So we've seen that mathematical models of
reaction-diffusion systems are capable of
forming stable spatial patterns.
In varying these parameters a and b, the
different diffusion rates of u and v or
chemical species a and b, gives different
sorts of stripes and dots, sometimes fixed
patterns, sometimes periodic oscillation.
So they're capable of doing a wide range
of things. And this is surprising, I
think, because these are models based on
diffusion, based on a process through
which particles should spread out and
reach some sort of uniformity.
So, this is a pretty amazing mathematical
result and it is, people think it's a
mechanism behind, certainly not all
patterns, of course, but many patterns
that are observed in the natural world,
particularly in biological systems. But
what's maybe even more amazing than this
mathematical result is that one can
actually do experiments with reaction-
diffusion systems and see the same
sorts of general patterns forming.
There are lots of experiments one can do
but I want to describe one just very, very briefly.
It's a reaction-diffusion type of system
known as a Belousov–Zhabotinsky reaction.
If one carries out this chemical reaction
just in a beaker, one sees colors that
oscillate. It's a couple of
different chemicals that interact with
each other, not too dissimilar from how
the a and b interacted with each other
in the system we just studied.
So, if we take this chemical system and
we put it not in a well mixed beaker,
but on a surface, usually it's in some
sort of a Petri dish, and one starts with
a more or less uniform distribution of
the two reactants, one can see patterns
emerging just like we did on the computer screen.
So what I want to do is show you a quick
video of one of those experiments, just
to give you a flavor of what these things
look like, and that this is a real
phenomenon, not just a mathematical phenomenon.
And then, I'll show you this video, talk
about it a little bit, and then in the
next subunit I'll draw a few conclusions.
Alright, so let's check out the Belousov–Zhabotinsky reaction.
So there are quite a few videos of
Belousov–Zhabotinsky pattern forming
reactions on YouTube and elsewhere,
in the ones I'd like to show you today
are posted by Stephen Morris, he's a
professor of geophysics at University
of Toronto. And I think these videos are
particularly nice and his entire YouTube
channel has all sorts of interesting pattern forming systems.
And we're lucky enough that we'll get to talk to him next week.
So in Unit 10 I'll have some recorded
conversations, interviews, with some
researchers and teachers in dynamical
systems, and Stephen Morris will be one of them.
Alright, the Belousov–Zhabotinsky reaction.
This is an example of reaction-diffusion.
What you'll see is two different chemicals
are poured into this Petri dish and,
presumably they are liquid and poured
right in and should be uniform but because
of this instability where they are a
little bit of an increase just at random
in one of these activators,
you'll see this reaction that leads to an
expanding wave front. You'll see patterns
appear seemingly out of nowhere.
So, let's run this. There he's pouring in
one liquid. There's another. It is well mixed.
But patterns form spontaneously.
And they grow outward. Looks a little
different from the Turing patterns we saw
in the program, but it's the same general phenomenon.
Let's look at that once more.
And I should mention that links to these
videos are in the additional resources
section of this unit.
So liquids are poured in, mixed together,
and out of nowhere, it's as if they
sort of, or not quite separate out,
but these reactions form these spatial
structures that persist and grow in time.
Alright, let me show you one more of these.
So this is a similar sort of thing.
Let me go back to the beginning.
This one has a soundtrack, I'll mute it.
Don't be alarmed if you watch it on your
own and suddenly the soundtrack jumps out
at you. So this is a Belousov–Zhabotinsky
reaction. Same thing. The video is sped
up a little bit so one can see what
happens. And you'll notice that a hand
will come in with a paperclip and it'll
break some of the circles that are
expanding outward and that has the effect
of turning these circles into spirals.
So let's watch that. So 8x normal speed.
So there they are. They're growing
outwards. And there, breaking some
of those circles and you can see spirals
forming in the middle, a little bit to the
right. And those spirals expand and then
one will end up with these interesting
spiral shapes that sort of merge into each other.
So I definitely recommend that you watch
these videos if you want on your own.
And there are lots of others on YouTube
or elsewhere if you search for "Belousov– Zhabotinsky reaction"
or "reaction-diffusion pattern" or "Turing pattern"
there are lots of nice demonstrations of these.
But the main point of this is that this
isn't just a mathematical phenomenon,
it's a real, physical phenomenon as well.
One can get these different chemicals to
interact this way. And this is believed to
be, not this exact equation, but this
general phenomena may be behind some
biological development, how structures
form out of initially homogeneous
tissues or some sort of substrate.
So, again, dynamical systems, now we're
looking at spatial dynamical systems, so
a different mathematical entity but again,
simple, deterministic, local dynamical
systems can produce a certain type of, it
can produce chaotic behavior, it can
produce a certain form of the butterfly
effect if one ran this experiment again
exactly where these spirals would be would vary from run to run.
But also, in addition to producing
chaotic behavior we see patterns,
structures forming out of nowhere.
Nobody is designing these spirals.
Nobody from above is making these spirals.
They are literally from the ground up.
A local rule with no global knowledge of
what is going on is able to produce these
large scale, and fairly intricate structures.