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Today's lecture is going to be
basically devoted to working out

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a single example of a nonlinear
system, but it is a very good

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example because it illustrates
three things which you really

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have to know about nonlinear
systems.

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I have indicated them by three
cryptic words on the board,

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but you will see at different
points in the lecture what they

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refer to.
Each of these represents

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something you need to know about
nonlinear systems to be able to

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effectively analyze them.
In addition,

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I am not allowed to collect any
more work, according to the

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faculty rules,
after today.

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And, of course,
I won't.

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But, nonetheless,
the final will contain material

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from the reading assignment G7,
and I will be touching on all

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of these today,
and 7.4, just a couple of pages

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of that.
You will see its connection.

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It discusses the same example
we are going to do today.

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And then I suggest those three
exercises to try to solidify

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what the lecture is about.

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Instead of introducing the
nonlinear system right away that

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we are going to talk about,
I would like to first explain,

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so that it won't interrupt the
presentation later,

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what I mean by a conversion to
a first-order equation.

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For that why don't we look at --

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Maybe I can do it here.
Our system looks like this,

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dx over dt.
For once, I am not writing x

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prime because I want to
explicitly indicate what the

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independent variable is.
It is a nonlinear autonomous

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system, so there is no t on the
right-hand side.

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But this is not a simple linear
function, ax plus by.

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It is more complicated,

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like you had in the
predator-prey robin-earthworm

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problem that you worked on last
night.

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And the other equation will be
g of (x, y).

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This is the system,
and explicitly it is nonlinear

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in general and autonomous,
as I have indicated on the

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right-hand side.
Now, remember that the

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geometric picture of this was as
a velocity field.

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You made a velocity field out
of the vectors whose components

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were f and g,
so that is (f)i plus (g)j,

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but what it looked like was a

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plane filled up with a lot of
vectors pointing different ways

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according to the velocity at the
point was.

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And side-by-side with that went
the solutions.

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A typical solution was x equals
x of t,

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y equals y of t
represented as a column vector.

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And that is a parametric
equation.

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And the geometric picture of
that was given by its

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trajectory.
When you plotted it,

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it was a trajectory.
It had not only the right

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direction at each point,
in other words,

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but it also had to have the
right velocity at each point.

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In other words,
sometimes the point was moving

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rapidly and sometimes it was
moving more slowly along that

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path.
And the way it moved was an

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important part of the solution.
Now, what I plan to do is --

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The conversation that I am
talking about takes place by

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eliminating t.

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Now, why would one want to do
that?

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t, after all,
is an essential part of the

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solution.
It is an essential part of this

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picture because without t you
would not know how long the

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arrows were supposed to be.
You would still know their

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direction.
And, of course,

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it occurs in here.
Now, let's take the first step

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and eliminate t from the system
itself.

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As you see, that is very easy
to do because the right-hand

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side has no t in it anyway and
the left-hand side has the t

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only as the denominators of the
differential quotients.

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The obvious way to eliminate t
is just to divide one equation

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by the other.
And since usually we think of

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not dx by dy,
but dy by dx,

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I will vow --
What is dy over dx?

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Well, if I divide this equation
by that equation,

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dy by dt divided by dx by dt,

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according to the chain rule,
or according to commonsense,

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you cancel the dt's and you get
dy over dx.

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And, on the right-hand side,
you get g of (x,

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y) divided by f of (x,
y).

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But what is that?
That is the equation of the

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first day of the term.
By taking that single step,

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I convert this nonlinear
system, which we virtually never

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find explicit solutions to,
into an ordinary first order

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equation which,
in fact, you also don't usually

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find explicit solutions to,
except with our narrow blinders

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on.
We only get problems at the

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beginning of 18.03 where there
will be an explicit solution to

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this.
I have converted the nonlinear

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system to a single first order
equation to which I can apply

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the usual first order methods
that include hope that I will be

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able to solve it.
Now, what do I lose?

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Well, what happens to this
picture?

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If you don't have the t in it
anymore then you don't have

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velocity, you don't have arrows
with a length to them because

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the length gives you how fast
the point is going.

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Without t I don't know at any
point how fast it is supposed to

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be going.
All I know at any point (x,

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y) is no longer dx over dt
and dy over dt.

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All I know is dy over dx.

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In other words,

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the corresponding picture,
if I eliminate t from this

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picture all that is left is the
directions of each of these

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arrows.
What disappears is their

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length.
And, in fact,

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since dy over dx is
purely a slope,

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I cannot even tell whether the
point is traveling this way or

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that way.
It doesn't make any sense to

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have the point traveling anymore
since there is no time in which

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it can do its traveling.
So that picture gets changed to

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the direction field.
The corresponding picture now,

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all you do is take each of
these arrows,

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you snip it off to a standard
length, being careful to snip

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off the little pointy end,
and it becomes nothing but a

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line element.
And all it has is a slope.

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What corresponds to the
velocity field here is now just

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our slope field or our direction
field.

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All we know is the slope at
each point.

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How about the solution?
Well, if I have eliminated

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time, the solution is no longer
a pair of parametric equations.

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It is just an equation
involving x and y.

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I can hope that the solution
will be explicit,

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y equals y of x,
but you already know from the

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first days of the term that
sometimes it isn't.

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Let's call that explicit
solution.

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Often you have to settle for
it.

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And sometimes it is not a pain.
It is something that is even

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better.
Settle for a solution which

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looks like this which is y
defined implicitly as a function

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of x.
And what does the picture of

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that look like?
Well, the picture of this is

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now simply an integral curve.
It is not a trajectory anymore.

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It is what we called an
integral curve.

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I am reminding you of the very
first day of the term,

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or maybe the second day.
And all it has at each point

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the right slope because that is
all the field is telling me now.

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It only has a slope at each
point.

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It doesn't have any magnitude
or direction.

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It has the direction,
but it doesn't have direction

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in the sense of an arrow telling
me whether it is going this way

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or the opposite way.
That is the picture with t in

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it.
These are the corresponding

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degeneration of that.
It is a coarsening,

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it's a cheapening of it.
It is throwing away

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information.
Throw away all the information

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that had to do with t,
and we are back in the first

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days of the terms with a single
first order equations involving

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only x and y with solutions that
involve only x and y with curves

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that are the graphs of these
solutions but again have no t in

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them.
In effect, they are just the

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paths of the trajectories.
Whereas, the trajectory is the

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point that shows actually how
the point is moving along in

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time faster or slower at various
points.

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Now, why would one want to lose
information?

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Well, the great gain is that
this might be solvable.

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Whereas, this almost certainly
is not.

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That is a big plus.
For example,

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let's illustrate it on the very
simplest possible case.

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I cannot illustrate it on a
nonlinear system very well,

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or not right now because,
in general, nonlinear systems

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are not solvable.
Let's take an easy example.

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Suppose the system were a
linear one, let's say this one

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that we have talked about
before, in fact.

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That is a simple linear
two-by-two system.

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This means the derivatives with
respect to time.

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Well, you know that the
solutions we talked about in

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fact, last time,
typical solutions that would be

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something like (x,
y) equals a constant times,

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let's say if x is cosine t
then y would be the

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derivative of that,
which is negative sine t.

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Or, another one would be c2

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times sine t,
cosine t.

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And those, of course,
are circles.

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They are parametrized circles,
so they are circles that go

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around, in fact,
in this direction.

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And they have a certain
velocity to them.

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Now what do I do if I follow
this plan of eliminating t?

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Well, if I eliminate t directly
from the system,

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what will I get?
I will get dy by dx.

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I divide this equation by that
one.

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-- is equal to negative x over.

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Oh, well, of course,
that is solvable.

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You were able to solve by 18.01
methods before you ever come

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into this course.
By separation of variables,

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it is y dy equals
negative x dx,

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which integrates to be one-half
y squared equals negative

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one-half x squared plus a
constant.

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And after multiplying through

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by two and moving things around
it becomes x squared plus y

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squared.
The nicest way to say it is

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implicitly, x squared plus y
squared equals some positive

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constant.
These are the circles.

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Now, can I eliminate t?
I could also take one of these

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solutions and eliminate t.
If I square this,

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the best way to do it is not to
use our cosines and arcsines and

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whatnot, which will just get you
totally lost.

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Square this,
square that and add them

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together.
And you conclude that x squared

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plus y squared is equal to one.

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Or, if c1 is not one it is
equal to c1 squared.

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You could eliminate t from the
solution the way you eliminate t

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from a pair of parametric
equations, and you get it the

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same way that x squared plus y
squared, in this case,

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would be c1 squared, I guess.

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In some sense,
what we are doing is cycling

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around to the beginning of the
term.

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In fact, this whole lecture,
as you will see,

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is about cycles of one sort or
another.

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But I keep thinking of that
great line of poetry,

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"in my end is my beginning" or
maybe it's "in my beginning is

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my end".
I think both lines occur.

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But that is what is happening
here.

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This is the beginning of the
course and this is the end of

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the course, and they have this
almost trivial relation between

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them.
But notice the totally

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different methods used for
analyzing this,

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where the t is included,
than from analyzing this where

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there is no t.
The goals are different,

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what you look for are
different, everything is

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different, and yet it is almost
the same problem.

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I promised you a nonlinear
example.

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I guess it is time to see what
that is.

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It is going to be another
predator-prey equation,

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but one which is,
in some ways,

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simpler than the one I gave you
for homework.

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The predator is going to be x
because you are used to that

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from the robins.
I am keeping the same

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predator-prey variables.
And the prey will be y.

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And now just notice the small
difference from what I gave you

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before.
For one thing,

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I am not giving you specific
numbers.

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I am going to,
at the beginning at least,

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do some of the analysis at the
beginning using letters,

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using parameters.
a, b, c, d are always going to

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be positive constants.
I am going to assume that x

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prime equals minus ax.

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This represents the predator
dying out if there is no prey

237
00:15:36,000 --> 00:15:42,000
there, but if there is something
to eat that term represents the

238
00:15:43,000 --> 00:15:49,000
predator meeting up with the
prey and gobbling it up.

239
00:15:48,000 --> 00:15:54,000
And b is the coefficient.
How about without predictors,

240
00:15:54,000 --> 00:16:00,000
the prey will multiply and be
fruitful.

241
00:16:00,000 --> 00:16:06,000
Unfortunately they get eaten,
and so there will be a term

242
00:16:04,000 --> 00:16:10,000
that looks like this.
Now, there are two basic

243
00:16:07,000 --> 00:16:13,000
differences between these simple
equations and the slightly more

244
00:16:12,000 --> 00:16:18,000
complicated ones you had with
the robin-earthworm equations.

245
00:16:16,000 --> 00:16:22,000
Namely, in the first place,
I am assuming that these guys,

246
00:16:20,000 --> 00:16:26,000
let's give them names.
I want you to remember which is

247
00:16:24,000 --> 00:16:30,000
x and which is y,
so we are going to think of

248
00:16:27,000 --> 00:16:33,000
these are "sharx".
[LAUGHTER]

249
00:16:31,000 --> 00:16:37,000
And these will be food fish,
so let's make them "yumfish".

250
00:16:50,000 --> 00:16:56,000
The fact that with your robins
you had a positive term here.

251
00:16:54,000 --> 00:17:00,000
I made this 2x.
And now it is minus a times x.

252
00:16:57,000 --> 00:17:03,000
What is the difference?
The difference is that robins

253
00:17:02,000 --> 00:17:08,000
have other things to eat,
so even if there are not any

254
00:17:06,000 --> 00:17:12,000
worms, a robin will survive.
It could eat other insects,

255
00:17:10,000 --> 00:17:16,000
grubs, Japanese beetle grubs.
I think it can even eat seeds.

256
00:17:15,000 --> 00:17:21,000
Anyway, the robins in my garden
seem to be pecking at things

257
00:17:20,000 --> 00:17:26,000
that don't seem to be insects.
And the other is that I am

258
00:17:24,000 --> 00:17:30,000
assuming a very naīve growth
law.

259
00:17:27,000 --> 00:17:33,000
For example,
if there are no sharks,

260
00:17:30,000 --> 00:17:36,000
how are the food fish fruitful
and multiplying?

261
00:17:35,000 --> 00:17:41,000
They multiple exponentially.
Now, obviously you cannot have

262
00:17:39,000 --> 00:17:45,000
unlimited growth like that.
With the worms we added the

263
00:17:43,000 --> 00:17:49,000
term minus a constant times y
squared to indicate that even

264
00:17:48,000 --> 00:17:54,000
worms cannot multiply purely
exponentially forever,

265
00:17:51,000 --> 00:17:57,000
but ultimately their growth
levels off because they cannot

266
00:17:56,000 --> 00:18:02,000
find enough organic matter to
plow their way through.

267
00:18:01,000 --> 00:18:07,000
Those are the two differences.
I am not assuming a logistic

268
00:18:05,000 --> 00:18:11,000
growth law.
This is less sophisticated than

269
00:18:08,000 --> 00:18:14,000
the one I gave you for homework.
This assumes that sharks have

270
00:18:12,000 --> 00:18:18,000
absolutely nothing to eat except
these fish, which is not so bad.

271
00:18:17,000 --> 00:18:23,000
That's true,
more or less.

272
00:18:25,000 --> 00:18:31,000
My plan is now,
with this model,

273
00:18:27,000 --> 00:18:33,000
let's start the analysis,
as you learned to do it.

274
00:18:30,000 --> 00:18:36,000
And we are going to run into
trouble at various places.

275
00:18:34,000 --> 00:18:40,000
And the troubles will then
illustrate these three points

276
00:18:38,000 --> 00:18:44,000
that I wanted to add to your
repertoire of things to do with

277
00:18:43,000 --> 00:18:49,000
nonlinear systems when you run
into trouble.

278
00:18:47,000 --> 00:18:53,000
And it will also increase your
understanding of the nonlinear

279
00:18:50,000 --> 00:18:56,000
systems.
The first thing we have to do

280
00:18:52,000 --> 00:18:58,000
is find the critical points.
I am going to assume that you

281
00:18:56,000 --> 00:19:02,000
are good at this and,
therefore, not spend a lot of

282
00:18:59,000 --> 00:19:05,000
time detailing the calculations.
I will simply write them on the

283
00:19:04,000 --> 00:19:10,000
board.
The first equation I am going

284
00:19:06,000 --> 00:19:12,000
to write down is x times minus a
plus by equals zero.

285
00:19:11,000 --> 00:19:17,000
And I assume you know why I am

286
00:19:13,000 --> 00:19:19,000
writing that down.
And the other equation will be

287
00:19:17,000 --> 00:19:23,000
y times c minus dx is zero.

288
00:19:20,000 --> 00:19:26,000
Those are the simultaneous
equations I have to find to find

289
00:19:24,000 --> 00:19:30,000
the critical point.
The first one is if the product

290
00:19:30,000 --> 00:19:36,000
of these is zero,
either x is zero or the other

291
00:19:34,000 --> 00:19:40,000
factor is.
So, from the second equation,

292
00:19:38,000 --> 00:19:44,000
if x is zero,
y has to be zero also.

293
00:19:42,000 --> 00:19:48,000
That is one critical point.
Now, if x is not zero then this

294
00:19:48,000 --> 00:19:54,000
factor has to be zero which says
that y must be equal

295
00:19:54,000 --> 00:20:00,000
to a over b.
And if y is equal to a over b,

296
00:19:59,000 --> 00:20:05,000
it is not zero here.
Therefore, this factor must be

297
00:20:02,000 --> 00:20:08,000
zero which says that x is equal
to c over d.

298
00:20:06,000 --> 00:20:12,000
You see right away that this
must be a simpler system because

299
00:20:10,000 --> 00:20:16,000
it is only producing two
critical points.

300
00:20:12,000 --> 00:20:18,000
Whereas, the system that you
did for homework had four.

301
00:20:16,000 --> 00:20:22,000
Here is one.
Well, let's just write them up

302
00:20:19,000 --> 00:20:25,000
here.
The critical points are zero,

303
00:20:21,000 --> 00:20:27,000
zero.
That doesn't look terribly

304
00:20:23,000 --> 00:20:29,000
interesting, but the other one
looks more interesting.

305
00:20:27,000 --> 00:20:33,000
It is c over d,
a over b.

306
00:20:31,000 --> 00:20:37,000
Well, let's take a look first
at the zero, zero critical

307
00:20:34,000 --> 00:20:40,000
point.
The origin, in other words.

308
00:20:37,000 --> 00:20:43,000
What does that look like?

309
00:20:44,000 --> 00:20:50,000
Well, at the origin,
the linearization is extremely

310
00:20:47,000 --> 00:20:53,000
easy to do because I simply
ignore the quadratic terms,

311
00:20:51,000 --> 00:20:57,000
which are the product of two
small numbers,

312
00:20:55,000 --> 00:21:01,000
where these only have a single
small number in it.

313
00:20:58,000 --> 00:21:04,000
It is minus ax,
cy.

314
00:21:01,000 --> 00:21:07,000
In other words,
the linearization matrix is

315
00:21:03,000 --> 00:21:09,000
minus a, zero,
zero and cy gives me a

316
00:21:06,000 --> 00:21:12,000
coefficient c there.

317
00:21:08,000 --> 00:21:14,000
Now, I don't think at any point
I have ever explicitly told you

318
00:21:12,000 --> 00:21:18,000
that I hope you have learned
from the homework or maybe your

319
00:21:16,000 --> 00:21:22,000
recitation teacher,
but for heaven's sake,

320
00:21:19,000 --> 00:21:25,000
put this down in your little
books, if you have a diagonal

321
00:21:23,000 --> 00:21:29,000
matrix, for god's sake,
don't calculate its

322
00:21:26,000 --> 00:21:32,000
eigenvalues.
They are right in front of you.

323
00:21:30,000 --> 00:21:36,000
They are always the diagonal
elements.

324
00:21:32,000 --> 00:21:38,000
The eigenvalue,
you can check this out if you

325
00:21:35,000 --> 00:21:41,000
insist on writing the equation,
but trust me it is clear.

326
00:21:40,000 --> 00:21:46,000
If I, for example,
subtract c from the main

327
00:21:43,000 --> 00:21:49,000
diagonal, I am going to get a
determinant zero because the

328
00:21:47,000 --> 00:21:53,000
bottom row will be all zero.
The eigenvalues are negative a

329
00:21:51,000 --> 00:21:57,000
and c.
In other words,

330
00:21:53,000 --> 00:21:59,000
they have opposite signs.
This is a negative number.

331
00:21:56,000 --> 00:22:02,000
That is a positive number --
-- because a,

332
00:22:00,000 --> 00:22:06,000
b, c and d are always positive.
And, therefore,

333
00:22:03,000 --> 00:22:09,000
this is automatically a saddle.
You don't have to calculate

334
00:22:07,000 --> 00:22:13,000
anything.
It is all right in front of

335
00:22:09,000 --> 00:22:15,000
you.
It must be a saddle and,

336
00:22:11,000 --> 00:22:17,000
therefore, unstable because all
saddles are.

337
00:22:13,000 --> 00:22:19,000
And, in fact,
you can even draw the little

338
00:22:16,000 --> 00:22:22,000
picture of what the stuff looks
like near the origin without

339
00:22:19,000 --> 00:22:25,000
even bothering to calculate
eigenvectors,

340
00:22:22,000 --> 00:22:28,000
although it is extremely easy
to do.

341
00:22:24,000 --> 00:22:30,000
Just from common sense,
these are the sharks and these

342
00:22:27,000 --> 00:22:33,000
are the yumfish.
Well, if there are zero

343
00:22:31,000 --> 00:22:37,000
yumfish, in other words,
if I am on the sharks axis,

344
00:22:35,000 --> 00:22:41,000
the axis of sharks,
I die out.

345
00:22:38,000 --> 00:22:44,000
Well, forget about this side.
It's on the positive side.

346
00:22:42,000 --> 00:22:48,000
That makes sense.
But I die out because the

347
00:22:45,000 --> 00:22:51,000
sharks have nothing to eat.
Whereas, if I am on the yumfish

348
00:22:50,000 --> 00:22:56,000
axis I go this way.
I grow because,

349
00:22:52,000 --> 00:22:58,000
without any sharks to eat them,
the yumfish increase.

350
00:22:56,000 --> 00:23:02,000
So it must look like that.
And, therefore,

351
00:23:00,000 --> 00:23:06,000
the saddle must look like this.
The saddle curves hug those and

352
00:23:04,000 --> 00:23:10,000
go nearby.
Now, the other critical point

353
00:23:07,000 --> 00:23:13,000
is the interesting one.
And for that the analysis,

354
00:23:10,000 --> 00:23:16,000
in order that you don't spend
the rest of this period writing

355
00:23:14,000 --> 00:23:20,000
a's, b's, c's and d's,
I am going to make the

356
00:23:17,000 --> 00:23:23,000
simplifying assumption.
But it doesn't change

357
00:23:20,000 --> 00:23:26,000
qualitatively any of the
mathematics.

358
00:23:22,000 --> 00:23:28,000
It just makes it a little
easier to write everything out.

359
00:23:26,000 --> 00:23:32,000
I am going to assume that
everything is one.

360
00:23:30,000 --> 00:23:36,000
Well, in fact,
even this would be good enough,

361
00:23:33,000 --> 00:23:39,000
but let's make everything one.
I am going to assume this.

362
00:23:37,000 --> 00:23:43,000
And it is only to make the
writing simpler.

363
00:23:40,000 --> 00:23:46,000
It doesn't really change the
mathematics at all.

364
00:23:44,000 --> 00:23:50,000
Well, if everything is one,
in order to calculate the

365
00:23:47,000 --> 00:23:53,000
linearized system,
I'd better use the Jacobian.

366
00:23:51,000 --> 00:23:57,000
But perhaps it would be better
to write out explicitly what the

367
00:23:56,000 --> 00:24:02,000
system is.
The system now is x prime

368
00:23:58,000 --> 00:24:04,000
equals minus x plus xy,
all the coefficients are

369
00:24:03,000 --> 00:24:09,000
one.
And y prime is equal to y minus

370
00:24:07,000 --> 00:24:13,000
xy.
What is the Jacobian?

371
00:24:10,000 --> 00:24:16,000
Well, the Jacobian is the
partial of this with respect to

372
00:24:14,000 --> 00:24:20,000
x which is minus one plus y,
the partial with respect

373
00:24:19,000 --> 00:24:25,000
to y which is x,
the partial of this with

374
00:24:22,000 --> 00:24:28,000
respect to x,
which is minus y,

375
00:24:24,000 --> 00:24:30,000
and the partial with respect to
y, which is one minus x.

376
00:24:28,000 --> 00:24:34,000
But I want to evaluate that at

377
00:24:32,000 --> 00:24:38,000
the point one, one,

378
00:24:35,000 --> 00:24:41,000
which is the critical point.
It is the critical point

379
00:24:40,000 --> 00:24:46,000
because I have assumed that all
these parameters have the value

380
00:24:46,000 --> 00:24:52,000
one.
And when I do that,

381
00:24:48,000 --> 00:24:54,000
evaluating it at one, one,

382
00:24:51,000 --> 00:24:57,000
I get what matrix?
Well, I get zero,

383
00:24:54,000 --> 00:25:00,000
one, negative one,
zero.

384
00:24:57,000 --> 00:25:03,000
In other words,

385
00:25:01,000 --> 00:25:07,000
the linearized system is x
prime equals y and y

386
00:25:07,000 --> 00:25:13,000
prime equals negative x.

387
00:25:11,000 --> 00:25:17,000
Well, that is just the one we
analyzed before in terms of --

388
00:25:23,000 --> 00:25:29,000
It's the one whose solutions
are circles.

389
00:25:25,000 --> 00:25:31,000
In other words,
what we find out is that the

390
00:25:28,000 --> 00:25:34,000
linearized system is what
geometric type?

391
00:25:32,000 --> 00:25:38,000
Saddle?
Node?

392
00:25:34,000 --> 00:25:40,000
Spiral?
None of those.

393
00:25:37,000 --> 00:25:43,000
It is a center.
The linearized system is a

394
00:25:44,000 --> 00:25:50,000
center.

395
00:25:52,000 --> 00:25:58,000
It consists,
in other words,

396
00:25:54,000 --> 00:26:00,000
of a bunch of curves going
round and round next to each

397
00:25:58,000 --> 00:26:04,000
other.
Concentric circles,

398
00:25:59,000 --> 00:26:05,000
in fact.
Well, what is wrong with that?

399
00:26:03,000 --> 00:26:09,000
Now, we are in deep trouble.
We are now in trouble because

400
00:26:07,000 --> 00:26:13,000
that is a borderline case.
Let me remind you of what the

401
00:26:12,000 --> 00:26:18,000
borderline cases are.
When we drew that picture,

402
00:26:15,000 --> 00:26:21,000
this is from last week's
problem set so you have a

403
00:26:19,000 --> 00:26:25,000
perfect excuse for forgetting it
totally.

404
00:26:22,000 --> 00:26:28,000
But I am trying to remind you
of it.

405
00:26:25,000 --> 00:26:31,000
That is another reason why I am
doing this.

406
00:26:30,000 --> 00:26:36,000
Remember that picture I asked
you about that appeared on the

407
00:26:34,000 --> 00:26:40,000
computer screen?
Let's make it a little flatter

408
00:26:38,000 --> 00:26:44,000
so that I can have room to write
in.

409
00:26:40,000 --> 00:26:46,000
This is a certain parabola
whose equation you are dying to

410
00:26:45,000 --> 00:26:51,000
tell me, but I am not asking.
There is the trace,

411
00:26:48,000 --> 00:26:54,000
this is the determinant,
and the characteristic equation

412
00:26:53,000 --> 00:26:59,000
is related to the values of
these numbers like plus d equals

413
00:26:57,000 --> 00:27:03,000
zero.
Then these were spiral synchs.

414
00:27:02,000 --> 00:27:08,000
This was the region of spiral
sources.

415
00:27:07,000 --> 00:27:13,000
That is the abbreviation for
sources.

416
00:27:11,000 --> 00:27:17,000
These were nodal synchs and
these were the nodal sources.

417
00:27:18,000 --> 00:27:24,000
And down here were the saddles.
And where were the centers?

418
00:27:24,000 --> 00:27:30,000
The centers were along this
line.

419
00:27:30,000 --> 00:27:36,000
The centers,
there weren't many of them,

420
00:27:32,000 --> 00:27:38,000
and they were the separation
for these two regions.

421
00:27:36,000 --> 00:27:42,000
I will put that down.
These are the centers.

422
00:27:39,000 --> 00:27:45,000
These other borderlines
correspond to other things.

423
00:27:42,000 --> 00:27:48,000
These are defective
eigenvalues, zero eigenvalues

424
00:27:45,000 --> 00:27:51,000
and so on.
But let's concentrate on just

425
00:27:48,000 --> 00:27:54,000
one of them.
You will find the others

426
00:27:50,000 --> 00:27:56,000
discussed in the notes,
GS7.

427
00:27:52,000 --> 00:27:58,000
But if you get this idea then
the rest is just details.

428
00:27:56,000 --> 00:28:02,000
I think it will be perfectly
clear.

429
00:28:00,000 --> 00:28:06,000
What is wrong with the center?
The answer is that if we are on

430
00:28:04,000 --> 00:28:10,000
the center, for example,
this system corresponds to the

431
00:28:08,000 --> 00:28:14,000
trace being zero and the
determinant being plus one.

432
00:28:12,000 --> 00:28:18,000
It corresponds to t equals zero
and d equals one.

433
00:28:16,000 --> 00:28:22,000
This point, in other words.
Now, if I wiggle the

434
00:28:19,000 --> 00:28:25,000
coefficients of the matrix just
a little bit,

435
00:28:23,000 --> 00:28:29,000
just change them a little bit,
what I am going to do is move

436
00:28:27,000 --> 00:28:33,000
off this pink line.
And I might move to here or I

437
00:28:31,000 --> 00:28:37,000
might move a little way to
there.

438
00:28:34,000 --> 00:28:40,000
But, if I do that,
I change the geometric type.

439
00:28:37,000 --> 00:28:43,000
In other words,
being a borderline,

440
00:28:39,000 --> 00:28:45,000
a slight change of the
parameters can change what it

441
00:28:42,000 --> 00:28:48,000
changes to.
And, in fact,

442
00:28:44,000 --> 00:28:50,000
that is geometrically clear.
What does a center look like?

443
00:28:48,000 --> 00:28:54,000
A center looks like this,
a bunch of curves going around

444
00:28:51,000 --> 00:28:57,000
all in the same direction,
like concentric circles or

445
00:28:55,000 --> 00:29:01,000
maybe ellipses or something like
that.

446
00:28:59,000 --> 00:29:05,000
If I deform the picture just a
little bit, well,

447
00:29:03,000 --> 00:29:09,000
I might change it into
something that looks like this

448
00:29:07,000 --> 00:29:13,000
where, after they go around they
don't quite meet up with where

449
00:29:12,000 --> 00:29:18,000
they were to start with.
And I am going to get a spiral

450
00:29:17,000 --> 00:29:23,000
synch.
Or, I might do the deformation

451
00:29:20,000 --> 00:29:26,000
by going around once and I'm a
little outside of where I was.

452
00:29:25,000 --> 00:29:31,000
In which case it's going to be
a spiral source.

453
00:29:31,000 --> 00:29:37,000
The fact that just changing
these curves a little bit can

454
00:29:35,000 --> 00:29:41,000
change the picture to this or to
that corresponds to the fact

455
00:29:39,000 --> 00:29:45,000
that if you are on here with
this value of t and d,

456
00:29:43,000 --> 00:29:49,000
zero and one,
and just change t and d a

457
00:29:46,000 --> 00:29:52,000
little bit, you are going to
move off into these regions.

458
00:29:51,000 --> 00:29:57,000
You might, of course,
stay on the pink line.

459
00:29:54,000 --> 00:30:00,000
It is not very likely.
Where does this leave us?

460
00:29:59,000 --> 00:30:05,000
Well, if the linear system is
not stable in the sense that if

461
00:30:02,000 --> 00:30:08,000
you change the parameters a
little bit it doesn't change the

462
00:30:06,000 --> 00:30:12,000
type.
This is, after all,

463
00:30:07,000 --> 00:30:13,000
only an approximation to the
nonlinear system.

464
00:30:10,000 --> 00:30:16,000
If in this approximation you
cannot really predict the

465
00:30:13,000 --> 00:30:19,000
behavior of very well when you
change the parameters,

466
00:30:17,000 --> 00:30:23,000
then from it you cannot tell
what the original system looked

467
00:30:20,000 --> 00:30:26,000
like.
In other words,

468
00:30:23,000 --> 00:30:29,000
the nonlinear system at one,
one might be any one

469
00:30:28,000 --> 00:30:34,000
of the possibilities,
still a center or it might be a

470
00:30:32,000 --> 00:30:38,000
spiral synch or it might be a
spiral source.

471
00:30:36,000 --> 00:30:42,000
Any one of those three is a
possibility.

472
00:30:39,000 --> 00:30:45,000
It couldn't be a saddle because
that is too far away.

473
00:30:44,000 --> 00:30:50,000
It couldn't be a node.
That is too far away,

474
00:30:48,000 --> 00:30:54,000
too.
But it could wander into either

475
00:30:51,000 --> 00:30:57,000
of these regions and,
therefore, the picture you

476
00:30:55,000 --> 00:31:01,000
cannot tell which of these three
it is just from this type of

477
00:31:00,000 --> 00:31:06,000
critical point analysis.
Well, that was Volterra's

478
00:31:06,000 --> 00:31:12,000
problem.
By the way, the person who

479
00:31:08,000 --> 00:31:14,000
introduced these equations and
studied them systematically in

480
00:31:13,000 --> 00:31:19,000
the way in which we are doing it
here was Volterra.

481
00:31:17,000 --> 00:31:23,000
And, in fact,
he was interested in sharks and

482
00:31:21,000 --> 00:31:27,000
food fish, as they were called.
What do we do?

483
00:31:24,000 --> 00:31:30,000
Well, you have to be smart.
What Volterra did was he went

484
00:31:29,000 --> 00:31:35,000
to method number one.
Let's do it.

485
00:31:38,000 --> 00:31:44,000
Volterra said I got my
equations x prime equals minus x

486
00:31:45,000 --> 00:31:51,000
plus xy.
y prime is equal to,

487
00:31:51,000 --> 00:31:57,000
these are the food fish,
y minus xy.

488
00:31:58,000 --> 00:32:04,000
Let's eliminate t.
And my problem is,

489
00:32:03,000 --> 00:32:09,000
of course, I am trying to
determine what type of critical

490
00:32:07,000 --> 00:32:13,000
point one, one is.
And the method we have used up

491
00:32:11,000 --> 00:32:17,000
until now has failed because it
gave us a borderline case,

492
00:32:16,000 --> 00:32:22,000
which is from that we cannot
deduce.

493
00:32:19,000 --> 00:32:25,000
He said let's eliminate t.
And we get dy over dx

494
00:32:23,000 --> 00:32:29,000
equals, I am going to factor,
y times one minus x on top.

495
00:32:29,000 --> 00:32:35,000
And on the bottom factor x
negative one plus y.

496
00:32:34,000 --> 00:32:40,000
You could solve that before you
came into 18.03,

497
00:32:39,000 --> 00:32:45,000
right?
This you can separate

498
00:32:42,000 --> 00:32:48,000
variables.
Let's separate variables.

499
00:32:46,000 --> 00:32:52,000
The y's all go on one side,
so y goes down here.

500
00:32:52,000 --> 00:32:58,000
It is (y minus one) over y dy
equals --

501
00:33:00,000 --> 00:33:06,000
On the other side the dx gets
moved up, one minus x over x dx.

502
00:33:03,000 --> 00:33:09,000
Now, of course,
you wouldn't dream of using

503
00:33:06,000 --> 00:33:12,000
partial fractions on this.
It would be illegal because,

504
00:33:09,000 --> 00:33:15,000
even though it is a rational
function, the quotient of two

505
00:33:13,000 --> 00:33:19,000
polynomials, the degree of the
top is not smaller than the

506
00:33:16,000 --> 00:33:22,000
degree of the bottom.
In other words,

507
00:33:18,000 --> 00:33:24,000
it is a partial fraction so
this degree must be bigger than

508
00:33:22,000 --> 00:33:28,000
that one.
And it isn't so you would have

509
00:33:24,000 --> 00:33:30,000
to first divide.
And if you divide by that then

510
00:33:27,000 --> 00:33:33,000
there is no point in using
partial fractions at all.

511
00:33:37,000 --> 00:33:43,000
Of course, you would have done
this by basic instant,

512
00:33:41,000 --> 00:33:47,000
I know.
What is the solution?

513
00:33:43,000 --> 00:33:49,000
It is y minus log y equals log
x minus x plus some constant.

514
00:33:49,000 --> 00:33:55,000
That is not the final constant

515
00:33:53,000 --> 00:33:59,000
I want so I will give it a
subscript one to indicate I want

516
00:33:58,000 --> 00:34:04,000
more.
This is very hard to see what

517
00:34:02,000 --> 00:34:08,000
that curve looks like.
We can make it look better by

518
00:34:05,000 --> 00:34:11,000
exponentiating.
If I exponentiate it,

519
00:34:08,000 --> 00:34:14,000
going back to high school
mathematics, but I know from

520
00:34:11,000 --> 00:34:17,000
experience that many of you are
not too good at this step,

521
00:34:16,000 --> 00:34:22,000
so that is another reason I am
doing it, it will be

522
00:34:20,000 --> 00:34:26,000
e to the y, times,
that part is easy because

523
00:34:23,000 --> 00:34:29,000
pluses and minuses change into
times, times e to the negative

524
00:34:27,000 --> 00:34:33,000
log y.
Well, e to the log y is y.

525
00:34:31,000 --> 00:34:37,000
If I put a minus sign in front

526
00:34:35,000 --> 00:34:41,000
of that, that sends it into the
denominator, so instead of being

527
00:34:40,000 --> 00:34:46,000
y it is one over y
Equals x, e to log x is x,

528
00:34:44,000 --> 00:34:50,000
e to the negative x

529
00:34:48,000 --> 00:34:54,000
is, therefore,
times one over e to the x.

530
00:34:51,000 --> 00:34:57,000
And that is times the

531
00:34:53,000 --> 00:34:59,000
exponential of c1,
which I will call c2.

532
00:34:58,000 --> 00:35:04,000
And now if I combine them all
and put them all on one side it

533
00:35:04,000 --> 00:35:10,000
is x over e to the x times y
over e to the y is equal to yet

534
00:35:10,000 --> 00:35:16,000
another constant,
one over c2.

535
00:35:14,000 --> 00:35:20,000
This is my final constant so

536
00:35:18,000 --> 00:35:24,000
let's call that c.
In other words,

537
00:35:21,000 --> 00:35:27,000
the integral curves are the
graphs of this equation for

538
00:35:27,000 --> 00:35:33,000
different values of the constant
c.

539
00:35:32,000 --> 00:35:38,000
Well, of course you've graphed
an equation like that all the

540
00:35:37,000 --> 00:35:43,000
time.
What am I going to do with this

541
00:35:41,000 --> 00:35:47,000
stupid thing?
Well, I deliberately picked

542
00:35:44,000 --> 00:35:50,000
something which involved all
your learning up until now.

543
00:35:50,000 --> 00:35:56,000
We are now going into 18.02,
right?

544
00:35:53,000 --> 00:35:59,000
You looked at that in 18.02 you
would say this is the contour

545
00:35:59,000 --> 00:36:05,000
curve, these are contour curves
of the function --

546
00:36:05,000 --> 00:36:11,000
Well, let's write it out the
other way.

547
00:36:07,000 --> 00:36:13,000
It doesn't make any difference,
but you are more likely to have

548
00:36:10,000 --> 00:36:16,000
seen the function in this form.
x over e to the x

549
00:36:13,000 --> 00:36:19,000
you won't recognize,
but this you will.

550
00:36:15,000 --> 00:36:21,000
In fact, we have had it before
this term.

551
00:36:18,000 --> 00:36:24,000
It is one of the kinds of
solutions you can add to second

552
00:36:21,000 --> 00:36:27,000
order equations.
Times y e to the negative y

553
00:36:23,000 --> 00:36:29,000
equals c.
It is the contour curves of

554
00:36:26,000 --> 00:36:32,000
this function.
Let's call that function,

555
00:36:29,000 --> 00:36:35,000
let's say, h of (x,y).

556
00:36:32,000 --> 00:36:38,000
Well, of course you could throw
it on Matlab,

557
00:36:35,000 --> 00:36:41,000
as you did in 18.02 maybe,
and ask Matlab to plot the

558
00:36:39,000 --> 00:36:45,000
function.
But Volterra didn't have that

559
00:36:42,000 --> 00:36:48,000
luxury.
He was smart,

560
00:36:43,000 --> 00:36:49,000
too.
So let's be smart instead.

561
00:36:53,000 --> 00:36:59,000
What is the function x?
Let's use a neutral variable

562
00:36:56,000 --> 00:37:02,000
like u and plot u e to the
negative u.

563
00:37:00,000 --> 00:37:06,000
18.01.
At zero, it has the value zero.

564
00:37:03,000 --> 00:37:09,000
When u is small this has
approximately the value one and,

565
00:37:08,000 --> 00:37:14,000
therefore, it starts out like
the function u.

566
00:37:12,000 --> 00:37:18,000
As u goes to infinity,
you know by L'Hopital's rule

567
00:37:16,000 --> 00:37:22,000
that this goes to zero so it
ends up like this.

568
00:37:20,000 --> 00:37:26,000
Well, what on earth could it
possibly do except rise to a

569
00:37:25,000 --> 00:37:31,000
maximum?
But where is that maximum?

570
00:37:30,000 --> 00:37:36,000
It is easy to see there is a
unique maximum just by 18.01.

571
00:37:34,000 --> 00:37:40,000
Take the derivative,
find out where the maximum

572
00:37:37,000 --> 00:37:43,000
point is and you will find,
perhaps you have already done

573
00:37:41,000 --> 00:37:47,000
it, this is at one.
The maximum occurs at one,

574
00:37:44,000 --> 00:37:50,000
that is where the derivative is
zero and the value there is one

575
00:37:49,000 --> 00:37:55,000
times e to the minus one.

576
00:37:52,000 --> 00:37:58,000
It is one over e,
in other words.

577
00:37:55,000 --> 00:38:01,000
That is what the curve looks
like.

578
00:37:57,000 --> 00:38:03,000
What do the contour curves of
this look like?

579
00:38:02,000 --> 00:38:08,000
Well, if we are looking at the
contour curves of the function

580
00:38:07,000 --> 00:38:13,000
h, so here is x,
here is y, and we want the

581
00:38:11,000 --> 00:38:17,000
contour curves of the function h
of (x, y),

582
00:38:16,000 --> 00:38:22,000
what do I know?
Where is its maximum?

583
00:38:20,000 --> 00:38:26,000
The maximum point of h of x,
y is where?

584
00:38:24,000 --> 00:38:30,000
Well, h is the product x e to
the negative x times y e to the

585
00:38:30,000 --> 00:38:36,000
minus y.

586
00:38:35,000 --> 00:38:41,000
This has its maximum at one,
this has its maximum at one,

587
00:38:39,000 --> 00:38:45,000
so where does h of (x,
y) head?

588
00:38:43,000 --> 00:38:49,000
Well, you have two factors.
One makes each of them the

589
00:38:48,000 --> 00:38:54,000
biggest they can be.
The maximum must be exactly at

590
00:38:52,000 --> 00:38:58,000
that critical point one,
one.

591
00:38:55,000 --> 00:39:01,000
That is our maximum point.
Let's make it conspicuous.

592
00:39:01,000 --> 00:39:07,000
This is the maximum point.
It is the point one,

593
00:39:05,000 --> 00:39:11,000
one.
It is the point that makes the

594
00:39:09,000 --> 00:39:15,000
function biggest.
Now, how about the contour

595
00:39:13,000 --> 00:39:19,000
curves?
Well, this is the top of the

596
00:39:16,000 --> 00:39:22,000
mountain.
What is it along the axes?

597
00:39:19,000 --> 00:39:25,000
Along the axes,
when either x or y is zero,

598
00:39:23,000 --> 00:39:29,000
the function has the value
zero, so it is biggest there.

599
00:39:28,000 --> 00:39:34,000
It has the value zero here.
So it is zero value.

600
00:39:33,000 --> 00:39:39,000
What are the contour curves?
Well, we must have a mountain

601
00:39:38,000 --> 00:39:44,000
peak here.
This is the unique maximum

602
00:39:41,000 --> 00:39:47,000
point.
In fact, it is easy to see the

603
00:39:44,000 --> 00:39:50,000
contour curves just surround it
like that.

604
00:39:47,000 --> 00:39:53,000
Now, the reason they cannot be
spirals, well,

605
00:39:51,000 --> 00:39:57,000
in the first place,
you never see contour curves

606
00:39:55,000 --> 00:40:01,000
that were spirals.
That is a good reason.

607
00:40:00,000 --> 00:40:06,000
It is a mountain.
That is the way contour curves

608
00:40:03,000 --> 00:40:09,000
of a mountain look.
But, all right,

609
00:40:06,000 --> 00:40:12,000
that is not a good argument.
But notice that along each

610
00:40:10,000 --> 00:40:16,000
horizontal line here,
I want to know how many times

611
00:40:14,000 --> 00:40:20,000
can it intersect the contour
curve?

612
00:40:16,000 --> 00:40:22,000
That is the same as asking
along one of these lines how

613
00:40:20,000 --> 00:40:26,000
many times could it intersect?
See, this is a graph of the

614
00:40:25,000 --> 00:40:31,000
function along this horizontal
line.

615
00:40:29,000 --> 00:40:35,000
And, therefore,
how many times can it intersect

616
00:40:32,000 --> 00:40:38,000
one of the contour curves the
same number of times that a

617
00:40:36,000 --> 00:40:42,000
horizontal line can intersect
this curve?

618
00:40:39,000 --> 00:40:45,000
Twice.
Only once if you are exactly at

619
00:40:42,000 --> 00:40:48,000
that height and after that
never.

620
00:40:44,000 --> 00:40:50,000
But here it can intersect each
contour curve only twice which

621
00:40:48,000 --> 00:40:54,000
means these things cannot be
spirals.

622
00:40:51,000 --> 00:40:57,000
Otherwise, they would intersect
those horizontal and vertical

623
00:40:55,000 --> 00:41:01,000
lines many times instead of just
twice.

624
00:41:00,000 --> 00:41:06,000
That is what it looks like.
In fact, we can even put in the

625
00:41:04,000 --> 00:41:10,000
direction without any effort.
We know that the sharks die out

626
00:41:09,000 --> 00:41:15,000
without any food fish and the
food fish grow.

627
00:41:13,000 --> 00:41:19,000
Well, I think this has to be
clockwise.

628
00:41:16,000 --> 00:41:22,000
Of course, the guys nearby must
be going the same way.

629
00:41:21,000 --> 00:41:27,000
And, therefore,
the guys near them must be

630
00:41:24,000 --> 00:41:30,000
going the same way.
And it is sort of a domino

631
00:41:28,000 --> 00:41:34,000
effect.
The direction spreads and there

632
00:41:32,000 --> 00:41:38,000
is only one compatible way of
making these.

633
00:41:35,000 --> 00:41:41,000
They all must be going
clockwise.

634
00:41:37,000 --> 00:41:43,000
What is happening,
actually?

635
00:41:39,000 --> 00:41:45,000
At this point there are a lot
of sharks but not so many food

636
00:41:44,000 --> 00:41:50,000
fish.
The sharks eat the few

637
00:41:46,000 --> 00:41:52,000
remaining food fish,
and now the sharks start to die

638
00:41:49,000 --> 00:41:55,000
out because they have nothing to
eat.

639
00:41:52,000 --> 00:41:58,000
When they are practically gone,
the food fish can start growing

640
00:41:56,000 --> 00:42:02,000
again and grow and grow.
For a while they grow happily,

641
00:42:02,000 --> 00:42:08,000
and then the sharks suddenly
start being able to find them

642
00:42:06,000 --> 00:42:12,000
again.
And the few remaining sharks

643
00:42:09,000 --> 00:42:15,000
start eating them,
the population of sharks grows,

644
00:42:13,000 --> 00:42:19,000
the food fish start dying out
again, and the cycle starts all

645
00:42:18,000 --> 00:42:24,000
over again.
Now, I wanted to talk about the

646
00:42:21,000 --> 00:42:27,000
qualitative behavior.
This is, in some ways,

647
00:42:24,000 --> 00:42:30,000
the most important part of the
lecture.

648
00:42:29,000 --> 00:42:35,000
I wanted to discuss,
just as we did at the beginning

649
00:42:35,000 --> 00:42:41,000
of the term, the effect of
fishing with nets at a constant

650
00:42:43,000 --> 00:42:49,000
rate k.

651
00:42:51,000 --> 00:42:57,000
Just as near the beginning of
the term, we talked about

652
00:42:54,000 --> 00:43:00,000
fishing a single population.
The only difference is,

653
00:42:57,000 --> 00:43:03,000
now there are two populations.
Both sharks and the food fish.

654
00:43:02,000 --> 00:43:08,000
Now, what happens to the
equations?

655
00:43:05,000 --> 00:43:11,000
Well, the equations start out
just as they were.

656
00:43:10,000 --> 00:43:16,000
I am now reverting to a and b.
You will see why.

657
00:43:14,000 --> 00:43:20,000
It's because we need the
general coefficients.

658
00:43:18,000 --> 00:43:24,000
Plus bxy.
But, if we are fishing with

659
00:43:21,000 --> 00:43:27,000
nets, then a certain fraction of
the sharks in the sea get hauled

660
00:43:27,000 --> 00:43:33,000
in by the net.
And say the fishing rate is k.

661
00:43:32,000 --> 00:43:38,000
There is a fishing term.
We remove minus k times a

662
00:43:36,000 --> 00:43:42,000
certain fraction of the sharks
in the sea.

663
00:43:40,000 --> 00:43:46,000
How about the food fish?
Well, we remove them,

664
00:43:44,000 --> 00:43:50,000
too.
And we don't distinguish

665
00:43:47,000 --> 00:43:53,000
between sharks and food fish.
This is cy minus dxy.

666
00:43:51,000 --> 00:43:57,000
But we also remove a certain

667
00:43:54,000 --> 00:44:00,000
fraction of them.
Minus k times y.

668
00:43:58,000 --> 00:44:04,000
What, therefore,
is the new system?

669
00:44:03,000 --> 00:44:09,000
The new system is therefore x
prime equals minus a plus k.

670
00:44:10,000 --> 00:44:16,000
I am combining those two x
terms.

671
00:44:14,000 --> 00:44:20,000
That is times x.
Plus bxy.

672
00:44:17,000 --> 00:44:23,000
And
the y prime is c minus k times y

673
00:44:24,000 --> 00:44:30,000
minus dxy.

674
00:44:30,000 --> 00:44:36,000
To solve these,
I do not have to go through the

675
00:44:33,000 --> 00:44:39,000
analysis.
All I have to do is change the

676
00:44:37,000 --> 00:44:43,000
numbers, change these
coefficients from a to a plus k

677
00:44:41,000 --> 00:44:47,000
and c minus k.
The old critical point was the

678
00:44:46,000 --> 00:44:52,000
point c over d,
a over b.

679
00:44:49,000 --> 00:44:55,000
The new critical point with

680
00:44:52,000 --> 00:44:58,000
fishing is what?
Well, the parameters have been

681
00:44:56,000 --> 00:45:02,000
changed by the addition of k.
The new critical point is c

682
00:45:02,000 --> 00:45:08,000
minus k over d and a plus k over
d.

683
00:45:07,000 --> 00:45:13,000
What is the effect of fishing?

684
00:45:11,000 --> 00:45:17,000
Well, if the old critical point
was over here,

685
00:45:15,000 --> 00:45:21,000
let's say this is the point c
over d and this is the

686
00:45:21,000 --> 00:45:27,000
point a over b,
that is the old critical point

687
00:45:27,000 --> 00:45:33,000
that was over here.
The result of fishing is to

688
00:45:33,000 --> 00:45:39,000
lower the value of x and raise
the value here.

689
00:45:38,000 --> 00:45:44,000
The new critical point is,
this gets lowered to c minus k

690
00:45:44,000 --> 00:45:50,000
over d,
and this gets raised to a plus

691
00:45:50,000 --> 00:45:56,000
k over b.
In other words,

692
00:45:54,000 --> 00:46:00,000
the new point is there.
The effect of fishing is to --

693
00:46:02,000 --> 00:46:08,000
It does not treat the sharks
and the yumfish equally.

694
00:46:07,000 --> 00:46:13,000
The effect of fishing lowers
the shark population.

695
00:46:11,000 --> 00:46:17,000
See, the critical point gives
sort of the average shark

696
00:46:17,000 --> 00:46:23,000
population.
Of course, it cycles around

697
00:46:20,000 --> 00:46:26,000
these.
But, on the average,

698
00:46:23,000 --> 00:46:29,000
this gives how many sharks
there are and how many food fish

699
00:46:28,000 --> 00:46:34,000
there are.
The new critical point with

700
00:46:33,000 --> 00:46:39,000
fishing lowers the shark
population and raises the food

701
00:46:39,000 --> 00:46:45,000
fish population.
That is not intuitive.

702
00:46:42,000 --> 00:46:48,000
And, in fact,
that was observed

703
00:46:45,000 --> 00:46:51,000
experimentally at a slightly
different context.

704
00:46:50,000 --> 00:46:56,000
And that is why Volterra
started working on the problem.

705
00:46:55,000 --> 00:47:01,000
I will need three more minutes.
The most interesting

706
00:47:01,000 --> 00:47:07,000
application of all is not to
sharks and food fish.

707
00:47:03,000 --> 00:47:09,000
I cannot assume that you are
dramatically interested in how

708
00:47:07,000 --> 00:47:13,000
many of them there are in the
ocean, but you might be more

709
00:47:10,000 --> 00:47:16,000
interested in this.

710
00:47:20,000 --> 00:47:26,000
That thing about lowering is
called Volterra's principle.

711
00:47:25,000 --> 00:47:31,000
Put that in your books.
Volterra's principle.

712
00:47:29,000 --> 00:47:35,000
Volterra is spelt over there.

713
00:47:43,000 --> 00:47:49,000
-- has found more modern
applications than sharks.

714
00:47:48,000 --> 00:47:54,000
Suppose you consider fish in a
pond.

715
00:47:53,000 --> 00:47:59,000
You have mosquito larvae which
breed in the pond.

716
00:47:58,000 --> 00:48:04,000
This happens.
And then suddenly there is a

717
00:48:03,000 --> 00:48:09,000
plague of mosquitoes and
concerned citizens.

718
00:48:07,000 --> 00:48:13,000
And this is what happened in
the '50s.

719
00:48:11,000 --> 00:48:17,000
That was before you were born,
but not before I was.

720
00:48:16,000 --> 00:48:22,000
What happened was a lot of
mosquitoes.

721
00:48:19,000 --> 00:48:25,000
Everybody said the mosquitoes
breed in the little stagnant

722
00:48:25,000 --> 00:48:31,000
ponds so spray DDT on them.
DDT them.

723
00:48:30,000 --> 00:48:36,000
Dump it in the ponds.
That will kill all the larvae

724
00:48:34,000 --> 00:48:40,000
and we won't get bitten anymore.
When you put DDT in the pond,

725
00:48:39,000 --> 00:48:45,000
as people did not realize at
the time because these things

726
00:48:44,000 --> 00:48:50,000
were new, of course you kill the
mosquitoes, but you also kill

727
00:48:49,000 --> 00:48:55,000
the fish because DDT is
poisonous to fish.

728
00:48:53,000 --> 00:48:59,000
What, in effect,
you are doing mathematically is

729
00:48:57,000 --> 00:49:03,000
the same as harvesting the fish
population with the sharks and

730
00:49:02,000 --> 00:49:08,000
the food fish.
The result is,

731
00:49:06,000 --> 00:49:12,000
the fish are the predators
because they eat the mosquito

732
00:49:11,000 --> 00:49:17,000
larvae, big food,
there was a certain

733
00:49:15,000 --> 00:49:21,000
equilibrium, according to
Volterra's principle.

734
00:49:20,000 --> 00:49:26,000
With DDT that equilibrium moves
to here.

735
00:49:24,000 --> 00:49:30,000
In other words,
the effect of indiscriminately

736
00:49:28,000 --> 00:49:34,000
spraying the pond with DDT is to
increase the number of

737
00:49:34,000 --> 00:49:40,000
mosquitoes and kill fish.
And, in fact,

738
00:49:39,000 --> 00:49:45,000
that is exactly what was
observed.

739
00:49:42,000 --> 00:49:48,000
The same thing was observed
with the bird population and

740
00:49:48,000 --> 00:49:54,000
insects.
Spraying trees for insects to

741
00:49:51,000 --> 00:49:57,000
get rid of some pests ends up
killing more birds than it does

742
00:49:58,000 --> 00:50:04,000
insects and the insects
increase.

743
00:50:01,000 --> 00:50:07,000
Thanks.