cc >> Okay well welcome to the
Chaos and Complex Systems Seminar.
The talk I'll be giving today is on the topic I became interested
in exactly five weeks ago to the day.
Before that, I knew absolutely nothing about what I'm going to
be talking about. But this is not unlike a
physicist, to jump into a field about which he knows little and
proceeds to try to make some progress.
I became interested in this when I was asked to review a paper
on mathematical models of the ecology of Easter Island.
Now I hadn't paid it much attention until that point but I
figured I'd better learn a little bit about it.
The more I learned about it, the more fascinated I became about
Easter Island and about some of the mathematical models that
have been applied to the population dynamics.
Easter Island is one of the most remote spots in the world.
It's located in the South Pacific, 27 degrees South of the
equator, subtropical. It's 3,500 kilometers from the
coast of South America and it's 2,000 kilometers from the
nearest inhabited island to the East-- sorry, to the West.
And, so, it's often been described as the most remote
inhabited spot in the world. It's a relatively small island,
triangular shape. It has just about the same area
as the city of Madison, about 170 square kilometers,
triangular. It's dominated by three extinct
volcanoes, the largest and tallest of which is about 500
meters above sea level. Easter Island is perhaps most
famous for the statues that were built somewhere around, as long
ago as 1,000 years ago before it was discovered by Westerners.
And here some of the locations for these statues, there's about
900 of them that were built and constructed out of the volcanic
lava that exists in abundance on the island.
It's a popular tourist destination.
There's an airport runway here that planes land on every day
and bring plane loads of tourists to visit.
I'd like to say I've been there and this is a picture I took but
it's not. I grabbed it off the web but you
see almost the entirety of the island here, presumably from the
plane which is about to land on the runway there, or maybe
taking off. You see it's relatively
unvegetated, there are few trees, there's agriculture that
goes on there and there's a lot of sheep herding, raising that
is done there. But primarily these days it's
mainly just a tourist mecca. Most of the population, which
amounts to about 5,000 people now, serves the tourist
industry. These are some of the statues
that, as I said there are some 900 of around the island, that
were built before Westerners visited the island and after
which no more were built and in fact they came into a state of
some disrepair by the time the island was first discovered by
Westerners. We know from pollen records that
long ago the island was densely vegetated with a type of palm
tree that is similar to the Chilean palm tree that I've
indicated here. As you saw from the picture,
however, there are essentially none of these left.
There are some trees but not of the variety that once dominated
the island. The conventional wisdom is that
when the people came, they cut down all the trees and
thereafter they had no way to make fires, to build houses, to
build boats to fish from and they basically used up their
resources and the population decayed to a relatively small
value. The history, there's a lot of
uncertainty when the first people came to Eastern Island.
I've seen numbers actually as early as 300 A.D., but more
modern texts give later dates, some as late as 1200 A.D.
The first inhabitants arrived presumably on boats from the
Polynesian islands to the West. Although Thor Heyerdahl became
famous for the Kon-Tiki expedition, showing that
in principle, it would have been possible for them to have come
from Peru and there is some evidence that their culture
had some connection with the South American culture
in the early days. But largely that's discredited
and most people now think that the inhabitants that came that
2,500 miles, 2,000 miles from the closest Polynesian island
which is quite a feat especially since they presumably didn't
know there was an island there. (LAUGHTER)
I find that quite remarkable that as early as that,
they would've done that but apparently they did.
The island was first visited by the Europeans in the year 1722
when a Dutch navigator Roggevee came.
He estimated the population to be about 3,000 at that time.
It's now thought that the population peaked at something
at the vicinity of 10,000. I've seen numbers as high as
17,000, maybe 7,000. Not very well known but 100 to
200 years prior to Roggevee being there, the island was
largely devegetated at this point and the population was
clearly in decline. The statues were there but there
was no evidence that there was any building going on.
About 50 years lapsed until the next visitors came to the
island. Captain Cook visited in the
1770s and the population had decayed further, the statues
were largely torn down, and in considerable disrepair.
Then there were a number of foreign visitors up through the
1860s, at which point there was slave trading, the Catholic
missionaries arrived about this time and the population reached
as low as 110. >> Were they enslaving the
Eastern Islanders? >> Sprott: Yes.
Yes, that's right. The Peruvians were taking the
Eastern Island natives to be slaves.
In 1888, it was annexed by Chile and then the population slowly
began to grow, and the latest census is just about 5,000.
So the interesting thing is what happened here, from presumably a
very small initial population up to something like 10,000, down
to 110, and then slowly recovering.
Are the 5,000 people now the direct descendants of the 110
people or are they outsiders who moved in from Peru or Chile or
wherever? >> Sprott: Well, I'm sure
they're some of each. And I'm not prepared to say what
proportion. I think there certainly are
descendants of the 110 without question.
But what fraction, I don't know. Now, in trying to understand the
population of Eastern Island, I want to emphasize that what I'm
going to attempt to do today is not so much a talk on the
ecology, I'm not going to really provide any new insight into
what happened on Eastern Island beyond what I've already told
you. But rather what I'm going to do
is try and show you some mathematical models that can
predict the kind of population dynamics that was observed and
for any of you who are biologists in the audience,
you'll be the first to realize that the models are vastly
oversimplified. And this is of course is what we
do in physics and what mathematicians like to do.
They like to simplify things and construct the simplest model
that somehow captures the relevant dynamics of the system.
And that's the spirit in which I am hoping you will accept this
talk, not as a treatise on what happened on Easter Island but
rather as an example of how one would perceive to model a
situation like this and some of the ways in which these models
would behave. And so Einstein said it quite
well, "Things should be explained as simple as possible,
but not more simply." So, the idea is to make the
models as simple as you can, but to somehow capture the essence
of what it is you're trying to explain.
And these models do that. Well, here's another quote by a
local Professor Box in our statistics department who wrote
a book on time series analysis. He says, "All models are wrong;
some models are useful." And that's another thing to
keep in the back of your mind that the things I'm going to
show you are almost certainly not a very good description,
but they may be useful. Now, when you want to model
something you first have to choose some variables.
And the simplest kind of model is the so-called linear model.
Linear models, is probably another way of saying it's a
wrong model, because if it's linear, it's almost certainly
going to give the wrong results. But it's a starting point and
understanding the behavior of the linear model is a useful
first step in understanding a more complicated model.
So, the simplest kind of linear model would involve a single
variable, the population "P". And it would describe how that
variable changes in time. So the little "t" is the time.
And in the linear model, you have the time derivative,
for those of you who are rusty on Calculus.
This means the rate at which P is changing.
This is a positive number, P is increasing.
If it's a negative number, it's decreasing.
That derivative is proportional to the population.
And the proportionality constant, which I've written as
a gamma is the growth rate of the population, which is
essentially the birth rate minus the death rate.
So, if the birth rate exceeds the death rate, the gamma is a
positive number. If the death rate exceeds the
birth rate, gamma is a negative number.
So, this is the simplest model you would construct for
population dynamics. Well, whenever you have a
dynamical model like that, the first thing you do is to look
for the equilibrium. Now one of the things I'm hoping
to do with this talk is to develop some terminology that we
can all use in future seminars. Usually we caution speakers when
they come to talk not to use a lot of jargon, and I'm going to
in some sense ignore my own admonition, because one of the
things I want to do is introduce some jargon and some terminology
that hopefully we can then use to talk amongst ourselves when
other speakers come and talk about their dynamical system.
So, an equilibrium is defined for a dynamical system like this
as the condition under which the time derivative is equal to 0.
That is to say, the condition where the population is not
changing. So, if the left hand side is 0,
the right hand side is 0, and you can accomplish that in one
of two ways. Gamma can be equal to 0
or P can be equal to 0. Well, P is the dynamical
variable and gamma is what we would call a parameter.
And in a model of this sort, what we would typically do is
allow the dynamical variable to change in time, because that's
the thing we're trying to predict and understand.
And the parameters, we're going to hold fixed, not allow them to
change in the model. Now in a more sophisticated
model, of course the parameters may themselves change and they
may obey equations. But in the simplest model,
especially in a linear model like this, we hold that quantity
fixed. So, in all the models you see,
there will be variables and there will be parameters.
And so the gamma is a parameter which, if you hold it fixed,
at any value other than 0, then the only equilibrium
is the one at P equals 0. That of course represents the
case where there are no people on the island.
So the trivial equilibrium, but it has two possibilities.
It can be either stable or unstable.
Now what does that mean? If gamma is greater than 0,
it means if you put a couple of people on the island, then the
number of people increase. And it goes away from the
equilibrium. If gamma is negative, however,
you put a few people on the island, you wait long enough,
they're not there anymore. So, it decays back to the
equilibrium at P equals 0. So that's the sense in which
it's stable. One of my demographer friends
just told me this is funny terminology to call a population
stable when it's 0. (laughter)
But mathematically, we call this a stable equilibrium, because
any perturbation away from it, returns to it.
Okay. Just some of the terminology
that I'm hoping to get across. Now we can make graphs of what
happens in these two cases. For the unstable case, as I
said, you put a few people on the island, their population
grows in time, that's gamma equals plus 1.
If you put a bunch of people on the island, and gamma is less
than 1, they die out. The growth rate
exceeds death rate, death rate exceeds growth rate.
These are the only two kinds of behavior you can get.
Mathematicians would of course worry about the gamma equal 0
case where of course the population would just be
constant. Demographers call that a
stationary population. >> Can we turn the lights off?
No. No, we cannot turn the lights
off, because of the video. So, I hope the slides are
legible to you. I apologize for that.
Now, I've looked at this special case of plus 1 and minus 1;
you could say, what about when gamma's 4 or 17?
Well, it turns out that nothing new happens, because you can
always rescale the time so the gamma is either plus 1
or minus 1. Just measure time in units
of 1 over gamma. So, mathematically, there's no
loss of generality in assuming the gamma is either
plus or minus 1, or 0. All of the dynamics is captured
under those conditions. So this is one of the nice
things about models. You can often simplify them
and strip them of a lot of parameters that would otherwise,
otherwise you might think would be important.
In this case, only the sign of this parameter is of
consequence. Well, that model is obviously
too simple especially the one where you have exponential
growth. We all know that things can't
grow exponentially forever. You run out of resources, so
something always has to stop it. So, the next order of complexity
we'd put into a model is to add a non-linearity.
Now the simplest way to add that is to just add a term to the
equation. The gamma P is the linear term
and then we just add a minus P. Now when P is small,
and in this case by small, I mean much less than 1.
You can ignore the P here and you can get back to the
linear model. As P gets bigger and bigger
approaching 1, then this basically turns the growth rate
down and in fact when P is equal to 1,
the right hand is 0 and the P stops increasing.
And so P equals 1 is called the carrying capacity
and obviously now P has to be measured in special units.
P can't be the number of people, otherwise you'd say
once you have one person, then it stops.
So P is measured in special units, units in carrying
capacity. So again, you often see the
logistic model written with another parameter in here.
Sometimes you put it in the denominator so that when P is
equal to some quantity, then the population stops increasing.
But there is no loss of generality in just redefining
the variables. We can in fact redefine the time
to make that one go away. We can redefine the P to make
that one go away. In this case, I've left the
gamma in there. But one can even eliminate that.
As a consequence, well, this system now has not a single
equilibrium, but two equilibria. There's still the one
at P equals 0. P is equal to 0,
when the right hand side is 0. And that indeed is stable if
gamma is negative, just as it was before.
But there is another equilibrium.
When P is equal to 1, namely when the population is at its
carrying capacity, then there is another equilibrium.
The right hand side is then 0 for any value of gamma.
Now, the one at 0 is stable when gamma is negative.
When gamma becomes positive, this one becomes unstable,
just as it did before. And simultaneously, the one at P
equals 1 then becomes stable. So the stability switches from
one to the other. This is what's called a
transcritical bifurcation. It's not a name you'll probably
ever hear again, but I'll say it once.
Okay, and so P equals 1 is the carrying capacity.
So we've scaled P. Here's a graph of what it looks
like, for gamma equals plus one, for any positive gamma, you
scale the time appropriately, you get the exponential growth,
and then it saturates. It saturates at a value
of P equals 1, at zero to two. P equals 1, it goes there.
Now, this point over here is called an attractor.
It's called an attractor because no matter where you start,
whatever population you start here, gamma is equal to
plus one, you will end up going there.
So it attracts all initial conditions, you can say.
There's another equilibrium down here, where there are no people,
and that's called a repellor, because it's unstable.
So if you move a little bit away from that, if you add a few
people, the dynamics push you farther away, and eventually
push you over to the attractor here.
So attractors and repellors. By the way, I've seen repellor
written with an "o" and "e" and I vacillate between
which I like better. I sort of like the "o," because
it agrees with attractor. I'll have to ask someone who's
better at language than I am. Well, that's the so-called
one dimensional model. There was a single variable,
people. In a real ecology, of course,
there's going to be lots of variables.
So let's go to the next level of complexity, and that is to add
another species. We know that the people were
highly dependent on the trees. That's presumably one reason why
they stopped making statues, because they needed the trees to
make sleds to bring the lava, the statues, from where they
were made to where they were erected.
And after a while, they could no longer do that.
In any case, the trees, you can think of as sort of a surrogate
for the resources. If you don't like the fact that
humans can't live without trees, just think of the trees as
encompassing a larger collection of things that the people are
reliant upon. Now, the simplest kind of model
with two parameters that people developed long, long ago is the
Lotka-Volterra model, named after these two people who
developed models. There's a whole class of models,
depending on just how you write them, and so forth.
But they involved a pair of variables, people and trees.
And sometimes these are called predator/prey models, because
you can think of the people as the predators and the trees as
the prey, even though they're not very good at getting away,
we're still reliant on them, and they are your resource.
You can see features of this that are very much like the
other model. In this case, if there are no
trees, we assume the population dies.
And so, gamma is a positive number and the negative sign
here means that they will die in the absence of trees.
You need to put some trees in, and you need to harvest them at
some rate. The more you harvest, the more
your population grows. The trees obey, essentially the
same equation I just showed you, the logistic growth equation,
except not only do the trees sort of compete with other trees
for the resources, but they're competing with the people.
The people are harvesting the trees.
So the more people you put in, the slower the growth rate of
the forest. Okay, well, this model has three
equilibria. It has a trivial one,
when P and t are both 0. Here's P and t.
P and t are 0, that's when the island is barren.
There's no way to get it started.
Whether it's stable or not is another issue.
There's one way there's lots of trees and no people, that's
presumably how the island was before the people came.
Then there's the interesting one, from our standpoint at
least, the so-called coexisting equilibrium, the one in which
the people and trees, and you can calculate where that is by
setting these two things equal to zero and solving for P and t.
I won't do the algebra, but there is such an equilibrium.
This is a model advanced by Brander and Taylor in 1998,
13 years ago. This is a graph similar to one
in their paper. They took time out to about this
point at about here. This is in units.
These of course are normalized units, so I have to tell you
what these are. I think these are hundreds of
years, so this is a thousand years all the way across here.
The population grows for a few hundred years, 300-500 years, at
which time the trees start to decrease, and the people then
start to decrease, because they're reliant on the trees.
And it eventually settles down at the coexisting equilibrium,
after it oscillates somewhat. So here's a case where there is
an attractor to the coexisting equilibrium.
Here's another way to plot the same thing.
In two-dimensional space, instead of plotting versus time,
here's now the number of trees and the number of people.
So we start with a lot of trees and very few people.
As the people come, the number of people increase, the number
of trees decrease. And here you see another way of
plotting the oscillations that occur, eventually reaching this
coexisting equilibrium. These numbers are chosen by the
biologists. These guys knew something about
the biology of the island and chose these parameters to be
realistic representations of what might have happened.
What happens on the right hand edge, when the line appears
to go off? >> Oh, I just cut it off.
It goes beyond it. >> Does it extend on the other
side of one? >> Yes, there's nothing to keep
it from overshooting and going beyond the carrying capacity.
In fact, that's what the people do.
For a while, they can exceed. In fact, the important thing is
they're beyond their steady state condition.
This is all that they can sustain in the steady state.
They considerably overshoot it while there are still some trees
there. Then they deplete the trees and
they oscillate back and forth, and eventually reach this.
This is another example of an attractor.
It's called a point attractor for obvious reasons.
It's a single point in the P-t plane, to which in fact all
initial conditions, at least all initial conditions in the
positive quadrant away from zero, all initial conditions
approach that point. So no matter how you start off
the model, you let it go, and you eventually end up at that
point if you wait long enough. Okay, now about three years ago,
a couple of guys came along that tried to do a little bit better
than that model. That model has a number of
faults. One is that it didn't bring the
population of people down to a sufficiently low value, down to
the low value that was actually observed.
And it did so rather too slowly. In fact, that model is incapable
of having a case where the population grows to some maximum
value and then crashes to zero. It can only come down sort of
exponentially decaying. It can't have a rapid decay,
like was observed. So they proposed a slightly
different model. It has many of the same
features. Here you can see the linear
growth of the people. Here you can see the linear
growth of the trees. You see the carrying capacity of
the trees set as one. You see now the harvesting of
the trees. The people are reducing the
growth rate of the trees by a simple linear term here.
Obviously, there's some scaling to make people and trees in the
same units, and to make the trees and units of their
carrying capacity. The most significant part of the
model is the fact that there's a P over t here.
So normally, the denominator here would be the carrying
capacity. And the bigger this denominator,
the more people you can sustain, because the bigger P can be
before it starts to approach one.
But by having the carrying capacity proportional to the
number of trees, you then have the feature that when the number
of trees is infinite, then there's an infinite carrying
capacity. This term is then always zero.
So with unlimited trees, the population would grow forever.
On the other hand, if the trees go to zero, then this term goes
to infinity. It gets very large.
So this term will become very large, even when this number of
people is quite small. So, this has a much more drastic
crash when the trees are depleted, when their number
decreases, this causes the growth rate of the people to die
much more quickly than the other model.
They argue it's more realistic biologically and it fits the
data better. It has three equilibria.
It has the equilibrium when the island is barren.
It also has the case where there are trees, but no people.
And it also has the coexisting equilibrium, so the same three
equilibria as before. But here's the solution they
showed in their paper three years ago, where the people
came, grew up to a very large value and then abruptly crashed.
The trees died at the same point.
So at this point, there are essentially no people and no
trees left. The crash is very abrupt, and
occurs at a particular time. The time here in real units,
these are millennia. So this is a thousand years.
I guess each of these is 200 years.
In the span of basically one human life span, the population
crashed from its maximum to zero.
The truth probably lies somewhere between these two
models. This one is too extreme.
And the previous one was a little too gentle.
So that's fine, you can make some kind of hybrid model.
Can you make another model, or just adjust the parameters?
Yes, you can adjust the parameters.
There are only two parameters in this model.
You can adjust them. For certain values of the
parameters, you can essentially recover the previous models.
Not exactly, but you can qualitatively recover their
behavior. But this model will do this,
whereas the other model, no matter how you choose the
parameters, will never do this. Well, they also point out in
that paper that there are periodic solutions.
Here's an example of trees and people, oscillating forever.
The units here, this is actually, each of these is 70
years. I think the period is every 80
years, you get a peak in the trees.
The people peak a little after the trees peak, because the
people are growing while there's a lot of trees.
When the trees die, the people sort of die with them, so it
oscillates. But it's a special case.
It requires a particular relationship between the growth
rate of the population with unlimited resources, and the
harvesting rate of the trees. It requires a particular ratio
of those two quantities. You can see that this satisfies
that equation. So this is a solution, but it's
not a solution you'd ever find in nature, because it's
so-called structurally unstable. That is to say that if you
change these parameters by the slightest amount, if you, let's
see, if you increase the harvesting rate, these
oscillations will get bigger and at some point, you'll have a
crash, and it will never recover.
If you reduce the rate, the oscillations will become
smaller, and it will damp out and approach the coexisting
equilibrium. So it's not a physically
interesting solution, but it does show that you can get
periodic solutions. >> If either one touches zero,
then that's the end? >> That's the end.
If you ever go to zero, you stay there.
It's fascinating and beyond my comprehension, but fascinating.
What's really happening? >> As population increases,
they start to harvest the trees at a younger and younger age.
The trees are smaller and smaller sized trees.
That's probably true. >> And that would account for
the speed of the crash, wouldn't it?
It could, in part. >> And that would account for
the decrease in the size of the species, as hunting expanded on
the North American continent. There used to be enormous
creatures, and they got smaller and smaller and smaller, as
every fisherman knows. >> Yeah, whether they cut down
the young trees or the old trees, preferentially, I don't
know. >> Well when they need the wood,
they cut younger and younger, they can't wait for it to build.
They can't wait, yeah. So these are the biological
details that the model doesn't care about.
Pardon? >> These are the biological
details that the model doesn't care about.
But that needs another model. >> If you wanted to make a
better model, which you can always do, then you do have to
think more about the details. >> So that changes the
definition of chaos. >> I haven't defined chaos yet.
( laughter ) Next slide, it's coming.
All right, I'll hold it. >> Be a little bit patient.
Now, I've shown you a one parameter model, where there's
only people and some fixed resources that they're
consuming, and a two parameter model where there's people and
trees. Sorry, a two variable model, a
two-dimensional model, because you can plot it in a
two-dimensional space. There's a theorem about
two-dimensional dynamical systems.
The standard reference I always use is Hirsch, Devaney and
Smale. Hirsch is sitting in the back
of the room, for the theorem, which in simple terms, maybe not
mathematically elegant, says that there's really only four
things that a two-dimensional dynamical system can do.
One, it can attract to an equilibrium, I showed you that.
In this case, it spiraled into it.
It can cycle periodically. I just showed you an example of
that. It goes on forever.
It can attract to a periodic cycle.
That is to say you can start with something that's not
periodic, but it goes and it eventually becomes periodic.
And it can go off to infinity. Not very physical, but it's
possible. And the reason is quite simple.
In a dynamical system, every point in the space of the
variables, the so-called state space, in this case, people,
trees. Every point in that space has a
unique direction in which it's going.
So if you advance time, those derivatives tell you which
direction to go in space. That means you can't have two
trajectories cross. Because at the crossing point,
there's be two directions you could go, and that's not
consistent with the equations. So, as long as you're away from
an equilibrium point, there is always, where there is no flow,
there is a unique direction. So if the trajectories can't
cross, all they can do is spiral into a point.
It's that case. They can go around and around in
a circle. That's that case.
They can start and spiral outward and eventually they're
going around in a circle. That's that case.
Or they can go off and never come back.
That's that case. So in particular, you can never
have chaos, because none of these are chaotic solutions.
Can I make a point? >> Yes, please.
In that one system, different trajectories can do different
things. It's not that all trajectories
do the same thing. >> Thank you.
Okay, so these same guys then proposed a three-component
model. There's some evidence that a lot
of the demise of the trees might have been not due to the humans,
but due to rats. There's archeological evidence
for this. They found seeds from the trees
in caves and buried under the ground, and they're almost all
eaten by rats. Apparently when the rats eat the
seeds, they're no longer viable. So that obviously would decrease
the rate of growth of the forest.
Now, where did the rats come from?
Well, they probably came with the humans, either as stowaways
on their boats. Or maybe, another speculation is
that the people actually brought them as a source of food.
We know they brought chickens. They might've also brought rats
to have something to eat. But anyway, there's good
evidence that there were lots of rats on the island at some
point. I think there are none anymore,
because all the trees are gone. I guess they were dependent on
the seeds. So here's the three component
model. People, rats, trees.
Three variables. It's a three-dimensional
dynamical system. The state space now is
three-dimensional. You can visualize it.
Hard to draw on the screen, but there are three variables that
are all changing in time, in some fashion.
The equations are quite similar. The people are dependent on the
trees. The rats are also dependent on
the trees. So the carrying capacity for the
people and the rats both are proportionate with the number of
trees. The people harvest the trees.
The rats suppress the growth of the trees, by eating the seeds.
So that term is put down here. Now we have a four parameter
model. The growth rate of the people,
the growth rate of the rats, the rate at which the people are
harvesting the trees, and the rate at which the rats are
eating the seeds, you might say. So it's a three parameter model.
It shows that it can give all of the right dynamics that I've
previously shown you. They more or less left it at
that point. One of my hobbies is whenever I
find a three-dimensional dynamical system in the
literature, I always look for chaos in that system.
And indeed in their paper, which was just published three years
ago, they don't show or even talk about the possibility of
chaotic solutions. So I got interested in whether
this three-dimensional system has chaotic solutions.
And it does. First of all, the equilibria.
There are now four equilibria for this system.
It's a little more complicated. There's the one with no people
and no rats, just trees. There's the one with trees and
people, and the one with trees and rats, and then there's one
with all three species coexisting, which is the
interesting one. Here is a chaotic solution that
I was able to find for a particular choice of these four
parameters. The name of the game is you play
with these parameters until you get a solution that isn't any of
the things you've previously seen.
It doesn't attract to a point. It doesn't attract to a periodic
cycle. It is a-periodic, it never
repeats. So if you look at this
oscillation, and the time here, let's see, I think these are, I
forget the units, but these are probably like years, actually.
I think these are exactly years. >> Does that matter, if it keeps
cycling, even if they're irregular?
Yeah, they're irregular cycles, but it keeps going.
You might say it's harmless biologically, except they're
pretty extreme, the population of people varies a factor of
three or something from one cycle to another.
It's hard to predict long-term, because a small change in how
you start it here will have a big change out here.
So there clearly is a dominant frequency associated with the
dynamics. I think it's 80 years or
something. But each cycle is a little bit
different from the cycle before it.
That's what we mean by chaos. This is not a very extreme
example of chaos. It's almost periodic.
But it has all of the characteristics of chaos,
including sensitive dependence on initial conditions.
So that's what we call chaos. Chaos is not necessarily a bad
thing. Maybe you like it.
I like it. Now, there are other ways to
plot this, and other ways to show that it's chaotic, in fact.
If you look at the minimum of the population of people, versus
the previous minimum, you can make a graph like this.
It's a so-called return map, because when the trajectory in
the state space returns to the minimum value of the people,
then there is a new value of the minimum that is a function of
the old value. It has this peculiar kind of
shape. It's the so-called unimodal map.
That means it has a single minimum in this case.
So, it looks like it's quite predictable.
It looks like, from the previous minimum, you can predict exactly
where it's going to be next, except that there are two values
from which it could have come. So this value could've come from
here, and it could've come from here.
Furthermore, what looks like a line is not really a line, if
you zoom in on it a factor of 1,000, you can see this is
really two lines. They're very close together.
And if you zoom in on each of these another factor of 1,000,
you'll see that they are two lines.
You can keep doing that forever and ever, and that produces
what's called a fractal. It means when the trajectory,
they usually use it in the sense of a -- section, where the
trajectory returns to the plane of the section.
It's the section at which the time derivative of t is a
minimum. P is local minimum.
So it's a particular section through the three-dimensional
space. It's where you cross that, where
you return after once around. Okay, here's yet another way to
think about it. There are four parameters in
this system. Let's hold three of them
constant and vary the fourth. The one I've chosen to vary is
the harvest rate, because that's the thing over which,
presumably, the humans had some control.
So if you harvest less, you're equilibrium population is
probably going to be less. You're reliant on the trees for
life. So with a smaller harvest rate,
you can support fewer people. But as you increase the harvest
rate, what happens? Well, the first thing that
happens, I actually don't show over here.
But that equilibrium point, the coexisting equilibrium becomes
unstable, and something called a Hopf bifurcation.
Remember, I showed a stable equilibrium point where it
spiraled into it. The rate of which it spirals in
depends critically on these parameters.
So as you change the parameters here, it spirals in slower and
slower and slower, until eventually you reach a point
where it's neither spiraling in or out, and it's just going
around in a circle. That's that structurally
unstable periodic solution I showed you.
Then you turn the parameter up a little bit more, and you spiral
outward, okay. That's called a Hopf
bifurcation. That happened over here at about
.4611, I happen to know. At which point you get a stable
periodic oscillation, or so-called limit cycle.
So a limit cycle is another kind of attractor.
It's not a point attractor, it's a line attractor.
In this three-dimensional space, trajectories going around
periodically, round and round. At a point here, that periodic
trajectory bifurcates in what's called a period doubling
bifurcation, where when it goes once around, it doesn't quite
close on itself, but the second time around, it closes on
itself. So every two times around the
cycle, it repeats. So there are then two minimums.
This is now the minimum of the population.
First time around the cycle, you're here.
The next time, you're here. Here, here, here, here.
You keep turning up the harvest rate, it bifurcates again.
Every fourth cycle, every eighth cycle, and so forth.
This is the classic period doubling route to chaos.
And after infinitely many of these, most of which are crammed
in here where you can't see them, it becomes chaotic, and it
takes an infinite number of cycles before it repeats, which
is tantamount to say it never repeats.
So, here is the region of chaos, over a very narrow range of this
parameter. Then the chaos grows, it gets
more chaotic, you might say, in the sense that the minimum takes
on a larger and larger range of values, until eventually,
it reaches a point here where dramatically it's called a
crisis, where the attractor touches its basin of attraction,
and the orbit is lost. It goes to zero in this case,
and thereafter remains there. That's where the system crashes,
as you begin to turn up the harvesting rate of the trees.
Okay, so plotted up here is the so-called Lyapunov exponent,
which is the quantitative measure of chaos.
When a system is purely chaotic, the Lyapunov exponent is zero.
So this is zero, minus .01 to .01.
At this critical value where the chaos unsets, the Lyapunov
exponent becomes positive. The Lyapunov exponent is the
rate at which two nearby initial conditions are separated, so it
is a measure of the sensitivity to the initial conditions.
The bigger this is, the less predictable the system is.
So, it generally increases. Notice there's some periodic
windows, you can even see them here, where it, for certain very
narrow ranges of this parameter, you have periodic behavior.
This is all very well known. In fact, for those of you seeing
the logistic map, a very simple population model that goes back
a long way, to the early '70s, this behavior is identical.
The period doubling route to chaos is a very common route.
Okay, so this is called a bifurcation diagram.
And this is the Lyapunov exponent.
This is the period doubling route to chaos.
Now you can look at these different dynamics also in the
P-t plane. Remember, there's also a third
variable here, the rats. So you can think of that as a
coordinate that's coming out of the page.
I've sort of tried to illustrate that, it's hard to see here, by
having a shadow, and one part of the trajectory goes behind the
other part, so you get a little bit of the sense of the third
dimension, but it's not all that important.
As you begin to turn up the harvesting rate, which is what
this A is, at a small harvesting rate, you have a stable
equilibrium. As you turn it up, this point,
which originally is stable, you spiral into it.
You then spiral out of it. So if you start here, you come
near that point, and then you get bigger and bigger, and you
approach this limit cycle. So there's the stable periodic
orbit that you approach from wherever you start.
So, this thing right here is an attractor.
It's a line attractor, a so-called limit cycle.
Well, here's what happens when the period doubles.
You start from this point, and you go through some
oscillations, and you eventually end up going here on one cycle
and here on the next cycle. Then here, then here, then here,
then here. Forever thereafter, you're just
winding around. Every other time, you're on one
or the other of these. Here, is a chaotic case, so I
just plotted a piece of it. If I kept going, it would
fill in where it would look very dense.
But it never repeats. If you turn it up a little more,
you get what's called transient chaos.
Transient chaos is a situation where it behaves sort of
chaotically for a while. It looks a lot like this.
It goes around four or five times in a fairly complicated
unpredictable manner, and then eventually it dies.
If you turn it up a little more, you don't even get that.
You just start out, and the people increase, and then the
people die. This is more like what happened
on Easter Island. But it shows you that by turning
the harvest rate down a little bit, you can get these other
kinds of behavior. >> Was "A" the harvest tree?
Yes, sorry about the change in notation.
It is very sensitive to A. >> Very sensitive.
Here's another way to look at it.
Here, I've plotted the three quantities at the equilibrium.
Well, I plotted the attractor for the three variables, the
people, the rats and the trees, as a function of the harvesting
rate. So if you're not harvesting any
trees, the population of people is essentially zero, because
you're depending on the trees for your survival.
But the rats are doing just fine.
The trees are doing just fine. But as you then begin to turn up
the harvesting rate, the people, of course, benefit from that.
The trees, of course, don't benefit from harvesting.
When the population of trees goes down, the population of
rats goes down with it. They're reliant on the trees.
By the way, the people and the rats don't interact.
The people, in this model, are assumed not to interact.
They only interact by way of the trees.
They're both competing for this resource, the trees.
And as you turn things up, all of the behavior I was just
talking about occurs in this little range of harvesting way
up here. Now, you could say therefore
this isn't very interesting. Or you could be pessimistic, and
you could say well, the humans are naturally going to increase
their harvesting rate, until all of a sudden, a crisis occurs, a
tipping point, I guess we'd call it.
You reach a point where the whole system dies.
But remember, this is not time, this is harvesting rate.
And this is not what the population is doing versus time.
These are all the possible solutions that occur after a
long time, for that particular harvesting rate.
So if you're able to watch the system evolve, presumably, you
know that you are approaching this point, well before you
actually get there, in time to make some changes.
So this is that Hopf bifurcation I referred to, where the
equilibrium point, these three variables, suddenly becomes
unstable, and you develop this limit cycle, which eventually
period doubles into chaos. This is the boundary crisis,
where the attractor touches its boundary and goes away.
So I looked at this three component model and I looked at
a case where you might call it over consumption, or you're
harvesting trees at a faster rate, and let's just look at the
people. The people increase and then the
people die. And the trees and the rats right
along with them. This is a lot like the previous
solution. But as I said, if you're
observant, and if you're mathematically modeling what's
happening to your population, you suddenly realize that the
trees are dying rather quickly, we'd better do something.
So you can play games now with a model like this.
You can say well, what if we let the trees be harvested until 80%
of them are gone, and then we pass a law, or somehow we tell
people that you've got to reduce your harvesting rate a factor of
two, let's say. Well, you can ask what happens.
Here's what happens. So right at this point, you
reduce the harvesting rate a factor of two.
It's not good for the people in the short term.
You notice the people, in the short term, die at a faster
rate. The population of people
increases faster than it would have if you hadn't have passed
that law, but it does allow you to eventually reach a coexisting
equilibrium. So by making a short term
sacrifice, you might say, your population, you achieve a good
long term end. Of course the people, the rats
and the trees are covered. Of course, this may not be the
optimum thing to do. There are other things you might
do. For example, you could eradicate
the rats. That might be a better thing.
Suppose at this point, you magically, when the trees got
down to 20% of their initial value, you suddenly turn off the
rats. You develop Warfarin, or
something, and all the rats are gone.
Well, then the people continue to decrease, but not at such a
fast rate, and they achieve a new equilibrium that is actually
considerably higher, because you're no longer competing with
the rats for the trees. So these are the kinds of games
you play. I'm not suggesting that this has
much to do with what really happened on Easter Island, or
even what might've happened if somebody had said, oh, that's
the last tree, don't cut it down, you're going to regret
that. Conclusions.
Simple models can produce certainly complex and arguably
realistic results. How realistic this all is, is
for the biologists to argue about.
At the very least, it's interesting to see how a model
with just three species can have a considerable amount of
complexity. A common route to extinction of
a species is this route of Hopf bifurcation, equilibrium
becoming unstable, followed by a limit cycle, which then period
doubles, producing chaos. The chaos increases in its
magnitude, its intensity, until eventually the system crashes.
Now you may be inclined to say that chaos isn't very
interesting because it occurs over only a very narrow range of
harvesting rate. But there may be reasons that
systems evolve to be in that state.
It's human nature to keep turning up your consumption
until something bad is about to happen.
Then if you're smart enough, you turn it down a little bit.
That sort of automatically keeps you right at that edge, right in
the state of chaos, where the chaos is there, but not extreme.
And the final point is that careful and prompt slight
adjustment of a single parameter can potentially prevent
extinction. I wanted to leave on a positive
note, because some people think Easter Island is a microcosm of
the fate of the world. A lot has been said about that.
I want to say that we're smarter, hopefully, than the
people on Easter Island. And if we realize the dynamics
of what's going on ecologically, there's small tweaks that we can
make. And in fact, a characteristic of
a chaotic system is not only that it's sensitive to initial
conditions, but it can be quite sensitive to small variations of
parameters. So that's a hopeful thing, that
by changing things just a little bit, we may be able to avoid the
kind of disasters that some of the rest of you worry a lot
about. Thank you very much.
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