can. And if they can, they're going to get energy. And
if they get enough energy they can breed. But if they don't find enough grass, then
they're going to die. And so if we let it go for a second, it goes really, really quickly.
So, let me stop it there for a second. So let's watch what happened. And so this red
line right here is going to represent the number of rabbits over time. And so again
we started with 5. And pretty soon we had 350 rabbits. What happened to the grass? Well
the grass started to reproduce as well, but it crashed because the rabbits were eating
all the grass. So let's watch what happens now if we let it roll for a second. So we're
going to see a big drop in the number of rabbits. And then it's going to finally hit a limit.
So we're going to find like a perfect amount of rabbits that we have. So we're reaching
what's called carrying capacity. And you can see that if we let it go over and over and
over, the grass and the rabbits are going to go up and down, but we're going to reach
a limit. Now let's say that we gave the grass more energy. So if there's more energy in
the grass, let's increase that. We're going to see exponential growth again. And then
we're going to have even more rabbits. Or let's say we said that the grass has less
energy, as we decrease that, then we're going to have way fewer rabbits. And so that rabbit
population is going to drop. But it's going to find another carrying capacity. So what
happens as your population size increases? You run up against these limiting factors.
And those limiting factors are going to slow your growth. So let me quit that. So let's
get to the math now. So if we're looking at exponential growth, so exponential growth,
the equation is going to be this right here. And dN over dt means the change in N over
the change in time. And so it's going to be r times N, where r is the growth rate and
N is the population size. And so if we started like we did at the beginning with exponential
growth, you're going to have two rabbits. What's our change in N over change in t? It's
going to be two. How did we get that? We're multiplying r which is 1 remember. Times N
which is 2. And I'm going to get 2. What happens to that change in N? Well that's going to
give me my new population. So I'm taking 2 here plus 2 here and now I have 4. What's
my change in N over time for the next one? Again, since our r is 1. It's going to be
4 again. And so we're going to have 8. We're going to have 16. We're going to have 32.
So if we were to graph that that's going to give us that "J" shaped curve. But what I
want you to do is look at this equation right here and watch as I change it to an equation
that shows carrying capacity. And so the only thing that's going to change is what's in
the parentheses right here. And so that's going to be be the factor that's related to
carrying capacity. And so I'm going to choose an arbitrary carrying capacity. Let's say
it's 10. That the area can only support 10 rabbits. Let's watch what happens. And so
we start with two rabbits. And so I'm going to show you what's in the parentheses here.
And so we're going to have K minus N divided by K. Well what's K? What's our carrying capacity?
That's 10. Minus N. So that's minus 2. So we're going to have 8. And then what's our
carrying capacity? It's 10. So it's going to be 8 divided by 10, or it's going to be
0.8. Okay. Now we're going to figure out this. And so what's this? It's R which we said was
1. Times N, which we said was 2. Times what's in this parentheses, which is 0.8. And so
that's going to be 1.6. And so this whole equation tells us how many new rabbits we're
going to add. And we're not going to round. So I can show you what happens as we increase
this. But know this. That we're going to increase by 1.6 rabbits. So let's look at time 1. Now
we have 3.6 rabbits. Where did I get that? It's the 2 original rabbits we had plus the
1.6. Okay. What's going to be in the parentheses now? It's going to be K, which is 10 - 3.6.
So that's 6.4 divided by 10. And now it's .64. You can see that the number became a
smaller number. We're going to multiply that times 3.6. Again it's 1 times 3.6 times that
value and we get 2.3. I'm going to add that to our our original 3.6 and now we get 5.9.
What's going to be our K minus n over K? It's going to be 0.41. And so what do we get? 2.4.
And so you can see that as the population is increasing, this factor is becoming smaller
and smaller and smaller. And so now we're only adding 1.4. And now we're only adding
0.29. And now we're only adding 0.009. And so what's happening to N? You can see that
it's increasing quickly right away, but the closer we get to that carrying capacity, it's
kind of leveling off. And it's reaching that limit at that one point. So this is just mathematical.
It's not a model. But it shows that we're eventually going to approach that point. And
so again, in review. What's r? r is going to be the growth rate. What's K? K is going
to be the carrying capacity. So if our r is ever greater than 0, remember we're going
to show exponential growth. And if we ever have K, if we ever have a carrying capacity,
we're going to show logistic growth. But we also have species that specialize in these
two. We've got r selected species and K selected species. So what does that mean? An r selected
species is going to be a species that loves to grow as quickly as it can. Look how many
babies this frog is going to have. Thousands of those. And so if I were to take this frog
and introduce it into a pond where there are no frogs, we're going to have a bunch of frogs
really really quickly. We're going to have an r that's really really large. And they're
going to fill that area. They're going to run out of resources and crash. And they're
going to boom and bust, but they're going to fill it really really quickly. If we were
to look at this which is a chameleon. A chameleon is going to only have two offspring. So it's
growth rate is going to be really really slow. What is it going to do? It's going to give
them a lot of parental care. It's going to invest a lot of energy in just its few offspring.
And they're all going to survive. If we look up here with r selected species, most of those
are going to die eventually. But with this chameleon, since it's taking so much care
of them, it's going to take very good care of it, and they're all going to survive. This
is going to gradually increase and find a nice carrying capacity. So what are we? What
are humans? Well we invest a lot in our young. A lot of my students are 17 years old and
they still live at home. Their parents take care of them. So we're really investing in
them. So what kind of population curve are we going to see in humans? We're going to
see a gradual exponential increase and then a nice carrying capacity which we're eventually
going to reach. And we don't want to have this boom and bust cycle. So it's not so cut
and dry as that. There are obviously species that are somewhere in the middle. So what
about a tree do you think? A tree grows really slow but they're going to produce a lot of
offspring. And so what are they? There's probably no right answer for that. And so again, what
is logistic growth? It's growing quickly but eventually reaching a carrying capacity. And
that's based on limits in your environment. And I hope that was helpful.