- [Justin] We've seen how creatures
that replicate can have their numbers
grow exponentially without limit,
but in the real world, there are limits,
so, a more realistic growth curve
would look something like this.
Sorry, buddy.
(peaceful music)
As we've built up our model
in the last few videos,
we've been running simulations
where the computer steps through time,
and at each time step, it
decides which creatures live,
die, and reproduce
according to certain odds,
and we built an equation
to help us predict
what we expect to happen from one instant
to the next in the simulation.
The expected change in the
number of creatures is equal
to the creature's replication
chance minus its death chance,
all times the current number of creatures,
and we can graph this equation
to help us visualize our prediction.
The most interesting case
is when R is greater than D,
for example, with a replication chance
of 10% and a death chance of 5%.
Then, this graph becomes
a straight line with a positive slope.
The more creatures there are,
the more new creatures we expect
to appear from one time to the next.
This leads to exponential growth.
We went over this pretty quickly,
but an earlier video in the series
called How to Grow Exponentially
goes through it in more detail.
Okay, so, that's what the world looks like
when growth is completely unchecked,
but what should it look like
if we want growth to
level off at some point?
To figure this out, let's work backward.
We want the population curve
to look something like this.
It's like an exponential
curve toward the beginning,
but it levels out at a certain
point, at 50 creatures.
For the population to level out here,
we need the expected change per time step
to go to zero when there are 50 creatures,
so, this curve is gonna
need to bend downward.
How can we change the
equation to make that happen?
This function equation already gives us
delta equals zero when N is zero,
but we want delta to also
be zero when N is 50.
One way to do this is to make
the creatures more likely
to die when there are
lots of creatures around.
There's only so much space
and food in the environment,
so, when it's crowded,
a creature might starve.
To do this, we'll leave the
base death chance alone,
but we'll include an extra term to adjust
the overall death chance
based on crowding.
What should this term be?
Well, we want the term to be small
when there aren't many creatures,
and we want it to be big when
there are a lot of creatures.
A simple way to achieve
this is to write it as
the current number of creatures
multiplied by a constant.
When N is small, the effect
of crowding will be small,
and when N is large, the
effect will be large.
Let's call this constant
the crowding coefficient,
just to give it a short name.
Its value specifies how much
the death chance goes up
for each creature when
we add a new creature,
so, if the value is, say, 0.001,
that means adding another
creature increases
the death chance of all
creatures by 1/10 of a percent.
The new creature is eating
food and taking up space,
so, there's less to go
around for everyone else,
and when we have a lot of creatures,
this term really adds up,
and because I looked ahead
when picking these numbers,
a crowding coefficient
of 0.001 does cause delta
to be zero when N is 50.
This is because the death chance
when adjusted for crowding becomes equal
to the replication chance per creature,
so, each creature is just as likely
to die as it is to reproduce.
The replication and death
chances balance each other out,
and we've found equilibrium again.
To give you some of the usual terminology,
this equilibrium number is
called the carrying capacity
because it's the largest
number of creatures
that the environment
can sustainably support,
and this number over time curve
is called a logistic growth curve,
as opposed to an exponential growth curve.
Now that we've decided
how to tweak the equation
and seen how it affects the graph,
let's double-check that this actually does
predict this S-shaped
logistic growth curve.
When N is small, the delta
curve is pretty similar
to that upward-sloping line
from the exponential case,
so, we'll expect the population to look
like it's growing about
exponentially, at first.
In this middle region, the delta curve
is near its maximum, and
it's mostly horizontal,
so, the overall expected growth
rate doesn't change much.
The growth rate is still high,
but it's just not speeding up anymore.
And finally, in this last region,
the growth rate is actually
slowing down toward zero,
so, we'll expect the
population to level off,
and if N goes above the carrying capacity,
which, again, is an equilibrium number,
the growth rate goes
negative, pushing N back down.
All right, let's run a simulation
to see whether this prediction works.
It sort of works,
and remember, this is all
based on chance, though,
so, to really see how
good this prediction is,
we need to look at many
simulations at once.
Next, let's look at what
happens if new kinds
of creatures appear through a mutation.
This green creature will come out
of 1% of blue's replications,
and it'll be slightly less good
at replication than the blue creature is,
with a replication chance of 8%,
but its replication chance is still
higher than its death chance,
and this orange creature will also
come out of 1% of blue's replications,
and this one will have
a lower death chance.
All three of these creatures
will share the same resources,
so, their delta equations
would have a crowding term
that includes the total number
of all kinds of creatures.
If we start a simulation
with a few blue creatures,
how would you expect things to go?
As you might have guessed,
orange eventually takes over.
It's not enough anymore for blue
to be good at surviving in isolation.
It now needs to be better
than its competitors to maintain numbers.
One surprising thing in this simulation is
that green is doing better
than blue after 500 time steps.
You wouldn't expect that,
since it has the worst
stats of all the creatures,
but this is a good example of
how luck is a big part
of evolving systems.
The most likely outcome
doesn't always happen.
All right, that's it for the fundamentals
of limited growth and competition,
but before we say goodbye in this video,
let's take stock of where we are.
We've seen how replication can lead
to exponentially growing populations.
We've seen how mistakes in replication
can lead to new kinds of creatures,
leading to diversity, and just now,
we saw how a finite pool of resources
puts a cap on populations
and causes competition
between different types of creatures.
Replication, mutation, and competition
make up the core of evolution.
Anywhere replicators exist,
even if there's life on other planets,
everything we've said so far would apply.
We're not done yet, though.
So far, we've been making
all the decisions ourselves.
You could say that we've been
artificially selecting
successful creatures.
In the next video, we'll
let go of the reins
and let the selection
happen a bit more naturally.
See you then.
