Cassandra has moved. Ugo Bardi publishes now on a new site called "The Seneca Effect."

Sunday, August 28, 2011

The Seneca effect: why decline is faster than growth

"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." Lucius Anneaus Seneca, Letters to Lucilius, n. 91

The subject of the Seneca Effect"is treated in detail in two books by Ugo Bardi. "The Seneca Effect" (2017, a little technical) and "Before Collapse" (2019, simpler).

Don't you stumble, sometimes, into something that seems to make a lot of sense, but you can't say exactly why? For a long time, I had in mind the idea that when things start going bad, they tend to go bad fast. We might call this tendency the "Seneca effect" or the "Seneca cliff," from Lucius Annaeus Seneca who wrote that "increases are of sluggish growth, but the way to ruin is rapid."

Could it be that the Seneca cliff is what we are facing, right now? If that is the case, then we are in trouble. With oil production peaking or set to peak soon, it is hard to think that we are going to see a gentle downward slope of the economy. Rather, we may see a decline so fast that we can only call it "collapse." The symptoms are all there, but how to prove that it is what is really in store for us? It is not enough to quote a Roman philosopher who lived two thousand years ago. We need to understand what factors might lead us to fall much faster than we have been growing so far. For that, we need to make a model and see how the various elements of the economic system may interact with each other to generate collapse.

I have been working on this idea for quite a while and now I think I can make such a model. This is what the rest of this post will be about. We'll see that a Seneca cliff may indeed be part of our future if we keep acting as we have been acting so far (and as we probably will). But let's go into the details.

Models of growth and decline

The paradigm of all models of growth and decline is the Hubbert model. Here is how it appeared for the first time, in a paper published by Marion King Hubbert in 1956 where he showed his prediction for crude oil production in the 48 US lower states.

If you are interested in this subject, you probably have seen this plot many times and you also know that it worked nicely as a prediction. Oil production in the US did peak when Hubbert had said it would, in 1970. The Hubbert model has been shown to be a good description of many historical cases of oil producing regions, as reported, for instance, by Adam Brandt in his 2007 paper "Testing Hubbert". It works not just for oil, also for other mineral resources and for slowly renewable biological resources, such as whales (Bardi 2007).

We can take the Hubbert model as a first step for the description of an economic system based on the exploitation of a non renewable resource. The idea underlying the model is that exploitation starts with the best, highest return resources. Then, depletion slowly forces the industry to move to lower return resources. Profits fall and the capability of the industry to invest in new extraction falls as a consequence. This slows down growth and, eventually, causes production to decline (Bardi et al. 2010). So, it is a very general model that could describe not just regional cases but whole civilizations. Most of the agrarian civilizations of the past were based on a depletable resource, fertile soil, as I discussed in a post of mine in 2009.

However, the Hubbert model does not generate the Seneca effect. Not only the production curve is normally assumed to be symmetric, but there are several historical cases where it is skewed backwards; something that we could call the "anti-Seneca" effect. The prevalence of these cases in oil production led Brandt (2007) to state that (p. 27) ".... there is simply no evidence in the historical data that rates of decline will be generally sharper than rates of increase. This should be taken as comforting news for those concerned about a quick decline in production causing additional disruption beyond that already anticipated for the transition from conventional oil to substitutes for conventional oil."

Fine, but there is a problem. The results reported by Brandt are all for regional cases and it couldn't be otherwise. But, in a regional case, when extraction costs go up, operators simply move to regions where costs are lower. What happens when there is no new region to move to? That is, what happens when you examine the worldwide trend? Do people simply give up extracting, as it is implicitly assumed in the Hubbert model, or do they try harder? And in the latter case, what happens?

Of course, we don't have historical data for the whole cycle of oil production, worldwide. But there exist models which are more sophisticated than the Hubbert model and which can tell us more about worldwide trends. One is the "World3" model used for "The Limits to Growth" study, first published in 1972. The model is based on assumptions not unlike those at the basis of the Hubbert model (see this post of mine comparing the two models), but it considers the world's economy as a whole. Here are the results of the "base case" scenario of the 2004 version.

Here, we clearly see that the curves for food production and industrial production are skewed forward. It is the Seneca effect; something that appears to be a general trend of these models. For an even clearer view of this trend, here is a graph taken from the front cover of the 2004 edition of "The Limits to growth"

Now, what creates the Seneca effect in a complex model such as "World3" but not in a simpler model as Hubbert's one? In order to understand this point, I'll try now to build simple ("mind sized") world models and see what parameters are the cause of the forward leaning curves. We'll see that the asymmetry is mainly caused by a factor that we may call "pollution."

Mind Sized World Modelling

"Mind Sized" is a term invented by Seymour Papert in his book "Mind Storms" (1980). The idea is that, in order to be convinced that a certain phenomenon is real, or that it may happen for real, you need to understand what makes it real. For this, a model must be simple enough that you can make sense of it within your mind. This was one of the problems with "The Limits to Growth" study of 1972; the model was so complex that people tended to disbelieve its results mostly because they didn't understand how the model worked, as I argue in my book on this subject (Bardi 2011). So, let's see if we can make mind sized world models, trying to explicit their relation with thermodynamics. This was the gist of a talk that I gave in Spain this year, "Entropy, peak oil and stoic philosophy".

For building these models, I'll use "system dynamics," the same method used for "The Limits to Growth" study. It is a method of simulation based on describing systems as composed as "stocks" linked to each other by "flows" controlled by "valves". The classic example of this kind of systems is that of a bathtub. The bathtub is the stock; you can fill it by means of a flow of water, or you can empty it letting water flow out of it. This is called "bathtub dynamics" and you can read a nice paper on this subject by Linda Sweeney and John Sterman. It shouldn't be necessary to say that a bathtub has to obey the laws of physics, but at times it is. You have to remember that mass must be conserved in order to understand how a bathtub fills or drains out. More in general, energy has to be conserved - this is the first law of thermodynamics. You have to remember also the second law of thermodynamics, which says that in everything that happens spontaneously, entropy must increase. Ultimately, the fact that water flows away from the drain of a bathtub has to do with increasing the entropy of the universe.

So, let's try to make a simple, mind-sized model that describes how an economic system exploits a non-renewable resource. We start with a stock that we call "Resources". We assume that it is a stock of energy on the basis of the idea that energy can be transformed into other kinds of resources (say, metals) but not the reverse. The resource could be, for instance "crude oil", which is the main resource on which our civilization is based. Then, we have another box that we call "Capital" that represents energy stocked in forms that can be utilized. We could say that this stock is a section of the economy; say, "the oil industry" or that it represents a whole civilization. Then we draw the flows of energy from the resource stock to the capital stock and to dissipation as low temperature heat, as the second law of thermodynamics commands. Here is the model.

This is the same model that I showed in previous posts (e.g. here) but, here I turned it 90 degrees clockwise in order to emphasize the fact that energy goes "down" from higher thermodynamic potentials to lower thermodynamic potentials; just like what water does in a bathtub or in a fountain. Unlike the case of a fountain or a bathtub, however, here the flow is governed by feedback: resources are transformed into capital in proportion to the amount of both Resources and Capital. Note also that the resource partly decays without producing anything (Rate3). That's due to the inefficiency of the production process; think of oil spills or of natural gas vented and burned.

As you see, the production curve (Rate1 in the figure) is bell shaped and symmetric. This model, indeed, is equivalent to the Hubbert model (Bardi and Lavacchi 2009). The problem is that you can play as much as you like with the model, changing the values of the three constants, but the curves will not show the Seneca effect; that is, decay will not be faster than growth. So, are we missing something, here?

It seems that, indeed, we are missing an element that, instead, is present in the world models of "The Limits to Growth" study. What we are missing is pollution or, better said, the effects of pollution. In the simple model above, degraded energy is harmlessly dissipated to space; it has no effect on the other elements of the model. But we know that, in the real world, that is not true. Pollution has a cost: money and resources must be spent to fight it; be it water or air poisoning or effects such as global warming.

In order to simulate the effects of pollution we can define it as a third stock that drains energy from the capital stock in proportion to the size of both the capital and the pollution stocks. Note that, since there are several constants, I grouped with the name of "l" (from the term "loss") those which go directly from a stock to outer space (l1, l2, l3). I kept the letter "k" for the flows that go from one stock to another. Here is the model. I am showing you a sample output where I chose parameters which emphasize the Seneca effect.

The parameters for this  run are k1=0.03. k2=0.3, l1=0, l2=0.01, l3=0.015, Resources (t0)=1, Capital(t0) =0.001, Pollution(t0)=0.001

Here is the production curve, alone, from a different run.

So,the model can generate a "Seneca-like" production curve which clearly shows the "Seneca cliff". It goes up slowly, then it collapses quickly. As Seneca says, "the way to ruin is rapid."

Now, can we say in words what generates the Seneca cliff? Yes, we can. It goes like this: first, consider that the effect of pollution is to drain economic capital. Secondly, consider that the pollution stock grows by feeding on the economy stock - so it has to wait for the economy to have grown before it can grow itself. It is this delay that causes an increase in the rate of energy draining from the economy as the process goes on. Since the size of the economy stock determines the production rate, we see also that parameter going down rapidly after the peak. This is the essence of the Seneca effect.

Let's now go more in depth in the model. What is exactly this "pollution" that causes so much trouble? It is what the authors of "The Limits to Growth" called "persistent pollution" to show that it is something different from infrared radiation harmlessly disappearing into space. It is a very general concept that includes anything that is generated by capital and will drain resources from capital. The Fukushima disaster is a good example of pollution coming back to bite at the industry that produced it. It could be poisoning of the air or of water. It could be global warming and it could also be wars. Wars are great producers of pollution and a nuclear war would make the Seneca effect take place almost instantly.

Now that we understand how the model works, we can go back to to Brandt's study and explain why in the majority of historical cases of oil production the curves are symmetric or show "anti-Seneca" shapes. We said that the Seneca effect is generated by pollution; so, does this result mean that oil extraction produces no pollution? Not at all, of course. It only means that those who extract oil don't have to pay for the pollution they produce. To make a practical example, in the case of oil extraction from the 48 US lower states, persistent pollution has mainly taken the shape of CO2 and other greenhouse gases added to the atmosphere. This is a factor that has not yet bitten us back, but, eventually, someone will have to pay for the damage done in the form of global warming. When the bill comes - and it is coming - we may discover that it is more expensive than what we can afford to pay.

Would technological progress save us from the Seneca cliff? Well, not automatically. Actually, it could make the cliff steeper! One way to simulate technology is to assume that the constants in the model are not constant but vary as the cycle proceeds. For instance, an increasing value of the "k1" constant corresponds to technological improvements in the capability of exploiting the resource. That will increase the total amount produced at the end of the cycle, but will also generate a steeper fall after the peak, as I discussed in a paper of mine (Bardi 2005). A more interesting idea would be to tweak the model by a making the "k2" constant gradually smaller. That would simulate the development of technologies that lower the production of pollution. In other words, the model tells us that "clean production" is a good idea in the sense that it would tend to make the production cycle more symmetric.

You might try other ways to modify the model, for instance increasing its complexity by adding more stocks. How about a "bureaucracy" stock that accumulates and then dissipates energy? Well, it will act just as the "pollution" stock; perhaps we might say that bureaucracy is a form of pollution. Incidentally, anyway, with this added stock the model becomes more similar to Tainter's model that has that civilizations decline and collapse because of an increase in complexity that brings more problems than benefits. If you keep adding more elements to the model, in the end you get to something that may be similar to the "World3" model used in "The Limits to Growth" study. We saw earlier on that this model does generate forward skewed curves.

There are many ways to modify these models and the Seneca effect is not the only possible outcome. Fiddling with the constants you may also generate the opposite behavior; that is the "anti-Seneca" curve, with decline slower than growth. As you would expect, that happens using constants that minimize the accumulation of persistent pollution. But, in general, the Seneca effect is a "robust" feature of this kind of models and it comes up for a variety of assumptions. You ignore the Seneca cliff at your own risk.

The Seneca effect in the real world

Do we have historical examples of the Seneca effect? Well, several, but not many for which we have quantitative data. The Roman civilization, for instance, took about seven centuries to peak and just about three centuries to fall, at least in its Western side (and Seneca himself may have perceived the Roman decline at his time). However, the data we have on such parameters as the Roman population are not good enough to see the effect in the form of a forward skewed curve.

We seem to have such data, instead, for the Mayan civilization. Here is an image taken from Dunning et al (1998).  The horizontal scale is very long: 10.000 years from the Pleistocene/holocene boundary.

In this case, pollution takes the shape of soil erosion that drains capital resources and generates population collapse. We should be careful with this interpretation, because some authors believe that the Mayan collapse was caused by climate change. But the world model developed here seems to be compatible with the historical data.

For something closer to us, here is a figure taken from Dmitry Orlov's paper "Peak oil is history". It shows Russian oil production.

The Soviet Union was a nearly closed economy before collapsing; a "mini-world" in itself. Notice how Russian oil production went down rapidly after the peak; a classic Seneca cliff. Note also how production picked up again afterwards. At some point, the Soviet Union ceased to exist as an isolated economic system and it became part of the whole world's economic system. At that point, the simple model that we have been using does not work any longer; most likely because the capital stock received an influx of resources that came from a region outside the model.

Conclusion: a banquet of consequences

Very often, we fail to understand the delayed effects of our actions. John Sterman reminds us of this point in a talk on global warming quoting Robert Louis Stevenson as saying, "Everybody, sooner or later, sits down to a banquet of consequences." The models shown here tell us that the Seneca cliff is the result of delayed consequences.

As always, the future is something that we build with our actions and the models can only tell us what kind of actions will lead us, eventually, to a certain outcome. Used in this way, models can be extremely useful and can even be applied to systems which are much more modest than an entire civilization, for instance to a single company or to our personal relationships with other people. In all cases, the Seneca effect will be the result of trying hard to keep things running as usual. In that way, we may run out faster of the resource that keeps the system running: be it a physical resource or a reserve of goodwill. The way to avoid this outcome may be to let the system run the way it wants, without attempting to force it to go the way we want it to go. In other words we need to take things in life with some stoicism, as Seneca himself would probably have said.

Thinking of the worldwide situation and of the problems involved, global warming and resource depletion, what the models tell us is that the Seneca cliff may be the inevitable result of putting too much strain on already badly depleted natural resources. We should try, instead, to develop alternative stocks of resources such as renewable (or nuclear) energy. At the same time, we should avoid to exploit highly polluting and expensive resources such as tar sands, oil shales, deepwater oil, and, in general, applying the "drill, baby, drill" philosophy. All those strategies are recipes for doom. Unfortunately, these are also examples of exactly what we are doing.

I don't know what Seneca would say if he could see this planet-wide effort we are making in order to put into practice the idea that he expressed in his letter to his friend, Lucilius. I can only imagine that he would take it with some stoicism. Or, maybe, he would comment with what he said in his "De Providentia" "Let Nature deal with matter, which is her own, as she pleases; let us be cheerful and brave in the face of everything, reflecting that it is nothing of our own that perishes."

Thanks to Dmitry Orlov for having been the source of inspiration for this post with his article "Peak Oil is History".


Bardi, U., 2007, Energy prices and resource depletion: Lessons from the case of whaling in the nineteenth century” Bardi U. Energy Sources, part B- Economics Planning and Policy Volume: 2 Issue: 3  Pages: 297-304

Bardi, U.  and Lavacchi, A., 2009, "A Simple Interpretation of Hubbert’s Model of Resource Exploitation” Energies 2009, 2(3), 646-661; doi:10.3390/en20300646

Bardi, U. 2011 "The Limits to Growth Revisited", Springer,  ISBN 978-1-4419-9415-8

Bardi, U., Lavacchi, A., Yaxley L., 2011 “Modelling EROEI and net energy in the exploitation of non renewable resources” Ecological Modelling, In Press.

Brandt, A.R. (2007). Testing Hubbert. Energy Policy, 35(May):3074-3088. DOI: 10.1016/j.enpol.2006.11.004

Dunning, N., D. Rue, T. Beach, A. Covich, A. Traverse, 1998, "Human - Environment Interactions in a Tropical Watershed: the Paleoecology of Laguna Tamarindito, Guatemala," Journal of Field Archaeology 25 (1998):139-151.


  1. Thank you for this very insightful and relatively simple yet revealing approach to understanding our situation.

  2. Thank you KotR, and keep going with the sunglasses!

  3. Nice (and scary) article Ugo, and so amazing that the "classical" economic growth discourse is still the prevalent one ...

    Just a small remark : The USSR wasn't a closed economy at all, and its main export (and currency input) was oil and gas (as today for Russia).

    And in fact the main "weapon" used to bring it down has been Reagan having the Saudis increase their prod in beginning eighties to cut the USSR export revenus, see below starting around 28 min for instance (with a Gorbatchev interview about it):

    note : the "reopen911" logo has nothing to do with original documentary, added by the poster, and a pity there is no English version of this doc (to my knowledge), really a good summary of "oil history"

  4. I add my thanks for this summary description - most memorable.

    Industrialised 20thC wars at the largest scale actually seem to have enhanced economic activity, at least during their build-up and early phases, but the immediate aftermath has been something else.

    I also have been thinking about modern famines or 'Seneca cliffs in food production'. I do not pretend to have studied these in any detail, but their importance is without doubt. Leaving aside famines in Africa, I am thinking of famines or, short of actual famine, big and sudden reductions in food production in very large heterogenous countries that were rapidly industrialising. Examples come to mind; Stalin's USSR in the 1930s; FSU in the 1990s; Mao's 'Great Leap Forward'. Somewhat different examples come to mind; Germany post-WWII (and perhaps Italy?), or even parts of USA in the Great Depression, or in UK (parts of Ireland) in the 1840s. In situations where agricultural production was barely sufficient, monetised distribution failed to supply stranded populations.

    In my main three selections, social disturbance imposed by a forced change in economic organisation seems a common factor. (A break down of 'valves' and 'taps' in your model perhaps?)

    Another factor could be that aggregates of food production even at large scale, (even at 'world scale), vary significantly year to year in any case. Storage capacity does not seem to 'iron-out' this variation too much.

    I saw some of the aftermath of agricultural decline in ex-Soviet bloc Eastern and Southern Europe as a background to my work in a number of these countries, 1997-2006. My tentative conclusion is that very rapid decline in agricultural production can follow from natural variation, e.g. drought or high temperature, but is very severe when there is a sudden loss of webs of input and output feedbacks - taps and valves again. Draining off food from areas of agricultural production in order to feed large industrial urban populations is very 'technical', needs high levels of maintenance of complex feed-back loops?

    thanks again - I am going to read your book!

  5. I've generated a Seneca-type effect in an economic model of an industrial economy exploiting ever more marginal energy resources, which require ever more capital intensive extraction. It relies on economies of scale in the manufacturing sector (which is a reasonable assumption). As the energy supply is constrained, the scale of this sector declines, leading to sudden collapse.

    if you're interested.

  6. David
    Fascinating paper.
    Can not really use Ugo's blog for comments or questions (leave that to him and other modellers), except that 'economies of scale' have been fundamental for competitive development throughout industrial period. Japanese 'miracle' post-WWII used super-bulk carriers, for example, to open up 'difficult' sources of iron-ore in Brazil, boot-strap fashion, for steel and ship building and etc etc see studies by Stephen G. Bunker et al.

  7. I agree with David that a declining EROEI may be much of Ugo's pollution. This obviously does not dismiss the idea of "pollution" but enlarges it; we should probably regard "pollution" in a wide sense, as any effect undermining available capital with respect to the extracted resources, namely any effect making resource exploitation more capital intensive (that's exactly what David says). And it seems to me that declining EROEI is perhaps the most important of those factors right now, and it's going to be the main player in our possible demise...

  8. I hope to don’t annoy you too much, but I wish to propose a little scale, ordinary life metaphor of the Seneca’s effect.
    Think about to walk upright, how we humans do continuously, without to think too much about it.
    It is a an hugely complex way to move, so complex that no robot got yet to imitate that strange “inverted pendulum” gait.
    It requires the perfect coordination of tens of muscles, under the rule of an even more complex nervous/sensorial system, a terrain sufficiently even and a lot of energy
    We spend many years to learn and perfect this gait, then we use it automatically for the rest of our life. In our existence, to walk upright looks much more likely than to fall, also if we risk it at every step..
    But then accident happens: it is sufficient a ankle distortion, a little brain damage, a back pain, a broken bone or even a uneven terrain and the gait become slow, painful and tiring. To walk in a not usual way, require more energy and put strain on body systems not evolved to sustain it.
    By the way we can resist and compensate, for example diverting more energy, attention and nervous power to different muscle systems, or using technology (a crutch), continuing to move, also if not smoothly as before.
    But if the energy input is not anymore sufficient (in a situation where you have no food and you must move, a simple distortion can be deadly) or if our new, abnormal gait, finally compromises other body systems, sooner or later we’ll fall. And more we strained our body and exhausted our energy reserve, with that unnatural gait, and more dangerous the fall will be, because it could prevent us to return on our feet for ever.
    In every case the fall will be more sudden and much more rapid that the long period when we learnt to walk properly and used the normal gait.

    I think I don’t need to explain you that the upright walk, is a metaphor of the current state of our society: an extremely complex balance in the working of many infrastructures, finely coordinated by government and corporation, using a lot of (human, fossil and other kind) energy to make them run smoothly and synchronised.
    If a infrastructure is damaged, we can always find a way to go around the problem, using more energy to get the same resource or service, from an other, less convenient infrastructure or source. But the “society gait” become less smooth.
    If we persevere in adding strains to the society, trying to hiding the growing limping, moving around a shrinking cover of energy and resources, finally we’ll fall, suddenly and ruinously as Seneca predicted....

  9. Yves, you are right about the Soviet Union. It was not a completely closed system; the model is approximated anyway. In any case, in the old Soviet Union, they did try to manufacture everything they needed inside. So, it was much more closed before the fall than after the fall.

  10. David, interesting paper but the formalism is completely different than the one I used. It would take some work to see what points the models have in common and what points are seen differently. Would be worth doing; I think.

  11. Antonio, declining EROEI is built in the dynamic model. Declining EROEI generates the bell shaped, Hubbert curve. To have the "sawtooth shaped" curve, one needs further assumptions; that is that EROEI declines faster than proportionally to the remaining amount of resources, or - as the model I have built assumes - that there is a different draining entity which I have identified as pollution. In any case, we might construct a form of EROEI that includes pollution; which I think is the correct way of defining it. But it is more work; I'll have to crank up the cloning machine....

  12. Ugo - it's not just the formalism that is completely different - the mechanism is completely different too. So it's not about what we have in common and what is different but about the fact that there are at least 2 mechanisms that can generate such an effect.

    Your mechanism is grounded in the real world with stocks and flows of physical quantities. My model is to do with prices: a social planner who allocates resources to different sectors can avoid collapse in my model, whereas they'd be just as susceptable to collapse as a market allocation mechanism in your model.

    My point was not to disagree with you - I believe your model says something important. But naysayers will counter your model with something along the lines of "scarcity will drive prices higher which will provide incentives for technology and substitutions". My model says that the price mechanism might not work in this situation.

  13. My name and email are below.

    In you note, the first graph of the seneca effect looks suspiciously lilke a graph which is simply exponential growth until it suddenly stops and the population goes to zero. That is in fact essentially what it is. The key statement in this article is the last:

    "Very often, we fail to understand the delayed effects of our actions. John Sterman reminds us of this point in a talk on global warming quoting Robert Louis Stevenson as saying, "Everybody, sooner or later, sits down to a banquet of consequences." The models shown here tell us that the Seneca cliff is the result of delayed consequences."

    There is a very simple mathematical model which demonstrates the way overshoot is caused by delayed consequences. Start with the classic model of a system with an ultimat sustained value of 1, the growthy of which isgradually curtailed as the value approaches 1, but at first is essentially exponential. A common very simple model of this is calculated from the equation,

    d/dt s(t) = s(t)(1-s(t)),

    which in English says the rate of change of s is equal to s times (one minus s). As longf as s is much less than 1, the rate of growth of s is approximatelty equal to s itself, which is classic exponential growth. But as the numbers approach a sustaionable value, the growth rate slows until it becomes zero when the sustainable level is reached.

    Now let's impose a delayed response. We'll use a very simple model in which we replace the s(t) inside the parentheses with zero if t is less than some time T, and with s(t) if t is later than T. In other words, we've delayed the response of the system to crowding until time T. I've made this example so simple that anyone can plot out the curve with a plain old calculator. If you do, you'll see that the bigger is T, the greater is the overshoot, because the system doesn't begin to warn itself in time that there are limits. The curve goes up exponentially until s = exp(T) (the exponential function for time t=T), and then drops precipitously, the greater the overshoot, the faster the drop. Try different values of T (including negative ones) you'll see that an eafrly-warning system results in a smooth transition to a sustainable number for s, but the more the transition is delayed, the greater is the overshoot and the more skewed is the drop.

    That is precisely what is happening. We were warned about overshoot around 1970 but ignored it and tried to avoid it. More or less the same thing happened with the"dot com bust." There continue to be attempts to postpone the time of transition, which will only make things worse. The job of the people of the world is (1) to start behaving as if we need to curtail our growth immediately even if it doesn't feel that way, because the longer we wait the worse things will be, and (2) to replace the ponzi-schemers who run the world with people who recognize that T is right around the corner and behave that way.

    One simple example of a time delay (of about 30 years) is the population explosion. Babies don't use as many resources as adults, so the effect of adding to the population isn't felt until the babies become adults.- an automatic "Seneca effect" even if we're not confronting exhaustion of resources.

    Incidentally, there is one and there are probablhy many population systems in nature whose populations look like the Seneca Effect. One example is lady bugs, whose populations grow exponentially until their food supply is almost completely gone, and then crash almost to zero, almost instantaneously. You watch the population grow year by year until finally everything is cvered with lady bugs, and then the next year you can't find a single one. You'd think we'd know better, but we apparently don't. We have to start learning FAST, because the longer we wait, the harder will be the fall.

    Nicholas C. Arguimbau

  14. Or another classic exemple is the reindeer population on St Matthew Island :

  15. About "Declining EROEI generates the bell shaped, Hubbert curve." seems to me that the symetric bell shaped Hubbert curve implies no declining EROEI. Declining EROEI could be modeled as a drain from the economy to the ressource, having same effect as the pollution stock and associated drain

  16. Yves, two comments to your comments. I had mentioned the Reindeer island in an earlier version of this post, but I removed it because it is, really, a simpler system and it is difficult to evidence a "Pollution" factor.

    About EROEI, look at my 2011 paper cited in the references. Declining EROEI is the essence of the Hubbert curve!

  17. Thks for the comment Ugo, yeah got it (although couldn't look at the paper, $31 a bit much ;) )

    But isn't it the case that we can say that Hubbert curve (and logistic function base model) assumes that the energy (or cost) used to extract the ressource doesn't come from the ressource itself, so that it works in a given region prior to global peak. But if you consider the global system (earth), then you also have to take the added energy required for extraction from the "global stock" itself (from the energy you extract from it), so that it makes the curve assymetrical ?

  18. The prices that editors charge for papers is outrageous; especially considering that they get them for free. Another unsustainable practice - I hope the Seneca Effect destroys it soon!! If you send me your email address, I'll send you a copy of the paper. Mine is my

    About the assumptions hidden in Hubbert, the symmetric curve results from assuming that the industry invests an approximately constant fraction of profits throughout the whole cycle. If this fraction is raised, it is equivalent to increasing the k2 or l2 constants in the model. The result is - again - the Seneca effect!!!

  19. @Nicholas. Interesting comment, the equation you cite is well known, often under the name of "May equation". The quadratic "friction" term could be incorporated in the dynamic model I use; it is possible but I think it is somewhat arbitrary and not really necessary. In my opinion, the "three-stock" model incorporates the friction effect. But, of course, there are various ways of looking at the same problem. Eventually, it is a question of common sense.

  20. @David. It is fine, I didn't mean that I understood that your approach is a criticism to mine. As I said in a previous comment, there are several different ways to look at the same phenomenon; so it is good that different models give similar results. But. as I was saying, the formalism is different, of course, but I believe that the mechanism is not so different as well. It is implicit in the dynamic model that people make choices which are based on their immediate interest. So, I figure that if we were to go in some depth in the comparison, we would find that the two models are saying the same thing, but using different languages. Or so I think....

  21. Signore Bardi, you have a wrong link to ;-)

  22. I agree with Antonio that declining EROI must be taken into account and that Hubbert's classical curve does not imply it. David Murphy has made a well-known net energy curve that shows similar cliffs than Hispanic Seneca's.

  23. The net energy curve I mentioned can be seen at

    You can see another interesting net energy curve at

    As far as I know, Hubbert worked on the quantitative side of oil extraction, i.e. his curve measures barrels of oil extracted each year. When you apply the energy content of those barrels along with the energy needed to extract them (the EROI), which follows its own curve, you have a product: quantities x net energy, and then you have as a result the net energy curve.

    You can see the three curves here:

    BTW, your dynamic system approach to peak oil predicament reminds me the one by George Mobus, I think you know his blog "Question everything"

  24. This comment has been removed by the author.

  25. Regarding the mentioned net-energy curve from Dave Murphy and Cutler Cleveland.

    As yet the EROI group working with Charlie Hall on this has yet to arrive at a consensus on how to define EROI. For a given energy source do you count the total energy consumed by the economy (call it EROI-S for whole system energy cost), or just count the mechanical energy consumed to get energy to market.

    My study, to be published with Charlie's upcoming EROI papers collection, is called Systems Energy Assessment(SEA). My basic finding is that the economy spends nominally four times as much energy for the business operating services needed to deliver energy to market than mechanical energy. The mechanical energy is mostly all that people have been counting, though. So it's a very big difference to consider what to do with. Implicitly the EROI's in Dave & Cutler's curve might possibly need to be reduced by 80%.

  26. Ugo, I recently posted a similar finding on my blog, from a natural systems physics view. It’s called "How natural system bankruptcy works"

    I'm basically saying that to produce net-energy for all its parts , a system needs to maintain an environmental productivity (EROI) sufficient to support its system productivity (SROE), maintaining EROI*SROE > 1.

    The parallels are partly the finding, likely precipitous system decline as resources become expensive and the complexity of the economy combine, causing whole sectors to become unprofitable to operate and collapse. There’s also a similarity between your definition of the key internal system factor, that you call economic "pollution", and my defining SROE as (Net GDP)/GDP, reflecting the ballooning costs a adapting an unmanageably complex growth system designed for cheap resources to ever more expensive ones.

  27. OK, pfh, it seems that my post is reverberating in a lot of earlier work. I'll read your post with attention as soon as I have a moment. Thanks!

  28. You know I have talked about all this in my book The Oil Conundrum. Rapid growth and fast decline happens all the time at the localized levels. The effect is often referred to as Gold Rush dynamics. Every gold discovery with easy access to the gold met with the same behavior, fast growth due to accelerating greed and then followed by collapse when the gold runs out.

    But this is localized dynamics. Dispersion in the effects leads to a superposition of all these curves and one can get back precisely the symmetric Hubbert logistic shape. It doesn't have to happen but with the maximum entropy principle applied to all sources of dispersion and an exponential rate modeling technology acceleration, it will drop out of the analysis.

    None of those World Dynamics models cover this and it's a pity because people have to realize that disorder and randomness and dispersion effects play a huge role in how things will play out. Anybody reading this that thinks the Seneca Effect can happen at a global scale has to reconcile the effects of statistical variation in their own mind.

    People talk about entropy without realizing that the most striking manifestation of entropy is disorder in the world we live in. If we don't model this correctly we are really missing out on some valuable insight.

    Otherwise thanks for the post.

  29. The problem never was too little oil, too little gas or too little coal; the size of the high-grade resources is quite small but the low grade stuff exists in unimaginable quantities.

    If you want to know what failure looks like, mirror the "seneca cliff" graph in the y-axis.

  30. I'm not sure of the precise mechanism that would be the driver of my theory here, but I'm thinking that there should be some factor involving the concept that growth is deliberate and planned while collapse is unplanned and thrust upon the affected entity.

  31. I am a biologist and I know something about ecology and envirementalism.

    Ithink I know what you want to define with "pollution". A better definition you can use is externalities. It is a conecpt from economy we biologist use with relation to enviroment degradation, that sometimes not come from "pollution". For example, a population of fish that get explorated until the extintion, because that fisheries are at international seas and are "commons". The "tragedy of commons" are an example of negative externality. Pollution too.

    A society for not have the "Seneca effect" need control the negative externalities and find a way to promote the positive externalities.

    João Carlos

  32. Is thermodynamically more 'difficult and expensive to build than destroy. Just leave something to herself and this degrades rapidly. Seneca did not know thermodynamics but it was probably a keen observer.

  33. Thank you for sharing this, sir. I hope to see it and other articles in Energy Bulletin, Oil Drum, and others, as well as in Facebook.

  34. Ugo,

    Very interesting post. As a non-academic, I wonder how debt enters the equation here. I'm thinking it's as an artificially-inflated source of capital, the support of which (interest on principal) becomes more onerous as time goes on and total debt to production ratio increases (interest = pollution?). This could account for the Seneca effect in all kinds of markets--stocks, tulip bulbs, cash inverted, and so on. Thoughts?


  35. Bella, I am not sure. I tend to think in terms of physical entities; energy, mainly. But maybe the model can be adapted to describe financial systems. It may be a very interesting idea - thanks. I'll be working on that

  36. One thing to mention might be another "pollution" effect causing supply curves to not have symmetric Hubbert logistic shapes. Resource users develop one habit when taking more makes it easier to take still more, and change behavior when the demand response reverses.

    At that point what happens to a system like ours, designed for ever faster use of ever cheaper resources, is the profits dry up. That kills demand, because the investors then pull out as their rule is to only invest for multiplying profits.

    That's something you can see the real experience of in the present pattern of resource prices rising exponentially as demand exceeds supply. It's followed by reaching points of economic failure for whole economic sectors at a time these days. My article on that is: A decisive moment for Investing in Sustainability

  37. Remarkable article and an incisive discussion to boot. Thanks. Steve Salmony

  38. I am not sure from the exact mechanism that might be the trucker associated with our theory right here, yet I am just convinced that there ought to be some factor relating to the concept that growth is actually deliberate and also organized while fall is improvised as well as thrust upon your affected entity.

  39. Interesting post and responses. There are many factors all coming together that will make the fall great and the longer we kick the can down the road putting off unpopular solutions the farther the fall will be. We built our modern day societies on cheap plentiful oil and the model of GROWTH to sustain the progress. Like bacteria in a petri dish, when the finite food is consumed the bacteria die off from their own consumption. Civilizations in the past have likewise done themselves in, then reestablished to new feeding grounds. Unfortunately we have no new feeding grounds left, which brings us to the bacteria petri dish scenario. Why do we think this time will be different in our dogmatic growth model approach as the solution? It is amazing to me how much carrying capacity momentum we are experiencing in our overshoot condition.

  40. Well, I read the article, and all the comments, but they still didn't dispel the notion that all these models ignore human motivation and the "viral" effect demonstrated by YouTube videos.

    Suppose, for an instance, that some shortage happens in a country like Bangladesh, and a 100 million people die from a cause that is easily understood to be common to all nations.

    Suppose the real tipping point is in the very complex minds of those zombies that are dismissed as stable in these Cassandra models, and that the population of a large country reacts intelligently, in a reverse of the silliness that caused Germany to dump its nuclear power plants.

    Suppose that India builds out thorium reactors with no regard for any outside nation's opinion, as they embraced neutrality and then built nuclear weapons.

    There are more examples, but you catch my drift. None of you could have predicted ten years ago the domination of the communications market by the smartphone you can talk to. 400 watt-hour/kg Lithium batteries. 40% efficiency in broadband multilayer solar panels.

    You seem to ignore solar energy input completely. Why is that?

    Cassandra may be right. Or wrong. I'll hang in there, living in my teeny house, eating a little.

  41. Well, Ormond, I can only hope that you are right. It is no fun being Cassandra!

  42. L'ho riletto oggi, Ugo. La Storia ha già galoppato da allora e sembra proprio che sia arrivato il momento di farci un paio di ali, per poter vedere, almeno per un poco, la scogliera di Seneca dall'alto...
    Pietro C.

  43. Great work! Yet I think it is still too technically to be understood by the majority of people and decision makers. Apart from that I am afraid that the notion that environmental damages are to be paid for isn't shared. However if you insert the financial system instead, with its rational and its psychological implications, you can nicely show convincing chain reaction that can be easily understood. I did so in my latest book. Bringing in the financial system may also lead to some balancing feedback loops but that would indeed make the model more complicated.
    One word to the idea of intensifying nuclear power. Wouldn't it be a sure mistake to think about nuclear energy as simple minded (monocausal) as the decision makers from the past and the present did when they formed our civilization that so heavily depends on oil and gas? There is not enough uranium to provide for more of 15 percent of the current energy need of the world. So let us go thorium? It is not the people that prevent us from using thorium, it is the financial market and a number of engineers. Pro thorium is a lobby of industries and people who need a simple solution to feel good again. (...) There need to be considered many, many factors with the use of thorium, probably not in a simple system dynamics model but rather in a qualitative cause and effect model. And that is the problem: we are looking for simple solutions instead of thinking of complicated, orchestrated solutions that could easily save 75 percent of today's energy consumption without any loss of living standard.
    So let us continue to provide simple illustrations of a complicated and complex reality but please let us not say that others didn't understand the complexity and now we know the solution that we haven't reflected on either.
    Keep modeling

  44. I didn't understand how this stuff works. And until now Seneca's effect is still a question for me why decline is faster than growth.

  45. I developed a simple Android app to visually explore the proposed model:

  46. I have no expertise in these fields, but nevertheless, my tuppence:

    We should be careful with this interpretation, because some other authors believe that the Mayan collapse was caused by climate change

    I'm intrigued by the thought that, if that's the case, might the climate change (local, presumably, as opposed to global) have itself been brought on by the 'pollution' (soil depletion)? If so, the similarity to our current situation is startling.

    At some point, the Soviet Union ceased to exist as an isolated economic system...

    I worked in Moscow on two occasions covering several months during 1994. I can vouch for the dire state of the society at that time. Your 'mind sized' model here would seem to imply that the Soviet Union's collapse was brought about by their inability to extract/ produce/ source oil -- a situation that, while it may have been exacerbated by the Cold War, may also have been incidental. If the latter, it would tend to put the lie to the popular 'communism was defeated by capitalism' meme -- something that some people would go to extraordinary lengths rather than admit (especially to themselves).

    PS typo alert:
    "... but, here I turned it [of?] 90 degrees clockwise..." (extraneous word?)
    "Now, can we say in words what [it] is that generates the Seneca cliff?" (missing word)
    "... similar to Tainter's model that has that civilizations decline and collapse because of..." (something wrong at 'that has that', though I'm not quite sure what)
    "... the Seneca effect is a "robust" feature of this kind of [model] and it..." (not 'models')

    1. Thanks, pedantry. These post would need a professional proof-reading, but I haven't got the resources to do that. Anyway, I'll see to correct as you suggest!

    2. About your other points, yes, the story of climate change bringing down the Mayans is intriguing. We could build a case for climate change being created, or at least exacerbated, by human actions. But we don't have enough data to say much.

      About the Soviet Union, yes, I was in Moscow, too, in the mid 90s. My interpretation (today, not then) is that the whole system collapsed Seneca-like not for a single cause, but for the accumulation of a negative feedback between increasing extraction costs and the heavy demand created mainly by the military build-up. It is typical of empires, anyway

  47. Development is disorderly (entropic) except that someone may have planned for whatever profit was to be had.

    Development is an as yet unaccounted for form of Pollution that will steepen the descent.

    Pavement is a huge unaccounted for source of CO2 loading (Raciti et al, 2012). It also indiscriminately covers the best farmlands; reduces groundwater recharge and creates heat islands...yet, socially and culturaly pavement is practically invisible. Depaving will also require huge investments of energy. Most of the world's populations and governments have not even figured this out yet. And here we are in the final minutes of the game...

    1. Again population is pollution.

      As we know the world's population is still growing rapidly (225,000/day I read earlier today).

      I think of my last visit to Rome, compared to my first visit in 1980. All those damn tourists messing it up for the rest of us! Commute times are getting worse, 3x slower now than 20 years ago where I live.

  48. Many thanks Ugo giving great speech at the 1. Dresdener Energie- und Rohstoffkonferenz (#DDerk the hashtag on Twitter). Here with links to your slides:

    2006 while working at BMW's new plant in Leipzig under its then active plant manager Peter Claussen, I got invite to the ISDC in Nijmegen by Prof. John Sterman. Since then system dynamics, and the (more) operational approach of lean thinking has become part of my work.

    I'd be very much interested to bring the matter forward on a broader basis in the economic context. Please let me know what you think, or whom to approach for a possible research work.

    1. Thank you, Ralf. Glad to hear that you found, my talk interesting. About your questions, please write to me at ugo.bardi(whirlet)

  49. I read with pleasure your paper on the seneca cliff. It was link at a recent article of Gail Tverberg ( which treat of PeakOil for years in an interesting multi factor manner.

    I translated in French some of it's post and today it was your turn. See here the result.

    I will push this translation on different web sites to help to spread your work in France

  50. Hi,
    The Hubbert curve relies on a substitution being found. In the diagram below, nuclear (notice the height!) is the substitute for oil, thus allowing a gradual downslope. If there is no oil substitute, well...

  51. My thinking on this: As you and the Anon above mentioned, the Hubbert curve is only symmetric under the assumption of a replacement because you can take your sweet time letting it decline naturally. Now that worldwide conventional crude has peaked we need substitutes. Each of those substitutes is subject to its own Hubbert curve, however those substitutes are harder to get and decline faster, so their Hubbert curves are narrower and shorter. If you add several of those you fill out the right side of the graph of the sum but in the end the right edge becomes steeper - just like you model a step function with a series of higher frequencies and lower amplitudes in a Fourier transform (or DCT, wavelet transform etc.) Or simple calculus - the integral under the curve can not exceed some total (which you can increase with technology and money, but not too much) and the production rate will only decline to zero asymptotically if you let the derivative decline naturally. If you try to keep it up, it'll try to intersect the x-axis, i.e. drop hard at some point.

    1. "the integral under the curve can not exceed some total"

      I have a hunch that our exponential growth is composed of two primary drivers or inputs: cheap fossil fuels and debt. If money is a type of energy token, and debt a form of energy from the future, then perhaps allocating future value (via debt) to the present will also increase the steepness of the seneca cliff outcome.

      Thus we have peak energy consumption and peak debt.

      Is it possible to add another feedback box to the diagram above for debt?

    2. "the integral under the curve can not exceed some total"

      I suspect that are primary inputs for our exponential growth is cheap energy and debt. If debt is a from of an energy token, then perhaps we can see it as increasing the current value of production at the expense of the future. That is, it would lead to the steepness of the downslope.

      Thus we have peak energy and peak debt as important events.

      Hmm, is it possible to add debt to the above diagram? Just curious :-)

  52. One form of "pollution" is compliance costs.
    That bureaucratic expansion is what destroys empires, as they increase size and economy then the initially bureaucracy has a modest cost and good governing input. As "economic" growth increases so does the bureaucratic "pollution". When things start to decline, often in real terms (eg wood to Rome), political upheaval etc. Then the bureaucratic pollution increases, and doesn't decrease with the loss of useful production, as the system struggles to adapt itself. Result: Seneca curve, the feedback loop enhances the failure, not slows it - same with many huge companies, the very systems they put in place to become big, are what consumes them and prevents them from being able to adapt to new opportunities.

    PS What you said about Cassandras predictions always being true? I don't believe you ;)

    1. Yes. Cassandra was always right. She had been cursed by Apollo that her predictions would always turn true but nobody would believe her! And what Gods say, happens

  53. I like how it was pointed out the delayed effect of our actions. Makes me think outside the box.

  54. If people believed Cassandra's predictions, then they would plan ahead, and perhaps ruin would not be so sharp... By the same token, the sharpness of the peak may be proportional to the level of our denial.

  55. "the first graph of the seneca effect looks suspiciously lilke a graph which is simply exponential growth until it suddenly stops and the population goes to zero."

    it reminds me of the "tragedy of the commons" - growing exploitation of limited resources, and growing dependence on those resources, until the limit is reached. one sees the same dynamic in a debt/credit economy - growing claims on future productivity as a basis for present activity.

  56. Thank you for this fascinating article!!!
    I'm planning to write a nice mathematical activity involving three interdependent sequences based on your "Resources-Economy-Pollution" model for my high school pupils...
    A great alternative, in that regard, to the predator-prey model - with some similarities, for that matter, when I think of its delayed peaks and valleys, but this would be another story!...

  57. Just a pedantic critique, but the second law of thermodynamics does not state that all spontaneous reactions increase entropy, because they don't (under the technical definition of spontaneity and Gibb's free energy). The law states that the entropy of the universe as an isolated system is always increasing. Your analogy still holds, but I just had to make that correction.

  58.  Contoh Email Thank you for this fascinating article!!!
    I'm planning to write a nice mathematical activity involving three interdependent sequences based on your "Resources-Economy-Pollution" model for my high school pupils...
    A great alternative, in that regard, to the predator-prey model - with some similarities, for that matter, when I think of its delayed peaks and valleys, but this would be another story!...



Ugo Bardi is a member of the Club of Rome, faculty member of the University of Florence, and the author of "Extracted" (Chelsea Green 2014), "The Seneca Effect" (Springer 2017), and Before the Collapse (Springer 2019)