In the last video, we analyzed the logistic equation
with harvest for different values of h.

The fruits of our labor was a bunch of phase lines.

So we looked at different h's. Here's h=0,
40, 65 and for each one, we used this graph

to make a phase line.

In this video, I'll show how we can combine
all this information to produce something called

a bifurcation diagram, that's a really important
quantity in the study of dynamical systems,

and lets us see how a system's behavior
changes as the parameter is varied.

Okay, so back to this pile of work, we've
got 0, 40, 65, 75 and 85, and then for the

quiz, which you might have done, here's h=30,
so we have one more, one that I didn't do in the

last video, and what I want to do, is focus
on these phase lines.

So we're not going to look at this, but I
just want to look at all these phase lines

to see how they change.

So in order to do that, I'm going to cut the
phase line off of each of these diagrams.

So I did that for h=30, and then I'll do
that for the rest of them.

So I've cut the phase lines off of those graphs,
and now I have a pile of phase lines.

And what I'm going to do next, is put these
in order, from, maybe, smallest h to largest h,

let's see, so we have 30, 40, 65, 75 and 85.

So there they are in order, and we can see 
here, this was the original logistic equation

without any harvest at all, and then as we start
harvesting more, the fixed points move in

and then the two fixed points are getting
closer together.

It's not perfect, because these are not
perfect phase lines, they're just sketches,

but there it is, you can see them kind of
moving together, here there's just one, and

by the time you get to here, there's just
no phase line at all.

So this is one way of viewing what happens
for different parameter values, different h values.

And it's traditional, though, to orient this
differently.

So typically, instead of having it like this,
we kind of rotate it like this, and so let me

do that, I'm going to move these like this.

Okay, so I've organized all of the phase lines
from left to right, so h now is running horizontally

left to right.

This is h, it's going in this direction.

So I'm increasing h, start at 0, 30, 40, 65,
the two fixed points get closer together,

they merge into one at 75 and disappear by
the time I'm over here.

And then on this axis, I can think of this.
I'm plotting X, this is my phase line, and the

phase line is X, or I'm calling it X now,
maybe I should have called it,

P, I'll call it P for population. And this was
up here, just as a reminder that was 100.

So this is a way, we can kind of begin to see
what's going on here, kind of follow these dots.

It's a little hard to see, maybe there's some
sort of shape emerging.

And we could do more experiments, do more
things like this, I can give you more of these,

you make more phase lines, cut them out, put
them in here, and basically build up an entire

plot of these.

And if one were to do so, one would end up
with something that looks like this.

So we have two fixed points. This one is stable,
and this is unstable. As h increases, this is h,

down here, write this as P for population instead
of X, the variable name doesn't really matter,

So as h increases, these two fixed points
get closer and closer together.

At 75, there's only one fixed point, it's
sort of a funny, half-stable one, and then

any harvest rate above 75, there's no stable
fish population at all, it'll just decrease to 0,

and then, mathematically, it would even give
you negative numbers, negative fish, whatever

that would mean.

So these blue lines are just phase lines,
which we normally draw horizontally, I just

rotated them up this way. So this is just
a phase line, here's another phase line,

here's a phase line for h=30, and so on.

So this type of diagram is known as a bifurcation diagram.

It shows us how the long term behavior of a system
changes as we change a parameter.

So the parameter that we're changing is h,
we can imagine increasing h, slowly increasing

the amount of fishing, and the long term
behavior of that is this light blue curve here.

So the long term behavior here is we have
two fixed points, this is unstable and that's stable.

We increase the amount of fishing
from 0, and we go to here.

And it's really, it's the same story. There's
still two fixed points, one of them is stable,

and one is unstable. The value of the fixed
point has changed a little bit, but that's

just a small quantitative change, not an overall
qualitative change in the feature of this system.

We continue to increase the rate of fishing
when we're here, again we have a stable fixed

point here, arrows are pushing towards it,
and an unstable fixed point here, by my thumb

So, yes, the values of the fixed points have
changed. but that doesn't change the overall

character of the equation. There's still one
stable and one unstable fixed point,

and we keep increasing the value of h, we
increase the amount of fishing, and here

we have a qualitative change. You'd say at
this point here is a bifurcation.

Right at this point, we're changing from two
fixed points, one stable one unstable to no

fixed points.

So this point here would be where the bifurcation
occurs.

It's a sudden change in the qualitative, sort of,
overall global features of the system as a

parameter is changed.

So as we vary h, from one side of 75 to another,
the system jumps from having two fixed points

of different stabilities to zero fixed points.

So this is an example of a bifurcation, and this 
type of diagram is known as a bifurcation diagram.

So let me summarize this example.

The type of graph we ended up with is a
bifurcation diagram.

We were plotting the phase lines for the
population as a function of h, the parameter.

So this is just a different view of this thing
in blue that I had a moment ago.

We have a line of equilibrium points that 
decreases, those are stable equilibriums,

then we have unstable equilibrium, this is
an attractor, this is a repeller, and they

disappear suddenly right here. So this is
where the bifurcation would occur.

So, again, a bifurcation is a sudden qualitative
change in a dynamical system as a parameter

is varied continuously.

What I mean by sudden qualitative change,
a qualitative change is not the location of

a fixed point, but a change in the number of
fixed points and or their stability.

So that's a sudden qualitative change in
the overall character, the global behavior

of the system.

And this change happens all of a sudden.
We go from two on one side of the transition

to zero on the other.

And so what I mean by varied continuously,
is that we get a jump in the qualitative behavior,

the number of fixed points and their stability
without having to jump the parameter.

So a very small change from one side of
this transition to the other gives rise to this

sudden qualitative change.

In the next several videos, I'll say more about
bifurcation diagrams and why they're important

and what lessons I think this general phenomena
of bifurcations holds for complex systems.

But first I'll suggest you practice interpreting 
bifurcation diagrams on the next quiz.

So, when in doubt about bifurcation diagrams,
remember that it begins its life as a series of

phase lines, and phase lines you're familiar
with by now, from the beginning of this unit

and from Unit 2.

So, for example, I can use this bifurcation
diagram, say, to look up what the behavior is

for this dynamical system for this particular
h value.

And I do it by just ignoring, just focusing
right in on this and I can see, it might not

be quite clear on the video, that I just have
a single phase line and then I know how to

interpret that single phase line.

We're used to orienting things this way.

And so this would show us that there is a stable
fixed point here and an unstable fixed point here.

So give the quiz a try and practice interpreting
bifurcation diagrams, and then we'll talk a little

bit more about this phenomena of bifurcations
and what it means.