(slow music)

- [Instructor] Hi.

This is the first video in a series about biology.

There's a lot to learn in biology.

Most courses start out with water,

then you learn about organic molecules

and then you learn about how those molecules come together

to form living things and you go from there.

And that's all great but we're not going to bother

with any of that, at least not right away.

Instead, we're gonna focus on evolution

because whenever you ask a why question in biology,

the answer always comes back to evolution

and we're going to spend a while on evolution.

It's gonna be 10 to 15 videos.

We're gonna take it slow and we're gonna use simulations

and math to really understand it.

So in the spirit of taking it slow,

let's zoom out and talk about things in general,

including nonliving things.

Why do things exist

You could answer this question in a number of ways

but the answer that we'll use has two parts.

First things, that exist have started existing

or they've been born.

Second, they haven't stopped existing yet

or they haven't died.

It kind of seems like we haven't really said anything,

but breaking it into two parts

does help us look at some patterns.

For example, raindrops.

Why do raindrops exist?

Well, they're good at this first part,

they're good at being born.

They don't last very long but they form pretty often,

often enough where it's not unusual to see them.

A different example is planets or stars.

Unlike raindrops, they don't form very often at all,

but they more than make up for it by lasting a long time.

When we look up in the sky or just at the ground,

we see planets and stars.

They're also common,

even though they do it in a very different way.

Any kind of thing that exists strikes some balance

at being good at one or both of these two things.

So this is all well and good but as promised,

we can make it a lot sharper

by building some simulations and an equation.

This blob creature will be the star of our first simulation.

At each frame in the simulation,

one of these blob creatures will form

and each living creature

will have a one in 10 chance of dying.

For comparison, let's make a second simulation

with this other type of blob creature.

Compared to the first kind,

this blob creature will be more like a planet.

It'll form less often but it will also die less often.

Each frame, there will only be a one in 10 chance

of one of these forming but each blob creature

will only have a one in 100 chance of dying.

After watching for a while,

we can see that even though these two

kinds of blob creatures are quite different,

on average, there's about the same number

of them at any given moment.

You might be able to guess what happens

when a kind of blob creature has a low birth rate

like a planet and also a high death rate like a raindrop.

There just aren't very many of 'em.

At the other extreme, if a kind of blob creature

has a high birth rate like raindrops and a low death rate,

there will be a lot of them.

So why do these simulations seem to stabilize

around a certain number of creatures?

You might already have some intuition for this,

but we can translate that intuition into an equation

which will let us predict equilibrium number

for any birth rate and any death rate.

If the total birth rate is equal to the total death rate,

we'll expect the population size

to stay the same from one frame to the next.

And since each creature individually has a chance of dying,

the total death rate depends

on the current number of live creatures.

Using numbers from our first simulation,

where one creature was born each frame,

and each creature had a one in 10 chance

of dying each frame, we can see that the expected

birth rate and death rates should be equal

when there are 10 creatures

and that's what we saw in the simulation.

But we also saw in the simulation

that the number of creatures fluctuated all over the place.

It didn't just stick at 10.

It's possible for all of the creatures

to get lucky and not die, in which case,

the number of creatures rises to 11 in the next frame

but then the expected death rate

will be higher than the birth rate

and then on average, we'd expect more than one creature

to die in the next frame,

which would push the number back toward 10.

Of course, the creatures could keep getting lucky

but the more creatures there are, the less likely that is.

It's also possible that more than one of them

will get unlucky and die, leaving us with fewer

than 10 creatures but if this happens,

the expected death rate is suddenly lower

than the birth rate, so in the next frame,

we'd expect the population to drift back up toward 10.

The fancy word for a balancing situation like this,

is equilibrium and 10 is the equilibrium number

of creatures for the first simulation.

Going back to the more general version of the equation,

we can shorten it up by using the letter symbols

instead of the full words and we can solve for N

to get a formula we can use

to predict the equilibrium number.

So now if we set up a new simulation

with a birth chance of 80% and a death chance of 2%,

what equilibrium numbers should we expect?

Well, the formula tells us it should be 40,

so let's see what the simulation does.

Looks about right.

All right, so what does this have to do with living things?

From what we said so far,

it kind of seems like we shouldn't exist.

We're too complex to form

spontaneously the way raindrops do.

Imagine all the right atoms and molecules

just happening to come together to form a rabbit.

It's pretty unlikely and we also don't live all that long,

but somehow, living organisms are still pretty common,

so what's going on?

Now would be a good time to pause and think for a second.

All right, ready for the big reveal?

As you may have guessed, living things are special

because we can make more of ourselves.

We have an extra parameter in our simulation,

a chance to replicate.

So let's add replication to our equilibrium equation.

Just like before, we're looking for a situation

where the birth rate is equal to the death rate.

The overall death rate is just like it was before,

it's the number of live creatures,

times the likelihood of each creature to die.

The overall birth rate is different though.

We still have this B, which stands for the likelihood

that a new creature will spontaneously pop into existence

like a raindrop but we also have this extra piece

to account for reproduction.

This works just like the overall death rate.

Each creature has its own chance to reproduce,

so we multiply that chance by the number

of living creatures to get the total number of births

we expect from reproduction, each frame.

The significant thing about this equation

is that N is on both sides.

Before, just the death rate went up

as the number of creatures grew,

but now the birthrate also grows as N gets bigger.

To see this in action, let's solve for N

and look at a simulation.

We'll start our simulation with two creatures.

The spontaneous birth chance each frame will be 10%.

That's a lot higher than it would be for an actual rabbit

but we only have so much time in this video.

And the death chance per creature, each frame, will be 5%.

For now, we'll just leave the replication chance at zero.

Our formula tells us that the equilibrium number

of creatures should be two, though it's a bit hard

to see in this simulation because of the fluctuation.

Anyway, if we bump the replication chance up to 1%,

we see the equilibrium number go up to 2.5.

Not a huge difference but as we push it higher,

the equilibrium number goes up by more and more each time

and we can start to see the effect in the simulation.

And you might notice that we're about to run into an issue.

If we raise the replication chance to 5%,

we'll be dividing by zero

and there won't be an equilibrium at all anymore.

The population will get bigger and bigger without limit

and if the replication chance goes higher

than the death chance, our formula

gives us a negative number for N, which makes no sense

because we can't have a negative number of creatures.

And even if we could, the number of creatures

is clearly going up and up and up

and not getting closer to that negative number.

Reproduction quite literally breaks our equation.

This is why living things are special.

They follow their own set of rules,

which makes it possible for the complexity of life to exist.

The rest of the videos in this series

are going to explore the consequences of this,

which are collectively called the theory of evolution.

See you in the next video.