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So let's summarize Unit Nine.
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As usual I'll begin with a topical summary
of what we covered in this unit
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and then speak a little bit more broadly
and generally about key themes and ideas.
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So, this unit was on pattern formation.
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It began by pointing out that we've seen
that dynamical systems are capable of
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exhibiting chaos; that's unpredictable,
aperiodic behavior, butterfly effect.
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But there's a lot more to the story.
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Dynamical systems can do much more
than produce chaos or disorder,
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they can also produce patterns,
structure, organization, and so on.
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And in this unit we looked at one
particular example of a pattern forming
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dynamical system, a class of systems known
as reaction-diffusion systems.
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So first we talked about diffusion,
and I tried actually to demonstrate it
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by dropping a few drops of food coloring
in a glass of water.
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Diffusion is the tendency of a substance
to spread out as a result of random motion
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so the random ink molecules or food
coloring molecules bounce around and
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eventually they'll spread themselves
out evenly throughout the glass of water.
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Another way to say this is that chemicals
tend to move from regions of high
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concentration to low concentration and the
result of this is that diffusion tends to
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smooth out differences in concentration,
leading to a homogeneous, or uniform
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distribution of the food coloring
or whatever it is that's diffusing.
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So diffusion is described mathematically
by the diffusion equation.
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Here it is.
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In this equation D is the
diffusion constant.
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It's related to how fast the
substance diffuses.
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Different substances will diffuse at
different rates in different media.
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Often this is something that one could
measure experimentally.
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This is the rate of change of
concentration, and this is the Laplacian.
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This is a certain type of spatial
derivative of the system.
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So I should say that, u, here is the
concentration of the chemical and
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it's a function of x and y.
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We might have, say, on this sheet of paper
different concentrations at different
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points on the page.
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So this describes the diffusion of
substances once
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you know D and the initial conditions.
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In the equilibrium state, when this stops
changing, if I set this equal to zero,
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then I'm, in effect, setting this quantity
equal to zero, and that has the effect of
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picking out the most boring
function possible.
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It'll just be a flat, level distribution
if the boundary conditions allow it.
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If there's more chemicals flowing
in on this side and out on this side
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then we get a smooth distribution and
interpolation between those two values.
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Again, the main point is that diffusion
leads to boring functions;
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functions that are as
homogeneous as possible.
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But things get more interesting
in reaction-diffusion systems.
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In this we now have two different
types of chemicals,
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often called "u" and "v" or "a" and "b".
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The details may vary, and u will be the
concentration of one chemical as a
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function of x and y, so different
concentrations at different points in
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space, same story for v.
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And both of these chemicals diffuse they
spread out across the surface
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but they also interact with each other.
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So the equation of motion, what determines
their values, the rule of the dynamical
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system are these two equations.
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So, this is just diffusion for u.
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But then this says there's some added term
that's an interaction, typically between
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u and v, so it's diffusion, plus something
else that depends on u and v.
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Same thing for the v equation, v will
diffuse plus there's some other
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different interaction, a function g,
but it's a function of u and v.
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So it's diffusion, plus something else.
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This is a deterministic, dynamical system
there's no element of chance,
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no stochasticity in the rules, and it's a
spatially-extended dynamical system.
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So if we give the initial condition,
initial values of u and v everywhere
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in the lattice or the rectangle,
it doesn't have to be a lattice,
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and we specify what the conditions are at
the edges, then that determines the future
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values everywhere on this piece of paper.
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Another point that is important is that
this rule is local. Let me explain that.
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This use of local is not an entirely
standard one, but I think it's justified.
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The idea is that if I want to determine
the next value of u so I know the present
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value of u and v at some particular point
on our surface, if I want to figure out
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the next value using something like
Euler's Method, all I need to know is the
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value of u here, the value of v here, and
the derivatives of u and v at this point.
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So what I mean by local is if I want to
know the next value of u, how u changes,
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at this point, I don't need to know
the value of u over here.
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So that these equations are all local and,
in any given equation, x and y
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gives reference to the same point.
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So another way to say this is that there
aren't any long range interactions here.
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The next value of u is determined by
the present value of u, and a little bit
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about it's curvature given by the
Laplacian at that point.
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So we have a local, deterministic,
spatially-extended dynamical system.
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So in reaction-diffusion systems or
specifically, activator-inhibitor systems
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the following setup typically is the case:
u is usually an activator.
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It's something that catalyzes its own
growth, so the presence
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of u gives rise to more u.
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Rabbit growth, exponentially growing
rabbits, are an example of this.
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We don't usually talk about rabbits
catalyzing their own growth,
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but in a sense that's what they do.
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Now v is some sort of an inhibitor.
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It's something that typically would
grow in the presence of u,
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but it would also inhibit u.
It prevents u from growing too much.
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In the example I gave, these were foxes,
foxes grow in the presence of u;
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foxes eat rabbits, there get to be more
foxes but the foxes inhibit the rabbits.
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They prevent them from
growing and growing.
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So we have these two things
and that v diffuses faster than u.
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This can lead to stable
spatial structures.
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We saw some examples of that. I'll show
you one again in a second.
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The particular shapes determined depend
on a number of different things:
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the relative diffusion rate, how much
faster one diffuses than the other,
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and also the geometry of the system.
In some sense, the initial values as well.
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Of course it will also depend on the
particular functions you choose for f, g.
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There are lots of different possibilities
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and one can do three
components, u, v, and w.
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There are lots of different models here
and the mathematics of analyzing them
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analytically gets pretty complex
pretty quickly.
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In the additional reading section for this
unit I have some suggestions for places
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you can go to learn more.
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So these are reaction-diffusion systems,
one particular type of them.
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Here's some results. We experimented
with the excellent program at
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Experimentarium Digitale.
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Here's the url. There's a link to this
on the links to program page.
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This was the Guépard setting.
So this makes Cheetah spots.
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There it is. It doesn't show up quite that
well in black and white.
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It's a more compelling picture on the
screen.
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Here the diffusion rate of u is 3.5, that
may be hard to see, and b was 16.
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And that program specifies what the f and
g functions are as well.
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The main point is that even though we have
a diffusive system, where things should be
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spreading out, you wouldn't expect to have
a higher density of u here than here and
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have that be a permanent situation.
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It should diffuse away, that's what
diffusion does; it smooths things out.
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But, if we have a reaction-diffusion
systems where things interact in the way
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I describe, one can get a variety of
different stable spatial structures.
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These systems form patterns stable spatial
structures, seemingly out of nowhere.
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This isn't just a mathematical result,
lots of physical systems do this.
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I showed a little bit of a video of a
Belousov Zhabotinsky experiment taking
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place on a Petri dish. Here's a link to
that YouTube video. This link is also in
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the additional resources section.
This video is by Stephen Morris of the
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University of Toronto who we'll be talking
to next week.
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So this looks a little different than the
cheetah spots but it's the same general
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thing; reaction-diffusion, we get these
propagating wave fronts that move out and
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then the wave fronts collide with each
other and interact in interesting ways.
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So to summarize once more,
pattern formation.
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There's more to the study of dynamical
systems than just chaos. Lot's more.
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It's very often the case that simple,
spatially-extended dynamical systems
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with local rules, like these reaction-
diffusion systems, are capable of
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producing stable, global patterns
and structures.
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This begins to give some insight into how
patterns might emerge in an otherwise
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structureless world.
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The reaction-diffusion systems that we
study here are just one of many, many
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examples of pattern forming systems.
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There's lots of different classes of
models; the partial differential equations
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of reaction-diffusion systems is
just one of them.
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In general, there's a certain creativity
to these simple dynamical systems that
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can not only produce chaos but can also
produce these interesting structures.