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PROFESSOR: The
Phase Lines mathlet

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helps us to understand
the behavior

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of nonlinear
autonomous equations.

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The large graphing window
shows a direction field.

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The horizontal axis is t,
and the vertical axis is y.

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As you see, the direction
field is independent of t.

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This is the significance
of an autonomous equation.

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The applet opens with the
equation 1 minus y times y

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minus a, but I want to talk to
you about a different equation,

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namely this one, which can
be written as the quantity

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a minus y, times y, minus 1/4.

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This is a logistic
equation with harvesting.

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Specifically, this is a
model of the oryx population

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in a certain game
preserve in Kenya,

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measured in kilooryx and years.

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The Kenyan government
wants to sell permits

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to kill 1/4 kilooryx
per year, and it

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wants to know how
large a game preserve

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it should create to guarantee
a stable oryx population.

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The size of the
preserve determines

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what would be the limiting
population of oryx which

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is the number a.

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We can use this
applet to explore

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the options to recommend
to the Kenyan government.

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We can control the parameter
a using this slider down here.

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Right now, it's
set to a equals 0,

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so the equation
is y prime equals

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minus 1/4 minus y squared,
always negative, so all

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the solutions decrease.

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With a game preserve
of zero area,

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the population is
guaranteed to collapse.

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We can draw some solution curves
by clicking on this window.

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You'll notice that any
horizontal translate

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of the solution is another
solution to this differential

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equation.

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This is another
consequence of the fact

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that it's an autonomous
differential equation.

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We can increase the value
of a using this slider.

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And when I do, notice that
the change in the direction

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field-- it starts to flatten
out, especially in an area just

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above the t-axis.

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And when a takes on
the value 1, a solution

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appears, a constant
solution in blue.

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This is an equilibrium solution.

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We can measure the
value of y along it.

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There's a readout
below the screen.

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It seems that y is about 1/2.

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And if we take a equals
1 and y equals 1/2,

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and plug those values
into this equation,

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you will get y prime equals 0.

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In other words, you get
a constant solution.

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Now notice this parabola in
the lower left of the screen.

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As I change a, it moves.

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That parabola is the graph of
y prime as a function of y.

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So it depends upon a.

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When a moves up to
the value a equals 1,

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the parabola moves
up and touches

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the y-axis at the
value y equals 1/2.

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Now this blue curve represents
a steady-state solution,

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but the Kenyan
government would be

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ill-advised to use a
preserve only this big.

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Because random fluctuations
will occasionally

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drive the population of
oryx below y equals 1/2,

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and as soon as that
happens, disaster ensues.

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The population is
guaranteed to collapse.

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So let's take a bigger game
preserve, make a larger,

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and when you do that,
the parabola moves up,

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the blue double root
splits into two roots,

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and the equilibrium solution
bifurcates into two equilibrium

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solutions.

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The red equilibrium is unstable.

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Solutions near to
it diverge from it.

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In fact, solutions below
it approach infinity.

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These curves become asymptotic
to vertical straight lines.

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They blow up in finite time.

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Solutions between
the two equilibria

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move from the lower
equilibrium up

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to the green upper
equilibrium, and solutions

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above the green equilibrium
decay towards it.

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The green solution is
a stable equilibrium.

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You can see this behavior
by looking at the parabola

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as well.

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If the value of y is
less than the red zero,

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the value of the
parabola is negative.

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Solutions are falling.

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Between the two roots, the
value of y prime is positive.

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Solutions are growing.

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And to the right of
this equilibrium,

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the value of y
prime is negative.

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Solutions are falling again.

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You can see all this very neatly
using the phase line, which I

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can invoke using this key here.

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The phase line carries all
the information present

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in the direction field, but
in a much more compact form.

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You can see the two
equilibria marked,

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and below this equilibrium,
all the solutions are decaying.

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Between the two, the
solutions are all increasing,

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and above the upper equilibrium,
the solutions are all decaying.

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These are indicated by
these yellow arrows.

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Now the Kenyan government
has some leeway.

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It may predict an
accidental variation

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of, say, 1/2 kilooryx.

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In that case, it should
arrange a game preserve large

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enough so that the distance
between the two critical points

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is at least 1/2.

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Then, assuming that
the initial population

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is close to this equilibrium,
a deviation of 1/2

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won't be a disaster,
because the population

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will recover and return
to the stable equilibrium.

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Well, we've come a long way.

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We're now looking at a family
of differential equations,

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when indexed by the parameter a.

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Each value of a has its
own direction field,

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its own set of critical
points, it's own solutions,

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and its own phase line.

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But as we see, for
practical policy reasons,

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it's important to be
able to consider them

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all simultaneously.

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The bifurcation diagram
lets you do exactly that.

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Let's invoke it
with this check box.

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The diagram has appeared
above the a slider.

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In it is marked a curve.

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This curve consists of
all the critical points

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for these differential
equations for all values of a,

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marked simultaneously.

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The critical points
for a given value of a

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appear above that value
of a in the slider.

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So here's the phase line for
this value of a containing

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an unstable critical point
and a stable critical point.

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And when I change
the value of a,

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that phase line
changes and moves.

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And as the value of a
decreases to a equals 1,

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those two critical
points collide and form

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a single semi-stable
critical point.

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For smaller values of a,
there are no critical points,

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until you reach the
value a equals minus 1,

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when a single semi-stable
critical point appears and then

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bifurcates into two critical
points for smaller values of a.

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The critical curve
is color coded

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according to the type
of critical point

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that it represents.

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The bifurcation diagram
takes place in the a-y plane.

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It's the curve defined by the
equation y prime equals 0.

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In this case, that's minus
1/4 plus a*y minus y squared.