WEBVTT
00:19.920 --> 00:25.720
Okay, so the title of this first talk is slightly different than
00:25.720 --> 00:27.080
what's on your schedule.
00:27.420 --> 00:32.100
This came about because of conversations with John Finnegan and Harpy
00:32.100 --> 00:37.140
and the desire to have several team projects that do not require
00:37.140 --> 00:42.200
enormous amount of background knowledge about the specifics of the
00:42.200 --> 00:42.480
topic.
00:42.480 --> 00:46.760
So with this, we decided that perhaps the first lecture could be
00:46.760 --> 00:52.540
almost a math review of some of the tools that will be needed in the
00:52.540 --> 00:53.200
next lectures.
00:54.240 --> 00:57.640
And I have tried to prepare a lecture along those lines.
00:58.280 --> 01:01.320
And so the title of this lecture is The Theoretical Minimum.
01:02.000 --> 01:07.040
The title is stolen from an exam that is given by the famous physicist
01:07.040 --> 01:08.360
Lev Landau.
01:08.360 --> 01:13.420
And Landau, who is a great Russian physicist, used to give his
01:13.420 --> 01:19.040
students an exam, basically, to assess whether it's worth for him to
01:19.040 --> 01:20.640
spend time working with them or not.
01:21.260 --> 01:24.140
And he used to call the exam The Theoretical Minimum.
01:24.520 --> 01:29.700
And that exam was a 13-part exam, by the way, so it was not a light
01:29.700 --> 01:29.920
one.
01:30.540 --> 01:34.240
And basically, that exam covers everything that Lev Landau knows.
01:34.240 --> 01:37.720
So basically, to work with Lev Landau, you had to know as much as Lev
01:37.720 --> 01:38.680
Landau about physics.
01:39.460 --> 01:43.820
And apparently, something like 20 students passed that exam over the
01:43.820 --> 01:46.620
entire history of Russian physics.
01:47.780 --> 01:50.180
Now, this is not what we want to achieve here.
01:50.800 --> 01:55.760
We are actually closer to a second level of Theoretical Minimum that
01:55.760 --> 01:58.800
was suggested by Leonard Susskind, who is a professor at Stanford
01:58.800 --> 02:02.040
University, also theoretical physics.
02:02.040 --> 02:07.220
But in fact, what Leonard Susskind decided to do is to offer some
02:07.220 --> 02:11.760
tools and some background about physics for researchers who want to
02:11.760 --> 02:12.840
move into physics.
02:13.260 --> 02:18.760
And that basically cut down on Landau's list from something like 13 to
02:18.760 --> 02:21.900
something like 6 items that now you need to know about to do physics.
02:22.660 --> 02:27.660
The Theoretical Minimum here, as articulated by John Finnegan, is
02:27.660 --> 02:31.000
that, hey, look, if you like nature and you like to study it from some
02:31.000 --> 02:35.040
perspective of mathematical tools, you're in the right place.
02:35.320 --> 02:36.580
So that's our Theoretical Minimum.
02:36.740 --> 02:39.780
But just to make sure, we're going to cover some few points on that.
02:42.740 --> 02:46.680
So let me, before I start, let me introduce the speakers.
02:47.580 --> 02:51.400
We are fortunate to have a group of speakers from outside and inside
02:51.400 --> 02:52.680
the KIT.
02:52.680 --> 02:58.440
The cluster of speakers that you see from outside KIT include John
02:58.440 --> 03:01.660
Finnegan from the Commonwealth Science and Industrial Research
03:01.660 --> 03:04.060
Organization, and more recently from the Australian National
03:04.060 --> 03:04.680
University.
03:05.640 --> 03:09.840
John has probably worked for the last 15 years or so now, almost, no?
03:10.080 --> 03:11.920
10, 15 years on complex system science.
03:13.020 --> 03:15.960
He's been leading, actually, the Complex System Science Initiative in
03:15.960 --> 03:21.160
Australia and has saw enormous amounts of projects and topics that
03:21.160 --> 03:22.500
have dealt with complex systems.
03:23.420 --> 03:27.660
And he was kind enough to agree to give an extra lecture instead of my
03:27.660 --> 03:31.780
seed dispersal lecture on topics that are pertinent to complex system
03:31.780 --> 03:35.120
science, population dynamics, collapses, and so forth.
03:36.020 --> 03:39.540
As you will see, social systems share a lot with turbulence, and
03:39.540 --> 03:42.020
that's already an interesting phenomenon about flows.
03:43.900 --> 03:49.380
So John also is perhaps more known to many of us on his work in fluid
03:49.380 --> 03:50.940
mechanics and canopy turbulence.
03:51.700 --> 03:56.920
In fact, micrometeorology as a field was born from plant science, for
03:56.920 --> 03:57.700
obvious reasons.
03:58.080 --> 04:01.020
The microclimate does affect plant productivity.
04:01.320 --> 04:05.520
And for those of you who have studied some basic atmospheric science
04:05.520 --> 04:09.140
and have read the paper by Monin Anaboukov, the famous paper that has
04:09.140 --> 04:12.720
laid out almost the foundations of micrometeorology, the paper
04:12.720 --> 04:16.200
actually starts by talking about the substrate on the land surface.
04:16.580 --> 04:18.860
Carbon, temperature, everything.
04:19.960 --> 04:23.680
And in fact, many of the physicists that were working at the Russian
04:23.680 --> 04:28.580
Institute where Monin Anaboukov worked, later bifurcated to other
04:28.580 --> 04:31.520
topics, but they all started working on turbulence.
04:32.000 --> 04:35.900
And my colleague Amilkari Porporato from Duke University summed it up
04:35.900 --> 04:36.060
well.
04:36.060 --> 04:40.120
He said, if you work on turbulence, you are really training well for
04:40.120 --> 04:40.660
everything else.
04:40.880 --> 04:42.200
So that was kind of his statement.
04:42.320 --> 04:45.640
And that's why he has all his students take some of the most serious
04:45.640 --> 04:46.540
turbulence courses.
04:47.680 --> 04:53.320
John's contribution basically was to take a field that had a lot of
04:53.320 --> 04:58.020
empirical results in terms of transport processes near the canopy
04:58.020 --> 05:02.500
-atmosphere interface and transform it into a field that is now
05:02.500 --> 05:06.200
basically contributing to fundamental understanding of physics, of
05:06.200 --> 05:07.840
transport phenomena next to interfaces.
05:08.260 --> 05:11.640
And I think this is one of the areas where environmental sciences
05:11.640 --> 05:14.720
actually is now in a position, thanks to people like John and Mike
05:14.720 --> 05:18.040
Ropach, to actually contribute to physics of turbulence rather than
05:18.040 --> 05:18.840
the other way around.
05:20.520 --> 05:26.320
Kari Jensen, who will be arriving later in the course, is a professor
05:26.320 --> 05:28.060
at the Technical University of Denmark.
05:28.980 --> 05:33.140
Kari actually finished his PhD quite recently, I think in 2010.
05:34.160 --> 05:36.980
And since then, he has been doing probably one of the most innovative
05:36.980 --> 05:41.520
and leading work, both experimental and theoretical, on water movement
05:41.520 --> 05:42.080
in plants.
05:42.920 --> 05:46.520
Microfluidetics, how water moves in the xylem, how water moves in the
05:46.520 --> 05:52.280
phloem, how all this network operates, how plants respond to pressures
05:52.280 --> 05:56.480
based on very innovative ideas that are coming from theoretical
05:56.480 --> 05:56.880
physics.
05:56.880 --> 06:01.260
And just to show you that our field now can be labeled truly as
06:01.260 --> 06:05.740
physics, that would be the sign if you are able to publish a paper in
06:05.740 --> 06:06.700
Reviews of Modern Physics.
06:07.200 --> 06:09.000
And that's what Kari Jensen just did.
06:09.140 --> 06:14.520
So in 2016, he had a very nice review paper on water movement in the
06:14.520 --> 06:17.000
xylem and the phloem in Reviews of Modern Physics.
06:17.120 --> 06:22.060
So now we are officially in the physics realms of our research.
06:22.620 --> 06:25.720
And certainly Kari is spearheading a lot of that effort.
06:27.080 --> 06:30.600
By the way, also, if you look at Kari's CV, this guy can publish
06:30.600 --> 06:34.260
easily in Plant Cell and Environment, Neophytologies, Physical Review
06:34.260 --> 06:34.680
Letters.
06:34.900 --> 06:37.480
Physical Review is actually where Albert Einstein got his first
06:37.480 --> 06:40.120
introduction to the peer review process, if you are wondering.
06:41.920 --> 06:45.320
Science, Nature, Proceedings of the National Academy.
06:45.660 --> 06:46.360
With great ease.
06:46.440 --> 06:48.860
I think when he finishes a paper, he debates, should it go to an
06:48.860 --> 06:52.060
ecology literature or should it go to probably a physics literature.
06:52.220 --> 06:53.600
So very, very sharp guy.
06:54.260 --> 06:59.120
Now Wilfred Conrad, on the other hand, who is here, had a very
06:59.120 --> 07:02.240
interesting trajectory to get to the point where he is.
07:02.500 --> 07:05.600
He started, actually, also he finished his PhD in Theoretical Physics.
07:06.820 --> 07:11.800
His PhD work was on deriving, basically, analytical solutions to
07:11.800 --> 07:13.540
Einstein's field equations.
07:14.320 --> 07:17.200
But then discovered that if you derived an analytical solution for
07:17.200 --> 07:19.960
Einstein's field equations, there is nothing else to be done.
07:20.800 --> 07:22.220
So he had to change.
07:24.220 --> 07:27.920
And at that point, he decided that if he is going to change, he is
07:27.920 --> 07:31.720
going to work on a topic where the equations for the problem have not
07:31.720 --> 07:32.340
yet been derived.
07:33.040 --> 07:36.460
So he connected with Anita Roth.
07:37.780 --> 07:42.540
And with Anita, he basically discovered that there is a whole field in
07:42.540 --> 07:46.360
ecology that is still struggling to even come up with the basic
07:46.360 --> 07:49.080
equations to describe the mechanisms in it.
07:49.700 --> 07:53.600
And I have to say that Wilfred has done a fabulous job at spearheading
07:53.600 --> 07:57.500
a lot of the initiatives that try to understand why plants look like
07:57.500 --> 07:57.800
that.
07:58.320 --> 08:03.800
Can we learn about the past climate from the shape and the function of
08:03.800 --> 08:06.220
what we think plants have done or are doing now?
08:06.380 --> 08:11.300
So basically, Wilfred's research connects quite a bit to climate
08:11.300 --> 08:14.900
reconstruction, but from a physics, physiology perspective.
08:16.460 --> 08:20.200
And because of his background, it's always fun to chat with Wilfred.
08:20.340 --> 08:23.880
He will find something different even if you give him a quadratic
08:23.880 --> 08:24.140
equation.
08:24.640 --> 08:28.860
He will give you an insight that you did not think about when you walk
08:28.860 --> 08:29.820
to talk to him about that.
08:30.360 --> 08:34.240
Costantino Manis is a professor at Polytechnic of Turin in Italy.
08:35.300 --> 08:39.060
Costantino or Costa has done his PhD actually in Aberdeen in
08:39.060 --> 08:43.460
experimental hydraulics with Vladimir Nikora, one of the best labs
08:43.460 --> 08:45.440
actually to do experimental hydraulics.
08:45.680 --> 08:50.060
And he focused quite a bit on a similar problem to John Finnegan's,
08:50.400 --> 08:53.180
but rather than look at plants, he was looking at gravel beds.
08:53.680 --> 08:58.520
And in fact, if you talk to hydrologists, a major question is what's
08:58.520 --> 09:01.900
happening in the hyphaeric zone between streams and the land surface?
09:02.280 --> 09:05.220
And that actually is where a lot of biogeochemistry happens,
09:05.980 --> 09:10.380
temperature gradients happen, and in fact, a lot of life is impacted
09:10.380 --> 09:11.140
by that zone.
09:11.400 --> 09:14.380
So Costantino has been working extensively on figuring out the
09:14.380 --> 09:18.700
connection between a free-moving fluid and what is happening inside
09:18.700 --> 09:19.380
the gravel bed.
09:19.860 --> 09:25.980
And he later extended his work to focus on basically how life, fish,
09:26.060 --> 09:29.700
and so forth capitalize on the exchange processes within the hyphaeric
09:29.700 --> 09:33.280
zone to enhance their chances of survival and so forth.
09:33.420 --> 09:36.540
So he's going to be giving two lectures as well on this topic.
09:37.280 --> 09:41.340
Stefano Manzoni is one of our own from Duke University.
09:42.300 --> 09:43.520
What can we tell you about Stefano?
09:43.600 --> 09:48.500
He finished his PhD, and the day he defended, he had a paper in
09:48.500 --> 09:48.880
science.
09:50.540 --> 09:54.760
And it was about decomposition of leaves or something like this.
09:54.860 --> 09:55.680
I don't remember.
09:57.340 --> 10:01.280
But basically, Stefano has studied with Amilcare Porporato, who is
10:01.280 --> 10:03.720
perhaps one of the leading experts on ecohydrology.
10:03.720 --> 10:07.100
And what Stefano has brought to the table is innovative tools about
10:07.100 --> 10:12.840
stochastic processes, dynamical systems theory, physiology, hydrology,
10:12.920 --> 10:15.840
and he's able to merge them all together to attack some of the
10:15.840 --> 10:17.680
toughest problems in the field.
10:17.920 --> 10:23.220
And he's been recently moving more and more towards the use of
10:23.220 --> 10:25.780
stoichiometry as a constraint on flows.
10:25.980 --> 10:29.280
So basically, how would dimensionless numbers that are set by
10:29.280 --> 10:33.220
stoichiometry basically constrain the entire biogeochemical cycle
10:33.220 --> 10:38.360
budgets in landscapes as big as the planet or as small as the root
10:38.360 --> 10:38.560
zone?
10:39.980 --> 10:42.760
So this is a snapshot of the external speakers.
10:43.040 --> 10:45.500
Now we can go to our own from KIT.
10:46.780 --> 10:48.620
We'll start with Matthias Moder.
10:48.960 --> 10:52.180
Matthias finished his PhD with Thomas Fokken.
10:52.620 --> 10:56.360
And just like Leblendau has the theoretical minimum, if you want to
10:56.360 --> 10:59.500
work with Thomas Fokken, you better study the energy balance closure
10:59.500 --> 10:59.940
problem.
10:59.940 --> 11:02.780
There is just no way around it.
11:02.880 --> 11:04.480
You have to do something there.
11:04.660 --> 11:07.400
That's the theoretical minimum of Thomas Fokken.
11:07.960 --> 11:09.600
But Matthias was able to do way more.
11:10.060 --> 11:15.020
And he was able to basically set the stage for a lot of the data
11:15.020 --> 11:19.200
processing machinery that is now being used in Fluxnet, thanks to his
11:19.200 --> 11:19.500
effort.
11:19.960 --> 11:23.400
And then he decided that he had enough of this, so he went to Canada,
11:23.520 --> 11:24.600
did a postdoc, came back.
11:25.240 --> 11:28.400
And it is safe to state that Matthias is probably now one of the
11:28.400 --> 11:33.440
leading experts on computational methods on flows on complex terrain
11:33.440 --> 11:36.360
where the biosphere and the atmosphere are talking to each other or
11:36.360 --> 11:37.400
intersecting.
11:38.320 --> 11:42.940
And unlike some of the work that we have done cheapishly with flat
11:42.940 --> 11:47.080
surfaces and so forth, actually Matthias is dealing with real systems.
11:48.100 --> 11:48.920
And both.
11:49.060 --> 11:53.800
He's bringing innovative measurements as well as simulation tools to
11:53.800 --> 11:54.720
these problems.
11:55.220 --> 11:57.960
And basically in the last five or six years, the simulations that are
11:57.960 --> 12:00.100
coming out of Matthias' group are just pretty impressive.
12:00.280 --> 12:03.240
I mean, that was unbelievable in terms of what is being done now.
12:04.600 --> 12:07.200
Okay, moving to Nadine Ruchot.
12:07.400 --> 12:10.080
Nadine, I'm sorry, I forgot the two dots.
12:11.320 --> 12:11.900
Where is Nadine?
12:14.920 --> 12:19.780
Nadine actually finished her PhD with Nina Buchmann from Switzerland.
12:21.200 --> 12:24.720
Nina is probably one of the top-notch grassland ecologists, I would
12:24.720 --> 12:24.980
say.
12:25.100 --> 12:25.860
It's safe to say that.
12:25.960 --> 12:27.020
And cropland ecology, actually.
12:27.380 --> 12:30.980
And she did a lot of work on carbon stocks, carbon fluxes.
12:31.380 --> 12:35.600
Then went to one of the better-kept secrets in US academia, Oregon
12:35.600 --> 12:40.140
State University, where she worked with Beb Law and many others in
12:40.140 --> 12:40.760
that group.
12:41.320 --> 12:45.740
And continued work on how climatic perturbations impact water and
12:45.740 --> 12:47.580
carbon fluxes in the soil-plant system.
12:48.600 --> 12:53.580
And Nadine, very much in line with Matthias, is bringing innovative
12:53.580 --> 12:54.580
measurement techniques.
12:55.160 --> 13:00.100
She is actually heading the greenhouse that you see when you enter
13:00.100 --> 13:01.100
KIT.
13:01.420 --> 13:03.760
All the experiments that you see there, all these high-tech
13:03.760 --> 13:05.600
experiments are actually Nadine Ruchot's.
13:05.980 --> 13:09.160
But at the same time, Nadine has several papers where she is using
13:09.160 --> 13:13.720
innovative models to also understand the coupling between radiation,
13:14.880 --> 13:18.300
micro -climate, soil processes, and plant responses to them.
13:18.560 --> 13:23.340
And she has delineated an area at KIT which is looking at
13:23.340 --> 13:25.820
physiological stress, basically.
13:26.100 --> 13:31.220
How elevated temperature or reduced soil moisture content impairs
13:31.220 --> 13:32.660
plant operation.
13:34.520 --> 13:36.040
We're leaving the best for last.
13:36.800 --> 13:38.200
No, it's just alphabetical order.
13:39.460 --> 13:40.620
Hans-Peter Schmidt.
13:41.040 --> 13:45.120
And Hapi has mentioned our meeting at the FluxNet, but he didn't
13:45.120 --> 13:47.620
mention a couple of other things that I'm going to fill the gaps for.
13:48.920 --> 13:53.900
When we met Hapi, there was on a table discussion about how to
13:53.900 --> 13:55.040
interpret fluxes.
13:55.680 --> 14:00.380
And Dave Hollinger, among others, was suggesting that maybe we should
14:00.380 --> 14:01.920
be using footprint models.
14:02.500 --> 14:06.460
And went to proceed to describe what the footprint model is to a more
14:06.460 --> 14:10.100
general audience and Department of Energy managers who were at that
14:10.100 --> 14:10.340
meeting.
14:11.420 --> 14:15.380
And then this guy from the room that nobody has seen before asks a
14:15.380 --> 14:17.480
question about the footprint models.
14:18.520 --> 14:20.840
And it was a very deep question.
14:21.020 --> 14:24.700
And we were looking at this guy saying, who is this?
14:25.640 --> 14:27.980
And then Dave Hollinger says, and who are you?
14:29.440 --> 14:33.680
And perhaps Dave Hollinger did not link the two that the model he was
14:33.680 --> 14:36.680
presenting is actually Hans-Peter Schmidt's model.
14:38.700 --> 14:41.360
And Hapi was the person asking him about his model.
14:42.980 --> 14:46.220
So that's how we got to know Hapi in the first meeting of the
14:46.220 --> 14:47.180
Department of Energy.
14:47.720 --> 14:51.020
But of course, footprint models now are widely in use.
14:51.540 --> 14:53.160
At that time, that was not the case.
14:53.400 --> 14:56.360
And the idea of connecting what was on the land surface with what you
14:56.360 --> 14:59.740
typically measure in the atmosphere was an open problem.
14:59.900 --> 15:04.100
But it was definitely needed because flux towers were proliferating
15:04.100 --> 15:04.480
everywhere.
15:05.200 --> 15:08.580
And being able to link them to what actually the sensor sees on the
15:08.580 --> 15:10.640
land surface was an open problem.
15:10.820 --> 15:12.120
But Hapi already had the solution.
15:12.120 --> 15:15.180
So he was always ahead of everybody else.
15:15.320 --> 15:18.520
And by the time we appreciated the solution, everybody was talking
15:18.520 --> 15:21.500
about aggregation and integration and so forth.
15:21.600 --> 15:23.460
But by then, Hapi already had papers on that.
15:23.500 --> 15:24.660
So you get the trend.
15:26.160 --> 15:29.480
So this is the group of speakers that are going to be offering
15:29.480 --> 15:29.920
lectures.
15:31.000 --> 15:34.040
I cannot say that when I was a graduate student, I had the opportunity
15:34.040 --> 15:36.300
to interact with this prolific group of speakers.
15:36.420 --> 15:38.140
So certainly take advantage of it.
15:39.480 --> 15:41.040
Any questions about the speakers?
15:42.720 --> 15:42.960
Okay.
15:45.400 --> 15:48.000
So now we are back to the theoretical minimum and the lecture.
15:48.700 --> 15:55.840
Thinking about projects for group effort, we thought that it might be
15:55.840 --> 15:59.720
important to have at least several topics that do not require high
15:59.720 --> 16:02.100
technical skills about the science of the topic.
16:02.320 --> 16:07.120
But perhaps if we can frame these problems in more general settings
16:07.120 --> 16:11.540
that require some mathematical skills, but at the same time, not
16:11.540 --> 16:15.520
technical skills about the specifics of the problem, then perhaps one
16:15.520 --> 16:20.060
might argue that sustainability research is probably a good one to
16:20.060 --> 16:20.500
start with.
16:20.960 --> 16:24.040
And perhaps a major question confronting sustainability research today
16:24.040 --> 16:28.640
is to what extent our planet with a finite environmental resource base
16:28.640 --> 16:33.880
can accommodate a faster than exponentially growing population.
16:33.880 --> 16:38.020
And this is of course a question that is on everybody's mind.
16:38.780 --> 16:41.940
Report after report from the United Nations are attempting to
16:41.940 --> 16:46.400
understand whether the planet can handle the faster than exponential
16:46.400 --> 16:47.260
growing population.
16:47.560 --> 16:50.920
So you are now wondering what are the connections between what we are
16:50.920 --> 16:52.300
going to talk about and this problem.
16:52.660 --> 16:53.540
They will become apparent.
16:53.880 --> 16:54.620
So be patient.
16:55.940 --> 17:00.700
Now these concerns are often attributed to Reverend Thomas Robert
17:00.700 --> 17:04.700
Malthus, his famous essay on the principle of population basically
17:04.700 --> 17:07.900
suggested that if the population is growing exponentially but the food
17:07.900 --> 17:10.220
supply is growing linearly, there is a problem.
17:10.680 --> 17:14.400
And that problem is basically attributed to him.
17:15.420 --> 17:19.080
But by no means he was the first to appreciate this problem.
17:19.320 --> 17:21.820
In fact, the Dutch were ahead of the Brits on this one.
17:23.600 --> 17:26.980
Early attempts to estimate how much the carrying capacity of the
17:26.980 --> 17:32.120
planet is were done by a very famous Dutch microbiologist by the name
17:32.120 --> 17:33.400
of Van Leeuwenhoek.
17:33.680 --> 17:35.240
I hope I'm pronouncing his name correctly.
17:36.420 --> 17:41.540
Van Leeuwenhoek is attributed to perhaps creating the field of
17:41.540 --> 17:45.860
microbiology and using microscopes to study microbes.
17:46.080 --> 17:47.720
So he was among the first to do that.
17:48.720 --> 17:52.280
He's also famous for being friends with Vermeer, the famous Dutch
17:52.280 --> 17:52.600
painter.
17:54.600 --> 18:00.460
But on his own, in terms of accomplishments, in 2004, the Dutch
18:00.460 --> 18:03.780
population was surveyed about who they think is the most influential
18:05.220 --> 18:07.520
Dutch person, so to speak.
18:08.720 --> 18:14.040
And Van Leeuwenhoek ranked number four, ahead of Johan Cruyff, for
18:14.040 --> 18:16.180
those of you who know who Johan Cruyff is.
18:17.080 --> 18:19.680
Which is speaking a lot about the Dutch.
18:19.940 --> 18:22.660
They actually value the scientists over a football player.
18:24.420 --> 18:29.780
Now, Van Leeuwenhoek, we will describe now what he did, came up with
18:29.780 --> 18:33.480
an estimate that the carrying capacity of the planet, that was in 1632
18:33.480 --> 18:38.280
or just in the 1600s, came up with an estimate that the planet can
18:38.280 --> 18:40.520
handle 13 billion people at that time.
18:41.400 --> 18:46.420
Now you may wonder how he did that, the crafty devil.
18:47.060 --> 18:48.060
This is what he did.
18:49.080 --> 18:52.760
He said that if we assume that the inhabited part of the earth is as
18:52.760 --> 18:57.880
densely populated as the Netherlands, Holland at the time, which had
18:57.880 --> 19:02.520
about a million people, and of course, if we're managing the land on
19:02.520 --> 19:07.620
the planet as efficiently as the Dutch, and since the Netherlands was
19:07.620 --> 19:13.560
13,000 times smaller than the inhabitable landmass, he estimated that
19:13.560 --> 19:16.680
the planet can handle 13 billion people.
19:17.800 --> 19:19.160
As simple as that.
19:21.100 --> 19:24.660
So where does that estimate sit relative to the most modern estimates
19:24.660 --> 19:27.900
that we have from the United Nations in terms of projections?
19:29.540 --> 19:30.380
Well, here's one.
19:30.560 --> 19:31.400
This is the most recent.
19:31.500 --> 19:38.260
I literally downloaded this graph maybe two weeks ago, and it shows on
19:38.260 --> 19:43.160
the x-axis time, and on the y-axis the population in billion, and it
19:43.160 --> 19:50.320
shows roughly the trends from 1950 to 2000 and roughly 16, and then it
19:50.320 --> 19:54.500
offers a bunch of projections, and these projections have some
19:54.500 --> 19:59.220
prediction interval uncertainties depending on what you assume as an
19:59.220 --> 20:00.140
intrinsic growth rate.
20:00.620 --> 20:05.060
And you can see that the projections are beginning to saturate or
20:05.060 --> 20:11.720
flatten at about maybe 11 billion or so, and I've put for reference
20:11.720 --> 20:15.480
the prediction by Van Leeuwenhoek in red line at the top.
20:16.580 --> 20:20.040
Now, to go back to Malthus so we don't leave the guy alone, now
20:20.040 --> 20:22.540
Malthus actually predicted that the world population would add a
20:22.540 --> 20:23.860
billion every 25 years.
20:24.020 --> 20:29.040
If we analyze the record that was published by the United Nations, you
20:29.040 --> 20:32.560
find that in fact we are adding 2 billion people every 25 years.
20:32.880 --> 20:36.520
So certainly double of what Malthus has predicted.
20:37.680 --> 20:38.180
Interesting.
20:40.280 --> 20:44.200
So how is the carrying capacity determined in general either through
20:44.200 --> 20:45.400
projections and so forth?
20:45.820 --> 20:48.540
Well, the workhorse model actually comes from the work of Pierre
20:48.540 --> 20:49.580
-Francois Verhulst.
20:51.100 --> 20:54.600
And in it, this is probably known to everyone, but it's worth putting
20:54.600 --> 20:58.240
on the board, in it he argues that the population can be exponential,
20:58.520 --> 21:02.760
but that exponential at some point faces some limitations that are
21:02.760 --> 21:03.560
density dependent.
21:04.140 --> 21:07.620
And so you could see basically that the density dependent correction
21:07.620 --> 21:12.640
is here, the exponential growth is here, and the outcome of this
21:12.640 --> 21:15.900
analysis is that you get a population that saturates at some carrying
21:15.900 --> 21:16.660
capacity K.
21:17.840 --> 21:20.480
Now you could write this equation a little bit differently, and there
21:20.480 --> 21:24.700
are many forms of it in the literature, but one form is shown here,
21:25.040 --> 21:27.860
where you have a proportionality constant multiplied by the population
21:27.860 --> 21:30.260
multiplied by the carrying capacity minus the population.
21:31.520 --> 21:35.300
And when you see a differential equation, it's always interesting to
21:35.300 --> 21:40.440
ask the obvious question, what's going to happen if I wait for a very,
21:40.500 --> 21:41.580
very, very, very long time?
21:42.940 --> 21:45.760
So what happens typically if you wait for a very long time?
21:48.850 --> 21:52.090
Well, you are likely to reach steady state or equilibrium.
21:53.710 --> 21:59.190
Okay, so you could say that fine, the obvious point to start studying
21:59.190 --> 22:01.370
differential equations is the equilibrium points.
22:02.650 --> 22:05.950
And these are typically viewed as special cases of steady state.
22:06.150 --> 22:08.710
Thermodynamically, they are special cases of steady states, even
22:08.710 --> 22:11.950
though in the dynamical systems literature, the usage of steady state
22:11.950 --> 22:13.990
and equilibrium is a little bit loose.
22:14.530 --> 22:16.750
But from a thermodynamic perspective, there is a difference.
22:17.150 --> 22:19.550
And we can discuss that if you're interested in the difference.
22:20.430 --> 22:22.970
A system that is in equilibrium must be in steady state, but the
22:22.970 --> 22:23.690
opposite is not true.
22:23.750 --> 22:25.810
A system that is in steady state need not be in equilibrium.
22:29.790 --> 22:33.170
So I'll leave it up to you to see how we might figure out which is
22:33.170 --> 22:34.510
which, if I give you some information.
22:36.090 --> 22:40.610
Of course, at equilibrium, the rate of change goes to 0, and suddenly
22:40.610 --> 22:43.050
rather than studying a differential equation, now you're studying the
22:43.050 --> 22:44.570
properties of an algebraic equation.
22:44.770 --> 22:46.110
Well, that is easier.
22:47.370 --> 22:52.450
And basically, what happens at equilibrium is that you get the rate
22:52.450 --> 22:57.810
going to 0, and you get two possible solutions that satisfy this
22:57.810 --> 22:58.730
algebraic equation.
22:59.150 --> 23:01.630
One of them is the population going to 0.
23:02.110 --> 23:05.990
If I multiply... if p goes to 0, this quantity all becomes 0.
23:06.510 --> 23:10.070
Or, if p goes to k, also it makes its derivative 0.
23:10.690 --> 23:13.390
Okay, fine.
23:13.530 --> 23:15.470
So we found the equilibrium points.
23:15.490 --> 23:15.910
Now what?
23:16.610 --> 23:19.990
The next thing you do in any theoretical analysis is you try to assess
23:19.990 --> 23:23.230
whether these equilibria are stable or unstable.
23:24.050 --> 23:29.050
Meaning that if you reach this equilibrium point, and you kick the
23:29.050 --> 23:31.530
system a little bit around this equilibrium point, what happens?
23:32.470 --> 23:36.270
Do I go back to equilibrium, or do I jump to another state?
23:37.450 --> 23:41.630
So to do that, we have to figure out ways to assess the stability of
23:41.630 --> 23:42.110
equilibrium.
23:42.410 --> 23:44.790
And this is all preparation for John Finnegan's talk.
23:47.170 --> 23:48.130
I think.
23:49.930 --> 23:54.490
So as we shall see in a minute, that extinction in the logistic
23:54.490 --> 23:56.570
equation turns out to be an unstable equilibrium.
23:56.830 --> 24:03.530
So the minute you create population somehow, then you're going to
24:03.530 --> 24:05.390
propagate all the way up to the carrying capacity.
24:06.550 --> 24:09.150
The carrying capacity turns out to be a stable equilibrium.
24:09.470 --> 24:12.390
And the way you typically assess that is by what is called linear
24:12.390 --> 24:13.330
stability analysis.
24:14.150 --> 24:18.730
And the way you do this is you basically start with the dynamical
24:18.730 --> 24:24.210
system, in this case the population, and you equate it to 0, and you
24:24.210 --> 24:25.710
evaluate the equilibrium points.
24:25.810 --> 24:29.290
Then what you do is you introduce a small perturbation from
24:29.290 --> 24:29.730
equilibrium.
24:29.930 --> 24:33.630
So in this case, delta is some small perturbation around the
24:33.630 --> 24:34.770
equilibrium point, p-star.
24:35.610 --> 24:39.190
In the case of the logistic equation, p-star was either extinction or
24:39.190 --> 24:39.970
the carrying capacity.
24:40.210 --> 24:43.090
So it's two values.
24:44.490 --> 24:49.010
Then what you do next in this analysis is you expand this f of p
24:49.010 --> 24:53.170
around the equilibrium point, and you butcher all the higher-order
24:53.170 --> 24:54.750
terms in the Taylor series expansion.
24:55.030 --> 24:59.050
So you are left with the first order Taylor series expansion of f of p
24:59.050 --> 25:00.990
around the equilibrium point f of p-star.
25:01.530 --> 25:05.630
And if you do that, you find that the second correction from p-star is
25:05.630 --> 25:09.470
d dp of this function evaluated at the equilibrium multiplied by the
25:09.470 --> 25:10.030
perturbation.
25:11.450 --> 25:12.230
So far so good?
25:14.390 --> 25:14.910
Okay.
25:16.830 --> 25:20.290
So we have just neglected the higher-order terms, and that has an
25:20.290 --> 25:21.850
important implication.
25:22.170 --> 25:26.350
Basically, we are assuming that this derivative at equilibrium is
25:26.350 --> 25:26.670
finite.
25:28.430 --> 25:33.450
This could be a problem that will put a monkey wrench in linear
25:33.450 --> 25:34.310
stability analysis.
25:34.670 --> 25:35.310
But let's proceed.
25:35.470 --> 25:37.290
Let's assume that it is finite.
25:40.810 --> 25:44.730
So let's now shift a little bit from the logistic equation and keep
25:44.730 --> 25:46.530
the discussion in the most general sense.
25:46.690 --> 25:51.510
So let's assume that x is our state variable, and f of x represents
25:51.510 --> 25:53.390
the derivative of x with respect to time.
25:53.950 --> 25:55.710
And we have found the equilibrious x-star.
25:55.990 --> 26:00.770
So basically, now we perturb x-star by delta.
26:01.590 --> 26:06.730
And if you differentiate, basically, dx dt, you find that it is d
26:06.730 --> 26:08.890
delta dt plus dx-star dt.
26:09.070 --> 26:11.330
And of course, dx-star dt is zero at equilibrium.
26:11.850 --> 26:15.310
And that has to be the Taylor series expansion of f of x.
26:15.350 --> 26:18.950
So that would be f of x-star plus the derivative of x with respect to
26:18.950 --> 26:23.010
x evaluated at x-star multiplied by the perturbation multiplied by all
26:23.010 --> 26:23.750
the higher-order terms.
26:23.890 --> 26:24.630
And that has to be zero.
26:25.770 --> 26:30.290
Now, because we are at equilibrium, f of x-star is zero and dx-star dt
26:30.290 --> 26:31.490
is by definition zero.
26:31.590 --> 26:32.990
That's how we evaluated the equilibrium.
26:33.510 --> 26:37.250
So we get rid of those, we get rid of this, and we are left with d
26:37.250 --> 26:41.930
delta dt is now the derivative of x evaluated at x-star multiplied by
26:41.930 --> 26:42.190
delta.
26:42.690 --> 26:46.250
So this is a first-order differential equation that describes how the
26:46.250 --> 26:48.110
perturbation will grow in time.
26:48.110 --> 26:52.770
And the direction of that growth is going to be dependent on what this
26:52.770 --> 26:55.970
derivative will do at equilibrium.
26:58.710 --> 27:01.130
So if it is zero, you have a problem.
27:02.890 --> 27:05.870
Linear stability analysis is going to choke a little bit.
27:05.970 --> 27:07.750
There are ways around it, but it will choke.
27:08.610 --> 27:12.550
If it is negative, then you have something that looks like d delta dt
27:12.550 --> 27:17.090
is minus some number multiplied by delta, and the solution of this is
27:17.090 --> 27:18.570
a negative exponential in time.
27:18.770 --> 27:22.030
So you know that the perturbations are going to decay in time, and so
27:22.030 --> 27:25.450
we can call the equilibrium stable, because whatever we do, if we
27:25.450 --> 27:28.070
perturb it a little bit, it's going to relax back to this equilibrium.
27:28.850 --> 27:34.530
On the other hand, if f-prime of x-star is positive, something else
27:34.530 --> 27:34.830
happens.
27:34.990 --> 27:38.730
Now, if we perturb the system, in fact, these perturbations grow, and
27:38.730 --> 27:42.810
very soon they will stop abiding by linear stability analysis.
27:43.670 --> 27:47.270
But we know one thing, that these perturbations are going to move us
27:47.270 --> 27:48.990
away from this equilibrium point.
27:49.110 --> 27:51.930
We are not going to relax back to this equilibrium point.
27:53.150 --> 27:56.910
Again, this whole analysis is making the assumption that the
27:56.910 --> 28:01.630
derivative of f of x, this generic function that describes the
28:01.630 --> 28:05.530
derivative of x with respect to time, is finite at equilibrium.
28:07.190 --> 28:11.290
So let's now specialize again into our logistic equation and see if we
28:11.290 --> 28:14.710
can recover the intuitive results that we know about it, that
28:14.710 --> 28:18.570
basically extinction is unstable, carrying capacity is stable.
28:19.310 --> 28:24.310
If we evaluate the derivative of f of p that is given here with
28:24.310 --> 28:27.330
respect to p, we find that this derivative is alpha multiplied by k
28:27.330 --> 28:28.010
minus 2p.
28:28.450 --> 28:33.330
So if we insert the extinction equilibrium, we get alpha k to be
28:33.330 --> 28:35.130
positive, so that's clearly unstable, right?
28:35.170 --> 28:38.610
Because if p goes to 0, we are left with alpha k, and that's a
28:38.610 --> 28:40.050
positive derivative.
28:40.050 --> 28:44.970
Whereas if we go to population reaching carrying capacity, we have k
28:44.970 --> 28:49.070
minus 2k, so we get a negative number here, and that equilibrium is
28:49.070 --> 28:49.670
unstable.
28:50.150 --> 28:52.570
Stable, sorry, because the derivative is negative.
28:53.630 --> 28:56.990
So basically we just showed that this is the case.
28:58.410 --> 28:58.690
Fine.
28:59.590 --> 29:02.790
There is one more thing that you could learn about this analysis,
29:03.110 --> 29:05.650
which is a relaxation timescale to equilibrium.
29:06.050 --> 29:07.550
And that's another important concept.
29:08.510 --> 29:12.570
Relaxation timescale, basically, you could show very quickly that 1
29:12.570 --> 29:17.770
over the derivative of p evaluated at equilibrium does describe, as
29:17.770 --> 29:21.030
you might expect, the timescale at which you decay or you move away
29:21.030 --> 29:21.650
from equilibrium.
29:22.270 --> 29:26.050
And again, in the logistic equation, if you plot the value of f prime
29:26.050 --> 29:30.450
of p star at the equilibria, the carrying capacity, you find that you
29:30.450 --> 29:33.610
would relax the equilibrium at a rate of 1 over r.
29:33.670 --> 29:39.070
But in general, in any dynamical system, you could think of f prime of
29:39.070 --> 29:42.930
p star, where p star is the equilibrium point, as almost a relaxation
29:42.930 --> 29:43.530
timescale.
29:46.050 --> 29:51.950
Now we come to the discussion about, basically, what is going to
29:51.950 --> 29:53.410
happen to the population in the future.
29:54.050 --> 29:58.490
Now, the United Nations analysis has taken a short-term view of about
29:58.490 --> 30:03.090
50 years or so to understand or project what will happen to the
30:03.090 --> 30:08.670
population in the future, and perhaps use the data from 1950 to 2016
30:08.670 --> 30:12.050
or so to try to estimate the carrying capacity of the planet,
30:12.170 --> 30:14.250
depending on different growth projections.
30:14.610 --> 30:19.190
But if you take a much longer viewpoint of human population, and that
30:19.190 --> 30:22.550
is shown on the graph here, where the y-axis is the world population,
30:23.070 --> 30:27.810
the x-axis is year, now we have, so to speak, the ability to
30:27.810 --> 30:33.070
reconstruct world population starting from almost 1 AD, roughly.
30:34.550 --> 30:39.210
And if you look at the human population since that period of time up
30:39.210 --> 30:42.470
to today, it looks that it is growing faster than exponential.
30:43.410 --> 30:46.670
Which brings the sustainability question that we started this lecture
30:46.670 --> 30:47.770
with into focus.
30:47.990 --> 30:49.250
This is exactly what we mean.
30:49.710 --> 30:53.630
How can we explain the faster-than-exponential growth in human
30:53.630 --> 30:57.190
population if we are anticipating to see carrying capacity
30:57.190 --> 30:58.530
restrictions kicking in?
31:00.370 --> 31:04.730
This point was not missed by the famous physicist Heinz von Forrester.
31:05.210 --> 31:08.990
And he had a wonderful paper in 1960 where he actually brought this
31:08.990 --> 31:09.430
point up.
31:09.510 --> 31:11.310
He said, hey, there is something fishy.
31:11.490 --> 31:14.390
If you have faster-than-exponential growth in human population, that
31:14.390 --> 31:16.350
cannot be compatible with the logistic equation.
31:17.050 --> 31:20.130
The fastest growth you could see in the logistic equation is
31:20.130 --> 31:20.550
exponential.
31:20.770 --> 31:23.450
So if something is growing faster than exponential, that means that
31:23.450 --> 31:26.910
the whole assumption of your logistic equation and any projections
31:26.910 --> 31:28.630
from it comes into question.
31:30.910 --> 31:31.390
Right?
31:32.270 --> 31:32.750
Okay.
31:33.870 --> 31:38.090
Now the paper was quite catchy, very stylish also.
31:38.550 --> 31:41.050
Very well written and very humorous if you read it.
31:41.890 --> 31:44.150
And I'll explain a little bit of the humor as we go along.
31:45.190 --> 31:46.290
This was science.
31:46.490 --> 31:49.930
This is one of the days where you could publish a paper in science and
31:49.930 --> 31:52.710
have a little bit of fun and still do serious work.
31:53.490 --> 31:55.690
So the paper was titled Doomsday.
31:57.590 --> 32:00.350
Friday the 13th November, AD 2026.
32:01.030 --> 32:04.330
At this date, human population will approach infinity if it grows as
32:04.330 --> 32:05.730
it had grown in the last two millennia.
32:08.690 --> 32:13.090
Now, the crafty devil, this guy, Heinz von Forster, with few data
32:13.090 --> 32:14.810
points, he was able to do miracles.
32:15.030 --> 32:17.030
A little bit like Wilfred Conrad in that sense.
32:17.270 --> 32:19.990
Give him a few data points, he could do a lot.
32:21.090 --> 32:25.450
And incidentally, the co-authors with Heinz von Forster happened to be
32:25.450 --> 32:27.350
two undergraduate students who were taking his class.
32:32.820 --> 32:36.520
He actually decided that, you know what, to really try to understand
32:36.520 --> 32:39.740
the limitations of the logistic equation, there are two ways to look
32:39.740 --> 32:40.040
at it.
32:40.200 --> 32:45.460
As in life, you could be an optimist or you could be a pessimist.
32:46.500 --> 32:49.580
So he was trying to interpret what it means if the population is
32:49.580 --> 32:50.760
growing faster than exponential.
32:52.040 --> 32:54.960
Which is pretty much what the people in stable isotopes do.
32:55.160 --> 32:57.860
They look at the two end members and they come up with a mixing model.
32:58.120 --> 33:00.860
So his argument is, if we can identify the two end members, then we
33:00.860 --> 33:02.020
can figure out everything in between.
33:03.100 --> 33:07.040
So he pitted the pessimist against the optimist, basically, in this
33:07.040 --> 33:07.300
paper.
33:08.320 --> 33:11.480
And he said, if you are a pessimist, you'd say, hey, sooner or later,
33:11.700 --> 33:14.720
the carrying capacity is going to kick in, environmental degradation
33:14.720 --> 33:18.880
is happening, we're going to be able to sustain less and less, and
33:18.880 --> 33:21.440
still, at steady state, we're going to reach a carrying capacity
33:21.440 --> 33:24.060
pretty fast that is going to depreciate, even though we don't see it
33:24.060 --> 33:25.760
now in the human population.
33:25.900 --> 33:27.100
But it is looming in the background.
33:27.260 --> 33:28.580
So that was his pessimist view.
33:29.080 --> 33:32.380
He actually made it sound like this view is so boring that it's not
33:32.380 --> 33:34.360
even worth exploring beyond what I've just said.
33:35.180 --> 33:37.800
The more interesting one, according to him, is the optimist.
33:38.580 --> 33:39.660
What is the optimist?
33:40.620 --> 33:43.840
The optimist basically will make the claim that the carrying capacity
33:43.840 --> 33:48.440
can increase with population due to technological innovations.
33:49.600 --> 33:52.660
If you think about the problems now on the planet, you could distill
33:52.660 --> 33:57.980
them pretty much to water, food, energy, and disease.
33:59.040 --> 34:01.280
We all know what the big problems are.
34:01.560 --> 34:06.320
I think it's safe to say that they are pretty much decided upon.
34:07.080 --> 34:09.580
And we all know that they are interlinked.
34:09.620 --> 34:11.340
They are not entirely independent problems.
34:12.020 --> 34:15.020
The challenge is how do you draw the boundaries around the inquiry of
34:15.020 --> 34:15.640
these connections.
34:17.340 --> 34:21.120
But I think it is safe to state that we are well aware of what the big
34:21.120 --> 34:23.300
problems facing humanity are.
34:24.140 --> 34:27.720
And so he said that if you're an optimist, you could say well, every
34:27.720 --> 34:32.140
time we have had carrying capacity limitations kicking in, we have
34:32.140 --> 34:36.000
seemed to have found a technological solution to overcome it.
34:36.380 --> 34:37.560
So that's the optimistic view.
34:38.800 --> 34:42.160
So if it was food limitations, we found agriculture, we found
34:42.160 --> 34:42.400
fertilizers.
34:42.400 --> 34:46.660
If it is quote-unquote health, we found vaccines, we found
34:46.660 --> 34:47.320
antibiotics.
34:48.460 --> 34:51.260
Energy, we have fossil fuel, we have fire, you name it.
34:51.320 --> 34:57.360
So it seems that every time we hit a bottleneck, the population as a
34:57.360 --> 35:01.080
whole was able to find a technological innovation and eliminate that
35:01.080 --> 35:01.520
bottleneck.
35:01.980 --> 35:02.940
So that's the optimism.
35:04.320 --> 35:07.980
Which is pretty much what you would see now in most political debates,
35:08.340 --> 35:09.160
more or less.
35:09.340 --> 35:11.600
You could distill it to these two simplified views.
35:12.680 --> 35:16.280
So then let's explore what happens if we pursue the line of arguments
35:16.280 --> 35:17.040
of the optimist.
35:17.140 --> 35:18.120
Where does that take us?
35:18.600 --> 35:20.020
And that's what von Forster just did.
35:20.920 --> 35:24.540
He said that okay, well if this is the case, then a macroscopic
35:24.540 --> 35:24.840
description.
35:24.980 --> 35:29.180
So we're not really concerned about a society in this area or that
35:29.180 --> 35:29.760
area or whatever.
35:29.860 --> 35:31.000
We're looking at the whole globe.
35:31.880 --> 35:35.460
Collectively, our collective knowledge, how it gets transferred, how
35:35.460 --> 35:38.460
we do things, that is not really the issue here.
35:38.460 --> 35:43.580
But collectively, we have seen that the carrying capacity limitations
35:43.580 --> 35:46.700
have been overcome because of the population growing.
35:46.960 --> 35:50.920
So the more people we have, according to the optimist, the more
35:50.920 --> 35:54.420
probability we have to find a technological fix to our limiting
35:55.080 --> 35:57.540
carrying capacity case.
35:57.940 --> 35:58.100
Okay?
35:58.420 --> 35:59.080
So fair enough.
35:59.140 --> 36:00.400
That's a very optimistic view.
36:00.740 --> 36:03.360
So he said, well, if you are really, really optimistic, then the
36:03.360 --> 36:05.180
carrying capacity will scale with the population.
36:06.460 --> 36:07.480
Raise to some power.
36:08.200 --> 36:09.540
And that power would be bigger than one.
36:09.880 --> 36:14.300
So that was the most optimistic you could get according to him.
36:14.600 --> 36:15.880
You cannot get better than this.
36:17.160 --> 36:18.640
So where does that take us?
36:19.100 --> 36:23.520
So he decided to go back and revisit the logistic equation.
36:23.620 --> 36:26.640
So he said, if you go back to the logistic equation with this model of
36:26.640 --> 36:30.380
the carrying capacity that is now not constant, but grows with human
36:30.380 --> 36:33.900
population, because you are able to find all these technological
36:33.900 --> 36:37.640
fixes, it grows with the human population, then the logistic equation
36:37.640 --> 36:38.840
reduces to this form.
36:39.140 --> 36:42.700
We have dp dt, rather than being a constant proportional to the
36:42.700 --> 36:45.940
carrying capacity here, it's actually dependent on the population
36:45.940 --> 36:50.000
raised to some exponent, delta plus one, minus the term that is
36:50.000 --> 36:53.440
actually causing the reduction in population, p squared.
36:54.680 --> 36:57.760
Now he said, if delta is bigger than one, so this exponent here is
36:57.760 --> 37:02.460
bigger than one, and as the population increases, this term that is
37:02.460 --> 37:08.120
here will dominate the solution, because this term is, say, cubical,
37:08.220 --> 37:09.580
or fourth power, or fifth power.
37:09.900 --> 37:12.380
This term is reducing the population as quadratic.
37:12.840 --> 37:16.760
For sure, for big populations, the bigger the power, the more dominant
37:16.760 --> 37:17.760
that term is going to be.
37:19.220 --> 37:20.260
Okay, fair enough.
37:20.580 --> 37:22.740
So let's do some simplifications.
37:22.940 --> 37:27.640
We're going to get rid of this term and just leave the big term, when
37:27.640 --> 37:29.300
we're looking at the difference between these two terms.
37:29.300 --> 37:34.560
So he did that, and basically you could simplify that for large
37:34.560 --> 37:35.100
populations.
37:35.820 --> 37:38.380
The logistic equation now looks more like that.
37:38.480 --> 37:40.900
dp dt, the rate of change of the population with respect to time,
37:41.020 --> 37:43.440
scales with the population raised to some delta plus one.
37:44.680 --> 37:47.500
And even I can solve this differential equation.
37:47.760 --> 37:53.360
So if you do that, you find that the solution turns out to be the
37:53.360 --> 37:57.820
exponent itself, delta, multiplied by this proportionality constant r'
37:57.960 --> 38:01.140
times t minus some integration constant raised to the minus one over
38:01.140 --> 38:03.660
delta, which we can write a little bit this way.
38:05.320 --> 38:09.240
So now we have something here that looks like time in the denominator.
38:10.540 --> 38:14.360
Fine, let's try to get rid of this integration constant by imposing
38:14.360 --> 38:16.220
some plausible condition.
38:16.540 --> 38:21.480
Say we know the population at some time t equal to zero, say 1 AD.
38:23.400 --> 38:25.820
So let's assume that P of zero is P naught.
38:25.900 --> 38:29.680
So that could be a reference point that you pick, but not the garden
38:29.680 --> 38:33.460
of Eden, because that would be a small population.
38:36.080 --> 38:40.320
So P of t then becomes when you impose this initial condition, and now
38:40.320 --> 38:42.780
you have P naught as the initial population, it becomes actually shown
38:42.780 --> 38:43.100
here.
38:43.520 --> 38:46.320
And automatically you see what's going to happen now with the
38:46.320 --> 38:47.240
optimistic solution.
38:48.980 --> 38:53.960
There is a chance that if t goes to P naught raised to the minus delta
38:53.960 --> 38:59.340
divided by delta r, the denominator of this quantity will go to zero.
39:00.220 --> 39:06.160
Raised to some positive exponent, and one over zero is infinity.
39:07.700 --> 39:08.120
Interesting.
39:08.980 --> 39:14.520
So at finite times, we can have an explosion in population.
39:16.300 --> 39:18.640
This is known as finite time singularities.
39:20.680 --> 39:22.460
The concept is not new.
39:22.780 --> 39:27.700
If you talk to engineers, they've already knew about critical time to
39:27.700 --> 39:28.040
failure.
39:28.720 --> 39:33.160
But it was an interesting idea in terms of population dynamics, that
39:33.160 --> 39:37.960
the population in fact, as von Forrester hinted at in the title of the
39:37.960 --> 39:43.740
paper of science, that this is in fact an explosion in infinite time
39:43.740 --> 39:44.320
of population.
39:45.740 --> 39:49.860
Now, what does that really mean when we say population goes to
39:49.860 --> 39:50.080
infinity?
39:50.260 --> 39:52.300
Well, the universe cannot handle that, no?
39:53.440 --> 39:54.420
What's happening?
39:55.320 --> 39:58.240
Well, this is a little bit of an interesting discussion.
39:58.340 --> 40:00.540
You actually never reach the critical time.
40:00.880 --> 40:04.460
What happens is that the system crashes and new dynamics have to
40:04.460 --> 40:04.820
emerge.
40:05.640 --> 40:11.900
So the history of the population dynamics that was describing the time
40:11.900 --> 40:15.060
evolution of the population after Tc will no longer hold.
40:15.440 --> 40:18.200
It's a little bit like the Big Bang.
40:18.440 --> 40:21.860
If you think about the Big Bang, you could say, hey, the universe
40:21.860 --> 40:23.920
started 13.6 billion years ago.
40:24.500 --> 40:25.500
What was there before?
40:27.840 --> 40:28.520
Who knows?
40:28.720 --> 40:29.560
There was no time.
40:29.860 --> 40:31.620
Time was created after the Big Bang.
40:33.180 --> 40:34.460
There was no space.
40:35.060 --> 40:36.920
So the whole question is wrong.
40:37.280 --> 40:38.640
What was before the Big Bang?
40:38.720 --> 40:39.740
You cannot ask that question.
40:39.780 --> 40:40.440
There was no time.
40:41.060 --> 40:44.980
Similarly, after the critical time, the dynamics of that population
40:44.980 --> 40:47.700
are no longer described by this differential equation.
40:48.060 --> 40:48.840
Does that make sense?
40:51.770 --> 40:58.170
Okay, so it appears, therefore, that yes, when you reach this finite
40:58.170 --> 41:02.230
time singularity or this critical time, the dynamics have to change,
41:02.350 --> 41:02.710
basically.
41:02.870 --> 41:07.430
And new dynamics, or new regime change, to borrow from our U.S.
41:07.510 --> 41:11.870
politicians when they want to change regimes and countries, same idea,
41:11.930 --> 41:13.810
but that is external rather than endogenous.
41:16.430 --> 41:20.890
Okay, so now what did von Forrester calculate based on what he knew
41:20.890 --> 41:22.350
out of curiosity?
41:23.050 --> 41:28.250
He used the data set, fitted the data set to his power law model, and
41:28.250 --> 41:31.590
estimated that the critical time would have been 2026.
41:33.390 --> 41:34.890
So that's a calculation.
41:36.610 --> 41:40.190
Now you could say, okay, we understand the 2026, but what about this
41:40.190 --> 41:40.950
13 November?
41:40.950 --> 41:43.950
Where did that come from?
41:45.010 --> 41:48.570
I mean, certainly the population dynamics was not that precise.
41:49.610 --> 41:51.490
Now I can only offer a conjecture.
41:52.930 --> 41:56.130
Von Forrester was born on November 13, 1911.
41:57.210 --> 42:00.430
And if you do the calculations, it turns out that November 13, 2026
42:00.430 --> 42:01.530
would have been his birthday.
42:02.990 --> 42:06.310
So it gives you a sense of the style of the paper.
42:07.850 --> 42:09.430
He snuck it in science.
42:10.050 --> 42:11.830
You have to give him credit for that.
42:14.190 --> 42:19.990
Okay, so with that, there are still a few questions that were pretty
42:19.990 --> 42:22.230
entertaining that was asked by von Forrester.
42:22.610 --> 42:23.930
It didn't stop here.
42:24.590 --> 42:25.610
There's more to come.
42:27.870 --> 42:33.010
But just to summarize the outcome, he says, hey, look, if you are an
42:33.010 --> 42:36.850
optimist or a pessimist, if you are a pessimist, you are Malthusian by
42:36.850 --> 42:37.390
profession.
42:38.770 --> 42:40.410
Your job is to be negative.
42:41.310 --> 42:43.650
If you're an optimist, you are Malthusian by heart.
42:44.410 --> 42:46.270
That was the conclusion of the paper.
42:47.190 --> 42:48.450
But there is actually more.
42:48.750 --> 42:53.990
So he said, hey, as much as it is fun to use mathematical models to
42:53.990 --> 42:57.890
predict the future, it's equally fun to try to go back in the past.
42:58.270 --> 43:00.030
And he actually asked an interesting question.
43:00.470 --> 43:05.430
He said, if we use the doomsday equation to go back all the way to the
43:05.430 --> 43:08.790
Garden of Eden, how long would it take us?
43:09.310 --> 43:12.050
How long would it take us to go back to a population of two?
43:13.870 --> 43:14.750
Time reversal.
43:15.630 --> 43:16.270
Clever guy.
43:16.690 --> 43:17.870
He didn't miss this point.
43:18.450 --> 43:19.390
He did not miss it.
43:22.710 --> 43:27.170
So he used the original doomsday equation and reversed time and said,
43:27.310 --> 43:30.050
okay, well, what would be the time to reach two?
43:30.150 --> 43:31.090
Adam and Eve, say.
43:31.090 --> 43:34.910
And it turns out that you would need more than 20 billion years.
43:35.990 --> 43:40.630
Now, if the age of the universe is 13.6 billion years, we don't know
43:40.630 --> 43:46.330
what's happening between 13.6 and the creation of the Garden of Eden.
43:47.410 --> 43:48.830
But the question is why?
43:48.950 --> 43:50.310
And he actually had a very good answer.
43:52.010 --> 43:52.950
Nothing misses him.
43:54.550 --> 43:57.350
This is all coming from some 20 data points about the human
43:57.350 --> 43:57.850
population.
43:57.850 --> 44:00.990
Just to show you what 20 data points can do in the hands of Heinz von
44:00.990 --> 44:01.350
Forster.
44:03.110 --> 44:05.350
I'm sure even Wilfred will agree this is impressive.
44:07.430 --> 44:11.630
So he actually says, well, to actually reach a phase where the
44:11.630 --> 44:15.170
carrying capacity depends on the population, you need at least a
44:15.170 --> 44:17.710
population base because you need information storage.
44:18.170 --> 44:21.650
You need actually the population to grow to a sufficient size before
44:21.650 --> 44:24.490
you could invest in scientists, in research.
44:24.870 --> 44:26.090
You need to store food.
44:26.090 --> 44:29.510
You need to be able to have a certain infrastructure in place before
44:29.510 --> 44:33.070
the carrying capacity actually grows as a function of the power law of
44:33.070 --> 44:33.610
the population.
44:34.530 --> 44:38.170
As a project, there are two interesting things that we could ask
44:38.170 --> 44:39.590
before we go next.
44:40.550 --> 44:43.570
First, how stable are those calculations of critical times?
44:44.330 --> 44:48.330
This is an important problem because basically what you are saying is
44:48.330 --> 44:52.350
that if you have a system, if you have any dynamical system that is
44:52.350 --> 44:56.410
growing as a power law with an exponent that is bigger than 1, this
44:56.410 --> 44:59.370
system will experience a crash, a finite time singularity.
45:00.990 --> 45:03.750
But how stable is the calculation of this finite time singularity?
45:03.770 --> 45:06.990
If I give you 10 points, if I give you 20 points, do I get the same
45:06.990 --> 45:11.250
answer or does this critical time seem to be a very moving target?
45:13.630 --> 45:17.670
One of the things that we would like to explore together is... I've
45:17.670 --> 45:21.330
put the data set of von Forrester and more recent updates up to 1998
45:21.330 --> 45:26.050
on the web and so we can actually play around fitting models like the
45:26.050 --> 45:31.050
one of von Forrester and see if we start adding every year from 1998
45:31.050 --> 45:35.550
or even starting from 1960 we start adding incrementally year by year
45:35.550 --> 45:40.370
information about the population and we track how well do we reach the
45:40.370 --> 45:40.810
critical...
45:40.810 --> 45:42.310
how stable is the critical time?
45:44.110 --> 45:45.450
Is it a moving target?
45:45.610 --> 45:48.130
Every time we add a little bit of data points, the critical time also
45:48.130 --> 45:48.470
shifts.
45:48.470 --> 45:51.230
It becomes almost like chasing ghosts.
45:53.010 --> 45:55.130
So we are going to do a little bit of that.
45:55.210 --> 45:57.870
But we're going to do this not just using von Forrester's equation but
45:57.870 --> 45:58.950
also some other approaches.
46:00.850 --> 46:06.330
Now the topic of von Forrester was picked up by another physicist by
46:06.330 --> 46:10.590
the name of Didier Sornet and Didier Sornet has worked with the
46:10.590 --> 46:14.190
European Space Agency for a while and was looking at material failure
46:14.190 --> 46:19.910
because they didn't want a repeat of NASA's explosion in 1986.
46:21.690 --> 46:25.950
So basically, Didier Sornet has looked at material failure for a while
46:25.950 --> 46:30.190
and noticed that material failure especially fatigue, shear crack
46:30.190 --> 46:37.070
propagation in materials tend to grow as a power law too but there are
46:37.070 --> 46:38.750
some interesting oscillations around it.
46:38.870 --> 46:43.710
And then when he revisited von Forrester's paper, he fitted a power
46:43.710 --> 46:48.330
law to the data set that he had and that was up to... well, the paper
46:48.330 --> 46:50.490
was published in 2001, so probably up to 1999.
46:51.770 --> 46:55.010
And he found that this critical time that von Forrester has calculated
46:55.010 --> 46:57.090
as 2020 suddenly becomes 2030.
46:57.590 --> 47:00.330
So that's the first hint that it could be a moving target.
47:01.350 --> 47:05.750
But more important, more important for our analysis now is that the
47:05.750 --> 47:09.290
residuals around this power law are not random.
47:09.570 --> 47:13.670
Now if you've ever taken a course in regression and you do a
47:13.670 --> 47:16.390
regression, the first thing you check is whether the residuals are
47:16.390 --> 47:18.310
random.
47:18.670 --> 47:21.030
Do they pass a certain randomness test?
47:23.950 --> 47:29.030
But Sornet seems to have noticed that these residuals are clearly
47:29.030 --> 47:29.490
structured.
47:29.790 --> 47:31.450
They seem to actually be even periodic.
47:33.610 --> 47:39.330
If you stare at this graph for a while, you almost see some
47:39.330 --> 47:41.650
periodicities around this power law.
47:42.330 --> 47:43.190
Does that make sense?
47:44.650 --> 47:45.530
Why is that?
47:45.690 --> 47:46.430
So he was curious.
47:49.530 --> 47:53.510
He, of course, suggested that, well, there may be a more general
47:53.510 --> 47:57.710
theory then that should be able to predict the power law and the
47:57.710 --> 47:59.090
oscillations around the power law.
48:00.470 --> 48:04.170
Is this part of a more general framework that allows us to actually
48:04.170 --> 48:08.070
see something more about the system than just the power law itself?
48:10.550 --> 48:13.610
And he tried to say, OK, well, can there be a more general theory?
48:13.710 --> 48:15.110
And the answer is a qualified yes.
48:15.450 --> 48:18.570
We will actually try to fine-tune it in the summer school.
48:22.150 --> 48:25.470
OK, so the idea that Sornet did is as follows.
48:26.150 --> 48:30.010
He said, OK, let's take the population, and that seems to scale as a
48:30.010 --> 48:33.110
power law with the critical time minus t raised to some exponent.
48:34.050 --> 48:38.030
Then he asked, what happens if this exponent is complex, meaning it
48:38.030 --> 48:40.230
has a real part and an imaginary part?
48:41.610 --> 48:43.710
Now, this did not come just out of the blues.
48:43.890 --> 48:46.950
He knew what he was looking for, and I will tell you why he knew what
48:46.950 --> 48:47.490
he was looking for.
48:47.530 --> 48:51.850
But let's proceed that, yes, maybe a complex exponent is a more
48:51.850 --> 48:55.210
general description than just assuming the exponent is real.
48:58.570 --> 49:01.910
He didn't say why it was complex, just to be clear.
49:02.150 --> 49:06.850
He did not offer an explanation why he would choose this exponent here
49:06.850 --> 49:07.370
to be complex.
49:07.490 --> 49:11.510
But let's humor him a little bit, and we proceed with the derivation
49:11.510 --> 49:13.050
that he had and assume it complex.
49:13.390 --> 49:14.470
What would that take us?
49:15.350 --> 49:18.630
So the idea is that you take this critical time minus t raised to this
49:18.630 --> 49:23.090
power z, and you break this z into a real part and an imaginary part,
49:23.230 --> 49:26.130
the real part being beta, the imaginary part being omega, and i
49:26.130 --> 49:27.310
squared here is minus 1.
49:29.390 --> 49:35.110
Then what you do is you write this quantity here as exponent of the
49:35.110 --> 49:36.070
log of that quantity.
49:37.030 --> 49:40.050
Basically, we have just used the obvious fact that if you have a
49:40.050 --> 49:44.330
number, you take its logarithm, then you exponentiate the outcome, you
49:44.330 --> 49:45.810
get back the same number.
49:49.280 --> 49:53.520
5, log 5, e to the log 5, you get back 5.
49:55.120 --> 49:57.900
Then what he did is he used, of course, the well-known property of the
49:57.900 --> 50:01.680
log that log x to the a is a log x.
50:02.640 --> 50:05.780
And then he basically separated this a into its real part and its
50:05.780 --> 50:09.500
imaginary part, so you have the beta log of tc minus d and the i omega
50:09.500 --> 50:10.600
log of tc minus d.
50:10.980 --> 50:12.200
This is all in the exponent.
50:13.760 --> 50:15.400
Then you notice something else.
50:16.940 --> 50:22.980
That e to the a plus b can be written as e to the a times e to the b.
50:24.700 --> 50:26.340
And that's what we did in this step.
50:26.580 --> 50:31.320
So now beta log of tc minus d plus i omega log of tc minus d can be
50:31.320 --> 50:36.680
written as e to the beta log tc minus d multiplied by exponential of i
50:36.680 --> 50:37.980
omega log of tc minus d.
50:38.040 --> 50:41.920
So all I have just done here is simply use the property that e to the
50:41.920 --> 50:44.280
a plus b is e to the a times e to the b.
50:46.180 --> 50:50.340
And then what he did is he brought the beta up here, so that's this
50:50.340 --> 50:53.960
part, and he used the well-known formula that e to the i theta is
50:53.960 --> 50:55.680
cosine theta plus i sine theta.
50:55.680 --> 50:57.640
And that's what you have here.
50:59.380 --> 51:00.120
So far so good?
51:01.040 --> 51:04.020
That's a quick review of complex numbers in case you forgot them.
51:04.780 --> 51:08.620
That's why I decided to put the steps rather than just show you
51:08.620 --> 51:09.360
quickly the result.
51:11.780 --> 51:15.600
Of course, we are seeing the population in real time, so we don't see
51:15.600 --> 51:16.480
the imaginary part.
51:16.560 --> 51:18.120
We just see only the real part.
51:18.220 --> 51:21.600
So we take the projection of the solution on the real axis, and
51:21.600 --> 51:25.500
basically you get rid of the i sine omega t, but the cosine omega of
51:25.500 --> 51:27.100
log tc minus t will survive.
51:28.380 --> 51:32.160
So in this case, if you have a population that is growing as a power
51:32.160 --> 51:37.680
law with an exponent z that is complex, the solution still has a power
51:37.680 --> 51:44.640
law component, but it will be multiplied by the cosine of omega log of
51:44.640 --> 51:45.360
tc minus t.
51:45.760 --> 51:49.620
And that is what is generating the log-periodic oscillations around
51:49.620 --> 51:51.980
that power law, according to Sornet.
51:52.960 --> 51:54.060
So far so good?
51:54.820 --> 51:58.280
Are you falling asleep or this is exciting enough?
52:01.440 --> 52:07.160
So with this, out of curiosity, I plotted some graphs to show you that
52:07.160 --> 52:13.580
yes, the power law, of course, dominates a big chunk of the time
52:13.580 --> 52:16.140
series when you plot population as a function of time.
52:16.200 --> 52:19.860
But when you plot the population on the log axis and tc minus t on the
52:19.860 --> 52:23.020
log axis, you already see these bumps that are coming from the log
52:23.020 --> 52:23.960
-periodic oscillations.
52:25.100 --> 52:27.940
One of the fun projects would be actually to go back and see did they
52:27.940 --> 52:33.320
occur during certain periods of time when there was climate change,
52:34.260 --> 52:35.720
the plague, whatever.
52:36.800 --> 52:39.160
Do these oscillations make sense?
52:39.820 --> 52:40.820
The fit is not bad.
52:41.200 --> 52:42.060
I've checked it, actually.
52:43.200 --> 52:48.100
So one of the project ideas that we could explore together is whether
52:48.100 --> 52:53.080
if we use Johansson and Sornet's log-periodic equations, we can also
52:53.080 --> 52:54.280
calculate the critical time.
52:54.500 --> 52:58.880
So is this extra information about log-periodic oscillations improving
52:58.880 --> 53:01.240
the fit, stabilizing a little bit the calculation of this critical
53:01.240 --> 53:01.940
time or not?
53:02.380 --> 53:03.440
So that's one thing.
53:03.540 --> 53:06.820
We are adding more parameters, of course, so there is a penalty to pay
53:06.820 --> 53:09.740
by having more tunable knobs.
53:10.740 --> 53:15.700
But maybe the tc that comes out of these calculations is more stable.
53:15.700 --> 53:19.000
So that's one of the projects.
53:19.140 --> 53:22.180
And I put some MATLAB script that does some nonlinear fitting.
53:23.120 --> 53:26.680
So in case you would like to pursue this, there is already some data
53:26.680 --> 53:29.340
set and some machinery to start the calculations.
53:30.720 --> 53:33.960
Another question that would be interesting to ask from Johansson and
53:33.960 --> 53:37.400
Sornet that they missed about von Forster's paper is what happens when
53:37.400 --> 53:38.760
we go back backwards in time?
53:38.920 --> 53:41.260
What do we calculate when we have log-periodic oscillations?
53:42.920 --> 53:45.640
It will be also interesting to repeat von Forster's calculations
53:45.640 --> 53:47.260
without the assumption of large population.
53:47.640 --> 53:48.680
What do we get out of that?
53:49.080 --> 53:53.740
So these are some ideas that would be an acceptable project,
53:55.500 --> 53:58.580
especially if you want to practice your skills in fitting data
53:58.580 --> 54:00.820
analysis and models and looking at residuals.
54:01.460 --> 54:02.980
So that's why I put also this.
54:03.100 --> 54:07.360
In case you are looking for some ideas you want to experiment with,
54:08.140 --> 54:09.080
the data is there.
54:10.640 --> 54:15.380
Now, to summarize the main point of the analysis of Sornet's paper is
54:15.380 --> 54:18.340
that complex exponents do generate log-periodic oscillations around
54:18.340 --> 54:22.700
the power law and they tend to intensify in amplitude and frequency as
54:22.700 --> 54:23.960
this critical time is approached.
54:24.980 --> 54:28.700
And that's kind of the message from Sornet's paper and in fact, he
54:28.700 --> 54:35.260
repeated this message, almost the same message, in about Avogadro's
54:35.260 --> 54:36.420
number of papers later.
54:37.660 --> 54:42.660
Perhaps the one that is most common is from the Proceedings of the
54:42.660 --> 54:43.680
National Academy of Sciences.
54:44.260 --> 54:49.940
At that time, PNAS had a whole special issue about what they call the
54:49.940 --> 54:51.600
signs of disasters, basically.
54:52.120 --> 54:55.460
And from Sornet's paper, I have a couple of graphs.
54:55.760 --> 55:00.080
One of them shows, for example, the energy release as a function of
55:00.080 --> 55:02.020
time during material rupture.
55:02.160 --> 55:06.780
So if you load a system and you keep loading it until time to failure,
55:06.840 --> 55:10.120
you see that the energy that is being released is actually spiking
55:10.120 --> 55:13.540
like that and it almost looks like it's a regular interval.
55:14.540 --> 55:18.000
And you could almost see some power law beginning to form.
55:18.660 --> 55:24.440
Similarly, stock market crashes, you have typical, almost power law
55:24.440 --> 55:25.820
rise followed by a crash.
55:26.780 --> 55:30.320
And the question is that, can these log-periodic oscillations around
55:30.320 --> 55:35.200
those power laws be a good way to fingerprint, perhaps, collapses in
55:35.200 --> 55:35.940
complex systems?
55:36.540 --> 55:39.140
And hence, the estimation of this critical time is telling you how
55:39.140 --> 55:42.280
much lead time do you have to make a decision to reverse the dynamics
55:42.280 --> 55:44.980
or do something different before the system crashes.
55:46.480 --> 55:50.240
So that's part of the analysis that we can do with the human
55:50.240 --> 55:50.700
population.
55:50.860 --> 55:54.860
But there are many, many things that one can do, especially any system
55:54.860 --> 55:58.260
that seems to be exhibiting very rapid rise, faster than exponential,
55:58.720 --> 56:03.160
with some oscillations around it, may actually be amenable to the
56:03.160 --> 56:05.100
general treatment that Sornet has proposed.
56:05.680 --> 56:08.520
And that was the message of this proceedings of the National Academy
56:08.520 --> 56:08.920
paper.
56:12.000 --> 56:14.620
One more thing, because we are going to talk a lot about coherent
56:14.620 --> 56:17.520
structures, and I would like to also show you that this type of
56:17.520 --> 56:21.220
insights also allows you to understand something about fluid
56:21.220 --> 56:21.680
mechanics.
56:23.260 --> 56:26.540
So this is nothing but what is called the vorticity equation in fluid
56:26.540 --> 56:26.960
mechanics.
56:27.380 --> 56:31.260
The vorticity is the curl of the velocity, but it is a useful quantity
56:31.260 --> 56:32.200
to look at.
56:32.300 --> 56:34.920
It is actually often used as a measure of coherency.
56:35.700 --> 56:38.980
If a vortex is coherent, it has a big vorticity.
56:40.460 --> 56:43.440
And basically, the vorticity equation has a time rate of change that
56:43.440 --> 56:43.920
is local.
56:44.220 --> 56:45.780
There is an effective time rate of change.
56:46.140 --> 56:49.300
There is an interaction between the vorticity and the velocity
56:49.300 --> 56:53.060
gradient, and then there is a viscous diffusion of vorticity, where nu
56:53.060 --> 56:54.000
here is the viscosity.
56:54.480 --> 56:56.680
This is internal friction in the case of fluids.
56:58.100 --> 57:02.140
Now, typically these two terms are combined together in fluid
57:02.140 --> 57:08.360
mechanics to what is called a local rate of change and an effective
57:08.360 --> 57:11.120
rate of change that is known sometimes as the material derivative or
57:11.120 --> 57:12.980
the substantial derivative or the total derivative.
57:14.660 --> 57:16.160
This term is the viscous diffusion.
57:16.280 --> 57:17.320
We're not going to tinker with it.
57:17.400 --> 57:19.340
And this term looks like an interesting term.
57:19.420 --> 57:22.760
It's a new term that arises from manipulating the Navier-Stokes
57:22.760 --> 57:25.100
equations in a way to get rid of the pressure.
57:25.300 --> 57:26.920
Notice that there is no pressure in this equation.
57:27.100 --> 57:30.500
So if you are a little bit aware of Navier-Stokes, one of the
57:30.500 --> 57:34.560
advantages of vorticity methods is that you bypass the need to compute
57:34.560 --> 57:34.980
pressures.
57:35.380 --> 57:38.560
And pressures are more tricky because there's, you know, hey, if you
57:38.560 --> 57:43.000
go to the bathroom and you open the door very quickly, the water in
57:43.000 --> 57:44.740
the toilet will oscillate.
57:44.880 --> 57:46.040
It's that non-local.
57:48.760 --> 57:52.760
And in fact, in turbulence, this is the term that is responsible to
57:52.760 --> 57:53.560
vortex stretching.
57:56.280 --> 58:01.200
But we have been able to eliminate from the Navier-Stokes equations by
58:01.200 --> 58:03.940
writing the vorticity equation of it the pressure term.
58:04.700 --> 58:06.040
So why am I saying all of this?
58:07.240 --> 58:10.880
Especially in connections to finite time singularities.
58:13.140 --> 58:13.920
It's coming.
58:15.760 --> 58:16.620
Here it is.
58:17.320 --> 58:20.820
In the limit of zero viscosity, so you have a frictionless fluid.
58:21.380 --> 58:23.060
This is an interesting limit.
58:23.300 --> 58:25.940
In the limit of zero viscosity, you kick this term out.
58:27.400 --> 58:31.420
Now, if the vorticity is nothing but the curl of the velocity, then it
58:31.420 --> 58:33.060
must be proportional to the velocity gradient.
58:33.760 --> 58:37.160
So I can get rid of this velocity gradient and basically put the
58:37.160 --> 58:37.880
vorticity here.
58:38.100 --> 58:42.560
So now I have an equation that looks like dw dt is proportional to the
58:42.560 --> 58:43.480
vorticity squared.
58:44.020 --> 58:44.840
What does that say?
58:46.600 --> 58:48.280
Explosion of vorticity in finite times.
58:48.380 --> 58:51.320
Now, of course, this is much more correct in the limit of two
58:51.320 --> 58:54.880
dimensions rather than three dimensions, but the idea is there.
58:55.040 --> 58:59.160
The insights that you get from analyzing equations, now that you have
58:59.160 --> 59:04.140
some perspective about what to look for, you could see why Euler's
59:04.140 --> 59:07.900
equations, which are the inviscid equations that describe vorticity,
59:09.440 --> 59:11.900
do exhibit singularities.
59:13.660 --> 59:18.580
Especially in 2D, where this connection, this term becomes
59:18.580 --> 59:21.980
proportional to the vorticity itself.
59:22.520 --> 59:24.620
This is being debated, by the way, by the mathematicians.
59:24.720 --> 59:29.580
There's a lot of debate going on in pure math whether the Euler
59:29.580 --> 59:33.000
equations do exhibit this type of singularity, but I just want to show
59:33.000 --> 59:38.220
you, without knowing anything, just basic definitions, you could
59:38.220 --> 59:42.440
reason that the vorticity equation might exhibit singularities.
59:44.480 --> 59:46.320
So that's the reason I bring this up.
59:46.520 --> 59:50.520
Just to show you one concrete example that we're going to see similar
59:50.520 --> 59:54.080
equations in this course, that if you do a scaling analysis on them,
59:54.120 --> 59:57.960
they do exhibit some information about perhaps finite time
59:57.960 --> 01:00:00.160
singularities and limits of zero viscosities.
01:00:00.500 --> 01:00:02.260
And these are the good limits to try to understand.
01:00:03.180 --> 01:00:03.820
See what happens.
01:00:05.500 --> 01:00:09.020
There is one more idea and then I will stop barraging you with ideas
01:00:09.020 --> 01:00:10.960
on methods of analysis.
01:00:11.140 --> 01:00:15.080
There is some interesting connection, and this will come a lot in
01:00:15.080 --> 01:00:19.980
Stefano Manzoni's talk, on critical points in complex systems.
01:00:20.440 --> 01:00:25.360
Now, if you have a population that is growing as a power law, the
01:00:25.360 --> 01:00:27.800
derivative of that population can be easily computed.
01:00:27.960 --> 01:00:32.000
And you see what happens is that as t approaches tc, not just that you
01:00:32.000 --> 01:00:35.720
have explosion of the state, but the derivative is also ill-defined.
01:00:35.980 --> 01:00:37.080
It's going to go to infinity.
01:00:38.720 --> 01:00:42.320
And basically, that sets the stage for what we call critical points.
01:00:42.460 --> 01:00:44.600
So what are critical points or stationary points?
01:00:44.820 --> 01:00:47.760
They are points where the derivative is either 0 or ill-defined.
01:00:50.460 --> 01:00:51.580
That's a critical point.
01:00:51.940 --> 01:00:54.900
Now, often than not, you typically associate critical points with
01:00:54.900 --> 01:00:56.740
maximas and minimas because they are 0.
01:00:57.100 --> 01:01:01.980
But there is a whole slew of other possibilities for critical points.
01:01:02.020 --> 01:01:05.320
And just for an illustration, I sketched a function that looks like
01:01:05.320 --> 01:01:05.660
this.
01:01:06.420 --> 01:01:10.060
So if y varies with x, and you're interested in the interval a to b,
01:01:11.360 --> 01:01:15.540
this function has a local maximum, so that's a critical point.
01:01:15.660 --> 01:01:16.680
It has a local minimum.
01:01:17.040 --> 01:01:19.380
That's also a critical point.
01:01:20.040 --> 01:01:21.320
This point is interesting.
01:01:21.500 --> 01:01:23.320
It looks like the derivative there is not defined.
01:01:23.480 --> 01:01:24.140
I have a corner.
01:01:25.080 --> 01:01:26.340
So what is the derivative there?
01:01:28.160 --> 01:01:30.560
The Italians have a word for it, bow.
01:01:33.160 --> 01:01:34.940
And this is a critical point.
01:01:35.460 --> 01:01:36.960
This is a critical point, too.
01:01:37.240 --> 01:01:38.300
Not a local minimum.
01:01:38.660 --> 01:01:39.580
Not a local maximum.
01:01:39.840 --> 01:01:41.020
But the derivative there is 0.
01:01:41.900 --> 01:01:45.040
And you might say, well, that is really a very boring point.
01:01:45.860 --> 01:01:46.320
Is it?
01:01:48.920 --> 01:01:51.640
Would I put it on the board if it was a boring point?
01:01:51.780 --> 01:01:55.000
I mean, honestly, what is so exciting about it?
01:01:57.320 --> 01:01:58.540
The suspense is building.
01:02:00.580 --> 01:02:02.940
Well, a new concept now.
01:02:03.700 --> 01:02:04.640
Critical slowdown.
01:02:05.820 --> 01:02:08.120
This point exhibits critical slowdown.
01:02:08.220 --> 01:02:09.520
What do we mean by critical slowdown?
01:02:10.200 --> 01:02:12.660
Remember the linear stability analysis I mentioned?
01:02:13.380 --> 01:02:18.280
What happens when the derivative goes to 0 that linear stability
01:02:18.280 --> 01:02:22.440
theory chokes or starts showing its own signs of wear and tear?
01:02:24.220 --> 01:02:27.440
Well, critical slowdown is one such idea.
01:02:28.020 --> 01:02:32.180
And we'll discuss in a minute why it's an important point for crashes
01:02:32.180 --> 01:02:32.720
of systems.
01:02:33.380 --> 01:02:36.240
But first, conceptually, what is a critical point?
01:02:37.200 --> 01:02:39.440
We show that the critical point basically is a point where the
01:02:39.440 --> 01:02:41.640
derivative is not defined or it is 0.
01:02:41.720 --> 01:02:46.020
So in this case, if you have a dynamical system where dx dt decreases
01:02:46.020 --> 01:02:50.100
as minus x cubed, surely equilibrium is at x equals 0, right?
01:02:50.980 --> 01:02:53.040
Steady state or equilibrium is at x equals 0.
01:02:53.880 --> 01:02:59.580
The derivative of minus x cubed at x equals 0 is 0 too.
01:03:00.080 --> 01:03:03.940
So already this is telling us that linear stability analysis might run
01:03:03.940 --> 01:03:05.280
into some issues.
01:03:07.980 --> 01:03:10.980
So how do we go about determining the stability of this point?
01:03:11.180 --> 01:03:13.320
Well, the easiest is a graphical way.
01:03:14.440 --> 01:03:20.200
So what you do is you look at the dx dt when x is negative, and you
01:03:20.200 --> 01:03:23.340
find that dx dt is in fact positive, so it is heading towards the
01:03:23.340 --> 01:03:24.000
equilibrium point.
01:03:24.400 --> 01:03:27.820
You also look at dx over dt on the positive side, and so when x is
01:03:27.820 --> 01:03:31.100
positive, dx dt is negative, so it is moving again towards the
01:03:31.100 --> 01:03:31.760
equilibrium point.
01:03:32.120 --> 01:03:34.720
So it looks like things are converging to this equilibrium point.
01:03:34.780 --> 01:03:37.140
So this equilibrium point is likely to be stable.
01:03:38.240 --> 01:03:40.200
But that's not why I'm putting this on the board.
01:03:40.600 --> 01:03:43.960
This is just to show you that graphically you could figure out the
01:03:43.960 --> 01:03:46.960
outcome of stability analysis without doing linear stability theory.
01:03:48.920 --> 01:03:52.480
The reason I'm putting this point is that if you remember from our
01:03:52.480 --> 01:03:56.640
analysis about relaxation timescales, what happens when you perturb
01:03:56.640 --> 01:03:59.740
something from equilibrium and you're watching how it relaxes back to
01:03:59.740 --> 01:04:00.100
equilibrium.
01:04:00.420 --> 01:04:02.860
Now linear stability theory, if you applied it blindly, would have
01:04:02.860 --> 01:04:06.100
predicted that the relaxation time is 1 over f prime of p star, if you
01:04:06.100 --> 01:04:06.380
remember.
01:04:06.940 --> 01:04:12.480
But this quantity in this particular system is 0.
01:04:13.560 --> 01:04:19.580
So 1 over 0 is so it takes infinite amount of time to relax to
01:04:19.580 --> 01:04:19.960
equilibrium.
01:04:20.180 --> 01:04:20.820
That's not true.
01:04:21.480 --> 01:04:23.020
Actually it turns out to be finite.
01:04:24.120 --> 01:04:27.240
If you solve this ordinary differential equation, you find that the
01:04:27.240 --> 01:04:30.300
solution, x of t, scales as t to the minus 1 half.
01:04:30.840 --> 01:04:32.820
And that's a very slow approach time.
01:04:33.420 --> 01:04:36.200
So yes, you will reach equilibrium if time goes to infinity.
01:04:37.680 --> 01:04:38.900
But very slowly.
01:04:39.260 --> 01:04:42.220
So that's known as critical slowdown.
01:04:43.120 --> 01:04:45.940
Because the approach to equilibrium is so slow.
01:04:45.940 --> 01:04:52.260
And there were lots of papers in magazines like Nature and Science
01:04:52.260 --> 01:04:58.340
using this whole idea of perhaps critical slowdown may be an early
01:04:58.340 --> 01:05:00.300
warning signal for critical transitions.
01:05:00.700 --> 01:05:05.600
And you know, Martin Sheffer and many others, Max Riedkirk, Stephen
01:05:05.600 --> 01:05:10.480
Carpenter and so forth, have harped a lot on this idea that perhaps
01:05:10.480 --> 01:05:15.520
near phase transitions, you may be experiencing critical slowdown.
01:05:16.260 --> 01:05:20.460
Phase transitions, regime change, you know, you get the picture.
01:05:25.620 --> 01:05:29.640
One more idea, and I'm going to go through this quickly, is what is
01:05:29.640 --> 01:05:31.080
called discrete scale invariance.
01:05:32.020 --> 01:05:36.480
This idea comes about that when you look at the exponent and it's
01:05:36.480 --> 01:05:41.880
complex, you find that the energy pulses in P are going to come when
01:05:41.880 --> 01:05:43.960
this cosine part is maximum.
01:05:43.960 --> 01:05:48.480
And those happen to be in discrete integers of 2pi.
01:05:49.400 --> 01:05:53.080
So that's why the pulses in the population or material failure or
01:05:53.080 --> 01:05:55.300
whatever, tend to have a unique scaling.
01:05:55.420 --> 01:05:59.360
In other words, the critical time minus any arbitrary time tn seems to
01:05:59.360 --> 01:06:01.820
scale as lambda to the minus n where n is integer.
01:06:02.120 --> 01:06:06.260
So basically these buildup and crash, buildup and crash, buildup and
01:06:06.260 --> 01:06:08.720
crash as the power law is approaching are quantized.
01:06:08.840 --> 01:06:10.640
They're not everywhere.
01:06:10.780 --> 01:06:15.660
They're actually happening at quantized times, and this is why this
01:06:15.660 --> 01:06:17.960
whole analysis is called discrete scale invariance.
01:06:18.840 --> 01:06:21.540
We're not going to go through a lot of this stuff, but I should say
01:06:21.540 --> 01:06:23.940
that there is a very interesting theoretical connection to
01:06:24.460 --> 01:06:27.160
renormalized group theory and discrete scale invariance.
01:06:27.440 --> 01:06:28.660
But we're not going to touch it.
01:06:29.120 --> 01:06:31.860
It does suggest that if you have a system where you have microscopic
01:06:33.300 --> 01:06:36.160
variabilities, you could coarse-grain them, come up with a macroscopic
01:06:36.160 --> 01:06:38.560
relation, and jump to the next scale and the next scale and the next
01:06:38.560 --> 01:06:41.820
scale, and you could do that indefinitely if you are guaranteed that
01:06:41.820 --> 01:06:44.040
the system exhibits discrete scale invariance.
01:06:45.020 --> 01:06:47.480
And that has been used in turbulence as well.
01:06:49.200 --> 01:06:55.100
So finally, how could complex exponents arise?
01:06:55.840 --> 01:07:00.640
In a separate paper by Ede and Sornet, this was to look at financial
01:07:00.640 --> 01:07:05.180
crashes, they proposed a dynamical model that has an acceleration
01:07:05.180 --> 01:07:07.960
term, yeah, d2x dt squared, this looks like an acceleration.
01:07:09.020 --> 01:07:13.480
And it seems to be related to a velocity that is a power law, and the
01:07:13.480 --> 01:07:15.720
state that is a power law, and it's looking at the difference.
01:07:16.240 --> 01:07:20.660
And it turns out that under some conditions, this dynamical system
01:07:20.660 --> 01:07:26.840
does exhibit complex exponents and finite time singularities.
01:07:27.120 --> 01:07:31.280
So, another project that, if you like to dig into this stuff a little
01:07:31.280 --> 01:07:35.720
bit deeper and capitalize on Wilfred Conrad's expertise and John
01:07:35.720 --> 01:07:41.340
Finnegan's expertise, and to build on Hoppe's idea that it's not just
01:07:41.340 --> 01:07:45.960
the students who should be experiencing new ideas outside the
01:07:45.960 --> 01:07:47.520
research, also the instructors.
01:07:49.040 --> 01:07:52.960
If you are interested in actually doing a formal analysis to look at
01:07:52.960 --> 01:07:56.980
how the system behaves, what happens when alpha is 0 and gamma is 0?
01:07:57.140 --> 01:07:59.620
Well, that's obviously 2x dt squared is 0.
01:08:00.160 --> 01:08:03.380
What happens if alpha is finite but gamma is 0?
01:08:03.420 --> 01:08:06.660
What happens when m is large relative to n?
01:08:06.760 --> 01:08:09.700
So there is a whole slew of possibilities to explore theoretically
01:08:09.700 --> 01:08:15.140
about this dynamical system and see under what conditions, what ranges
01:08:15.140 --> 01:08:19.140
of m and n you would see that will give you basically log-periodic
01:08:19.140 --> 01:08:20.220
oscillations and so forth.
01:08:20.320 --> 01:08:23.420
And this system does have all the necessary ingredients to do that.
01:08:23.700 --> 01:08:26.320
So, if you're interested in a project that has a little bit more of a
01:08:26.320 --> 01:08:31.300
theoretical flavor, feel free to do a systematic analysis, what is
01:08:31.300 --> 01:08:32.380
called bifurcation analysis.
01:08:32.580 --> 01:08:36.760
So you try to understand what range of parameters alpha, gamma, m and
01:08:36.760 --> 01:08:40.320
n will give you certain features of the solution.
01:08:41.940 --> 01:08:45.120
Okay, so we are almost running out of time.
01:08:45.360 --> 01:08:47.980
So the salient points to be covered that we have covered in this
01:08:47.980 --> 01:08:50.740
lecture, we've shown that power law acceleration leads to finite time
01:08:50.740 --> 01:08:51.380
singularities.
01:08:51.560 --> 01:08:53.700
This is true in population, in vorticity, etc.
01:08:54.380 --> 01:08:57.220
We have shown that complex exponents can lead to log-periodic
01:08:57.220 --> 01:09:00.820
oscillations that may be used to fingerprint collapses in complex
01:09:00.820 --> 01:09:01.260
systems.
01:09:01.780 --> 01:09:04.720
We have discussed relaxation timescales to equilibria.
01:09:04.860 --> 01:09:07.040
We have shown its connection to dynamical systems.
01:09:07.420 --> 01:09:10.640
We have also defined critical points, critical slowdown, and discrete
01:09:10.640 --> 01:09:13.140
-scale invariance, even though we didn't discuss much of the discrete
01:09:13.140 --> 01:09:13.860
-scale invariance.
01:09:15.000 --> 01:09:17.360
Okay, so this is basically what we have covered in this lecture.
01:09:18.480 --> 01:09:23.280
To conclude, on a philosophical note, we use the statistician George
01:09:23.280 --> 01:09:24.760
Box, very famous statistician.
01:09:25.560 --> 01:09:28.360
Milan, he also spent some time at NC State, if you are curious.
01:09:30.500 --> 01:09:33.740
And basically what George Box says is that all models are wrong,
01:09:33.920 --> 01:09:36.940
because, of course, all models are simplifications of real systems, so
01:09:36.940 --> 01:09:38.160
they are wrong by definition.
01:09:38.980 --> 01:09:42.760
But some are actually useful, so that's basically a good thing to
01:09:42.760 --> 01:09:43.100
remember.
01:09:44.800 --> 01:09:46.880
Okay, so more ideas about projects.
01:09:47.860 --> 01:09:52.600
I've put three papers on the cloud.
01:09:53.180 --> 01:09:57.740
One of them is coming from Gabriele Manoli and Marco Marani from Duke
01:09:57.740 --> 01:09:59.560
University, as well as myself.
01:10:00.620 --> 01:10:04.200
And in it we looked, well, actually Gabriele looked, I was cheering on
01:10:04.200 --> 01:10:06.680
the side more, about the human energy-climate system.
01:10:06.920 --> 01:10:11.260
And the idea that came from there is that we stumbled on a paper way
01:10:11.260 --> 01:10:14.300
back by Didier Sornet that was looking at what is called punctuated
01:10:14.300 --> 01:10:14.740
equilibrium.
01:10:14.920 --> 01:10:17.620
What happens if you have a system that seems to be flat, flat, flat,
01:10:17.700 --> 01:10:19.120
flat, flat, flat, flat?
01:10:19.780 --> 01:10:21.640
And that's known as a punctuated equilibrium.
01:10:22.140 --> 01:10:26.000
And it turns out that delayed differential equations, when they are
01:10:26.000 --> 01:10:28.840
applied to the logistic equation that we have just covered in great
01:10:28.840 --> 01:10:30.920
detail, exhibit exactly that pattern.
01:10:32.120 --> 01:10:36.480
So building on this idea and realizing that the emissions of
01:10:36.480 --> 01:10:41.260
anthropogenic CO2 per capita is exhibiting exactly this stairway
01:10:41.260 --> 01:10:47.600
pattern, Gabriele decided to put this idea along with a simple model
01:10:47.600 --> 01:10:52.760
for CO2 emissions and temperature, zeroth order model of the entire
01:10:52.760 --> 01:10:57.340
climate system, and study the consequence of this type of step-up and
01:10:57.340 --> 01:10:57.960
step -down.
01:10:58.440 --> 01:11:03.180
So in other words, if you think of the step-down as diffusion of green
01:11:03.180 --> 01:11:06.280
technology that are now dropping the consumption per capita, so you
01:11:06.280 --> 01:11:10.180
are doing this in step-down, he tried to figure out what should be the
01:11:10.180 --> 01:11:15.120
diffusion time, which would dictate the step-down, the diffusion time
01:11:15.120 --> 01:11:18.840
to actually meet the Paris Climate Agreement.
01:11:19.860 --> 01:11:23.820
So he did this and actually showed that this build-up of inertia is
01:11:23.820 --> 01:11:28.580
important, and in fact, if the step-up has periods of 60 years before
01:11:28.580 --> 01:11:31.620
you see the next jump and the next jump and the next jump, the step
01:11:31.620 --> 01:11:35.100
-down to reach the Paris Agreement, you would actually need to go down
01:11:35.100 --> 01:11:36.060
at a rate of six years.
01:11:36.320 --> 01:11:40.360
So that's how fast green technology should, at least in this model
01:11:40.360 --> 01:11:41.320
framework, should go.
01:11:41.680 --> 01:11:44.460
So if you're interested in pursuing this idea a little bit more, I've
01:11:44.460 --> 01:11:47.620
put the data set that Gabriele has assembled to run these types of
01:11:47.620 --> 01:11:51.920
models, and they're all on the cloud with the MATLAB code that repeats
01:11:51.920 --> 01:11:53.280
all the graphs in the paper.
01:11:54.700 --> 01:11:58.220
Other questions that pop up, of course, what happens if you change a
01:11:58.220 --> 01:11:59.100
little bit the formulations?
01:11:59.240 --> 01:12:02.920
What happens if there is a feedback between temperature and the
01:12:02.920 --> 01:12:03.520
emission rates?
01:12:03.800 --> 01:12:07.820
What happens if you add a mortality term because of warming causing
01:12:07.820 --> 01:12:08.600
more mortalities?
01:12:09.340 --> 01:12:13.280
So the opportunities of looking at a set of questions are almost
01:12:13.280 --> 01:12:13.940
infinite.
01:12:13.940 --> 01:12:15.780
But it is all there.
01:12:16.000 --> 01:12:17.860
It runs, it repeats the graphs in the paper.
01:12:18.540 --> 01:12:24.680
The model is detailed quite well and have fun if you want to learn
01:12:24.680 --> 01:12:26.620
also about logistic equations with delays.
01:12:26.840 --> 01:12:29.460
Logistic equations with delays are worse than partial differential
01:12:29.460 --> 01:12:30.760
equations, just so that you know.
01:12:31.180 --> 01:12:33.580
So there is no way you could do anything analytical with them.
01:12:34.580 --> 01:12:37.780
The other example that I've put is the Easter Island example.
01:12:38.320 --> 01:12:43.200
This was popularized by the book of Jared Diamond, Collapse, and it's
01:12:43.200 --> 01:12:44.120
a two-equation model.
01:12:44.860 --> 01:12:48.200
And the reason I picked this example is that this is... I don't know
01:12:48.200 --> 01:12:50.300
how many of you know about the Easter Island Collapse.
01:12:50.420 --> 01:12:51.680
Easter Island is a small island.
01:12:51.820 --> 01:12:56.460
It was settled way back by Polynesian settlers that had to travel
01:12:56.460 --> 01:13:00.160
thousands of miles by boat to get to it.
01:13:00.320 --> 01:13:01.920
And it's an isolated system.
01:13:02.080 --> 01:13:06.240
So this is it, you know, humans, environment, no other external
01:13:06.240 --> 01:13:06.640
agents.
01:13:09.140 --> 01:13:10.680
And, of course, the population collapsed.
01:13:10.780 --> 01:13:13.100
And that's a very advanced civilization that settled the Easter
01:13:13.100 --> 01:13:13.320
Island.
01:13:13.660 --> 01:13:15.640
And there are many, many reasons that are discussed.
01:13:16.120 --> 01:13:21.020
One of them is endogenous dynamics between the population and the
01:13:21.020 --> 01:13:21.800
natural resources.
01:13:22.960 --> 01:13:25.880
And it seems to exhibit a boom-bust cycle.
01:13:26.280 --> 01:13:28.400
So that's why I'm bringing it up to your attention.
01:13:28.660 --> 01:13:33.160
It has this interesting feature about the dynamics that you have a
01:13:33.160 --> 01:13:36.540
rapid explosion in the population despite the fact that the resource
01:13:36.540 --> 01:13:39.560
is being diminished extremely fast, faster than the growth of the
01:13:39.560 --> 01:13:39.880
population.
01:13:40.420 --> 01:13:43.420
So Easter Island is there, and the paper is there, and it's very
01:13:43.420 --> 01:13:44.180
clear, by the way.
01:13:44.340 --> 01:13:46.960
It's a very well-written paper by Brander and Taylor.
01:13:47.860 --> 01:13:50.620
That's not the only reason the Easter Island population collapsed.
01:13:50.860 --> 01:13:53.580
There are plenty of other counter-arguments to it.
01:13:53.740 --> 01:13:57.960
But we can certainly explore some of these ideas, whether they make
01:13:57.960 --> 01:14:01.720
sense in terms of the Easter Island information that we have.
01:14:01.980 --> 01:14:05.780
The population of Easter Island was reasonably reconstructed, and the
01:14:05.780 --> 01:14:09.540
vegetation amounts were reasonably reconstructed from many other
01:14:09.540 --> 01:14:10.400
pieces of information.
01:14:10.620 --> 01:14:12.260
So we can certainly look into it.
01:14:12.300 --> 01:14:13.740
We can also try to ask other questions.
01:14:13.800 --> 01:14:15.260
What happens if there is a climate anomaly?
01:14:15.480 --> 01:14:19.100
Would that have accelerated or decelerated the dynamics of the Easter
01:14:19.100 --> 01:14:20.400
Island population?
01:14:21.140 --> 01:14:25.340
The third one is a little bit more open, but John Finnegan has already
01:14:25.340 --> 01:14:26.120
thought about that.
01:14:26.640 --> 01:14:29.320
So that's why I'm putting it here.
01:14:31.060 --> 01:14:35.560
Joe Tainter has also written many papers and books about the collapse
01:14:35.560 --> 01:14:39.260
of complex societies, and not just human societies but also ant
01:14:39.260 --> 01:14:39.640
societies.
01:14:39.960 --> 01:14:44.160
That's why some people actually call the outcome of Joe Tainter's work
01:14:44.160 --> 01:14:45.440
as Tainter's Law.
01:14:47.100 --> 01:14:50.840
So if you Google Tainter's Law, you get actually a bunch of papers.
01:14:52.640 --> 01:14:54.380
It's a very interesting work.
01:14:54.440 --> 01:14:59.640
He actually studied how complexity arises in ant colonies as well as
01:14:59.640 --> 01:15:02.920
in the Roman Empire and many, many other empires, and how this
01:15:02.920 --> 01:15:07.460
addition of complexity reaches a certain phase where adding complexity
01:15:07.460 --> 01:15:10.880
actually doesn't buy you as much return, and that would be the tipping
01:15:10.880 --> 01:15:12.580
point at which the collapse starts.
01:15:14.040 --> 01:15:19.020
So I've put also a paper that describes this mechanism and as a
01:15:19.020 --> 01:15:24.480
project it would be fun to try to write a very simple dynamical system
01:15:24.480 --> 01:15:29.760
that reproduces the main features of Joe Tainter's Law.
01:15:29.760 --> 01:15:34.820
And John has already done some of that, but slightly more elaborate, I
01:15:34.820 --> 01:15:35.000
guess.
01:15:36.220 --> 01:15:41.780
So if we can reduce John's approach to a simpler one, we've achieved
01:15:41.780 --> 01:15:42.320
something too.
01:15:43.160 --> 01:15:44.940
And that could be a fun project as well.
01:15:45.160 --> 01:15:48.080
Any questions?
01:15:51.310 --> 01:15:56.790
So this was primarily intended to give you an overview of dynamical
01:15:56.790 --> 01:16:00.470
systems, but with a problem that I think we all appreciate and
01:16:00.470 --> 01:16:04.070
understand that has nothing to do with turbulence or cavitation in
01:16:04.070 --> 01:16:07.130
plants, even though cavitation in plants, the spread of embolism, does
01:16:07.130 --> 01:16:09.570
follow a logistic equation very similar to what we have covered.