So, let me step through an example rather qualitatively,
and then I'll do a little bit more of equations
and then in the next subunit we'll turn towards
a computer to solve these systems for us.
So here's reaction and diffusion.
We have two types of chemical or two things
u and v, or A and B,
and they diffuse - that's this part of the equation.
And they also react.
They interact with each other in somehow... in some way.
Typically, in this setup, one of the types of chemicals
is called an activator and the other is an inhibitor.
So, let me write that.
So, in this picture u will be the activator,
and v is an inhibitor.
Something that slows down or inhibits or kills whatever u is.
And B will be greater than A,
which means that the inhibitor diffuses faster than the activator.
So, we have two different chemical species,
two different types of chemicals playing different roles.
One is an activator, one is an inhibitor.
I'll describe than in a moment.
And they diffuse at different rates.
And typically the inhibitor will diffuse faster than the activator.
And this will be enough to give us some sort of patterns.
So, let me try and sketch... describe some sort of made up story
for how this might work.
So, in this picture, we're going to have an activator... will be rabbits.
An inhibitor will be foxes.
And let's see, let's have the foxes be red.
And rabbits... do I have any rabbit colors?
Let's have brown rabbits.
Rabbits can be brown. Why not?
Alright. So, imagine we have some initial condition.
And, alright, so here's the rectangle
the world where these rabbits and foxes live.
And, okay, obviously this is not supposed to be a serious rabbit and fox example.
I just want try to illustrate this general activator inhibitor and different diffusion rate idea.
So, we might have an initial condition that has rabbits
spread all over - choose it at random.
And then some foxes.
Try to draw the foxes a little bigger, so they're a little more visible.
Okay, so there's a possible initial condition -
rabbits and foxes spread out across this world.
And I'm imagining here that the rabbit fox interaction is not cyclic
the way it is in the Lotka-Volterra equations we studied earlier,
but that rabbits and foxes can somehow coexist.
So, there's some equilibrium population of rabbits and foxes,
some sort of happy proportion where they each have an equilibrium.
So maybe that's right, sort the equivalent of the right of the middle of those Lotka-Volterra circles.
Alright, so the foxes and rabbits diffuse around,
and mabye the foxes and rabbits both
they don't like to be crowded.
So, they're going to move away from too dense regions to farther out regions.
And we might expect that after a little while
if we just started this, we would see a uniform distribution
of rabbits and foxes throughout this world.
It would be counterintuitive, suprising to see clusters of rabbits and clusters of foxes,
because the picture here is that these rabbits or foxes or chemicals diffuse out.
Diffusion smoothes everything out.
However, for activator inhibitor dynamics that's not always the case.
And so let me describe a little bit about why that might be.
So let's say there's a region that just happens have unusually high density of rabbits.
Just a random fluctuation.
We don't start with a perfectly uniform distribution of rabbits.
So here are the rabbits.
And when there are a lot of rabbits there tend to be more rabbits.
So the rabbits are an activator here.
That if they're... rabbits make more rabbits.
And so all of a sudden we see a high density of rabbits here.
And we might expect that to diffuse out.
And it will be diffusing.
However, if there are a high density of rabbits,
there are going to be some foxes here too,
because foxes want to go where the rabbits are.
And so some foxes, they might move in,
or, okay, unrealistically in this model there might be foxes just added from above or foxes created out of nothing.
So, what's going on here is we have first a growth in the population of rabbits,
and then a growth in the population of foxes.
And the foxes are eating the rabbits, but the rabbits got such a big head start.
There were so many of them that the rabbits keep growing.
So now diffusion happens here.
Right? There's this cluster.
This highly dense region of both rabbits and foxes.
And they're going to diffuse out.
So it's expected to see the rabbits and the foxes spreading out.
However, in these systems foxes move faster than rabbits.
The inhibitor diffuses faster than the activator.
So what happens is -
okay, let's see, where am I going to draw this?
Let me, zoom in. So here I've got all of these rabbits.
And I've got some foxes, but now the foxes are spreading out faster.
And we might have a situation here where the foxes spread out faster.
And then there's sort of a wall of foxes around the rabbits,
preventing the foxes... excuse me, the rabbits from pushing out further.
Or we could have something a little bit more complicated still
where this rabbit region will tend to expand
with foxes always sort of pushing out on the boundary.
And then we might have high density rabbit regions collide into each other.
And then that's when the, sort of, growth would stop.
So, this has not been a precise mathematical argument at all.
But the main thing I'm trying to suggest is that when we have an activator and an inhibitor -
So an activator, it activates itself. You can get some sort of runaway behavior.
But then you also have an inhibitor that starts out around where all the rabbits are
but spreads out faster.
That those two sorts of things - an unstable growth in the activator
Unstable meaning that it's growing very rapidly.
But an inhibitor that moves faster can create spatial structures.
So it would be possible in a system where you have some interaction terms and diffusive behavior
to have a steady state, a stable equilibrium, that is not uniform.
Even though diffusion tends to be a leveling, smoothing out force.
We can have a situation where diffusion plus this interaction,
plus this dynamic, gives rise to a really interesting array of spatial patterns.