- [Justin] In the last video, we used a simple model
to predict the size of a population of creatures.
That model had two ways
for creatures to come into existence.
The first was a spontaneous birth chance,
which allowed creatures to pop into
existence without any parent.
The second was a replication chance,
which allowed a live creature to create a copy of itself.
In this video, we're going to make the model more realistic
by completely getting rid of that spontaneous birth chance,
which we know is unrealistic for complex organisms.
Along the way, we'll build some graphs to take a closer look
at how replicating populations behave with or without
a spontaneous birth chance.
We'll actually have two kinds of graphs.
The first kind will track the number of creatures
over time in a simulation.
These number-over-time graphs will be a bit different
each time we run a simulation, even if the settings
for that simulation are exactly the same.
Our second kind of graph will take a bit longer to build,
but it'll help us understand what these
number-over-time graphs have in common and even predict
what we'd expect the graph to look like
before we run a simulation.
For this second type, instead of plotting data
from a simulation, we're going to graph a function,
where one quantity depends directly on another
without any randomness.
The first step in creating this graph will be to tweak our
equation from before and make it into a function equation
that lets us plug values in and get values out.
The starting point from before was to set
the total birth rate equal to the total death rate.
When these are equal, we expect some creatures
to die each time step, but we expect the same number
to appear to replace them.
The overall expected change in the population would be zero.
How can we extend this thinking to make a function equation?
Instead of assuming the birth and death rate are the same,
we can subtract them to get the overall expected change.
For this, we'll use the symbol delta.
If you add the ones that are born, and subtract the ones
that die, you get the total change.
The total expected change will still be zero
some of the time, but now we have a foothold for thinking
about cases where we do expect an overall change,
which is good because, as we can see, the number
of creatures does change pretty constantly.
Our equation is starting to look like a function equation.
The last step is to write the birth and death rates
in terms of the total number of creatures
and then we'll be able to draw the graph.
We did this in the last video,
but let's run through it again.
We can break the total birth rate into
the spontaneous birth rate, which we're going to get rid
of later, plus the replication rate, which is the chance
of a single creature to replicate at a given time,
multiplied by the total number of creatures.
Similarly, the total death rate is the chance
for a single creature to die multiplied, again,
by the total number of creatures.
Finally, we can rearrange things to see more clearly
that the replication and death chances are competing
to determine the expected net change per creature.
This difference is going to have
a big impact on the population.
Now, we have the expected change
written as a function of the number of creatures.
For any number of creatures, N, we can determine
the expected change, delta.
Let's look at a specific situation.
Let's make the spontaneous birth rate one,
so each time step will see one brand new creature
pop into existence on its own.
The death chance for each creature will be 0.2,
so each living creature will have a one in five chance
of dying each time step.
To start out, let's leave the replication chance at zero.
Now we get that satisfying feeling
of plotting all the points and making a curve.
When we have zero creatures, the expected change is equal to
the spontaneous birth rate and as N increases,
the expected change decreases by 0.2 per creature.
Notice this point here where delta is equal to zero?
That's our old friend the equilibrium point from before.
At that point, which is five creatures, in this case,
we don't expect any change on average.
If the number of creatures goes below five,
we'll then expect a positive change at the next time step,
pushing the population back toward five.
If N goes about five, we'll expect a negative change,
again, pushing it back toward equilibrium.
To toss another vocabulary term at you,
this is known as a stable equilibrium.
At this point, the birth and death rates are in balance,
and it's stable because if we deviate from this point,
the system will tend to go back.
It's self-regulating, as if there's a rubber band pulling
the system back toward equilibrium whenever it strays away.
What kind of number-over-time graph will this produce?
If the simulation is at equilibrium, our best guess
is that there will be zero change at the next time step.
That's what this point means, the expected change is zero.
Over time, what we expect is for the simulation
to stay at the equilibrium number.
This straight line might seem naive, because we've seen
how these simulations can fluctuate like crazy,
but if we graph the data from a whole bunch of simulations
at the same time, we can start to see that this really
is a pretty good guess for what to expect on average.
What happens if we add a replication chance?
The slope of the line changes
and so does the equilibrium point.
Overall, though, things are pretty much the same.
Things get interesting when the replication chance
is the same as the death chance.
In this case, we get a flat line,
and the equilibrium disappears.
If you saw the last video, this is where we
divided by zero and our equation broke.
There is no equilibrium.
What about number-over-time graphs
in this kind of situation?
There's no more equilibrium, but we can still
say something about the growth rate.
That's why we built this function equation
in the first place.
No matter how many creatures there are,
the death and replication chances offset each other,
and we expect the population to constantly grow
at the spontaneous birth rate which, in this case,
is at one creature per time step.
Starting at five creatures again,
the expected curve looks like this.
Just like before, we can convince ourselves that this
is a good guess by plotting the results
of lots of simulations on top of each other.
There's something extra to notice here, though.
That rubber band effect of the stable equilibrium is gone,
so we see this spread develop as time passes.
Things get even more interesting when the replication chance
is higher than the death chance.
We get a positive slope,
and this reverses that rubber band effect.
When the number of creatures goes up,
the expected change also goes up,
causing the number to go up even more in the next time step.
The two feed off of each other, going up and up and up.
The result is exponential growth.
It's slightly more involved to show this precisely,
so I'll link to some other resources in the description
in case you want proof, but, once again,
we can look at a bunch of simulations to confirm the result.
Notice how the number of creatures is much higher here.
This explosive growth comes from that
tiny bump in the replication chance.
Also, the spread is even worse this time
because of the reversal of the rubber band effect.
Let's finally set that spontaneous birth rate to zero.
We still have that reverse rubber band effect,
where the expected change gets bigger and bigger and bigger,
but if the population is ever zero, it gets stuck.
There are no creatures to replicate
and none are going to be born spontaneously.
This is called extinction.
It's over for those creatures.
If we start with a few creatures, two in this case,
some simulations will grow exponentially
and quickly be in no danger of extinction.
But others will get unlucky and die out, never to come back.
Creatures that depend only on replication are on
a knife's edge, especially when their numbers are small.
If complex organisms can't appear spontaneously,
and they can't replicate unless they already exist,
how do they get their start?
We'll find out as we keep building our model
in the next video.
See ya then.
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