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

Tuesday, January 17, 2017

Amelie the Amoeba: How Things Grow

This academic year, I gave a lesson on the growth mechanism of complex systems. It is a fascinating subject that can be applied to several fields, from biology to economics. Since the students I was talking to were not specializing in complex systems (they were students of geology), I used a light tone and used "Amelie the Amoeba" an image for the growth mechanism of bacteria in a Petri dish of many other things dish. Then, the image above summarizes what I told them.

If you know about these matters, you can probably understand what the drawings show. If you don't, some notes are appropriate. So, here is a very brief summary of how things grow in the universe.

1. The "Solow" mode, or exponential growth. The name refers to the economist Robert Solow who proposed this model, but most economists today seem to argue that exponential growth is the natural, actually the only possible, mode of growth of the economy. They may not be completely wrong; after all, it is the way bacteria grow (for a while) in a Petri dish. So, Amelie the Amoeba is very happy to be growing exponentially, too bad that if she were to continues for a long time, she would eventually devour the whole universe.

2. The "Malthus" mode, also "Verhulst" or simply "sigmoid" mode. It takes into account the fact that the Petri dish contains a limited amount of nutrients and Amelie can't keep growing forever. Malthus was the first to apply this model to the human population, assuming that it would reach a certain limit and then stay there: contrarily to what commonly said, Malthus never predicted collapses. The concept of "collapse" was alien to him, but at least he was right in noting that all physical systems have limits.

3. The "Hubbert" mode or the "bell-shaped" curve. That's more like what could happen to Amelie in a Petri dish. Grow for a while, reach a "peak amoeba" size, and then shrink and die for lack of food. Hubbert applied the model to the oil production of the United States, predicting reasonably well the future of the extraction of "conventional" oil. And, if you try to do the test for bacteria (or amoebas) in a Petri dish, it works as well.

4. The "Seneca" mode. This is the name I gave to the kind of growth kinetics where the decline is much faster than the growth. It comes from something that the Roman philosopher Lucius Annaeus Seneca said in one of his letters ("increases are of sluggish growth, but the way to ruin is rapid") and it happens all the time, even to amoebas in a Petri dish.

5. The "Hokusai" mode. The Japanese painter Katsushita Hokusai never made mathematical models and he probably never knew what an amoeba is. But with his famous painting, "the wave", he provided a good visual impression of what happens when things get real bad. Not only decline is faster than growth, but the curve actually starts chasing you! Even amoebas can get nasty and eat your brain.


  1. Thanks for this, Ugo!

    A very accessible presentation of exponential growth (and decay), complete with a handy classification among its varieties...

    To my mind, you address perhaps THE barrier to popular understanding of our plight.

    Hear, hear!

  2. The denial of collapse is part of reality no one likes to face, like the denial of physical death, or denial of our animal nature and total dependence of natural systems. Having everything we need as consumers, provided for by the complex and dispersed economic system, of "invisible hand", is "too much magic". Death and the destruction of nature is mostly out of sight. Financial system manipulations of mountains of debt are also phantasmagorical. People do not want to hear that the party is over, and goodies have run out, and the hordes of deprived multitudes just outside are teeming to gate crash. Denying reality, political responses turn out to be increasing in grossness, since fairness of distribution of resources, and collaborative sharing of responsibility has never been a strong feature of diverse groups of peoples who won't get along.

  3. Good day to you Ugo, I think you might want one more mode in there: the Turchin mode. Peter Turchin shows (pertty convincingly, to my mind), that human agrarian societies have relatively rapid population growth compared to the social and political feedback dynamics. The result is a second order, or oscillatory population curve.

    The emphasis of that research isn't resource depletion as such, but it is highly relevant to collapse dynamics. One of the factors on the downswing is the reduction in available resource base as a result of increasing unrest / warfare, but clearly the same is true as a result of loss of technological capability in a declining industrial society.


  4. Ugo

    Perhaps the problem is that were mislead ourselves into looking at growth in isolation of everything else. We should look at growth in term of a part of a process of transformation of a system from one state to another. Example the growth of an infant into an adult, once the transformation is complete the growth stops, if it doesn't you get illness, deformity and eventually early death, continued growth becomes pathological.

    The promotion of growth at all cost is therefore pathological thinking.

  5. Hokusai curve(s) might show our American political system. (as in brain eating ameoba) I expect an equal and opposite, but just as damaging, reaction from our next Democratic president. We keep expending resources our grandkids will never see, believing we can create infinite growth. We will spend trillions fighting in the middle east and trillion creating new roads and bridges for oil that will no longer exist. Oh my poor little amoeba brain!



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)