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.
