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Hi, modelers.

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I've been really enjoying the
discussions around module two.

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The tools we're seeing right
now are completely different

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from those we saw in module one.

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We are looking at cellular automatons

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that have a lot of structure
and also random walks,

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which are now stochastic processes,

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meaning that when you rerun them,

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you can get different outcomes,

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because you're flipping random coins

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and making random decisions.

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That's very different
from compartmental models

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that had barely any structure in them,

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or ordinary differential equations

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that were deterministic by nature.

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You could rerun them and always,

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and always get the same results.

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And so what I'd like to do in this video

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is show you that even though
those two different toolboxes

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may seem very different, they
can sometimes join forces

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and help you answer complex questions.

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So the project we'll be looking at

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is "How the Forest Got Its Shape."

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That's one of those classic
scientific questions

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that appears super simple,
but it's hard to answer.

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It's a question that popped
in my mind around 2014

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when I joined the Santa Fe Institute.

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I saw a seminar by ecologists

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who were talking about all
the great data science tools

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that their field was developing.

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What you see here on the
screen is one of them.

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So this is from a paper by Hansen

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and colleagues in "Science" in 2013,

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and they show these high
resolution global maps

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that not only give us
great spatial information

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about forests, but also
temporal information.

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So the four panels here
are four different places

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in the world, so regions
of Paraguay in panel A,

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Indonesia in panel B, United
States in C, and Russia in D,

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and they not only show you
that there's a great diversity

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in terms of forest shape and structure,

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so the forest would be the
really bright green pixels here,

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but also different
dynamics across the globe.

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So you also see forest
loss in the red areas,

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forest gains in the blue areas,

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and a mixture of both in purple.

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And so during the seminar, I
was looking at the diversity

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of forest shapes and patches
and different dynamics,

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and I was trying to make sense of it.

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So, I raised my hand and
I say it a lot in class,

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there are no dumb
questions, and I just asked,

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"Well, what is the shape
of a typical forest?

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"Is what we're seeing, the
ridges or rivers we're seeing

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"in (indistinct) that typical

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"to have linear forests like that?"

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Or, "Are most forests like
what we're seeing in panel B

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"and have this dense shape,
almost like discs or squares,

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"which you would expect the
forest to grow outward?"

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And really, the ecologist didn't know,

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and it sparked a great collaboration.

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And the questions we were trying to answer

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is not only what is the
typical shape of a forest,

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'cause you can do that by just looking

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at these high resolution maps.

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You can do an observational
study on the data itself.

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But what we wanted to answer

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is really how the forest got its shape,

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and this how question gets at mechanisms.

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What are the key mechanism

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behind forest growth and forest loss?

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And if we understand these mechanisms,

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maybe we can better
understand forest stability

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and contribute to the
strength of forest ecosystems.

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And so to answer that
question, we're going to focus

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on a subset of these high resolution maps.

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We're gonna look at the Brazilian Cerrado.

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It's interesting and an
opportunity for two reason.

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One is that it's just a
very important region.

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So the Cerrado borders
the Amazon rainforest,

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which, as you may know,
is the long of the earth.

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But it also has this
really interesting feature

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that it's a mix, so
what we call an ecotone.

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When I say we, I learned that
word during that project.

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An ecotone is an interface

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between two different
ecosystems or environments,

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in this case between
rainforest and grassland,

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like savanna, like really dry grassland.

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And that's interesting to see
these two different systems

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cohabiting under the same
roof in the same region,

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because it makes you wonder
whether the Amazon rainforest

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could be destabilized
and collapse to savanna,

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which is just much more dry

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and obviously, less of
a lung for the planet,

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so that's an important question to answer.

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But also, one interesting thing
about the Brazilian Cerrado

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are all the protected areas shown in red

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where there is limited human activity,

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and so we're studying forests

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in their natural environment, if you will,

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with not too much human interference.

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So this is almost a roadmap
for what we wanna do.

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So we want to go from
reality which is here,

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represented by a picture of
the system we care about,

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and here, you see this mix of trees

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but also very, very dry grass.

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And we want to go from
this reality to the data,

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which is what's shown in panel B.

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So this is one of those high
resolution maps of tree cover

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where every pixel in this
case is about 30 meters

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by 30 meters, so not quite
like individual trees,

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but enough to get a high resolution map

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that gives us the shape
of the forest patches.

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And this is inferred tree cover,

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which is a continuous variable
between zero and 100%,

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telling us how much of
the canopy do we think

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is closed by the trees.

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But in most cases, it's
actually pretty bimodal.

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It's either green, so high tree cover

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or yellow, very low tree cover,

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so it feels natural to binarize it,

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and that's where we
already get close to a CA

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at least in terms of image in panel C.

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So this panel C is a
finalized version of panel B

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where we just apply a threshold.

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Everything that has more
than say 50% tree cover,

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we call forest, everything
that's below, we call grass,

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and then we get this binarized version

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of the data in panel B.

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And then what we want
to do is build a model

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that will grow synthetic forest,

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and that's what's represented in panel D.

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And hopefully, the statistics
are similar between C and D,

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such that we hope we've unlocked
some of the key mechanisms

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that govern forest growth
in this area and therefore,

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forest stability to help
us understand the stability

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of the Amazon rainforest.

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All right, so we wanna build our system.

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We're gonna follow our recipe,

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so the first big picture
questions about the system itself.

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What is our system?

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In this case, we care
about the forest grassland,

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ecotone, the interface
of forest and grass,

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so we're gonna ignore
water, like river and so on.

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We're gonna ignore topology, like hills.

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We're gonna ignore humans.
We're in protected areas.

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But we're also gonna ignore animals,

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which can play a big role in
shaping the growth of forests

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through seed dispersion or
destruction of young trees.

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Is the system open or closed?

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So we're gonna focus on a grid,
like in the previous image.

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So we're gonna assume that
we're gonna ignore everything

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that happens outside of that
grid, so it's a closed system.

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The topology or the space
of the system, again,

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is enforced by the data that we have.

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We wanna build a system and
then compare it to the data,

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so we're gonna use the same topology,

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which is a square grid of pixels,

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30 meter by 30 meter for each pixel.

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Is the state of the system
discrete or continuous?

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It's gonna be discrete.

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We're gonna employ a cellular automaton,

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so discrete state for
every cell in the grid.

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Is time discrete or continuous?

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This is a tricky one, and
here, I'm gonna answer both,

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and I'm gonna add a question

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that I rarely ask in those terms.

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What are the important timescales,

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because that's why I
have this weird answer

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to question number five.

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So there's a tricky thing here,

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which is that we have different processes

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that occur over a very
different timescale.

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A forest fire burns in a matter
of hours or days hopefully,

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whereas tree growth usually
happens over decades.

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And so, if you think about the
type of temporal resolution,

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either in a step of an
integrator for an ODE

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or time windows for a discrete model

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needed to fall forest fires,

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it's gonna take that discretization,

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a very long time to follow tree growth.

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If you're looking at minutes per minutes

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and following something
that happens over a decade,

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it's just gonna be a very slow model.

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So that's why we're going to be combining

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different temporal scale in the same model

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and different notions of time.

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And so now we have our basic setup,

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and we want to think about our
model in a compartmental way.

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So even if we're building a CA,

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it's useful to think of our
model in terms of parts.

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So boxes, what are the
different types of elements

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in our system or the different states

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that the cells in our
cellular automaton can take?

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In this case, we're gonna use four.

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So it might be useful for you all

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to visualize the diagram in
terms of boxes and arrows.

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As I present this, we're
gonna have four boxes.

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So cells can either be occupied by trees,

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so forest cell, or grass or fire or ashes.

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And the next step is
obviously to draw mechanisms

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or arrow between all of these states.

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So we'll start with
the final state, ashes.

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So ashes is what comes from fire,

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and what can happen to ash sites?

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So in our cellular automaton,

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we're gonna have some
sites that are just ashes

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that are assigned a specific value,

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and then what kind of
transition can they undergo?

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We're gonna say that ash
sites can turn to grass

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spontaneously at a rate lambda,

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so that's just grass being reborn

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from the ashes of the forest

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or the grassland that was there before.

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But it can also turn to trees

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by spreading mechanism at rate alpha.

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And when I say spreading
mechanism for trees

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and not for grass, what I mean is that

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regardless of whether there
is grass around or not,

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grass can come from ash,

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but trees will only spread to ash sites

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if there is a neighboring tree.

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So essentially for every ash site,

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you would look in your Moore
or von Neumann neighborhood,

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and you would ask,

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"Is there a tree there in
this neighboring cell?"

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If yes, there's a certain probability,

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alpha times time window delta T,

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that my ash site turns to tree.

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What happens to grass sites?

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Well, they can catch fire
hereby by lightning strike,

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so which will happen at
a rate f or by spreading.

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So if there is fire in a neighboring site,

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it can spread to the grass,

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and we're gonna use a probability
p subscript capital G,

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meaning the probability of
fire spreading in grass.

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If the grass sites don't catch
fire, it can turn to trees,

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again, either by seeding at rate alpha,

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so the same mechanism as before,

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but here also through spontaneous
seeding at a rate beta.

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This mechanism is meant to
represent a bird carrying seeds

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over a long distance.

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This mechanism doesn't
happen in ash sites.

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This is all expert advice.

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The idea being that to protect the seed

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and help foster the growth of the tree,

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it's good to already have
some vegetation present.

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So a lot of this is not just me

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coming up with a model out of nothing

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but is discussions with experts

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to decide what mechanisms
are at play here.

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The tree sides are what's
really interesting.

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So now we have a way to
catch fire, spread fire,

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regenerate grass and all that.

254
00:11:54,887 --> 00:11:57,630
The only thing that trees will do

255
00:11:57,630 --> 00:11:59,640
is catch fire by spreading.

256
00:11:59,640 --> 00:12:01,800
So a lightning strike
won't light up a tree.

257
00:12:01,800 --> 00:12:05,160
The idea is that this
is rainforest territory,

258
00:12:05,160 --> 00:12:07,980
so there's a lot of
humidity under the tree,

259
00:12:07,980 --> 00:12:10,260
that their canopy protects the humidity

260
00:12:10,260 --> 00:12:11,730
or traps the humidity.

261
00:12:11,730 --> 00:12:13,800
So the only way the trees
can really catch fire

262
00:12:13,800 --> 00:12:16,110
is if there's a fire
in a neighboring cell,

263
00:12:16,110 --> 00:12:18,120
and then it's spread
with some probability p

264
00:12:18,120 --> 00:12:20,400
under subscript capital T.

265
00:12:20,400 --> 00:12:23,130
And then we assume that this PT

266
00:12:23,130 --> 00:12:24,960
is much smaller than this pG,

267
00:12:24,960 --> 00:12:27,870
'cause dry grassland catches fire

268
00:12:27,870 --> 00:12:29,973
much more easily than trees.

269
00:12:31,560 --> 00:12:34,953
And then fire sites will turn
to ash with probability 1,

270
00:12:36,030 --> 00:12:39,960
just meaning that fire
lives for one time unit

271
00:12:39,960 --> 00:12:41,810
in the cellular automaton.

272
00:12:41,810 --> 00:12:43,350
So I call it the cellular automaton.

273
00:12:43,350 --> 00:12:48,350
Again, it's a stochastic one,
but it is a fixed set of rules

274
00:12:48,510 --> 00:12:53,510
and a finite number of states
that live on a discrete grid,

275
00:12:53,898 --> 00:12:57,423
and that gives us a
simple way to grow trees.

276
00:12:59,070 --> 00:13:02,640
So if we apply it with
some parameters informed

277
00:13:02,640 --> 00:13:05,820
by our friends and
experts in forest ecology,

278
00:13:05,820 --> 00:13:08,100
we might get something like panel D here.

279
00:13:08,100 --> 00:13:10,680
And here, I've drawn
everything that is not a tree,

280
00:13:10,680 --> 00:13:13,950
so whether it's ash or
or grassland in yellow

281
00:13:13,950 --> 00:13:17,763
and then trees in green to
again, represent the binarization

282
00:13:17,763 --> 00:13:19,200
that we did over the data.

283
00:13:19,200 --> 00:13:20,550
The data itself doesn't tell us

284
00:13:20,550 --> 00:13:21,800
whether it's actually grass or not.

285
00:13:21,800 --> 00:13:25,203
We just assume that everything
that was not trees was grass.

286
00:13:27,391 --> 00:13:30,780
Then we're gonna look
at a few observables.

287
00:13:30,780 --> 00:13:34,020
We now have the rules.

288
00:13:34,020 --> 00:13:37,020
If we think about our
recipes in terms of IOU,

289
00:13:37,020 --> 00:13:38,940
going backwards, we've defined the rules

290
00:13:38,940 --> 00:13:41,040
that allow us to update the system.

291
00:13:41,040 --> 00:13:43,110
The observables that
we're gonna be looking at

292
00:13:43,110 --> 00:13:47,460
is the total fraction of cells
that are covered in trees,

293
00:13:47,460 --> 00:13:49,980
but we're also gonna look at
individual forest patches,

294
00:13:49,980 --> 00:13:53,550
meaning patches of
neighboring forest cells

295
00:13:53,550 --> 00:13:55,800
that are interconnected,
and then we're gonna look

296
00:13:55,800 --> 00:13:59,340
at the area and the parameter
of these forest patches.

297
00:13:59,340 --> 00:14:03,000
So this is our O in the
IOU, what we observe.

298
00:14:03,000 --> 00:14:05,070
And typically for initialization,

299
00:14:05,070 --> 00:14:07,350
we're gonna start with
random initial values

300
00:14:07,350 --> 00:14:09,243
and let the forest evolve on its own.

301
00:14:11,070 --> 00:14:13,260
So we start with random initial values.

302
00:14:13,260 --> 00:14:16,050
We start with parameters
informed by the literature,

303
00:14:16,050 --> 00:14:19,620
but not fine tuned, and here's what we get

304
00:14:19,620 --> 00:14:21,573
in terms of comparison with the data.

305
00:14:22,920 --> 00:14:26,940
So on the left what you
have is a scatter plot

306
00:14:26,940 --> 00:14:30,270
of the area on the horizontal axis

307
00:14:30,270 --> 00:14:33,060
and meters squared for every forest patch

308
00:14:33,060 --> 00:14:34,950
and their parameter in meter.

309
00:14:34,950 --> 00:14:37,404
I apologize, the y label here is wrong,

310
00:14:37,404 --> 00:14:41,790
parameter in meters for
the same forest patch.

311
00:14:41,790 --> 00:14:45,420
You have six sets of data
points on the scattered plot.

312
00:14:45,420 --> 00:14:50,160
So in circles, you have
three subsets of the data,

313
00:14:50,160 --> 00:14:51,990
depending on whether
they're in the bottom third

314
00:14:51,990 --> 00:14:54,840
in terms of yearly precipitation,

315
00:14:54,840 --> 00:14:57,783
medium third or highest third.

316
00:14:59,460 --> 00:15:01,950
And you have three sets of simulation

317
00:15:01,950 --> 00:15:04,470
with somewhat different parameters,

318
00:15:04,470 --> 00:15:07,800
but none of them were
particularly selected

319
00:15:07,800 --> 00:15:09,930
or fine tuned, I should say.

320
00:15:09,930 --> 00:15:11,340
Well what's crazy about this plot

321
00:15:11,340 --> 00:15:14,340
is that there is very little scatter.

322
00:15:14,340 --> 00:15:18,450
Most of our simulations
and most of the actual data

323
00:15:18,450 --> 00:15:22,500
fall very steadily along the same axis,

324
00:15:22,500 --> 00:15:25,650
meaning that there is, because
we're in a log plot here,

325
00:15:25,650 --> 00:15:27,270
there's a power log relationship

326
00:15:27,270 --> 00:15:31,023
between the perimeter of a
forest patch and its area.

327
00:15:31,860 --> 00:15:34,530
And it's interesting to see
that the shape of the forest

328
00:15:34,530 --> 00:15:36,810
doesn't seem to depend on our parameters,

329
00:15:36,810 --> 00:15:38,160
the same way that it doesn't seem

330
00:15:38,160 --> 00:15:40,920
to depend too much on
precipitation either.

331
00:15:40,920 --> 00:15:42,810
Really, what might be important

332
00:15:42,810 --> 00:15:46,440
are the mechanisms underlying
the growth of these forests,

333
00:15:46,440 --> 00:15:47,823
not the exact parameters.

334
00:15:48,990 --> 00:15:52,470
On the right, we show the
probability distribution function

335
00:15:52,470 --> 00:15:56,040
for one of these dimensions,
so for forest patch area.

336
00:15:56,040 --> 00:15:59,423
And here again, you see
a parallel distribution

337
00:15:59,423 --> 00:16:02,802
or roughly parallel distribution,

338
00:16:02,802 --> 00:16:06,960
because we have this very straight decay

339
00:16:06,960 --> 00:16:09,483
of forest patch area as they grow bigger.

340
00:16:10,650 --> 00:16:12,630
And again, very similar values

341
00:16:12,630 --> 00:16:15,690
regardless of precipitation
value or simulation.

342
00:16:15,690 --> 00:16:18,240
I don't want to go too
much into the detail here,

343
00:16:18,240 --> 00:16:21,600
but the crazy thing is the fact

344
00:16:21,600 --> 00:16:23,160
that we get this similar outcome

345
00:16:23,160 --> 00:16:25,173
regardless of the parameters we put in.

346
00:16:26,708 --> 00:16:28,830
Then we can look at our global observable,

347
00:16:28,830 --> 00:16:32,133
the total fraction of
cells covered in trees.

348
00:16:33,330 --> 00:16:37,170
I showed two different
set of parameters here.

349
00:16:37,170 --> 00:16:40,800
So in both panels on the horizontal axis,

350
00:16:40,800 --> 00:16:44,070
what I'm varying is the
probability of fire,

351
00:16:44,070 --> 00:16:46,170
which was this f rate in my rules.

352
00:16:46,170 --> 00:16:49,233
So that's how often
grass sites catch fire.

353
00:16:50,640 --> 00:16:53,370
Notice that we have very
different scales here.

354
00:16:53,370 --> 00:16:58,370
On the left, it's very rare fire strikes,

355
00:16:58,530 --> 00:17:00,870
and on the right, is very
frequent fire strikes.

356
00:17:00,870 --> 00:17:03,220
They differ by about 10 to minus six

357
00:17:04,680 --> 00:17:07,290
or a factor of a million.

358
00:17:07,290 --> 00:17:08,850
And then on the horizontal axis,

359
00:17:08,850 --> 00:17:12,210
we look at the steady
state value of tree cover,

360
00:17:12,210 --> 00:17:14,823
so what fraction of the
system is covered in trees.

361
00:17:17,250 --> 00:17:19,830
If we start with the right panel, panel B,

362
00:17:19,830 --> 00:17:23,880
then we know what to expect
if there's no fire at all,

363
00:17:23,880 --> 00:17:27,241
because I said the only thing
that can happen to trees

364
00:17:27,241 --> 00:17:32,241
is to catch fire if a neighboring
cell is currently on fire.

365
00:17:33,930 --> 00:17:37,353
Well, that means that if trees
only have trees around them,

366
00:17:38,280 --> 00:17:40,200
they're not ever gonna change state,

367
00:17:40,200 --> 00:17:42,690
and if the entire system
is covered in trees,

368
00:17:42,690 --> 00:17:45,390
nothing's gonna happen,
that's a fixed point.

369
00:17:45,390 --> 00:17:47,730
So I know that for Pf equals zero,

370
00:17:47,730 --> 00:17:51,330
I have a fixed point at historical zero.

371
00:17:51,330 --> 00:17:54,120
And then as I start
having more and more fire

372
00:17:54,120 --> 00:17:56,043
during the evolution of my system,

373
00:17:57,060 --> 00:18:01,530
eventually I destabilize
my fully-forested state,

374
00:18:01,530 --> 00:18:06,530
T equal 1 and then my steady
state or equilibrium value

375
00:18:06,810 --> 00:18:08,823
of tree coverage starts falling,

376
00:18:09,750 --> 00:18:11,490
and of course, that still varies in time.

377
00:18:11,490 --> 00:18:14,700
So here I've colored one
of the data point in green,

378
00:18:14,700 --> 00:18:16,080
and that's what you see in the inset

379
00:18:16,080 --> 00:18:18,200
is the time series for that data point.

380
00:18:18,200 --> 00:18:19,650
So you still have fluctuation,

381
00:18:19,650 --> 00:18:21,720
because it is a stochastic process.

382
00:18:21,720 --> 00:18:24,870
You still have fluctuations
of the tree cover,

383
00:18:24,870 --> 00:18:26,910
the fraction of cells covered in trees,

384
00:18:26,910 --> 00:18:31,910
but there is steady values,
in this case around 85% or so.

385
00:18:33,960 --> 00:18:36,690
If you go on the left, you
get a very different picture.

386
00:18:36,690 --> 00:18:37,980
The one on the right, by the way,

387
00:18:37,980 --> 00:18:41,490
like looks a little bit
like epidemic transitions

388
00:18:41,490 --> 00:18:44,310
that we saw in the SSIS model or SIR model

389
00:18:44,310 --> 00:18:46,620
if we were looking at
the steady state value

390
00:18:46,620 --> 00:18:48,450
of susceptible individuals.

391
00:18:48,450 --> 00:18:51,630
If the disease or the fire is very weak,

392
00:18:51,630 --> 00:18:53,460
everyone's susceptible, everyone's happy.

393
00:18:53,460 --> 00:18:55,710
If the disease is strong
enough, it starts spreading,

394
00:18:55,710 --> 00:18:57,783
and you get this bifurcation.

395
00:18:58,860 --> 00:19:01,170
On the left, we have a
very different picture.

396
00:19:01,170 --> 00:19:04,800
So we still have that for very weak fires,

397
00:19:04,800 --> 00:19:05,910
so Pf equals zero,

398
00:19:05,910 --> 00:19:09,483
you expect a steady state
at a fully forested system.

399
00:19:10,641 --> 00:19:14,633
But what happens here is that
it actually seems to decay.

400
00:19:18,810 --> 00:19:19,770
And when I say decay,

401
00:19:19,770 --> 00:19:22,560
I mean that the forests get destabilized

402
00:19:22,560 --> 00:19:25,770
as Pf increases in a discontinuous way.

403
00:19:25,770 --> 00:19:29,850
So around Pf equals four
times 10 to minus six,

404
00:19:29,850 --> 00:19:31,290
you get this sudden jump,

405
00:19:31,290 --> 00:19:34,050
where instead of expecting
a fully forested state,

406
00:19:34,050 --> 00:19:36,690
you might expect only a few percent

407
00:19:36,690 --> 00:19:39,780
of your system to be covered in trees,

408
00:19:39,780 --> 00:19:42,153
and that's already very different.

409
00:19:43,170 --> 00:19:46,890
And there's a weird shading
effect on my markers here,

410
00:19:46,890 --> 00:19:49,530
because what the shading
represents, the alpha value,

411
00:19:49,530 --> 00:19:53,100
the transparency of my markers
represent the probability

412
00:19:53,100 --> 00:19:56,130
that I end up either on
top or at the bottom.

413
00:19:56,130 --> 00:19:58,050
So if my marker is very weak,

414
00:19:58,050 --> 00:20:00,120
that means there's a very low probability

415
00:20:00,120 --> 00:20:02,490
that my system settles at a steady state

416
00:20:02,490 --> 00:20:05,823
of about 3% or 5% of system size.

417
00:20:07,260 --> 00:20:11,640
And I'm talking about roughly
Pf equals four times 10

418
00:20:11,640 --> 00:20:15,960
to minus six and there is
therefore very high probability

419
00:20:15,960 --> 00:20:18,030
of ending up in the fully forested state.

420
00:20:18,030 --> 00:20:20,160
Pf increases the probability switch

421
00:20:20,160 --> 00:20:22,627
and now I'm much more likely
to be in the lower branch

422
00:20:22,627 --> 00:20:25,530
by the time I reach an equilibrium.

423
00:20:25,530 --> 00:20:26,610
What's interesting is that

424
00:20:26,610 --> 00:20:29,730
if we look at a typical time
series again in the inset,

425
00:20:29,730 --> 00:20:32,310
we now see very large fluctuations,

426
00:20:32,310 --> 00:20:34,980
so fluctuations of orders of magnitude

427
00:20:34,980 --> 00:20:37,560
more than the actual equilibrium value.

428
00:20:37,560 --> 00:20:41,220
And that's because sometimes,
stochastically, forests grow

429
00:20:41,220 --> 00:20:44,433
until there's a fire big enough
to shrink them back again.

430
00:20:48,360 --> 00:20:49,620
So we wanna understand this.

431
00:20:49,620 --> 00:20:54,620
It's very hard to study
bifurcation diagrams like this

432
00:20:54,690 --> 00:20:56,100
and stochastic processes

433
00:20:56,100 --> 00:20:58,290
and systems within an absorbing state.

434
00:20:58,290 --> 00:21:01,830
So as soon as our system
is fully covered in trees,

435
00:21:01,830 --> 00:21:03,510
it's frozen there forever,

436
00:21:03,510 --> 00:21:06,833
it's hard to study the fluctuations,

437
00:21:06,833 --> 00:21:11,820
just a lot of computational
firepower is needed.

438
00:21:11,820 --> 00:21:13,177
So we can turn back and say,

439
00:21:13,177 --> 00:21:16,447
"Well, what if we had built
differential equations

440
00:21:16,447 --> 00:21:17,977
"from our boxes and arrows

441
00:21:17,977 --> 00:21:20,070
"this same way we did in module one?"

442
00:21:20,070 --> 00:21:21,510
Of course now, we're not gonna be able

443
00:21:21,510 --> 00:21:25,170
to describe the shape of
forests, but we might be able

444
00:21:25,170 --> 00:21:28,953
to better understand the
stability of our fixed points.

445
00:21:29,790 --> 00:21:32,400
So instead of looking
at stochastic dynamics,

446
00:21:32,400 --> 00:21:34,620
we're gonna fall back on
mean field descriptions.

447
00:21:34,620 --> 00:21:37,440
Again, mean field means here
we're gonna ignore space

448
00:21:37,440 --> 00:21:40,170
and simply look or assume

449
00:21:40,170 --> 00:21:41,850
that everything is potentially connected,

450
00:21:41,850 --> 00:21:44,040
meaning that a cell is
defined by its state

451
00:21:44,040 --> 00:21:46,680
but then connected to the average

452
00:21:46,680 --> 00:21:49,173
or expected value of all neighbors.

453
00:21:50,460 --> 00:21:53,070
So let me describe just
one question in detail,

454
00:21:53,070 --> 00:21:57,480
and here, I'm switching some
of the rates a little bit,

455
00:21:57,480 --> 00:22:02,400
because it's almost impossible
to get a one-to-one matching

456
00:22:02,400 --> 00:22:06,480
between parameters in a
stochastic and discrete simulation

457
00:22:06,480 --> 00:22:08,880
and parameters in a
mean field description,

458
00:22:08,880 --> 00:22:10,620
so some of them are gonna
be a little different,

459
00:22:10,620 --> 00:22:11,453
and you'll see.

460
00:22:12,540 --> 00:22:15,090
So I'm just gonna describe this
first differential equation,

461
00:22:15,090 --> 00:22:17,700
which is B dot, so the time derivative

462
00:22:17,700 --> 00:22:19,803
for the number of burning trees.

463
00:22:21,108 --> 00:22:23,760
So the number of burning sites,

464
00:22:23,760 --> 00:22:25,950
so that could be burning
trees or burning grass,

465
00:22:25,950 --> 00:22:29,073
so burning sites in my
lattice, in my grid.

466
00:22:30,150 --> 00:22:32,760
So every grass site, G,

467
00:22:32,760 --> 00:22:36,720
can catch fire with
probability f, or with rate f.

468
00:22:36,720 --> 00:22:39,690
So I'm gonna increase of f times G

469
00:22:39,690 --> 00:22:41,853
in the number of burning sites.

470
00:22:43,380 --> 00:22:46,383
That's one arrow flowing from G to B.

471
00:22:47,460 --> 00:22:51,060
I also have another arrow
flowing from G to B,

472
00:22:51,060 --> 00:22:53,400
which is the probability that grass sites

473
00:22:53,400 --> 00:22:55,530
have a neighboring site
that is currently burning

474
00:22:55,530 --> 00:22:57,660
and that the fire spread.

475
00:22:57,660 --> 00:22:59,700
Why is there a factor of four here?

476
00:22:59,700 --> 00:23:03,210
Well, let's take the perspective
of a single grass site.

477
00:23:03,210 --> 00:23:07,530
So for every grass site
here, we have four neighbors

478
00:23:07,530 --> 00:23:09,960
that we're specifying
our neighborhood here.

479
00:23:09,960 --> 00:23:11,820
So we have four neighbors,

480
00:23:11,820 --> 00:23:14,520
which will be burning with probability B,

481
00:23:14,520 --> 00:23:16,380
because there is no structure,

482
00:23:16,380 --> 00:23:18,393
so every neighbor is a random neighbor.

483
00:23:19,890 --> 00:23:22,380
And for every of those potential

484
00:23:22,380 --> 00:23:24,090
four times B burning neighbors,

485
00:23:24,090 --> 00:23:27,900
the fire will spread with
probability row underscore G.

486
00:23:27,900 --> 00:23:30,690
And here we use row, because
it's not a probability,

487
00:23:30,690 --> 00:23:31,590
I just misspoke.

488
00:23:31,590 --> 00:23:34,980
It's not a p underscore G as
in the cellular automaton.

489
00:23:34,980 --> 00:23:37,230
This is an actual rate,
so we're using row,

490
00:23:37,230 --> 00:23:40,683
because it is a rate per
time unit, not a probability.

491
00:23:41,970 --> 00:23:45,210
Then we have a very similar
terms for trees catching fire.

492
00:23:45,210 --> 00:23:48,630
So for every tree site
T, we have four neighbors

493
00:23:48,630 --> 00:23:51,570
that are currently burning
with probability B,

494
00:23:51,570 --> 00:23:54,900
therefore an average four
times B burning neighbors,

495
00:23:54,900 --> 00:23:59,900
which spread the fire with
rate row subscript, capital T.

496
00:24:00,750 --> 00:24:05,750
And then the only arrow
leaving the burning box

497
00:24:06,000 --> 00:24:11,000
is the arrow leading
from burning B to ash A,

498
00:24:11,040 --> 00:24:14,673
and that happens with a rate
here that we're calling mu,

499
00:24:16,020 --> 00:24:17,730
which was previously one,

500
00:24:17,730 --> 00:24:20,013
but here we have to put
an actual value on it.

501
00:24:20,013 --> 00:24:21,810
It's just a very fast rate

502
00:24:21,810 --> 00:24:23,823
at which local fires stop burning.

503
00:24:25,260 --> 00:24:27,000
And you can go through the same process

504
00:24:27,000 --> 00:24:32,000
and understand every term
in the growth of the system.

505
00:24:33,420 --> 00:24:35,640
You'll notice that here in the grass term,

506
00:24:35,640 --> 00:24:39,270
we're using exponential growth
and not logistic growth.

507
00:24:39,270 --> 00:24:41,760
That's a choice, assuming
that at any given time,

508
00:24:41,760 --> 00:24:44,283
there won't be a lot of
ash present in the system.

509
00:24:47,520 --> 00:24:48,660
And then, we can solve

510
00:24:48,660 --> 00:24:51,510
for the bifurcation diagram of the system.

511
00:24:51,510 --> 00:24:54,000
I'll focus on panel A for now.

512
00:24:54,000 --> 00:24:56,490
So we have a lot of
parameters, so it's not as easy

513
00:24:56,490 --> 00:25:00,990
to analyze as the SSIS model, for example.

514
00:25:00,990 --> 00:25:05,010
We're gonna be looking at
panel A at the 3-D plot,

515
00:25:05,010 --> 00:25:10,010
so the steady state or the
fixed value of forest coverage

516
00:25:11,640 --> 00:25:16,293
as a function of fire rate on the x axis.

517
00:25:17,190 --> 00:25:20,730
On the depth axis, so let's call it y,

518
00:25:20,730 --> 00:25:23,910
we're gonna be looking at the
growth rate of trees beta,

519
00:25:23,910 --> 00:25:26,910
which if you remember, is the
seeding mechanism for trees.

520
00:25:26,910 --> 00:25:29,580
So this is how much do birds

521
00:25:29,580 --> 00:25:34,170
from outside of our
system carry seeds inside?

522
00:25:34,170 --> 00:25:36,780
And that's why I said that we
sort of capture an open system

523
00:25:36,780 --> 00:25:38,610
but not really, 'cause
we don't model the birds

524
00:25:38,610 --> 00:25:41,366
or what's out there, we
just have this parameter

525
00:25:41,366 --> 00:25:42,753
that accounts for that.

526
00:25:44,010 --> 00:25:45,930
And so what we see here,

527
00:25:45,930 --> 00:25:48,813
the blue lines are the usual fixed points.

528
00:25:50,057 --> 00:25:54,090
So, we looked at that
in terms of SSIS before.

529
00:25:54,090 --> 00:25:55,920
It's the the expected fixed point

530
00:25:55,920 --> 00:25:58,830
for a given value of f and beta.

531
00:25:58,830 --> 00:26:00,090
What happens in red

532
00:26:00,090 --> 00:26:02,883
is that sometimes those
fixed points are unstable.

533
00:26:04,020 --> 00:26:08,910
So what connects the fully
forested state T equal one

534
00:26:08,910 --> 00:26:11,940
to the lower branch of stable values

535
00:26:11,940 --> 00:26:14,553
is a branch of unstable fixed points.

536
00:26:16,380 --> 00:26:18,450
We've seen unstable fixed points

537
00:26:18,450 --> 00:26:20,460
when looking at flow diagram before,

538
00:26:20,460 --> 00:26:22,200
so that's for a given set of parameters.

539
00:26:22,200 --> 00:26:26,040
Here, we're changing the
parameters along the x and Y axis,

540
00:26:26,040 --> 00:26:27,720
and therefore, our fixed point is moving

541
00:26:27,720 --> 00:26:29,313
and tracing this red line.

542
00:26:30,360 --> 00:26:31,560
When it becomes wide here,

543
00:26:31,560 --> 00:26:36,000
it's simply because the red
line goes into a physical value

544
00:26:36,000 --> 00:26:39,840
for fire rate negative
and stuff like that,

545
00:26:39,840 --> 00:26:42,030
so it's just not a
region of parameter space

546
00:26:42,030 --> 00:26:44,640
that we wanna care about, so
we're just not plotting it.

547
00:26:44,640 --> 00:26:47,540
But, they're still connected,
and they're still out there.

548
00:26:48,480 --> 00:26:49,490
Well, what's interesting here

549
00:26:49,490 --> 00:26:52,500
is that if we have a lot
of spontaneous tree growth,

550
00:26:52,500 --> 00:26:54,060
which again, could be birds,

551
00:26:54,060 --> 00:26:56,670
it could also be me going
out and planting trees,

552
00:26:56,670 --> 00:26:58,890
we're stabilizing the system.

553
00:26:58,890 --> 00:27:01,080
We're helping not only the lower branch

554
00:27:01,080 --> 00:27:05,280
of stable values reach higher
levels of forest coverage,

555
00:27:05,280 --> 00:27:10,280
but we're getting rid of this
discontinuous instability,

556
00:27:11,820 --> 00:27:12,720
and that's good news,

557
00:27:12,720 --> 00:27:14,940
because these discontinuous instability

558
00:27:14,940 --> 00:27:16,530
are what we're really scared about,

559
00:27:16,530 --> 00:27:20,850
is for the Amazon rainforest

560
00:27:20,850 --> 00:27:23,070
to suddenly go from a
highly forested state

561
00:27:23,070 --> 00:27:26,280
to a barely forested state, a
sudden discontinuous collapse.

562
00:27:26,280 --> 00:27:29,010
What we want is a stabilized system.

563
00:27:29,010 --> 00:27:31,500
So one key mechanism already
that we could suggest

564
00:27:31,500 --> 00:27:32,910
is spontaneous tree growth,

565
00:27:32,910 --> 00:27:35,670
either by replanting in grassland

566
00:27:35,670 --> 00:27:38,520
or by increasing
biodiversity in the system.

567
00:27:38,520 --> 00:27:40,443
All of these would be a good option.

568
00:27:42,120 --> 00:27:45,870
We can also look at different
ways in terms of network.

569
00:27:45,870 --> 00:27:47,970
We can expect a disconnected transition

570
00:27:47,970 --> 00:27:51,570
or continuous transition, some
cases, no transition at all,

571
00:27:51,570 --> 00:27:54,600
meaning that if trees grow quickly enough,

572
00:27:54,600 --> 00:27:59,130
there's no fire strong enough
to destabilize the forest

573
00:27:59,130 --> 00:28:00,690
and then we can solve for all that.

574
00:28:00,690 --> 00:28:02,190
I won't go too much into detail.

575
00:28:02,190 --> 00:28:03,420
I'll give you the reference,

576
00:28:03,420 --> 00:28:07,830
but this is a good example of
chapter eight on bifurcation.

577
00:28:07,830 --> 00:28:10,030
There's a lot of rich
dynamics here at play.

578
00:28:10,920 --> 00:28:12,780
The reason I wanted to
talk about this project

579
00:28:12,780 --> 00:28:15,600
is that we can also make
interesting predictions.

580
00:28:15,600 --> 00:28:17,820
Once we believe that
we've captured something

581
00:28:17,820 --> 00:28:21,150
about the growth of
forests in this system,

582
00:28:21,150 --> 00:28:23,973
we can make predictions
about individual forests.

583
00:28:24,930 --> 00:28:27,450
So this is a cool experiment that we did.

584
00:28:27,450 --> 00:28:30,360
So we take our realistic
set of parameters,

585
00:28:30,360 --> 00:28:32,760
we run our cellular automaton,

586
00:28:32,760 --> 00:28:35,280
and at some point we freeze the system,

587
00:28:35,280 --> 00:28:37,560
and then we go in the system with scalpel,

588
00:28:37,560 --> 00:28:41,160
and we find a patch of
grassland somewhere,

589
00:28:41,160 --> 00:28:42,930
and in that patch of grassland,

590
00:28:42,930 --> 00:28:46,620
we inject a forest with
a very specific shape.

591
00:28:46,620 --> 00:28:48,840
So we try all possible shapes,

592
00:28:48,840 --> 00:28:53,840
meaning that on the x axis we have area

593
00:28:54,060 --> 00:28:55,890
in terms of number of cells.

594
00:28:55,890 --> 00:28:59,460
So if you have a forest that
covers an area of 100 cells,

595
00:28:59,460 --> 00:29:04,460
it could be a line with a
parameter of 100 on one side,

596
00:29:05,040 --> 00:29:07,560
100 on the other, and two on both ends,

597
00:29:07,560 --> 00:29:10,740
so a parameter of 202,
or it could be a square

598
00:29:10,740 --> 00:29:12,390
that's 10 cells by 10 cells,

599
00:29:12,390 --> 00:29:14,371
you can get very different shapes,

600
00:29:14,371 --> 00:29:16,920
and we're gonna try them all essentially

601
00:29:16,920 --> 00:29:19,353
over this little figure.

602
00:29:21,003 --> 00:29:22,980
The black lines in these figures

603
00:29:22,980 --> 00:29:24,930
are showing what shapes are possible.

604
00:29:24,930 --> 00:29:27,360
So you can't be any more compact,

605
00:29:27,360 --> 00:29:29,580
meaning you can't have a parameter lower

606
00:29:29,580 --> 00:29:31,230
than what you would have as a square.

607
00:29:31,230 --> 00:29:35,220
So that gives us this
square root relationship

608
00:29:35,220 --> 00:29:38,940
between the perimeter of the
forest patch and its area.

609
00:29:38,940 --> 00:29:41,490
And you can't have a perimeter higher

610
00:29:41,490 --> 00:29:43,200
than if you were a line.

611
00:29:43,200 --> 00:29:46,230
And so that gives us
this linear relationship,

612
00:29:46,230 --> 00:29:48,750
which is the top black
line that is steeper,

613
00:29:48,750 --> 00:29:50,940
so the top black line as a slope of one,

614
00:29:50,940 --> 00:29:54,150
the bottom one is slope of
one half, a square root,

615
00:29:54,150 --> 00:29:56,280
and all possible shapes exist

616
00:29:56,280 --> 00:30:00,540
in between these two
lines, so in this cone.

617
00:30:00,540 --> 00:30:04,080
In the red, you have the stable prediction

618
00:30:04,080 --> 00:30:05,310
that we make from our model,

619
00:30:05,310 --> 00:30:08,610
which is an exponent of
about three quarters,

620
00:30:08,610 --> 00:30:09,690
so meaning a perimeter

621
00:30:09,690 --> 00:30:12,903
that's about three quarters
the area of the forest patch.

622
00:30:14,100 --> 00:30:16,200
Then we try a bunch of individual patches,

623
00:30:16,200 --> 00:30:19,350
we see how they grow,
how does their area grow,

624
00:30:19,350 --> 00:30:20,670
how does their perimeter grow,

625
00:30:20,670 --> 00:30:22,800
and we average it over
thousands and thousands

626
00:30:22,800 --> 00:30:25,260
of realization of the same forest patch.

627
00:30:25,260 --> 00:30:27,510
So here, are probably
billions and billions

628
00:30:27,510 --> 00:30:30,090
of simulations of individual forest patch

629
00:30:30,090 --> 00:30:32,640
within a larger ecosystem.

630
00:30:32,640 --> 00:30:36,090
And the orange arrows are
showing you the expected growth

631
00:30:36,090 --> 00:30:37,500
or expected dynamics,

632
00:30:37,500 --> 00:30:39,390
because sometimes it's not growth at all,

633
00:30:39,390 --> 00:30:41,580
the expected dynamics of the forest patch.

634
00:30:41,580 --> 00:30:44,040
So if you start below the red line,

635
00:30:44,040 --> 00:30:48,210
you get forest patches that
grow a tiny bit in area.

636
00:30:48,210 --> 00:30:51,570
But remember, growing an
area is a slow process

637
00:30:51,570 --> 00:30:53,700
and they grow a lot in perimeter.

638
00:30:53,700 --> 00:30:56,070
Growing in perimeter means
that you're facing fire,

639
00:30:56,070 --> 00:30:58,410
and the fire is chipping at you,

640
00:30:58,410 --> 00:31:00,849
so you go from a square to
something with a (indistinct)

641
00:31:00,849 --> 00:31:03,213
and a higher perimeter,
so that's what happens.

642
00:31:04,290 --> 00:31:07,950
And as they do so, the
forests grow in perimeter,

643
00:31:07,950 --> 00:31:10,020
but the arrows get smaller and smaller.

644
00:31:10,020 --> 00:31:12,720
So around the red line,
there's actually an area

645
00:31:12,720 --> 00:31:15,150
where forests barely follow any dynamics.

646
00:31:15,150 --> 00:31:18,660
They're very stable. They grow
as much as they get burned.

647
00:31:18,660 --> 00:31:20,220
But if they burn too much,

648
00:31:20,220 --> 00:31:22,050
which will always happen at some point,

649
00:31:22,050 --> 00:31:25,680
because it's a stochastic
process, if they do burn too much,

650
00:31:25,680 --> 00:31:29,580
then they enter in this, I like
to call it, it's like river,

651
00:31:29,580 --> 00:31:34,580
but this flow of rapid decrease
in both area and perimeter

652
00:31:35,160 --> 00:31:36,420
that happens at the top.

653
00:31:36,420 --> 00:31:38,520
So once you're too linear, what happens,

654
00:31:38,520 --> 00:31:40,650
that fire can fragment forest patches

655
00:31:40,650 --> 00:31:43,830
and just make them collapse
really, really quickly.

656
00:31:43,830 --> 00:31:46,080
So stochastically, you'd expect forests

657
00:31:46,080 --> 00:31:47,980
to slowly go towards the right

658
00:31:49,710 --> 00:31:51,990
and up following the red line,

659
00:31:51,990 --> 00:31:54,060
but if they're unlucky
and there's a lot of fire,

660
00:31:54,060 --> 00:31:55,380
then they can suddenly collapse.

661
00:31:55,380 --> 00:31:57,480
That's the lifecycle of a forest patch.

662
00:31:57,480 --> 00:31:58,950
What's cool about this is that

663
00:31:58,950 --> 00:32:02,850
if you were to go cut down some trees,

664
00:32:02,850 --> 00:32:05,550
deforestation, part of an industry,

665
00:32:05,550 --> 00:32:07,230
well at least this gives you a map

666
00:32:07,230 --> 00:32:10,500
of what shape you should
leave the final patch in.

667
00:32:10,500 --> 00:32:13,320
Or if you're tempted to
leave the forest patch

668
00:32:13,320 --> 00:32:15,240
in a weird shape because it's convenient,

669
00:32:15,240 --> 00:32:17,760
this can give you an idea of the risk

670
00:32:17,760 --> 00:32:20,523
that you're putting this forest patch in.

671
00:32:22,095 --> 00:32:25,710
Here's a reference, the
paper I wanna point out.

672
00:32:25,710 --> 00:32:30,710
This was a great
multidisciplinary collaboration

673
00:32:30,810 --> 00:32:33,813
between physicists and
ecologists and more.

674
00:32:35,256 --> 00:32:38,820
I want to point out, especially the team

675
00:32:38,820 --> 00:32:40,530
that was with me at
the Santa Fe Institute,

676
00:32:40,530 --> 00:32:43,320
so Andrew M. Berdahl, we're
gonna hear more about him

677
00:32:43,320 --> 00:32:46,260
who does a lot of
collective behavior ecology.

678
00:32:46,260 --> 00:32:49,680
Sidney Redner, I already
recommended Sid's book,

679
00:32:49,680 --> 00:32:54,680
on "A Guide to First Passages Processes,"

680
00:32:55,182 --> 00:32:59,760
one classic if you're
interested in random walks

681
00:32:59,760 --> 00:33:01,395
and processes like this.

682
00:33:01,395 --> 00:33:04,560
Uttam Bhat, which was with
us at the Santa Fe Institute

683
00:33:04,560 --> 00:33:06,390
and then Adam Pellegrini
and Stephen Pacala,

684
00:33:06,390 --> 00:33:10,618
who were at Princeton
at the time in ecology.

685
00:33:10,618 --> 00:33:15,618
So, there are a lot of
directions this could take,

686
00:33:15,630 --> 00:33:16,831
like once you have a good model,

687
00:33:16,831 --> 00:33:18,300
there are a lot of cool questions to ask.

688
00:33:18,300 --> 00:33:20,010
We could think about comparing data

689
00:33:20,010 --> 00:33:22,170
within all those protected
areas that we looked at

690
00:33:22,170 --> 00:33:24,540
and outside to see what is the impact

691
00:33:24,540 --> 00:33:26,253
on the shape of forests of human-