cc
Chaos and Complex Systems >> Okay weSeminar.me to the
The talk I'll be giving today is
in exactly five weeks ago to the on the topic I became interested
Before that, I knew absolutely day.
be talking about. nothBut this is not unlike a to
physicist, to jump into a field about which he knows little and
proceeds to try to make some
I became interested in this when progress.
on mathematical models I was asked to review a paper
Now I hadn't paid it much attention until that point but I
figured I'd better learn a
The more I learned about it, the little bit about it.
Easter Island and about some of mthe mathematical models thatt
have been applied to the
Easter Island is one of the most population dynamics.
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
inhabited spot in the world. described as the most remote
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
meters above sea level. Easter Island is perhaps most
famous for the statues that were
ago as 1,000 years ago before it built somewhere around, as long
And here some of the locations was discovered by Westerners.
900 of them that were built and constructed out of the volcanict
lava that exists in abundance on
It's a popular tourist the island.
There's an airport runway here destination.
and bring plane loads of that planes land on every day
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
plane which is about to land on islthe runway there, or maybethe
taking off. You see it's relatively
unvegetated, there are few
goes on there and there's a lot tof sheep herding, raising that
is done there.
mainly just a tourist mecca. But primarily these days it's
amounts to about 5,000 people Mosnow, serves the touristich
industry. These are some of the statues
that, as I said there are some
were built before Westerners 900 of around the island, that
which no more were built and in fact they came into a state of
some disrepair by the time the
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
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
thereafter they had no way to make fires, to build houses, to
build boats to fish from and
resources and the population they basically used up their
value. decayed to a relatively small
uncertainty when the first The history, there's a lot of
I've seen numbers actually as pearly as 300 A.D., but mored.
modern texts give later dates,
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
from Peru and there is some pevidence that their cultureme
had some connection with
in the early days. the South American culture
and most people now think that But largely that's discredited
2,500 miles, 2,000 miles from the closest Polynesian islandt
which is quite a feat especially
know there was an island there. since the(LAUGHTER)ly didn't
I find that quite remarkable
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
It's now thought that the population peaked at something
at the vicinity of 10,000.
17,000, maybe 7,000. I've seen numbers as high as
200 years prior to Roggevee Nbeing there, the island wasto
largely devegetated at this point and the population was
clearly in decline. The statues were there but there
was no evidence that there was
About 50 years lapsed until the next visitors came to the
island. Captain Cook visited in the
1770s and the population had
were largely torn down, and in decayed further, the statues
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.
Eastern Islanders? >> Were they enslaving the
Yes, that's right. >> Sprott: Yes.
Eastern Island natives to be The Peruvians were taking the
In 1888, it was annexed by Chile slaves.
began to grow, and the latest acensus is just about 5,000.ly
So the interesting thing is what
very small initial population up happened here, from presumably a
to 110, and then slowly to somethrecovering.,000, down
Are the 5,000 people now the
people or are they outsiders who direct descendants of the 110
wherever? mo>> Sprott: Well, I'm sure or
they're some of each. And I'm not prepared to say what
proportion. I think there certainly are
descendants of the 110 without
But what fraction, I don't know. question.
population of Eastern Island, I want to emphasize that what I'me
going to attempt to do today is not so much a talk on the
ecology, I'm not going to really
what happened on Eastern Island beyond what I've already told
you.
is try and show you some But rather what I'm going to do
predict the kind of population mathematical models that can
for any of you who are dynamics that was observed and
you'll be the first to realize biologists in the audience,
oversimplified. And this is of course is what we
do in physics and what
They like to simplify things and mathematicians like to do.
that somehow captures the relevant dynamics of the system.
And that's the spirit in which I
talk, not as a treatise on what happened on Easter Island buts
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
of what it is you're trying to to somehow capture the essence
And these models do that. Well, here's another quote by a
local Professor Box in our
a book on time series analysis. He says, "All models are wrong;
some models are useful."
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
Linear models, is probably is the so-called linear model.
wrong model, because if it's another way of saying it's a
going to give the wrong results. linear, it's almost certainly
understanding the behavior of Bthe 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,
for those of you who are rusty you have the time derivative,
This means the rate at which on Calculus.
This is a positive number, P is changing.
If it's a negative number, P is increasing.
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
So, if the birth rate exceeds the death rate, the gamma is a
positive number.
birth rate, gamma is a negative If the deatnumber.exceeds the
So, this is the simplest model
population dynamics. you would construct for
dynamical model like that, the Well, whenever you have a
for the equilibrium. first thing you do is to look
to do with this talk is to Now one of the things I'm hoping
can all use in future seminars. Usually we caution speakers when
they come to talk not to use a
in some sense ignore my own ladmonition, because one of the
things I want to do is introduce
that hopefully we can then use some jargon and some terminology
other speakers come and talk to talk amongst ourselves when
So, an equilibrium is defined for a dynamical system like this
as the condition under which the
That is to say, the condition twhere the population is not0.
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.
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,
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.
model, of course the parameters may themselves change and they
may obey equations. But in the simplest model,
especially in a linear model
fixed. like this, we hold that quantity
there will be variables and Sothere will be parameters.ee,
And so the gamma is a parameter which, if you hold it fixed,
at any value other than 0,
is the one at P equals 0. That of course represents the
case where there are no people
So the trivial equilibrium, but it has two possibilities.
It can be either stable or
Now what does that mean? unstable.
it means if you put a couple of people on the island, then the
number of people increase.
equilibrium. And it goes away from the
you put a few people on the island, you wait long enough,,
they're not there anymore.
equilibrium at P equals 0. So, it decays back to the
it's stable. So that's the sense in which
just told me this is funny One of my demographer friends
stable when it's 0. terminology(laughter) population
But mathematically, we call this
any perturbation away from it, a stable equilibrium, because
Okay. Just some of the terminology
that I'm hoping to get across.
happens in these two cases. NFor the unstable case, as Iat
said, you put a few people on the island, their population
grows in time,
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
case where of course the worry about the gamma equal 0
constant. Demographers call that a
stationary population.
No. No, we cannot turn the lights?
off, because of the video.
legible to you. SoI apologize for that.re
Now, I've looked at this special
you could say, what about caswhen gamma's 4 or 17? 1;
Well, it turns out that nothing new happens, because you can
always rescale the time so the
or minus 1. Just measure time in units
of 1 over gamma.
loss of generality in assuming So, mathematically, there's no
plus or minus 1, or 0. the gamma is either
under those conditions. AllSo this is one of the niceed
things about models.
and strip them of a lot of You can often simplify them
otherwise you might think would parameterbe important.otherwise,
In this case, only the sign of
consequence. this parameter is of
too simple especially the one Wewhere you have exponentialy
growth. We all know that things can't
grow exponentially forever. You run out of resources, so
something always has to stop it.
we'd put into a model is to add So, the a non-linearity.mplexity
Now the simplest way to add that
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,
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
down and in fact basically turns the growth rate
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,
once you have one person, oththen it stops.ay
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
But there is no loss of generality in just redefining
the variables.
to make that one go away. WWe can redefine the P to makeme
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
equilibrium, but two equilibria. system now has not a single
at P equals 0. There's still the one
when the right hand side is 0. P is equal to 0,
gamma is negative, just as it And thatwas before.stable if
But there is another
When P is equal to 1, namely equilibrium.
carrying capacity, then there is when the population is at its
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,
just as it did before. this one becomes unstable,
equals 1 then becomes stable. ASo the stability switches fromP
one to the other. This is what's called a
transcritical bifurcation.
ever hear again, It's not a name you'll probably
Okay, and so P equals 1 is the but I'll say it once.
So we've scaled P. carrying capacity.
like, for gamma equals plus one, Here's a graph of what it looks
scale the time appropriately, you get the exponential growth,
and then it saturates.
of P equals 1, at zero to two. It saturates at a value
Now, this point over here is P equals 1, it goes there.
It's called an attractor because called an attractor.
whatever population you start nhere, gamma is equal tot,
plus one, you will end up going there.
So it attracts all initial
There's another equilibrium down conditions, you can say.
and that's called a repellor, here,because it's unstable.ople,
So if you move a little bit away
people, the dynamics push you farther away, and eventually
push you over to the attractor
So attractors and repellors. here.
written with an "o" and "e" By and I vacillate betweenllor
which I like better.
it agrees with attractor. II'll have to ask someone who's
better at language than I am. Well, that's the so-called
one dimensional model.
people. In a real ecology, of course,
there's going to be lots of variables.
So let's go to the next level of
another species. complexity, and that is to add
highly dependent on the trees. We know that the people were
they stopped making statues, That's presumably one reason why
make sleds to bring the lava, bethe statues, from where theyto
were made to where they were
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.
humans can't live without trees, If you don't like the fact that
encompassing a larger collection just think of the trees as
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
you can think of the people as predator/prey models, because
the prey, even though they're not very good at getting away,
we're still reliant on them,
You can see features of this and they are your resource.
other model. that are very much like the
trees, we assume the population In this casedies.there are no
And so, gamma is a positive number and the negative sign
here means that they will die in
You need to put some trees in, the absence of trees.
some rate. and you need to harvest them at
your population grows. The more you harvest, the more
same equation I just showed you, The trees obey, essentially the
except not only do the trees the logistic growth equation,
for the resources, but they're sort of compete with other trees
The people are harvesting the competing trees.he people.
So the more people you put in,
the forest. Okay, well, this model has three
equilibria. It has a trivial one,
when P and t are both 0.
P and t are 0, that's when Here's P and t.
There's no way to get it the island is barren.
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
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
I won't do the algebra, but tthere is such an equilibrium.t.
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.
These of course are normalized This is in units.
what these are. I think these are hundreds of
years, so this is a thousand
The population grows for a few years all the way across here.
which time the trees start to hdecrease, and the people thenat
start to decrease, because
And it eventually settles down they're reliant on the trees.
after it oscillates somewhat. So here's a case where there is
an attractor to the coexisting
Here's another way to plot the equilibrium.
In two-dimensional space, instead of plotting versus time,
here's now the number of trees
So we start with a lot of trees andand very few people.e.
As the people come, the number
of trees decrease. of people increase, the number
plotting the oscillations that And here you see another way of
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?
It goes beyond it. >> Oh, I just cut it off.
side of one? >> Does it extend on the other
it from overshooting and going >> Yes, there's nothing to keep
In fact, that's what the people beyond the cado.ing capacity.
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
there. while there are still some trees
they oscillate back and forth, Then they deplete the trees and
This is another example of an and evenattractor.ch this.
It's called a point attractor for obvious reasons.
It's a single point in the P-t
initial conditions, at least all plane, to which in fact all
positive quadrant away from zero, all initial conditions
approach that point.
the model, you let it go, and you eventually end up at thatf
point if you wait long enough.
a couple of guys came along that Okay, now about three years ago,
than that model. tried to do a little bit better
faults. One is that it didn't bring the
population of people down to a
the low value that was actually sufficientlobserved.ue, down to
And it did so rather too slowly.
of having a case where the In fact, that model is incapable
value and then crashes to zero. pIt can only come down sort ofum
exponentially decaying. It can't have a rapid decay,
like was observed.
different model. So they proposed a slightly
features. Here you can see the linear
growth of the people. Here you can see the linear
growth of the trees.
the trees set as one. YYou see now the harvesting ofof
the trees.
growth rate of the trees by a Thsimple linear term here.e
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.
model is the fact that there's a The most significant part of the
So normally, the denominator P over t here.
capacity. And the bigger this denominator,
the more people you can sustain,
before it starts to approach because the bigger P can be
But by having the carrying one.
number of trees, you then have capacity proportional to the
of trees is infinite, then the feature that when the number
capacity. This term is then always zero.
So with unlimited trees, the
On the other hand, if the trees go to zero, then this term goes
to infinity.
So this term will become very large, even when this number of
people is quite small.
crash when the trees are So, this has a much more drastic
decreases, this causes the depleted, when their number
much more quickly than the other growth rate of the people to die
They argue it's more realistic model.
data better. biologically and it fits the
It has the equilibrium when the It island is barren.ria.
It also has the case where there
And it also has the coexisting are trees, but no people.
equilibria as before. eBut here's the solution theye
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
So at this point, there are essentially no people and no
trees left. The crash is very abrupt, and
occurs at a particular time.
these are millennia. The time here in real units,
I guess each of these is 200 So this is a thousand years.
In the span of basically one years.
crashed from its maximum to human life spzero.he population
The truth probably lies somewhere between these two
models. This one is too extreme.
And the previous one was a
So that's fine, you can make little too gentle.
Can you make another model, or just adjust the parameters?
Yes, you can adjust the parameters.
There are only two parameters in
You can adjust them. this model.
parameters, you can essentially recover the previous models.
Not exactly, but you can
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
The units here, this is actually, each of these is 70
years.
years, you get a peak in the I think the period is every 80
The people peak a little after trees.
people are growing while there's the trees peak, because the
When the trees die, the people a lot of trees.
oscillates. soBut it's a special case.it
It requires a particular
rate of the population with reunlimited resources, and theh
harvesting rate of the trees. It requires a particular ratio
of those two quantities.
that equation. You can see that this satisfies
not a solution you'd ever find So this is a solution, but it's
so-called structurally unstable. in nature, because it's
change these parameters by the That is to say that if you
see, if you increase the slightest amount, if you, let's
oscillations will get bigger and harvesting rate, these
crash, and it will never at some point, you'll have a
If you reduce the rate, the oscillations will become
smaller, and it will damp out and approach the coexisting
equilibrium.
interesting solution, but it So it's not a physically
periodic solutions. >> If either one touches zero,
then that's the end?
If you ever go to zero, >you stay there.d.
It's fascinating and beyond my
What's really happening? comprehension, but fascinating.
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 could, in part. it?
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.
the young trees or the old >> Yeah, whether they cut down
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.
details that the model doesn't So these are the biological
Pardon? >> These are the biological
details that the model doesn't
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.
( laughter ) >> I haven't defined chaos yet.
All right, I'll hold it. Next slide, it's coming.
Now, I've shown you a one parameter model, where there's
only people and some fixed
consuming, and a two parameter resources that they're
trees. model where there's people and
two-dimensional model, because Sorry, a two variable model, a
two-dimensional space. you can plot it in a
two-dimensional dynamical There's systems.m about
The standard reference I always use is Hirsch, Devaney and
Smale.
of the room, for the theorem, which in simple terms, maybe not
mathematically elegant, says
things that a two-dimensional thadynamical system can do.ur
One, it can attract to an equilibrium, I showed you that.
In this case, it spiraled into
It can cycle periodically. it.
that. I justIt goes on forever.ple of
It can attract to a periodic
That is to say you can start cycle.
periodic, but it goes and it with something that's not
And it can go off to infinity. eventually becomes periodic.
possible. And the reason is quite simple.
In a dynamical system, every
variables, the so-called state point in the space of the
trees. space, in this case, people,
unique direction in which it's Every point in that space has a
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.
an equilibrium point, there is So, as long as you're away from
there is a unique direction. always, where there is no flow,
cross, all they can do is spiral So if the trajectories can't
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
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.
Okay, so these same guys then proposed a three-component
model.
of the demise of the trees might There's some evidence that a lot
but due to rats. have been not due to the humans,
for this. They found seeds from the trees
in caves and buried under the ground, and they're almost all
eaten by rats.
seeds, they're no longer viable. Apparently when the rats eat the
the rate of growth of the So that obviforest.ould decrease
Now, where did the rats come
Well, they probably came with the humans, either as stowaways
on their boats. Or maybe, another speculation is
that the people actually brought
We know they brought chickens. them as a source of food.
to have something to eat. TheyBut anyway, there's goodats
evidence that there were lots of
point. rats on the island at some
because all the trees are gone. II guess they were dependent on
the seeds.
model. So here's the three component
Three variables. People, rats, trees.
dynamical system. IThe state space now isl
three-dimensional.
Hard to draw on the screen, but there are three variables that
are all changing in time, in
The equations are quite similar. some fashion.
trees. The people are dependent on the
the trees. The rats are also dependent on
people and the rats both are So the carrying capacity for the
trees. proportionate with the number of
The rats suppress the growth of the trees, by eating the seeds.
So that term is put down here.
model. Now we have a four parameter
the growth rate of the rats, the The growth rate of the people,
harvesting the trees, and the rrate at which the rats aree
eating the seeds, you might say.
It shows that it can give all of So it's a three parameter model.
previously shown you. They more or less left it at
that point. One of my hobbies is whenever I
find a three-dimensional
literature, I always look for dchaos in that system.e
And indeed in their paper, which
ago, they don't show or even was just published three years
chaotic solutions. talk about the possibility of
this three-dimensional system So Ihas chaotic solutions.ther
And it does.
There are now four equilibria First ofor this system.ibria.
It's a little more complicated. There's the one with no people
and no rats, just trees.
people, and the one with trees There's the one with trees and
with all three species and rats, and then there's one
interesting one. coexisting, which is the
I was able to find for a Here is a chaotic solution that
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
It doesn't attract to a point. seen.
cycle. It doesn't attract to a periodic
repeats. It is a-periodic, it never
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.
cycling, even if they're >> Does that matter, if it keeps
Yeah, they're irregular cycles, but it keeps going.
You might say it's harmless biologically, except they're
pretty extreme, the population
three or something from of one cycle to another.r of
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
dynamics. freI think it's 80 years orhe
something.
different from the cycle before But each cyclit.s a little bit
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
So that's what we call chaos. Chaos is not necessarily a bad
thing.
I like it. Maybe you like it.
plot this, and other ways to Now, there are other ways to
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,
the state space returns to the because when the trajectory in
then there is a new value of the minimum value of the people,
the old value. minimum that is a function of
shape. It has this peculiar kind of
That means it has a single It's the so-called unimodal map.
So, it looks like it's quite minimum in this case.
It looks like, from the previous predictable.
where it's going to be next, minimum, you can predict exactly
from which it could have come. except that there are two values
here, and it could've come from So this valuehere.d've come from
Furthermore, what looks like a
you zoom in on it a factor of li1,000, you can see this isf
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
You can keep doing that forever and ever, and that produces
what's called a fractal.
they usually use it in the sense It means when the trajectory,
trajectory returns to the plane of a of the section.re the
It's the section at which the time derivative of t is a
minimum.
So it's a particular section P is local minimum.
space. through the three-dimensional
you return after once around. Okay, here's yet another way toe
think about it. There are four parameters in
this system.
constant and vary the fourth. The one I've chosen to vary is
the harvest rate, because that's
presumably, the humans had some the thicontrol.which,
So if you harvest less, you're
probably going to be less. equilibrium population is
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
But that equilibrium point, the coexisting equilibrium becomes
unstable, and something called a
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
where it's neither spiraling in or out, and it's just goingt
around in a circle. That's that structurally
unstable periodic solution I
Then you turn the parameter up a showed you.
outward, okay. little bit more, and you spiral
bifurcation. That's called a Hopf
.4611, I happen to know. That happened over here at about
periodic oscillation, or At which point you get a stable
So a limit cycle is another kind so-called limit cycle.
It's not a point attractor, it's of attractor.
In this three-dimensional space, a line attractor.
periodically, round and round. trajectories going around
trajectory bifurcates in what's At called a period doublingdic
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.
This is now the minimum of the So there are then two minimums.
First time around the cycle, population.
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
And after infinitely many of doubling route to chaos.
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,
parameter. over a very narrow range of this
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.
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,
So this is zero, the Lyapunov exponent is zero.
At this critical value where the minus .01 to .01.
exponent becomes positive. chaos unsets, the Lyapunov
rate at which two nearby initial The Lyapunov exponent is the
is a measure of the sensitivity conto the initial conditions.it
The bigger this is, the less
So, it generally increases. Notice there's some periodic
windows, you can even see them
narrow ranges of this parameter, here, where it, for certain very
This is all very well known. you have periodic behavior.
the logistic map, a very simple population model that goes backg
a long way, to the early '70s,
The period doubling route to this behavior is identical.
Okay, so this is called a chaosbifurcation diagram.ute.
And this is the Lyapunov
This is the period doubling route to chaos.
Now you can look at these different dynamics also in the
P-t plane.
variable here, the rats. Remember, there's also a third
coordinate that's coming out of So you canthe page. that as a
I've sort of tried to illustrate
having a shadow, and one part of that, it's hard to see here, by
other part, so you get a little bit of the sense of the thirde
dimension, but it's not all that
As you begin to turn up the harvesting rate, which is what
this A is, at a small harvesting
equilibrium. rate, you have a stable
which originally is stable, As yoyou spiral into it.point,
You then spiral out of it.
near that point, and then you get bigger and bigger, and you
approach this limit cycle.
orbit that you approach from So there's the stable periodic
So, this thing right here whis an attractor..
It's a line attractor,
Well, here's what happens when a so-called limit cycle.
You start from this point, the period doubles.
oscillations, and you eventually and you go through some
and here on the next cycle. Then here, then here, then here,
then here.
winding around. Every other time, you're on one
or the other of these.
just plotted a piece of it. HeIf I kept going, it would I
fill in where it would look very
But it never repeats. If you turn it up a little more,
you get what's called transient chaos.
Transient chaos is a situation
chaotically for a while. where it behaves sort of
It goes around four or five times in a fairly complicated
unpredictable manner, and then
If you turn it up a little more, eventually it dies.
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
kinds of behavior. bit, you can get these other
Yes, sorry about the change >> Was "in notation.st tree?
It is very sensitive to A.
Here's another way to look at >> Veryit.nsitive.
Here, I've plotted the three
Well, I plotted the attractor qfor 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.
the harvesting rate, the people, But as you then begin to turn up
The trees, of course, don't of benefit from harvesting.t.
When the population of trees
rats goes down with it. They're reliant on the trees.
By the way, the people and the
The people, in this model, rats don't interact.
They only interact by way of the are assumed not to interact.
They're both competing for this resource, the trees.
And as you turn things up, all
talking about occurs in this of the behavior I was just
up here. little range of harvesting way
this isn't very interesting. Now, you could say therefore
you could say well, the humans Or you could be pessimistic, and
their harvesting rate, until all are naturally going to increase
tipping point, I guess we'd call of a sudden, a crisis occurs, a
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
These are all the possible population is doing versus time.
long time, for that particular solutions that occur after a
So if you're able to watch the harvesting rate.
know that you are approaching sthis point, well before youou
actually get there, in time to
So this is that Hopf bifurcation make some changes.
equilibrium point, these three I referred to, where the
unstable, and you develop this variables, suddenly becomes
period doubles into chaos. limit cycle, which eventually
where the attractor touches its Thboundary and goes away.is,
So I looked at this three component model and I looked at
a case where you might call it
harvesting trees at a faster over consumption, or you're
people. The people increase and then the
people die. And the trees and the rats right
along with them.
solution. ThisBut as I said, if you'reous
observant, and if you're
happening to your population, you suddenly realize that thes
trees are dying rather quickly, we'd better do something.
So you can play games now with a
You can say well, what if we let model like this.
of them are gone, and then we tpass a law, or somehow we tell%
people that you've got to reduce
two, let's say. Well, you can ask what happens.f
Here's what happens.
reduce the harvesting rate a So right at this point, you
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
that law, but it does allow you to eventually reach a coexisting
equilibrium.
sacrifice, you might say, your So by making a short term
long term end. population, you achieve a good
and the trees are covered. Of course the people, the rats
optimum thing to do. Of course, this may not be the
do. For example, you could eradicate
the rats. That might be a better thing.
Suppose at this point, you
down to 20% of their initial magically, when the trees got
rats. valuYou develop Warfarin, or the
something, and all the rats are gone.
Well, then the people continue
fast rate, and they achieve a new equilibrium that is actually
considerably higher, because
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
the last tree, don't cut it sdown, you're going to regret
that. Conclusions.
Simple models can produce certainly complex and arguably
realistic results.
for the biologists to argue How realistiabout. all is, is
At the very least, it's interesting to see how a model
with just three species can have
complexity. A common route to extinction of
a species is this route of Hopf bifurcation, equilibrium
becoming unstable, followed by a
doubles, producing chaos. liThe chaos increases in itsod
magnitude, its intensity, until eventually the system crashes.
Now you may be inclined to say
interesting because it occurs over only a very narrow range of
harvesting rate.
systems evolve to be in that But there may be reasons that
It's human nature to keep turning up your consumption
until something bad is about to happen.
Then if you're smart enough, you
That sort of automatically keeps turn it down a little bit.
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
extinction. I wanted to leave on a positive
note, because some people think Easter Island is a microcosm of
the fate of the world.
I want to say that we're A lot has been said about that.
people on Easter Island. smarter, hopefully, than the
of what's going on ecologically, And if we realize the dynamics
make. And in fact, a characteristic of
a chaotic system is not only
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
kind of disasters that some of bit, we may be able to avoid the
about. the rest of you worry a lot
( applause Thank you very much.
)