The Seneca Cliff of Energy Production

Published on Cassandra's Legacy on Thursday, January 8, 2015

The graph above was created by Gail Tverberg on her blog "Our Finite World". It is, clearly, another case of what I called the "Seneca Cliff" (from the Roman philospher who said "the road to ruin is rapid). The Seneca Cliff takes this shape, when generated by a system dynamics model:

Gail's forecast of the future of energy production is not the result of a the same model I developed, but the reasons behind the steep decline are the same. Gail explains it in a post of hers as:



All parts of our economy are interconnected. If parts of the economy is becoming increasingly inefficient, more than the cost of production in these parts of the economy are affected; other parts of the economy are affected as well, including wages, debt levels, and interest rates.

Wages are especially being crowded out, because the total amount of goods and services available for purchase in the world economy is growing more slowly. This is not intuitively obvious, unless a person stops to realize that if the world economy is growing more slowly, or actually shrinking, it is producing less. Each worker gets a share of this shrinking output, so it is reasonable to expect inflation-adjusted wages to be stagnating or declining, since a stagnating or declining collection of goods and services is all a person can expect.

At some point, something has to “give”. 


Which is a good description of the mathematical model at the basis of the Seneca cliff idea. The burden on the economy of increasing costs becomes more and more heavy in times of diminishing returns (or, as Gail says, increasing inefficiency, which is the same). At some point, something "gives" and the whole thing comes down. Seneca rules.

Peak pyramids: the way to ruin is rapid

Originally published on Cassandra's Legacy on Friday, December 19, 2014

The graph above is a little exercise in cliodynamics, the attempt of quantitatively modeling historical data. Here, the size of the great Egyptian pyramids is plotted as a function of their approximate building date, taken as the last year of the reign of the Pharaoh associated to them. The data are fitted with a simple Gaussian, which approximates the cycle of the Hubbert model of resource depletion.

The great Egyptian pyramids built during the 3rd millennium BCE are the embodiment of the power and of the wealth of the Egyptian civilization of the time. But why did the Egyptians stop building them? Not lack of interest, apparently, since they kept building pyramids for a long time. But they never built again pyramids on such a giant scale.

Probably, we will never have sufficient data to understand the economics of the Egyptian pyramid building cycle of the 3rd and 4th Egyptian dynasties. But we can try at least to examine the quantitative data we have. So, I went toWikipedia and I found data for the size of pyramids and their approximate dates. The result is the graph above. Here, I show only the data for completed pyramids as a function of the last year of the reign of the Pharaoh associated for each one.

As you can see, it is possible to fit the data with a Gaussian curve, which approximates the Hubbert curve, known to describe the depletion of a limited, non renewable resource. This suggests that the Egyptians had run out of resources, possibly in the form of the fertile soil necessary to sustain the large workforce needed to build pyramids. Or, perhaps, in an age of increasing warring activity, they were forced to funnel more and more resources into the military sector, taking them away from pyramid building.

Another phenomenon we can note in the graph is the rapid collapse of the size of the pyramids at the end of the cycle. The last pyramid of this cycle, the one associated to Pharaoh Menkaure, is even smaller than the first one of the cycle, the "stepped pyramid" of Pharaoh Djoser. Perhaps, this rapid decline is a manifestation of the "Seneca Effect", a term that I coined to describe economic cycles in which decline is faster than growth. Unfortunately, however, the data are too scattered and uncertain to be sure on this point. But surely there was no "plateau" nor a slow decline after the construction of the largest pyramids andit is suggestive to think that even pyramid building may be described with Seneca's words "increases are of sluggish growth, but the way to ruin is rapid."

Seneca cliffs of the third kind: how technological progress can generate a faster collapse

Originally published on Cassandra's Legacy on Monday, December 15, 2014

The image above (from Wikipedia) shows the collapse of the North Atlantic cod stocks. The fishery disaster of the early 1990s was the result of a combination of greed, incompetence, and government support for both. Unfortunately, it is just one of the many examples of how human beings tend to worsen the problems they try to solve. The philosopher Lucius Anneus Seneca had understood this problem already some 2000 years ago, when he said, "It would be some consolation for the feebleness of our selves and our works if all things should perish as slowly as they come into being; but as it is, increases are of sluggish growth, but the way to ruin is rapid."

The collapse of the North Atlantic cod fishery industry gives us a good example of the abrupt collapse in the production of resources - even resources which are theoretically renewable. The shape of the production curve landings shows some similarity with the "Seneca curve", a general term that I proposed to apply to all cases in which we observe a rapid decline of the production of a non renewable, or slowly renewable, resource. Here is the typical shape of the Seneca Curve:

The similarity with the cod landings curve is only approximate, but clearly, in both cases we have a very rapid decline after a slow growth that, for the cod fishery, had lasted for more than a century. What caused this behavior?

The Seneca curve is a special case of the "Hubbert Curve" which describes the exploitation of a non renewable (or slowly renewable) resource in a free market environment. The Hubbert curve is "bell shaped" and symmetric (and it is the origin of the well known concept of "peak oil). The Seneca curve is similar, but it is skewed forward. In general, the forward skewness can be explained in terms of the attempt of producers to keep producing at all costs a disappearing resource.

There are several mechanisms which can affect the curve. In my first note on this subject, I noted how the Seneca behavior could be generated by growing pollution and, later on, how it could be the result of the application of more capital resources to production as a consequence of increasing market prices. However, in the case of the cod fishery, neither factor seems to be fundamental. Pollution in the form of climate change may have played a role, but it doesn't explain the upward spike of the 1960s in fish landings. Also, we have no evidence of cod prices increasing sharply during this phase of the production cycle. Instead, there is clear evidence that the spike and the subsequent collapse was generated by technological improvements.

The effect of new and better fishing technologies is clearly described by Hamilton et al. (2003)

Fishing changed as new technology for catching cod and shrimp developed, and boats became larger. A handful of fishermen shifted to trawling or “dragger” gear. The federal government played a decisive role introducing newtechnology and providing financial resources to fishermen who were willing to take the risk of investing in new gear and larger boats. 
 ...

Fishermen in open boats and some long-liners continued to fish cod, lobster and seal inshore. Meanwhile draggers  and other long-liners moved onto the open ocean, pursuing cod and shrimp nearly year round. At the height of the boom, dragger captains made $350,000–600,000 a year from cod alone. ... The federal government helped finance boat improvements, providing grants covering 30–40% of their cost.
....
By the late 1980s, some fishermen recognized signs of decline. Open boats and long-liners could rarely reach their quotas. To find the remaining cod, fishermen traveled farther north, deployed more gear and intensified their efforts. A few began shifting to alternative species such as crab. Cheating fisheries regulation—by selling unreported catches at night, lining nets with small mesh and dumping bycatch at sea—was said to be commonplace. Large illegal catches on top of too-high legal quotas drew down the resource. Some say they saw trouble coming, but felt powerless to halt it. 

So, we don't really need complicated models (but see below) to understand how human greed and incompetence - and help from the government - generated the cod disaster. Cods were killed faster than they could reproduce and the result was their destruction. Note also that in the case of whaling in the 19th century, the collapse of the fishery was not so abrupt as it was for cods, most likely because, in the 19th century, fishing technology could not "progress" could not be so radical as it was in the 20th century.

The Seneca collapse of the Atlantic cod fishery is just one of the many cases in which humans "push the levers in the wrong directions", directly generating the problem they try to avoid. If there is some hope that, someday, the cod fishery may recover, the situation is even clearer with fully non-renewable resources, such as oil and most minerals. Also here, technological progress is touted as the way to solve the depletion problems. Nobody seems to worry about the fact that the faster you extract it, the faster you deplete it: that's the whole concept of the Seneca curve.

So take care: there is a Seneca cliff ahead also for oil!


____________________________________

A simple dynamic model to describe how technological progress can generate the collapse of the production of a slowly renewable resource; such as in the case of fisheries. 

by Ugo Bardi

Note: this is not a formal academic paper, just a short note to sketch how a dynamic model describing overfishing can be built. See also a similar modeldescribing the effect of prices on the production of a non renewable resource


The basics of a system dynamics model describing the exploitation of a non renewable resource in a free market are described in detail in a 2009 paper byBardi and Lavacchi. According to the model developed in that paper, it is assumed that the non renewable resource (R) exists in the form of an initial stock of fixed extent. The resource stock is gradually transformed into a stock of capital (C) which in turn gradually declines. The behavior of the two stocks as a function of time is described by two coupled differential equations. 
R' = - k1*C*R C' = k2*C*R - k3*C, 
where R' and C' indicate the flow of the stocks as a function of time (R' is what we call "production"), while the "ks" are constants. This is a "bare bones" model which nevertheless can reproduce the "bell shaped" Hubbert curve and fit some historical cases. Adding a third stock (pollution) to the system, generates the "Seneca Curve", that is a skewed forward production curve, with decline faster than growth.  

The two stock system (i.e. without taking pollution into account) can also produce a Seneca curve if the equations above are slightly modified. In particular, we can write:  
R' = - k1*k3*C*R C' = ko*k2*C*R - (k3+k4)*C. 
Here, "k3" explicitly indicates the fraction of capital reinvested in production, while k4 which is proportional to capital depreciation (or any other non productive use). Then, we assume that production is proportional to the amount of capital invested, that is to k3*C. Note how the ratio of R' to the flow of capital into resource creation describes the net energy production (EROI), which turns out to be equal to k1*R. Note also that "ko" is a factor that defines the efficiency of the transformation of resources into capital; it can be seen as related to technological efficiency. 
The model described above is valid for a completely non-renewable resource. Dealing with a fishery, which is theoretically renewable, we should add a growth factor to R', in the form of k5*R. Here is the model as implemented using the Vensim (TM) software for system dynamics. The "ks" have been given explicit names. I am also using the convention of "mind sized models" with higher free energy stocks appearing above lower free energy stocks

If the constants remain constant during the run, the model is the same as the well known "Lotka-Volterra" one. If the reproduction rate is set at zero, the model generates the symmetric Hubbert curve. 

In order to simulate technological progress, the "production efficiency" constant is supposed to double stepwise around mid-cycle. A possible result is the following, which qualitatively reproduces the behavior of the North Atlantic cod fishery. 

Among other things, this result confirms the conclusions of an early paper of mine (2003) on this subject, based on a different method of modeling.

Let me stress again that this is not an academic paper. I am just showing the results of tests performed with simple assumptions for the constants. Nevertheless, these calculations show that the Seneca cliff is a general behavior that occurs when producers stretch out their system allocating increasing fractions of capital to production. Should someone volunteer to give me a hand to make better models, I'd be happy to collaborate!

The Seneca effect in the Easter egg hunt

Originally published on Cassandra's legacy on Saturday, April 19, 2014

For the Easter of 2014, let me repropose my post on the dynamic modelling of the Easter egg hunt which, for some reason, has been the most successful post ever to be published (in 2012) on the former "Cassandra's Legacy" blog, now known as "Resource Crisis." -Happy Easter, everyone!

Here is a little Easter post where I try to model the Easter Egg hunt as if it were the production of a mineral resource. A simple model based on system dynamics turns out to be equivalent to the Hubbert model of oil production. We can have "peak eggs" and the curve may also take the asymmetric shape of the "Seneca Peak." So, even this simple model confirms what the Roman Philosopher told us long ago: that ruin is much faster than fortune. (Image fromuptownupdate)

For those of you who may not be familiar with the Easter Bunny tradition, let me say that, in the US, bunnies lay eggs and not just that: for Easter, they lay brightly colored eggs. The tradition is that the Easter Bunny spreads a number of these eggs in the garden and then it is up to the children to find them. It is a game that children usually love and that can last quite some time if the garden is big and the bunny has been a little mean in hiding the eggs in difficult places.

A curious facet of the Easter Egg hunt is that it looks a little like mineral prospecting. With minerals, just as for eggs, you need to search for hidden treasures and, once you have discovered the easy minerals (or eggs), finding the well hidden ones may take a lot of work. So much that some eggs usually remain undiscovered; just as some minerals will never be extracted.

Now, if searching for minerals is similar to searching for Easter Eggs, perhaps we could learn something very general if we try a little exercise in model building. We can use system dynamics to make a model that turns out to be able to describe both the Easter Eggs search and the common "Hubbert" behavior of mineral production. The exercise can also tell us something on how system dynamics can be used to make "mind sized" models (to use an expression coined by Seymour Papert). So, let's try.

System dynamics models are based on "stocks"; that is amounts of the things you are interested in (in this case, eggs). Stocks will not stay fixed (otherwise it would be a very uninteresting model) but will change with time. We say that stocks (eggs) "flow" from one to another. In this case, eggs start all in the stock that we call "hidden eggs" and flow into the stock that we call "found eggs". Then, we also need to consider another stock: the number of children engaged in the search.

To make a model, we need to make some assumptions. We could say that the number of eggs found per unit time is proportional to the number of children, which we might take as constant. Then, we could also say that it becomes more difficult to find eggs as there are less of them left. That's about all we need for a very basic version of the model.

Those are all conditions that we could write in the form of equations, but here we can use a well known method in system dynamics which builds the equations starting from a graphical version of the model. Traditionally, stocks are shown as boxes and flows as double edged arrows. Single edged arrows relate stocks and flows to each other. In this case, I used a program called "Vensim" by Ventana systems (free for personal and academic use). So, here is the simplest possible version of the Easter Egg Hunt model:

As you see, there are three "boxes," all labeled with what they contain. The two-sided arrow shows how the same kind of stock (eggs) flows from one box to the other. The little butterfly-like thing is the "valve" that regulates the flow. Production depends on three parameters: 1) the ability of the children to find eggs, 2) the number of children (here taken as constant) and 3) the number of remaining hidden eggs. 

The model produces an output that depends on the values of the parameters. Below, there are the results for the production flow for a run that has 50 starting eggs, 10 children and an ability parameter of 0.006. Note that the number of eggs is assumed to be a continuous function. There are other methods of modeling that assume discrete numbers, but this is the way that system dynamics works.

Here, production goes down to nearly zero, as the children deplete their egg reservoir. In this version of the model, we have robot-children who continue searching forever and, eventually, they'll always find all the eggs. In practice, at some moment real children will stop searching when they become tired. But this model may still be an approximate description of an actual egg hunt when there is a fixed number of children - as it is often the case when the number of children is small.

But can we make a more general model? Suppose that there are many children and that not all of them get tired at the same time. We may assume that they drop out of the hunt simply at random. Then, can we assume that the game becomes so interesting that more children will be drawn in as more eggs are found? That, too can be simulated. A simple way of doing it is to assume that the number of children joining the search is proportional to the number of eggs found (egg production). Here is a model with these assumptions. (note the little clouds: they mean that we don't care about the size of the stocks where the children go or come from) 

This model is a little more complex but not so much. Note that there are two new constants "k1" and "k2" used to "tune" the sensitivity of the children stock to the rest of the model. The results for egg production are the following:

Now egg production shows a very nice, bell shaped peak. This shape is a robust feature of the model. You can play with the constants as you like, but what you get, normally, is this kind of symmetric peak. As you probably know, this is the basic characteristic of the Hubbert model of oil production, where the peak is normally called "Hubbert  peak". Actually, this simple egg hunt model is equivalent to the one that I used, together with my coworker Alessandro Lavacchi, to describe real historical cases of the production of non renewable resources. (see this article published in "Energies" and here for a summary)

We can play a little more with the model. How about supposing that the children can learn how to find eggs faster, as the search goes on? That can be simulated by assuming that the "ability" parameter increases with time. We could say that it ramps up of a notch for every egg found. The results? Well, here is an example: 

We still have a peak, but now it has become asymmetric. It is not any more the Hubbert peak but something that I have termed the "Seneca peak" from the words of the Roman philosopher Seneca who noted that ruin is usually much faster than fortune. In this example, ruin comes so fast precisely because people try to do their best to avoid it! It is a classic case of "pulling the levers in the wrong direction", as Donella Meadows told us some time ago. It is counter-intuitive but, when exploiting a non renewable resource, becoming more efficient is not a good idea. 


There are many ways to skin a rabbit, so to say. So, this model can be modified in many ways, but let's stop here. I think this is a good illustration of how to play with "mind sized" models based on system dynamics and how even very simple models may give us some hint of how the real world works. This said, happy Easter, everyone!





(BTW, the model shown here is rather abstract and not thought to describe an actual Easter Egg hunt. But, who knows? It would be nice to compare the results of the model with some real world data. My children are grown-ups by now, but maybe someone would be able to collect actual data this Easter!)

Mineral resources and the limits to growth.

Originally published on Cassandra's legacy on Wednesday, September 18, 2013

This is a shortened version of a talk I gave in Dresden on September 5, 2013. Thanks to Professor Antonio Hurtado for organizing the interesting conference there.

 

So, ladies and gentleman, let me start with this recent book of mine. It is titled "The Plundered Planet." You can surely notice that it is not titled "The Developed Planet" or "The Improved Planet." Myself and the coauthors of the book chose to emphasize the concept of "Plundering"; of the fact that we are exploiting the resources of our planet as if they were free for us for the taking; that is, without thinking to the consequences. And the main consequence, for what we are concerned here is called "depletion," even though we have to keep in mind the problem of pollution as well. 

Now, there have been many studies on the question of depletion, but "The Plundered Planet" has a specific origin, and I can show it to you. Here it is.  

It is the rather famous study that was published in 1972 with the title "The Limits to Growth". It was one of the first studies that attempted to quantify depletion and its effects on the world's economic system. It was a complex study based on the best available data at the time and that used the most sophisticated computers available to study how the interaction of various factors would affect parameters such as industrial production, agricultural production, population and the like. Here are the main results of the 1972 study, the run that was called the "base case" (or "standard run"). The calculations were redone in 2004, finding similar results. 

As you can see, the results were not exactly pleasant to behold. In 1972, the study saw a slowdown of the world's main economic parameters that would take place within the first two decades of the 21st century. I am sure that you are comparing, in your minds, these curves with the present economic situation and you may wonder whether these old calculations may be turning out to be incredibly good. But I would also like to say that these curves are not - and never were - to be taken as specific predictions. No one can predict the future, what we can do is to study tendencies and where these tendencies are leading us. So, the main result of the Limits to Growth study was to show that the economic system was headed towards a collapse at some moment in the future owing to the combined effect of depletion, pollution, and overpopulation. Maybe the economic problems we are seeing nowadays are a prelude to the collapse seen by this model, maybe not - maybe the predicted collapse is still far away in the future. We can't say right now.

In any case, the results of the study can be seen at least worrisome. And a reasonable reaction when the book came out in 1972 would have been to study the problem more in depth - nobody wants the economy to collapse, of course. But, as you surely know, the Limits to Growth study was not well received. It was strongly criticized, accused of having made "mistakes" of all kinds and at times to be part of a worldwide conspiracy to take control of the world and to exterminate most of humankind. Of course, most of this criticism had political origins. It was mostly a gut reaction: people didn't like these results and sought to find ways to demonstrate that the model was wrong (or the data, or the approach, or something else). If they couldn't do that, they resorted to demonizing the authors - that's nothing now; I described it in a book of mine "Revisiting the limits to growth".

Nevertheless, there was a basic criticism of the "Limits" study that made sense. Why should one believe in this model? What are exactly the factors that generate the expected collapse? Here, I must say, the answer often given in the early times by the authors and by their supporters wasn't so good. What the creators of the models said was that the model made sense according to their views and they could show a scheme that was this (from the 1972 Italian edition of the book):

Now, I don't know what do you think of it; to me it looks more or less like the map of the subway of Tokyo, complete with signs in kanji characters. Not easy to navigate, to say the least. So, why did the authors created this spaghettimodel? What was the logic in it? It turns out that the Limits to Growth model has an internal logic and that it can be explained in thermodynamic terms. However, it takes some work to describe the whole story. So, let me start with the ultimate origin of these models:

If you have studied engineering, you surely recognize this object. It is called "governor" and it is a device developed in 19th century to regulate the speed of steam engines. It turns with the engine, and the arms open or close depending on speed. In so doing, the governor closes or opens the valve that sends steam into the engine. It is interesting because it is the first self-regulating device of this kind and, at its time, it generated a lot of interest. James Clerk Maxwell himself studied the behavior of the governor and, in 1868, he came up with a set of equations describing it. Here is a page from his original article

 

I am showing to you these equations just to let you note how these systems can be described by a set of correlated differential equations. It is an approach that is still used and today we can solve this kind of equations in real time and control much more complex systems than steam engines. For instance, drones. 

You see here that a drone can be controlled so perfectly that it can hold a glass without spilling the content. And you can have drones playing table tennis with each other and much more. Of course they are also machines designed for killing people, but let's not go into that. The point is that if you can solve a set of differential equations, you can describe - and also control - the behavior of quite complex systems.

The work of Maxwell so impressed Norbert Wiener, that it led him to develop the concept of "cybernetics"

We don't use so much the term cybernetics today. But the ideas that started from the governor study by Maxwell were extremely fecund and gave rise to a whole new field of science. When you use these equations for controlling mechanical system, you use the term "control theory." But when you use the equations for study the behavior of socio-economic systems, you use the term "system dynamics"

System dynamics is something that was developed mainly by Jay Wright Forrester in the 1950s and 1960s, when there started to exist computers powerful enough to solve sets of coupled differential equations in reasonable times. That generated a lot of studies, including "The Limits to Growth" of 1972 and today the field is alive and well in many areas.

A point I think is important to make is that these equations describe real world systems and real world systems must obey the laws of thermodynamics. So, system dynamics must be consistent with thermodynamics. It does. Let me show you a common example of a system described by system dynamics: practitioners in this field are fond of using a bathub as an example:

On the right you have a representation of the real system, a bathtub partly filled with water. On the left, its representation using system dynamics. These models are called "stock and flow", because you use boxes to represent stocks (the quantity of water in the tub) and you use double edged arrows to indicate flows. The little butterfly like things indicate valves and single edged arrows indicate relationship.

Note that I used a graphic convention that I like to use for my "mind sized" models. That is, I have stocks flowing "down", following the dissipation of thermodynamic potential. In this case what moves the model is the gravitational potential; it is what makes water flow down, of course. Ultimately, the process is driven by an increase in entropy and I usually ask to my students where is that entropy increases in this system. They usually can't give the right answer. It is not that easy, indeed - I leave that to you as a little exercise

The model on the left is not simply a drawing of box and arrows, it is made with a software called "Vensim" which actually turns the model "alive" by building the equations and solving them in real time. And, as you may imagine, it is not so difficult to make a model that describes a bathtub being filled from one side and emptied from the other. But, of course, you can do much more with these models. So, let me show a model made with Vensim that describes the operation of a governor and of the steam engine.

Before we go on, let me introduce a disclaimer. This is just a model that I put together for this presentation. It seems to work, in the sense that it describes a behavior that I think is correct for a governor (you can see the results plotted inside the boxes). But it doesn't claim to be a complete model and surely not the only possible way to make a system dynamics model of a governor. This said, you can give a look to it and notice a few things. The main one is that we have two "stocks" of energy: one for the large wheel of the steam energy, the other for the small wheel which is the governor. In order to provide some visual sense of this difference in size, I made the two boxes of different size, but that doesn't change the equations underlying the model. Note the "feedback", the arrows that connect flows and stock sizes. The concept of feedback is fundamental in these models.

Of course, this is also a model that is compatible with thermodynamics. Only, in this case we don't have a gravitational potential that moves the system, but a potential based on temperature differences. The steam engine works because you have this temperature difference and you know the work of Carnot and the others who described it. So, I used the same convention here as before; thermodynamic potential are dissipated going "down" in the model's graphical representation

Now, let me show you another simple model, the simplest version I can think of a model that describes the exploitation of non renewable resources:

It is, again, a model based on thermodynamics and, this time, driven by chemical potentials. The idea is that the "resources" stock as a high chemical potential in the sense that it may be thought as, for instance, crude oil, which spontaneously combines with oxygen to create energy. This energy is used by human beings to create what I can call "capital" - the sum of everything you can do with oil; from industries to bureaucracies.

On the right, you can see the results that the model provides in terms of the behavior as a function of time of the stock of the resources, their production, and the capital stock. You may easily notice how similar these curves are to those provided by the more complex model of "The Limits to Growth." So, we are probably doing something right, even with this simple model.

But the point is that the model works! When you apply it to real world cases, you see that its results can fit the historical data. Let me show you an example:

This is the case of whaling in 19th century, when whale oil was used as fuel for lamps, before it became common to use kerosene. I am showing to you this image because it is the first attempt I made to use the model and I was surprised to see that it worked - and it worked remarkably well. You see, here you have two stocks: one is whales, the other is the capital of the whaling industry that can be measured by means of a proxy that is the total tonnage of the whaling fleet. And, as I said, the model describes very well how the industry grew on the profit of killing whales, but they killed way too many of them. Whales are, of course, a renewable resource; in principle. But, of course, if too many whales are killed, then they don't have enough time to reproduce and they behave as a non-renewable resource. Biologists have determined that at the end of this fishing cycle, there were only about 50 females of the species being hunted at that time. Non renewable, indeed!

So, that is, of course, one of the several cases where we found that the model can work. Together with my co-workers, we found that it can work also for petroleum extraction, as we describe in a paper published in 2009 (Bardi and Lavacchi). But let me skip that - the important thing is that the model works in some cases but, as you would expect, not in all. And that is good - because what you don't want is a "fit-all" model that doesn't tell you anything about the system you are studying. Let's say that the model reproduces what's called the "Hubbert model" of resource exploitation, which is a purely empirical model that was proposed more than 50 years ago and that remains a basic one in this kind of studies: it is the model that proposes that extraction goes through a "bell-shaped" curve and that the peak of the curve, the "Hubbert peak" is the origin of the concept of "peak oil" which you've surely heard about. Here is the original Hubbert model and you see that it has described reasonably well the production of crude oil in the 48 US lower states.

Now, let's move on a little. What I have presented to you is a very simple model that reproduces some of the key elements of the model used for "The Limits to Growth" study but it is of course a very simplified version. You may have noted that the curves for industrial production of the Limits to Growth tend to be skewed forward and this simple model can't reproduce that. So, we must move of one step forward and let me show you how it can be doing while maintaining the basic idea of a "thermodynamic cascade" that goes from higher potentials to lower potentials. Here is what I've called the "Seneca model"

You see that I added a third stock to the system. In this case I called it "pollution"; but you might also call it, for instance, "bureaucracy" or may be even "war". It is any stock that draws resource from the "Capital" (aka, "the economy") stock. And the result is that the capital stock and production collapse rather rapidly; this is what I called "the Seneca effect"; from the roman philosopher Lucius Anneaus Seneca who noted that "Fortune is slow, but ruin is rapid".

For this model, I can't show you specific historical cases - we are still working on this idea, but it is not easy to make quantitative fittings because the model is complicated. But there are cases of simple systems where you see this specific behavior, highly forward skewed curves - caviar fishing is an example. But let me not go into that right now.

What I would like to say is that you can move onward with this idea of cascading thermodynamic potentials and build up something that may be considered as a simplified version of the five main stocks taken into account in the "Limits to Growth" calculations. Here it is

Now, another disclaimer: I am not saying that this model is equivalent to that of the Limits to Growth, nor that it is the only way to arrange stocks and flows in order to produce similar results to the one obtained by the Limits to Growth model. It is here just to show to you the logic of the model. And I think you can agree, now, that there is one. The "Limits" model is not just randomly arranged spaghetti, it is something that has a deep logic based on thermodynamics. It describes the dissipation of a cascade of thermodynamic potentials.

In the end, all these model, no matter how you arrange their elements, tend to generate similar basic results: the bell shaped curve; the one that Hubbert had already proposed in 1956

The curve may be skewed forward or not, but that changes little on the fact that the downside slope is not so pleasant for those who live it.

Don't expect this curve to be a physical law; after all it depend on human choices and human choices may be changed. But, in normal conditions, human beings tend to follow rather predictable patterns, for instance exploiting the "easy" resources (those which are at the highest thermodynamic potential) and then move down to the more difficult ones. That generates the curve.

Now, I could show you many examples of the tendency of real world systems to follow the bell shape curve. Let me show you just one; a recent graph recently made by Jean Laherrere.

These are data for the world's oil production. As you can see, there are irregularities and oscillations. But note how, from 2004 to 2013, we have been following the curve: we move on a predictable path. Already in 2004 we could have predicted what would have been today's oil production. But, of course, there are other elements in this system. In the figure on the right, you can see also the appearance of the so-called "non-conventional" oil resources, which are following their own curve and which are keeping the production of combustible liquids (a concept slightly different from that of "crude oil) rather stable or slightly increasing. But, you see, the picture is clear and the predictive ability of these models is rather good even though, of course, approximate.

Now, there is another important point I'd like to make. You see, these models are ultimately based on thermodynamics and there is an embedded thermodynamic parameter in the models that is called EROI (or EROEI) which is the energy return for the energy invested. It is basically the decline in this parameter that makes, for instance, the extraction of oil gradually producing less energy and, ultimately, becoming pointless when the value of the EROEI goes below one. Let me show you an illustration of this concept:

You see? The data you usually read for petroleum production are just that: how much petroleum is being produced in terms of volume. There is already a problem with the fact that not all petroleums are the same in the sense of energy per unit volume, but the real question is the NET energy you get by subtracting the energy invested from the energy produced. And that, as you see, goes down rapidly as you move to more expensive and difficult resources. For EROEIs under about 20, the problem is significant and below about 10 it becomes serious. And, as you see, there are many energy resources that have this kind of low EROEI. So, don't get impressed by the fact that oil production continues, slowly, to grow. Net energy is the problem and many things that are happening today in the world seem to be related to the fact that we are producing less and less net energy. In other words, we are paying more to produce the same. This appears in terms of high prices in the world market.

Here is an illustration of how prices and production have varied during the past decades from the blog "Early Warning" kept by Stuart Staniford.

And you see that, although we are able to manage a slightly growing production, we can do so only at increasingly high prices. This is an effect of increasing energy investments in extracting difficult resources - energy costs money, after all.
So, let me show you some data for resources that are not petroleum. Of course, in this case you can't speak in terms of EROEI; because you are not producing energy. But the problem is the same, since you are using fossil fuels to produce most of the commodities that enter the industrial system, and that is valid also for agriculture. Here are some data.

Food production worldwide is still increasing, but the high costs of fossil fuels are causing this increase in prices. And that's a big problem because we all know that the food demand is highly anelastic - in plain words you need to eat or you die. Several recent events in the world, such as wars and revolutions in North Africa and Middle East have been related to these increases in food prices.

Now, let me go to the general question of mineral production. Here, we have the same behavior: most mineral commodities are still growing in terms of extracted quantities, as you can see here (from a paper by Krausmann et al, 2009 http://dx.doi.org/10.1016/j.ecolecon.2009.05.007)

These data go up to 2005 - more recent data show signs of plateauing production, but we don't see clear evidence of a peak, yet. This is bad, because we are creating a climate disaster. As you seee from the most recent data, CO2 are still increasing in a nearly exponential manner

 

But the system is clearly under strain. Here are some data relative to the average price index for aluminum, copper, gold, iron ore, lead, nickel, silver, tin and zinc (adapted from a graphic reported by Bertram et al., Resource Policy, 36(2011)315)

So, you see, there has been this remarkable "bump" in the prices of everything and that correlates well with what I was arguing before: energy costs more and, at the same time, energy requirements are increasing because of ore depletion. At present, we are still able to keep production stable or even slowly increasing, but this is costing to society tremendous sacrifices in terms of reducing social services, health care, pensions and all the rest. And, in addition, we risk to destroy the planetary ecosystem because of climate change.

Now I can summarize what I've been saying and get to the take-home point which, I think can be expressed in a single sentence "Mining takes energy"

Of course, many people say that we are so smart that we can invent new ways of mining that don't require so much energy. Fine, but look at that giant wheel, above, it used to extract coal in the mine of Garzweiler in Germany. Think of how much energy you need to make that wheel; do you think you could use an i-pad, instead?

In the end, energy is the key of everything and if we want to keep mining, and we need to keep mining, we need to be able to keep producing energy.  And we need to obtain that energy without fossil fuels. That's the concept of the "Energy Transition"

Here, I use the German term "Energiewende" which stands for "Energy Transition". And I have also slightly modified the words by Stanley Jevons, he was talking about coal, but the general concept of energy is the same. We need to go through the transition, otherwise, as Jevons said long ago, we'll be forced to return to the "laborious poverty" of older times.

That doesn't mean that the times of low cost mineral commodities will ever return but we should be able to maintain a reasonable flux of mineral commodities into the industrial system and keep it going. But we'll have to adapt to less opulent and wasteful life as the society of "developed" countries has been accustomed so far. I think it is not impossible, if we don't ask too much:

h/t ms. Ruza Jankovich - the car shown here is an old Fiat "500" that was produced in the 1960s and it would move people around without the need of SUVs