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GILBERT STRANG: OK, today is
about differential equations.
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That's where calculus
really is applied.
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And these will be equations
that describe growth.
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And the first you've
already seen.
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It's the most important and the
simplest. The growth rate
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dy dt is proportional
to y itself.
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Let's call c that constant
that comes in there.
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And all these problems will
start from some known point y
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at time 0 and evolve, grow,
to k, whatever they do.
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And this one we know
the solution to.
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We know that that will be an
exponential growth with a
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factor c, a growth rate c.
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Actually, we should know that
the solution has an e to the
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ct because when we take the
derivative of this, it will
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bring down that factor
c that we want.
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And at t equals 0
this is correct.
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We get started correctly because
at t equals 0 we have
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y of 0 is correctly given by
the known starting point.
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Because when t is 0, that's 1.
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So that's the one we know.
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The most fundamental.
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Now, the next step.
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Still linear.
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That allows a source term.
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This might be money in the bank
growing with an interest
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rate c per year.
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And the source term would be
additional money that you're
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constantly putting in.
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Every time step in
goes that saving.
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Or if s were negative, of
course, it could be spending.
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For us, that's a linear
differential equation with a
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constant right-hand
side there.
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And we have to be able
to solve it.
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And we can.
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In fact, I can fit it into
the one we already now.
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Watch this.
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Let me take that to be
c times y plus s/c.
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Remember, s and c
are constants.
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And let me write that right-hand
side, dy dt.
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I can put this constant in there
too and still have the
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same derivative.
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So dy dt is exactly the same
as the derivative of y plus
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s/c because that s/c is
constant; its derivative is 0.
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So that dy dt is the
same as before.
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But now look.
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We have this expression here
that's just like this one,
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only it's y plus s/c that's
growing at this growth rate c,
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starting from--
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the start would be
y at 0 plus s/c.
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You see, this quantity in
parentheses, I'm grabbing that
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as the thing that grows
perfect exponentially.
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So I conclude that at a later
time its value is its initial
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value times the growth.
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That is a rather quick
solution to this
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differential equation.
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You might want me to put s/c
on the right-hand side and
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have y of t by itself, and that
would be a formula for
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the correct y of t that solves
this equation that
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starts at y of 0.
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And of course, this one starts
at y of 0 plus s/c and that's
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why we see y of 0 plus
s/c right there.
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Do you see that that equation,
well, it wasn't really
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systematic.
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And in a differential equations
course you would
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learn how to reach that
answer without
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sort of noticing that--
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actually, I should
do that too.
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This is such a useful,
important equation.
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It's linear, but it's got some
right-hand side there.
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I should give you a system,
say something about linear
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equations before I go on.
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The most interesting equation
will be not linear, but let me
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say a word about linear
equations of which that's a
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perfect example with
a right-hand side.
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So linear equations.
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Linear equations.
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The solutions to linear
equations, y of t, is--
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to all linear equations.
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Actually, this is a linear
differential equation.
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When I'm teaching linear algebra
I'm talking about
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matrix equations.
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The rule, it's the linear part
that is important for these
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few thoughts.
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The solution to linear
equations--
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can I underline linear--
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always has the form of some
particular solution.
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Some function that does solve
the equation plus another
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solution with a right side 0.
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You will see what I
mean by those two
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parts for this example.
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Can I copy again this example
dy dt is cy plus s.
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So I'm looking for a solution
to that equation.
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First, let me look for a
particular solution.
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That means just any function
that solves the equation.
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Well, the simplest function I
could think of would be a
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constant function.
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And if my solution was a
constant, then its derivative
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would be 0.
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So that would be 0.
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And then c times that constant
plus s would be 0.
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I want the equation
to be solved.
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I claim that a particular
solution is the constant y--
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so when I set that momentarily
to 0, I'll discover I'll move
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the s over as minus s.
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I'll divide by the c and
I'll have a minus s/c.
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That's a particular solution.
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Let's just plug it in.
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If I put in y.
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That's a constant, so it's
derivative is 0.
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And c times that gives me
minus s plus s is the 0.
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OK.
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That's a particular solution.
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And then what I mean
by right side 0.
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What do I mean by that?
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I'm not going to wipe out all
this, I'm going to wipe out--
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for this right side 0 part,
I'm going to wipe out the
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source term.
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I mean, I keep the y's, but
throw away the source in the
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right side 0.
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Maybe a book might often write
the homogeneous equation.
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And the homogeneous equation
is where we started,
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dy dt equals cy.
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What's the solution to that?
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The solution to dy dt is cy.
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This is this solution,
e to the ct.
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Any a, any a times e to the ct
will solve my simple equation.
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Once I've knocked out this
s, then if y solves it, a
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times y solves it.
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I could multiply the
equation by a.
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So a is arbitrary, any number.
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But of course, I'm going to
find out what it is by
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starting at the right place.
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By starting at y of 0,
I'll find out what--
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by setting t equals 0.
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So put t equals 0.
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To find a, put t equals 0.
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Because we know where
we started.
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So we started at a known y of
0 equals minus s/c plus A.
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Setting t equals
0 makes that 1.
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You see we found
out what A is.
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A is y of 0 plus s/c.
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And that's the thing that
grows, that multiplies
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the e to the ct.
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And of course, it
did it up here.
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Up here what multiplied the e to
the ct was this y of 0 plus
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s/c, which is exactly our A.
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As I said, my main point in this
lecture is to bring in a
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nonlinear equation.
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That's quite interesting
and important.
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And it comes in population
growth, ecology, it appears a
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lot of places and I'll write
it down immediately.
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But just the starting point
is always linear.
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And here we've got the basic
linear equation and now the
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basic linear equation
with a source term.
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Ready for the population
equation.
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Population growth.
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Population P of t.
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OK, what's a reasonable
differential equation that
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describes the growth
of population?
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All right, so I'm interested
in, what is a model?
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So this is actually where
mathematics is applied.
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And many people will
have models.
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And people who have a census
can check those models and
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see, well, what are
the constants?
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Does a good choice of constant
make this model realistic over
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the last hundred years?
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Lots to do.
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Fantastic projects
in this area.
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If you're looking for a project,
google "population
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growth." See what they say.
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Several sites will say--
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Wikipedia, I did it
this morning.
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I looked at Wikipedia.
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I'll tell you some things
later about Wikipedia.
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And also, the Census Bureau has
a site and they all know
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the population now
pretty closely.
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Everybody would like
to know what it's
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going to be in 50 years.
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And to get 50 years out,
you have to have some
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mathematical model.
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And I'm going to pick this
differential equation.
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There's going to be a growth.
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That c would be sort
of birth rate.
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Well, I guess birth rate
minus death rate.
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So that would be the
growth rate.
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And if we only had that term
the population would grow
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exponentially forever.
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There's going to be another
term because that's not
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realistic for the population
just to keep growing.
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As the earth fills up, there's
some maybe competition term.
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Some slowdown term.
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So it's going to have
a minus sign.
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And maybe s for a slow
down factor.
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And I'm going to
take P squared.
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That reflects sort of population
interacting with
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population.
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There are many problems. This
is also a model for problems
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in chemistry and biology,
mass action it would
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be called in chemistry.
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Where I would have two different
materials, two
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different substances, two
different chemicals.
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And the interaction between the
two would be proportional
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to the amount of one and also
proportional to the
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amount of the other.
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So it'll be proportional
to the product.
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One concentration
times the other.
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Well, there I would have to have
two equations for the two
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concentrations.
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Here I'm taking a more basic
problem where I just have one
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unknown, the population P,
and one equation for it.
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And it's P times P. It's the
number of people meeting other
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people and crowding.
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So s is a very small number.
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Very small, like one over
a billion or something.
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Now, I want you to solve
that equation.
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Or I want us to solve it.
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And we don't right now
have great tools
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for nonlinear equations.
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We have more calculus to learn,
but this one will give
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in if we do it right.
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And let me show you what the
solution looks like because
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you don't want to miss that.
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We'll get into the formula.
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We'll make it as nice
as possible, but
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the graph is great.
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So the population.
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Well, notice, what would happen
if the population
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starts out at 0?
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Nobody's around.
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Then the derivative is 0.
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It never leaves 0.
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So this 0 would be a solution.
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P equals 0 constant solution.
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Not interesting.
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There's another case, important
case, when the
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derivative is 0.
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Suppose the derivative is 0.
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That means we're going to look
for a particular very special
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solution that doesn't move.
263
00:16:05,690 --> 00:16:11,080
So if that derivative is 0,
then cP equals sP squared.
264
00:16:11,080 --> 00:16:13,220
Can I do this in the corner?
265
00:16:13,220 --> 00:16:16,720
If cP equals sP squared.
266
00:16:16,720 --> 00:16:23,680
And if I cancel a P, I get
P is equal to c/s.
267
00:16:23,680 --> 00:16:25,660
So there's some solution.
268
00:16:25,660 --> 00:16:27,340
Can I call it that c/s?
269
00:16:27,340 --> 00:16:30,090
270
00:16:30,090 --> 00:16:33,220
Well, everybody would like to
know exactly what those
271
00:16:33,220 --> 00:16:35,190
numbers are.
272
00:16:35,190 --> 00:16:39,860
It's maybe about 10
billion people.
273
00:16:39,860 --> 00:16:42,840
So what I'm saying is
if I got up to--
274
00:16:42,840 --> 00:16:43,650
not me.
275
00:16:43,650 --> 00:16:49,120
If the world got up to 10
billion, then at that point,
276
00:16:49,120 --> 00:16:53,490
the growth and the competition
would knock each other out,
277
00:16:53,490 --> 00:16:55,310
would cancel each other.
278
00:16:55,310 --> 00:16:57,540
There wouldn't be any
further change.
279
00:16:57,540 --> 00:17:02,780
So that's a kind of point
that we won't go past.
280
00:17:02,780 --> 00:17:06,650
But now, let me draw the real
solution because the real
281
00:17:06,650 --> 00:17:08,940
solution starts with--
282
00:17:08,940 --> 00:17:10,430
who knows?
283
00:17:10,430 --> 00:17:12,319
Two people, Adam and Eve.
284
00:17:12,319 --> 00:17:13,910
100 people, whatever.
285
00:17:13,910 --> 00:17:15,880
Whenever we start counting.
286
00:17:15,880 --> 00:17:18,500
This is time.
287
00:17:18,500 --> 00:17:20,609
You know, that's the time
that we're starting.
288
00:17:20,609 --> 00:17:21,859
We could start today.
289
00:17:21,859 --> 00:17:24,599
290
00:17:24,599 --> 00:17:30,640
This is time starting at some
point where we know something.
291
00:17:30,640 --> 00:17:33,270
Well, we have pretty good
numbers back for
292
00:17:33,270 --> 00:17:34,350
some hundreds of years.
293
00:17:34,350 --> 00:17:37,590
So probably it wouldn't
be today we'd start.
294
00:17:37,590 --> 00:17:40,575
But wherever we start, we
follow that equation.
295
00:17:40,575 --> 00:17:44,560
296
00:17:44,560 --> 00:17:47,020
Let me just draw the graph.
297
00:17:47,020 --> 00:17:50,760
It grows.
298
00:17:50,760 --> 00:17:55,510
It grows and then at a certain
point, and it turns out to be
299
00:17:55,510 --> 00:18:00,670
the halfway point, c over 2s.
300
00:18:00,670 --> 00:18:02,510
It's beautiful calculus.
301
00:18:02,510 --> 00:18:06,540
It turns out that that's
a point of inflection.
302
00:18:06,540 --> 00:18:10,340
It's not only growing, it's
growing faster and faster up
303
00:18:10,340 --> 00:18:11,420
to this time.
304
00:18:11,420 --> 00:18:15,740
I don't know when that is, but
that's a very critical moment
305
00:18:15,740 --> 00:18:17,350
in the history of the world.
306
00:18:17,350 --> 00:18:20,400
I think we may be a little
passed it actually.
307
00:18:20,400 --> 00:18:23,050
And after that point,
the whole thing
308
00:18:23,050 --> 00:18:24,455
is just like symmetric.
309
00:18:24,455 --> 00:18:26,530
It's an s-shaped.
310
00:18:26,530 --> 00:18:34,090
It goes up and now slows down
and gets closer and closer to
311
00:18:34,090 --> 00:18:36,960
this c/s steady state.
312
00:18:36,960 --> 00:18:39,010
I would call c/s
a steady state.
313
00:18:39,010 --> 00:18:42,930
If it gets there it stays
there steadily.
314
00:18:42,930 --> 00:18:45,010
It won't quite get there.
315
00:18:45,010 --> 00:18:46,340
This would be the s curve.
316
00:18:46,340 --> 00:18:49,440
And if I went back in time, it
would be going back, back,
317
00:18:49,440 --> 00:18:52,000
back to nearly 0.
318
00:18:52,000 --> 00:18:53,580
Probably to 2.
319
00:18:53,580 --> 00:18:57,210
And up here, but not
quite there.
320
00:18:57,210 --> 00:19:04,700
That's the population curve that
comes from this model.
321
00:19:04,700 --> 00:19:07,510
I can't say that's the
population curve that's going
322
00:19:07,510 --> 00:19:14,080
to happen in the next thousand
years because the model is a
323
00:19:14,080 --> 00:19:14,840
good start.
324
00:19:14,840 --> 00:19:20,500
But of course, you could add in
more terms for epidemics,
325
00:19:20,500 --> 00:19:24,760
for wars, for migration.
326
00:19:24,760 --> 00:19:31,140
All sorts of things
affect the model.
327
00:19:31,140 --> 00:19:33,920
OK, now the math.
328
00:19:33,920 --> 00:19:36,940
The math says solve
the equation.
329
00:19:36,940 --> 00:19:39,790
OK, how am I going to do that?
330
00:19:39,790 --> 00:19:42,580
Well, there are different
ways, but here's
331
00:19:42,580 --> 00:19:44,250
a rather neat way.
332
00:19:44,250 --> 00:19:49,390
It turns out that if I try y.
333
00:19:49,390 --> 00:19:57,890
Let me introduce y as 1/P, then
the equation for y is
334
00:19:57,890 --> 00:19:59,140
going to come out terrific.
335
00:19:59,140 --> 00:20:02,240
336
00:20:02,240 --> 00:20:07,190
This is a trick that works
for this equation.
337
00:20:07,190 --> 00:20:10,520
Nonlinear equations, it's
OK to have a few tricks.
338
00:20:10,520 --> 00:20:15,500
All right, so this is
y of t is 1/P of t.
339
00:20:15,500 --> 00:20:19,060
So of course the graph of y
will be different and the
340
00:20:19,060 --> 00:20:21,030
equation for y will
be different.
341
00:20:21,030 --> 00:20:24,590
Let me say what that equation
looks like.
342
00:20:24,590 --> 00:20:27,810
OK, so I want to know,
what's dy dt?
343
00:20:27,810 --> 00:20:31,770
344
00:20:31,770 --> 00:20:34,010
I know how to take
the derivative.
345
00:20:34,010 --> 00:20:35,590
We get to use calculus.
346
00:20:35,590 --> 00:20:38,220
This is P to the minus 1.
347
00:20:38,220 --> 00:20:43,500
So the derivative of P to
the minus 1 is minus 1.
348
00:20:43,500 --> 00:20:47,905
P to the minus 2 times dP dt.
349
00:20:47,905 --> 00:20:51,050
350
00:20:51,050 --> 00:20:52,680
That's the chain rule.
351
00:20:52,680 --> 00:20:56,160
P to the minus 1, the derivative
is minus 1, p to
352
00:20:56,160 --> 00:21:00,060
the minus 2, and the derivative
of P itself.
353
00:21:00,060 --> 00:21:02,800
And now I know dP dt.
354
00:21:02,800 --> 00:21:05,180
Now I'm going to use
my equation.
355
00:21:05,180 --> 00:21:11,170
So that's cP minus sP squared
with a minus.
356
00:21:11,170 --> 00:21:13,950
Oh, I can't lose that minus.
357
00:21:13,950 --> 00:21:15,920
Put it there.
358
00:21:15,920 --> 00:21:17,600
Keep your eye on that minus.
359
00:21:17,600 --> 00:21:19,185
Divided by P squared.
360
00:21:19,185 --> 00:21:22,220
361
00:21:22,220 --> 00:21:24,470
So now I'll use the minus.
362
00:21:24,470 --> 00:21:28,170
That looks like an s to me.
363
00:21:28,170 --> 00:21:30,730
Minus minus is plus s.
364
00:21:30,730 --> 00:21:32,170
And what do I have here?
365
00:21:32,170 --> 00:21:34,340
Minus cP over P squared.
366
00:21:34,340 --> 00:21:42,580
I think I have minus c/P.
And you say OK, true.
367
00:21:42,580 --> 00:21:44,790
But what good is that?
368
00:21:44,790 --> 00:21:48,890
But look, that 1/P is our y.
369
00:21:48,890 --> 00:21:55,470
So now I have s minus cy.
370
00:21:55,470 --> 00:22:01,550
The equation for y, the dy dt
equation turns out to be
371
00:22:01,550 --> 00:22:07,710
linear with a source term
s just as in the
372
00:22:07,710 --> 00:22:09,180
start of the lecture.
373
00:22:09,180 --> 00:22:18,790
And the growth rate term has
a minus c, which we expect.
374
00:22:18,790 --> 00:22:24,450
Because our y is now 1 over.
375
00:22:24,450 --> 00:22:28,020
When this growth is going up
exponentially, 1 over it is
376
00:22:28,020 --> 00:22:29,630
going down exponentially.
377
00:22:29,630 --> 00:22:33,280
And it turns out that same c.
378
00:22:33,280 --> 00:22:36,280
In other words, we can
solve this equation.
379
00:22:36,280 --> 00:22:40,960
In fact, we have solved this
equation just for--
380
00:22:40,960 --> 00:22:44,370
shall I write down the answer?
381
00:22:44,370 --> 00:22:48,490
So the answer for y.
382
00:22:48,490 --> 00:22:54,240
You remember the answer
for y is--
383
00:22:54,240 --> 00:22:56,820
I'm going to look over here.
384
00:22:56,820 --> 00:22:59,400
Up there.
385
00:22:59,400 --> 00:23:03,130
This is our equation, the only
difference is c is coming in
386
00:23:03,130 --> 00:23:03,970
with a minus sign.
387
00:23:03,970 --> 00:23:06,730
So I'm just going to write
that same solution,
388
00:23:06,730 --> 00:23:09,110
just copy that here.
389
00:23:09,110 --> 00:23:11,280
y of t.
390
00:23:11,280 --> 00:23:15,920
Now c is coming in with
a minus sign, is
391
00:23:15,920 --> 00:23:20,940
y of 0 minus s/c.
392
00:23:20,940 --> 00:23:22,950
c has that minus sign.
393
00:23:22,950 --> 00:23:26,050
e to the minus ct.
394
00:23:26,050 --> 00:23:27,300
OK, good.
395
00:23:27,300 --> 00:23:30,560
396
00:23:30,560 --> 00:23:33,810
That's the solution for y.
397
00:23:33,810 --> 00:23:37,230
398
00:23:37,230 --> 00:23:40,060
So we solved the equation.
399
00:23:40,060 --> 00:23:42,160
You could say, well, you solved
the equation for y.
400
00:23:42,160 --> 00:23:45,890
But now P is just 1/y.
401
00:23:45,890 --> 00:23:54,130
Or y is 1/P. So I just go back
now and change from y back to
402
00:23:54,130 --> 00:23:59,070
P. 1 over P of t is--
403
00:23:59,070 --> 00:24:01,730
here I have 1 over P of 0.
404
00:24:01,730 --> 00:24:06,840
405
00:24:06,840 --> 00:24:10,950
Is it OK if I leave the
solution in that form?
406
00:24:10,950 --> 00:24:15,120
I can solve for P of t and get
it all-- you get P of t equals
407
00:24:15,120 --> 00:24:17,140
and a whole lot of stuff
on the right side.
408
00:24:17,140 --> 00:24:18,830
I move this over.
409
00:24:18,830 --> 00:24:21,850
I have to flip it upside down.
410
00:24:21,850 --> 00:24:25,040
It doesn't look as nice.
411
00:24:25,040 --> 00:24:26,390
Well, I guess I could.
412
00:24:26,390 --> 00:24:29,550
I could move that over
and then just put
413
00:24:29,550 --> 00:24:32,100
to the minus 1 power.
414
00:24:32,100 --> 00:24:33,290
That would do it.
415
00:24:33,290 --> 00:24:36,270
And that is the solution.
416
00:24:36,270 --> 00:24:37,590
Maybe I should do that.
417
00:24:37,590 --> 00:24:40,250
P of t is--
418
00:24:40,250 --> 00:24:41,410
I'm going to move this over.
419
00:24:41,410 --> 00:24:46,900
So this same parenthesis times
e to the ct plus s/c.
420
00:24:46,900 --> 00:24:48,600
I moved that over.
421
00:24:48,600 --> 00:24:52,400
And then I say I have
to flip it.
422
00:24:52,400 --> 00:24:55,210
So it's 1 over that.
423
00:24:55,210 --> 00:24:59,290
1 over all that stuff, where
this is just this
424
00:24:59,290 --> 00:25:02,240
stuff copied again.
425
00:25:02,240 --> 00:25:05,450
So we get an expression but
to me, that top one
426
00:25:05,450 --> 00:25:07,460
looks pretty nice.
427
00:25:07,460 --> 00:25:11,140
And graphing it is no problem.
428
00:25:11,140 --> 00:25:14,460
And that's what the
graph looks like.
429
00:25:14,460 --> 00:25:18,520
It's that famous s curve.
430
00:25:18,520 --> 00:25:23,960
What we've solved here was the
population equation and it's
431
00:25:23,960 --> 00:25:26,265
often called the logistic
equation.
432
00:25:26,265 --> 00:25:34,155
433
00:25:34,155 --> 00:25:37,790
I mention that word because you
could see it there as the
434
00:25:37,790 --> 00:25:41,270
heading for this particular
model.
435
00:25:41,270 --> 00:25:50,290
So let me let you leave open
everything we've done today.
436
00:25:50,290 --> 00:25:56,890
But maybe I can say the most
interesting aspect is the
437
00:25:56,890 --> 00:25:58,140
model itself.
438
00:25:58,140 --> 00:26:00,290
439
00:26:00,290 --> 00:26:01,820
To what extent is it accurate?
440
00:26:01,820 --> 00:26:03,080
We could try it.
441
00:26:03,080 --> 00:26:08,480
We could estimate c and s,
those numbers, to fit the
442
00:26:08,480 --> 00:26:12,980
model of what we know over
one time period.
443
00:26:12,980 --> 00:26:15,140
And then we could see,
does it extend over
444
00:26:15,140 --> 00:26:16,760
another time period?
445
00:26:16,760 --> 00:26:20,550
I was going to say
about Wikipedia.
446
00:26:20,550 --> 00:26:23,360
And generally, Wikipedia
is not too bad.
447
00:26:23,360 --> 00:26:26,775
They made one goof, which
I thought was awful.
448
00:26:26,775 --> 00:26:29,630
449
00:26:29,630 --> 00:26:34,950
The question was, what's
the growth rate?
450
00:26:34,950 --> 00:26:37,350
Actually, Wikipedia
doesn't discuss
451
00:26:37,350 --> 00:26:38,810
this particular equation.
452
00:26:38,810 --> 00:26:43,260
It tells you a lot of other
things about population.
453
00:26:43,260 --> 00:26:45,600
Of course, they talk about the
growth rate and at one point
454
00:26:45,600 --> 00:26:48,580
they say at one time,
it was 1.8%.
455
00:26:48,580 --> 00:26:51,890
456
00:26:51,890 --> 00:26:55,920
I say 1.8% is not
a growth rate.
457
00:26:55,920 --> 00:26:59,040
Growth rate, c, don't have--
458
00:26:59,040 --> 00:27:01,570
they're not percentages.
459
00:27:01,570 --> 00:27:04,380
Their units are 1 over time.
460
00:27:04,380 --> 00:27:08,370
And maybe I could make that
point, emphasize that point.
461
00:27:08,370 --> 00:27:13,940
When I see ct together, that
tells me that has the
462
00:27:13,940 --> 00:27:15,930
dimension of time, of course.
463
00:27:15,930 --> 00:27:20,010
So c must have the dimensions
of 1 over time.
464
00:27:20,010 --> 00:27:25,410
The growth rate is
1.8% per year.
465
00:27:25,410 --> 00:27:29,360
And I will admit that three
lines later on Wikipedia, when
466
00:27:29,360 --> 00:27:34,330
they're referring to an earlier
growth rate earlier in
467
00:27:34,330 --> 00:27:42,700
the last century, they
do say 2.2% per year.
468
00:27:42,700 --> 00:27:44,270
They get it right.
469
00:27:44,270 --> 00:27:48,570
The units are 1 over time.
470
00:27:48,570 --> 00:27:53,680
The solution is telling us
that and the equation is
471
00:27:53,680 --> 00:27:54,380
telling us that.
472
00:27:54,380 --> 00:27:57,400
This has the dimensions of
population, number of people.
473
00:27:57,400 --> 00:27:59,720
So does this.
474
00:27:59,720 --> 00:28:02,430
I'm dividing by a time.
475
00:28:02,430 --> 00:28:07,790
So c must be the source of
that division by a time.
476
00:28:07,790 --> 00:28:10,050
The units of c must
be 1 over time.
477
00:28:10,050 --> 00:28:15,850
You might like to figure out the
units of s because in an
478
00:28:15,850 --> 00:28:18,840
equation, everything has to have
the same units, the same
479
00:28:18,840 --> 00:28:23,260
dimension to make any
sense at all.
480
00:28:23,260 --> 00:28:28,860
So with that little comment,
let me emphasize
481
00:28:28,860 --> 00:28:32,740
the interest in just--
482
00:28:32,740 --> 00:28:37,460
if you have a project to do,
this could be a quite
483
00:28:37,460 --> 00:28:38,960
remarkable one.
484
00:28:38,960 --> 00:28:43,870
Now just I'll write down,
but not solve--
485
00:28:43,870 --> 00:28:46,720
since I have a little space
and mathematicians tend to
486
00:28:46,720 --> 00:28:49,140
fill space--
487
00:28:49,140 --> 00:28:53,150
I'll write down one
more equation.
488
00:28:53,150 --> 00:28:55,930
Actually, it will be two
equations because it's a
489
00:28:55,930 --> 00:29:00,150
predator and a prey.
490
00:29:00,150 --> 00:29:03,850
A hunter and a hunted.
491
00:29:03,850 --> 00:29:10,150
And in ecology, or modeling,
whatever.
492
00:29:10,150 --> 00:29:13,080
Foxes and rabbits, maybe.
493
00:29:13,080 --> 00:29:17,220
We might have these equations,
the predator
494
00:29:17,220 --> 00:29:21,520
and the prey equation.
495
00:29:21,520 --> 00:29:24,070
And what would that look like?
496
00:29:24,070 --> 00:29:25,510
So we have two unknowns.
497
00:29:25,510 --> 00:29:26,360
The predator.
498
00:29:26,360 --> 00:29:27,810
Can I call that u?
499
00:29:27,810 --> 00:29:29,060
So I have a du dt.
500
00:29:29,060 --> 00:29:31,760
501
00:29:31,760 --> 00:29:35,080
What's the growth of the
predators, the number or the
502
00:29:35,080 --> 00:29:35,890
population?
503
00:29:35,890 --> 00:29:40,300
These are both populations, but
of two different species.
504
00:29:40,300 --> 00:29:45,170
And the prey will be
v. So u is the
505
00:29:45,170 --> 00:29:46,515
population of the predator.
506
00:29:46,515 --> 00:29:49,220
507
00:29:49,220 --> 00:29:53,270
If the predator has nothing
to eat, the populations of
508
00:29:53,270 --> 00:29:55,840
predator is going to drop.
509
00:29:55,840 --> 00:29:59,650
But the more prey
that's around--
510
00:29:59,650 --> 00:30:03,150
so here we'll have a u times
v going positive.
511
00:30:03,150 --> 00:30:12,280
So we'll have something like a
minus cu as a term that if
512
00:30:12,280 --> 00:30:16,360
there's nothing to eat, if
that's all there is, if the
513
00:30:16,360 --> 00:30:19,660
prey is all gone, the predator
will die out.
514
00:30:19,660 --> 00:30:24,570
But when there is a predator,
there will be a source term
515
00:30:24,570 --> 00:30:31,540
proportional to u and v. The
more predators there are,
516
00:30:31,540 --> 00:30:33,460
they're all eating away.
517
00:30:33,460 --> 00:30:38,010
And the more prey there is
the more they're eating.
518
00:30:38,010 --> 00:30:40,800
And now what about dv dt?
519
00:30:40,800 --> 00:30:43,910
OK, v is a totally different
position.
520
00:30:43,910 --> 00:30:48,470
I guess it's getting eaten.
521
00:30:48,470 --> 00:30:53,930
So this term, well, I'm
not sure what all the
522
00:30:53,930 --> 00:30:57,490
constants are here.
523
00:30:57,490 --> 00:31:00,550
The prey, this is now the prey,
the little rabbits.
524
00:31:00,550 --> 00:31:02,470
They're just eating grass.
525
00:31:02,470 --> 00:31:04,620
Plenty of grass around.
526
00:31:04,620 --> 00:31:06,110
So they're reproducing.
527
00:31:06,110 --> 00:31:14,480
So they have some positive
growth rate, capital C.
528
00:31:14,480 --> 00:31:16,540
Multiple per year.
529
00:31:16,540 --> 00:31:24,410
But then they will have a
negative term as we saw in
530
00:31:24,410 --> 00:31:29,230
population coming from the
competition with a predator.
531
00:31:29,230 --> 00:31:34,860
So this would be a model, a
predator-prey model that also
532
00:31:34,860 --> 00:31:38,950
shows up in many applications
of mathematics.
533
00:31:38,950 --> 00:31:44,510
So there you see two linear
differential equations.
534
00:31:44,510 --> 00:31:48,310
The simple one dy
dt we'll see y.
535
00:31:48,310 --> 00:31:51,080
We totally know how
to solve that.
536
00:31:51,080 --> 00:31:56,460
Now with a source term s,
we're OK, and we got the
537
00:31:56,460 --> 00:31:59,900
solution there and we found
it a second way.
538
00:31:59,900 --> 00:32:06,040
Then thirdly was the main
interest, the population
539
00:32:06,040 --> 00:32:09,520
equation, the logistic equation
with a P squared,
540
00:32:09,520 --> 00:32:15,130
which we were able to solve
and graph the solution.
541
00:32:15,130 --> 00:32:17,810
That s curve is just
fantastic.
542
00:32:17,810 --> 00:32:22,980
If you want a challenge,
check that this--
543
00:32:22,980 --> 00:32:25,420
so what do I mean at that
inflection point?
544
00:32:25,420 --> 00:32:26,960
What's happening there?
545
00:32:26,960 --> 00:32:35,260
I claim that at this value of P,
the curve stops bending up,
546
00:32:35,260 --> 00:32:39,710
starts bending down,
which means second
547
00:32:39,710 --> 00:32:45,710
derivative of P is 0.
548
00:32:45,710 --> 00:32:48,110
You could check that.
549
00:32:48,110 --> 00:32:52,280
Take the equation, take its
derivative to get the second
550
00:32:52,280 --> 00:32:52,830
derivative.
551
00:32:52,830 --> 00:32:56,040
Then that'll involve
dP dts over here.
552
00:32:56,040 --> 00:32:58,640
But then you know dP dt,
so plug that in.
553
00:32:58,640 --> 00:33:01,810
And then finally, you'll
get down to P's.
554
00:33:01,810 --> 00:33:05,670
And I think you'll find that the
thing cancels itself out.
555
00:33:05,670 --> 00:33:10,200
The second derivative, the
bending in the s curve, is 0
556
00:33:10,200 --> 00:33:11,790
at c over 2s.
557
00:33:11,790 --> 00:33:15,170
So that's the moment
of fastest growth.
558
00:33:15,170 --> 00:33:22,780
And maybe that happened
relatively recently.
559
00:33:22,780 --> 00:33:26,430
And we're slowing down, but
we've got a long way to go.
560
00:33:26,430 --> 00:33:29,380
So roughly, I think the
population now is
561
00:33:29,380 --> 00:33:32,630
just under 7 billion.
562
00:33:32,630 --> 00:33:36,270
And oh boy, if we just left--
563
00:33:36,270 --> 00:33:39,850
if it was 7 billion at that
point, it would be 14 billion
564
00:33:39,850 --> 00:33:41,480
at the end.
565
00:33:41,480 --> 00:33:44,350
I think we're a better further
along now at 7 billion.
566
00:33:44,350 --> 00:33:45,840
7 billion's somewhere
about here.
567
00:33:45,840 --> 00:33:51,880
So maybe the end with this model
is more like 10 or 11
568
00:33:51,880 --> 00:33:58,700
billion as the population of the
earth at t equal infinity.
569
00:33:58,700 --> 00:34:00,820
OK, that's a mathematical
model.
570
00:34:00,820 --> 00:34:02,580
Thank you.
571
00:34:02,580 --> 00:34:04,390
ANNOUNCER: This has been
a production of MIT
572
00:34:04,390 --> 00:34:06,780
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Gilbert Strang.
573
00:34:06,780 --> 00:34:09,060
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574
00:34:09,060 --> 00:34:10,270
Foundation.
575
00:34:10,270 --> 00:34:13,400
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576
00:34:13,400 --> 00:34:16,480
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577
00:34:16,480 --> 00:34:18,040
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578
00:34:18,040 --> 00:34:20,166