WEBVTT

00:00:00.500 --> 00:00:02.090
PROFESSOR: The
Phase Lines mathlet

00:00:02.090 --> 00:00:03.870
helps us to understand
the behavior

00:00:03.870 --> 00:00:06.765
of nonlinear
autonomous equations.

00:00:06.765 --> 00:00:10.070
The large graphing window
shows a direction field.

00:00:10.070 --> 00:00:16.650
The horizontal axis is t,
and the vertical axis is y.

00:00:16.650 --> 00:00:21.550
As you see, the direction
field is independent of t.

00:00:21.550 --> 00:00:26.150
This is the significance
of an autonomous equation.

00:00:26.150 --> 00:00:30.080
The applet opens with the
equation 1 minus y times y

00:00:30.080 --> 00:00:34.260
minus a, but I want to talk to
you about a different equation,

00:00:34.260 --> 00:00:38.710
namely this one, which can
be written as the quantity

00:00:38.710 --> 00:00:44.210
a minus y, times y, minus 1/4.

00:00:44.210 --> 00:00:47.380
This is a logistic
equation with harvesting.

00:00:47.380 --> 00:00:50.730
Specifically, this is a
model of the oryx population

00:00:50.730 --> 00:00:53.320
in a certain game
preserve in Kenya,

00:00:53.320 --> 00:00:56.710
measured in kilooryx and years.

00:00:56.710 --> 00:00:58.970
The Kenyan government
wants to sell permits

00:00:58.970 --> 00:01:02.900
to kill 1/4 kilooryx
per year, and it

00:01:02.900 --> 00:01:04.879
wants to know how
large a game preserve

00:01:04.879 --> 00:01:10.090
it should create to guarantee
a stable oryx population.

00:01:10.090 --> 00:01:12.140
The size of the
preserve determines

00:01:12.140 --> 00:01:16.890
what would be the limiting
population of oryx which

00:01:16.890 --> 00:01:19.120
is the number a.

00:01:19.120 --> 00:01:21.110
We can use this
applet to explore

00:01:21.110 --> 00:01:26.500
the options to recommend
to the Kenyan government.

00:01:26.500 --> 00:01:31.050
We can control the parameter
a using this slider down here.

00:01:31.050 --> 00:01:34.250
Right now, it's
set to a equals 0,

00:01:34.250 --> 00:01:37.690
so the equation
is y prime equals

00:01:37.690 --> 00:01:43.600
minus 1/4 minus y squared,
always negative, so all

00:01:43.600 --> 00:01:46.260
the solutions decrease.

00:01:46.260 --> 00:01:48.230
With a game preserve
of zero area,

00:01:48.230 --> 00:01:52.100
the population is
guaranteed to collapse.

00:01:52.100 --> 00:01:57.200
We can draw some solution curves
by clicking on this window.

00:01:57.200 --> 00:02:00.170
You'll notice that any
horizontal translate

00:02:00.170 --> 00:02:03.330
of the solution is another
solution to this differential

00:02:03.330 --> 00:02:04.586
equation.

00:02:04.586 --> 00:02:06.210
This is another
consequence of the fact

00:02:06.210 --> 00:02:10.770
that it's an autonomous
differential equation.

00:02:10.770 --> 00:02:15.120
We can increase the value
of a using this slider.

00:02:15.120 --> 00:02:19.510
And when I do, notice that
the change in the direction

00:02:19.510 --> 00:02:24.040
field-- it starts to flatten
out, especially in an area just

00:02:24.040 --> 00:02:26.650
above the t-axis.

00:02:26.650 --> 00:02:33.620
And when a takes on
the value 1, a solution

00:02:33.620 --> 00:02:36.930
appears, a constant
solution in blue.

00:02:36.930 --> 00:02:39.460
This is an equilibrium solution.

00:02:39.460 --> 00:02:42.700
We can measure the
value of y along it.

00:02:42.700 --> 00:02:45.010
There's a readout
below the screen.

00:02:45.010 --> 00:02:48.930
It seems that y is about 1/2.

00:02:48.930 --> 00:02:53.760
And if we take a equals
1 and y equals 1/2,

00:02:53.760 --> 00:02:56.450
and plug those values
into this equation,

00:02:56.450 --> 00:02:59.050
you will get y prime equals 0.

00:02:59.050 --> 00:03:03.190
In other words, you get
a constant solution.

00:03:03.190 --> 00:03:07.260
Now notice this parabola in
the lower left of the screen.

00:03:07.260 --> 00:03:11.740
As I change a, it moves.

00:03:11.740 --> 00:03:18.360
That parabola is the graph of
y prime as a function of y.

00:03:18.360 --> 00:03:20.660
So it depends upon a.

00:03:20.660 --> 00:03:24.240
When a moves up to
the value a equals 1,

00:03:24.240 --> 00:03:28.030
the parabola moves
up and touches

00:03:28.030 --> 00:03:31.940
the y-axis at the
value y equals 1/2.

00:03:34.450 --> 00:03:38.350
Now this blue curve represents
a steady-state solution,

00:03:38.350 --> 00:03:39.880
but the Kenyan
government would be

00:03:39.880 --> 00:03:43.700
ill-advised to use a
preserve only this big.

00:03:43.700 --> 00:03:46.310
Because random fluctuations
will occasionally

00:03:46.310 --> 00:03:50.630
drive the population of
oryx below y equals 1/2,

00:03:50.630 --> 00:03:55.270
and as soon as that
happens, disaster ensues.

00:03:55.270 --> 00:03:58.410
The population is
guaranteed to collapse.

00:03:58.410 --> 00:04:02.370
So let's take a bigger game
preserve, make a larger,

00:04:02.370 --> 00:04:07.560
and when you do that,
the parabola moves up,

00:04:07.560 --> 00:04:12.910
the blue double root
splits into two roots,

00:04:12.910 --> 00:04:18.105
and the equilibrium solution
bifurcates into two equilibrium

00:04:18.105 --> 00:04:18.605
solutions.

00:04:21.480 --> 00:04:24.670
The red equilibrium is unstable.

00:04:24.670 --> 00:04:28.890
Solutions near to
it diverge from it.

00:04:28.890 --> 00:04:33.370
In fact, solutions below
it approach infinity.

00:04:33.370 --> 00:04:37.610
These curves become asymptotic
to vertical straight lines.

00:04:37.610 --> 00:04:41.260
They blow up in finite time.

00:04:41.260 --> 00:04:43.760
Solutions between
the two equilibria

00:04:43.760 --> 00:04:46.150
move from the lower
equilibrium up

00:04:46.150 --> 00:04:49.290
to the green upper
equilibrium, and solutions

00:04:49.290 --> 00:04:55.085
above the green equilibrium
decay towards it.

00:04:55.085 --> 00:04:59.330
The green solution is
a stable equilibrium.

00:04:59.330 --> 00:05:01.790
You can see this behavior
by looking at the parabola

00:05:01.790 --> 00:05:03.050
as well.

00:05:03.050 --> 00:05:10.250
If the value of y is
less than the red zero,

00:05:10.250 --> 00:05:12.260
the value of the
parabola is negative.

00:05:12.260 --> 00:05:14.180
Solutions are falling.

00:05:14.180 --> 00:05:19.070
Between the two roots, the
value of y prime is positive.

00:05:19.070 --> 00:05:21.150
Solutions are growing.

00:05:21.150 --> 00:05:24.110
And to the right of
this equilibrium,

00:05:24.110 --> 00:05:26.350
the value of y
prime is negative.

00:05:26.350 --> 00:05:28.450
Solutions are falling again.

00:05:31.200 --> 00:05:34.890
You can see all this very neatly
using the phase line, which I

00:05:34.890 --> 00:05:38.400
can invoke using this key here.

00:05:38.400 --> 00:05:42.390
The phase line carries all
the information present

00:05:42.390 --> 00:05:47.340
in the direction field, but
in a much more compact form.

00:05:47.340 --> 00:05:50.450
You can see the two
equilibria marked,

00:05:50.450 --> 00:05:55.130
and below this equilibrium,
all the solutions are decaying.

00:05:55.130 --> 00:05:58.670
Between the two, the
solutions are all increasing,

00:05:58.670 --> 00:06:03.170
and above the upper equilibrium,
the solutions are all decaying.

00:06:03.170 --> 00:06:06.010
These are indicated by
these yellow arrows.

00:06:10.370 --> 00:06:13.476
Now the Kenyan government
has some leeway.

00:06:13.476 --> 00:06:15.690
It may predict an
accidental variation

00:06:15.690 --> 00:06:18.480
of, say, 1/2 kilooryx.

00:06:18.480 --> 00:06:21.130
In that case, it should
arrange a game preserve large

00:06:21.130 --> 00:06:25.570
enough so that the distance
between the two critical points

00:06:25.570 --> 00:06:29.570
is at least 1/2.

00:06:29.570 --> 00:06:32.220
Then, assuming that
the initial population

00:06:32.220 --> 00:06:38.030
is close to this equilibrium,
a deviation of 1/2

00:06:38.030 --> 00:06:40.730
won't be a disaster,
because the population

00:06:40.730 --> 00:06:44.990
will recover and return
to the stable equilibrium.

00:06:44.990 --> 00:06:46.590
Well, we've come a long way.

00:06:46.590 --> 00:06:49.880
We're now looking at a family
of differential equations,

00:06:49.880 --> 00:06:52.790
when indexed by the parameter a.

00:06:52.790 --> 00:06:56.750
Each value of a has its
own direction field,

00:06:56.750 --> 00:07:03.930
its own set of critical
points, it's own solutions,

00:07:03.930 --> 00:07:07.290
and its own phase line.

00:07:07.290 --> 00:07:10.090
But as we see, for
practical policy reasons,

00:07:10.090 --> 00:07:11.990
it's important to be
able to consider them

00:07:11.990 --> 00:07:14.230
all simultaneously.

00:07:14.230 --> 00:07:18.180
The bifurcation diagram
lets you do exactly that.

00:07:18.180 --> 00:07:21.730
Let's invoke it
with this check box.

00:07:21.730 --> 00:07:25.540
The diagram has appeared
above the a slider.

00:07:25.540 --> 00:07:27.530
In it is marked a curve.

00:07:27.530 --> 00:07:30.830
This curve consists of
all the critical points

00:07:30.830 --> 00:07:33.900
for these differential
equations for all values of a,

00:07:33.900 --> 00:07:36.230
marked simultaneously.

00:07:36.230 --> 00:07:38.650
The critical points
for a given value of a

00:07:38.650 --> 00:07:42.760
appear above that value
of a in the slider.

00:07:42.760 --> 00:07:48.390
So here's the phase line for
this value of a containing

00:07:48.390 --> 00:07:52.210
an unstable critical point
and a stable critical point.

00:07:52.210 --> 00:07:54.400
And when I change
the value of a,

00:07:54.400 --> 00:07:58.110
that phase line
changes and moves.

00:07:58.110 --> 00:08:02.710
And as the value of a
decreases to a equals 1,

00:08:02.710 --> 00:08:05.850
those two critical
points collide and form

00:08:05.850 --> 00:08:08.860
a single semi-stable
critical point.

00:08:08.860 --> 00:08:12.590
For smaller values of a,
there are no critical points,

00:08:12.590 --> 00:08:15.980
until you reach the
value a equals minus 1,

00:08:15.980 --> 00:08:20.120
when a single semi-stable
critical point appears and then

00:08:20.120 --> 00:08:27.270
bifurcates into two critical
points for smaller values of a.

00:08:27.270 --> 00:08:30.850
The critical curve
is color coded

00:08:30.850 --> 00:08:33.909
according to the type
of critical point

00:08:33.909 --> 00:08:35.870
that it represents.

00:08:35.870 --> 00:08:41.059
The bifurcation diagram
takes place in the a-y plane.

00:08:41.059 --> 00:08:45.560
It's the curve defined by the
equation y prime equals 0.

00:08:45.560 --> 00:08:52.070
In this case, that's minus
1/4 plus a*y minus y squared.