University Place presents  "Chaos In Easter Island Ecology" Clint Sprott  January 25, 2011 UP 543  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. (applause)