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.