Q&A with Dr. William Basener on his Easter Island collapse model
I hope its OK to put a comment on this discussion of the press release about work I was involved with. I really like Rian’s discussion here though - you make some good points. Thanks for the grace about the wording of the press release…
The first point I should make is that other people have worked on this problem. However, all previous models used assumptions from neoclassical economics which a priori assumed away the equilibrium. People had done nice work on the economics and archeology, but all of their mathematical models predicted that the population of Easter Island, and its resources, would return to their high values, then collapse again, and repeat forever. In short, they predicted the behavior would be periodic. But the trees, and much of the ecosystem, were completely destroyed so a replenishing was impossible without outside help.
The original part of what we (David Ross and myself) did was to reform the differential equations model without these assumtpions. The new model can handle the possibility of a complete elimination of some of the resources.
The phrase “the first mathematical formula to accurately model the island’s monumental societal collapse” reads better in a press release than “the first model to avoid the funamental assumptions of neoclassical economics and predict the possibility of complete anhiliation of one of the resources in finite time.”
The complete paper is available from my website:
Q: What new have you discovered?
A: We developed a new way to model the growth of the human population and its use of resources.
Q: What are the conclusions?
A: As with anything in mathematics, we determined the implications of basic assumptions. The assumptions in this case where regarding the population’s use of their environmental resources. We showed that if a society uses its resources at quickly enough, if the resources replenish slowly enough, and, now this is the important part, if when the resources become scarce that the society continues to use the resources quickly enough then the resource will go extinct and the society will crash. We also developed equations that determine precisely what “enough” means here, and we showed that the situation on Easter Island was ripe for such a collapse.
This conclusion is quite obvious, as well it should be; I would be skeptical of a conclusion that could not be explained using common sense. The point is that the mathematics enables us to describe the process very precisely, allows us to determine if a given civilization is headed for a collapse. It helps us to determine the long-term sustainability of resource use. One of the interesting results is that a collapse due to environmental overuse can be extremely sudden; consequently a civilization can be headed for a collapse even when the population appears to be growing steadily. This is evident in the graphs in our paper.
It’s worth mentioning the limitations of the mathematics. Mathematics can determine exactly the end result of certain rates of resource use, but it does not determine precisely how the resources are being used by a particular society. Math is extremely good at predicting the future consequences of a given system, but it is only as good as our ability to describe a civilization in mathematical terms. Any model incorporates only some factors. The more factors you include, the more accurate the model is, but also the more difficult it becomes to analyze. It is important not to underestimate the mathematics; it is very good at what it does. But we shouldn’t overestimate it either; math cannot completely recreate a civilization in a few equations. I believe the math can be a very useful tool when combined with the work of archeologists and economists. Ecologist Peter Turchin has two very nice books, “Complex Population Dynamics: a Theoretical/Empirical Synthesis” and “Historical Dynamics: Why States Rise an Fall” about using differential equations to model human populations.
Q: So what does this tell us about the population crash on Easter Island?
A: Well, it tells us that it is possible, even perhaps likely, that the Polynesian population on Easter Island experienced a collapse because of overuse of their environment. There have been many proposed causes; environmental overuse, cannibalism, disease, war, slave trade, environmental disaster such as a hurricane, etc. Easter Island had particularly slow growing species of trees, and, according to our work, this would make a collapse more likely. Also, the use of the trees to build and transport the Moai would have made a collapse more probable. I should mention that previous work, by economists James Brander, and Scott Taylor arrived at a similar conclusion using a different but related model.
There are several well-known examples of societies that have experienced a collapse. Several of these are described in Jared Diamond’s recent book “Collapse.” In each case there are a number of theories for the cause of the collapse. For example, take the Maya Civilization. At their peak, the population was as dense as modern day Los Angeles, and they built the magnificent buildings, but their population very suddenly dropped to a small fraction of their peak. Many possible causes have been proposed, including war, disease, drought, slavery, and trade disruption. Almost certainly, most of these factors were present in their collapse, but the leading cause or causes is hotly debated. David Webster, who is an archeological anthropologist at Penn State, wrote a very nice book called “The Fall of the Ancient Maya” about this collapse. These causes tend to go in and out of vogue, but the oldest and most durable cause has been environmental overuse. However, recent focus has been on drought.
We are presently looking into the reproduction rates of the resources the Maya used, mainly Maize and breadnut, to determine if a collapse could be caused by environmental overuse alone. We also plan to investigate what the effects of a drought would be on a population using their resources. It would be nice to know if a drought would merely decrease the number of people a region could support, or if it could actually trigger a collapse. Interestingly, the Maya collapse began near the bodies of water. This seems contrary to a drought hypothesis, but this seems unlikely for any collapse.
Archeology suggests that the Maya did not collapse from a single catastrophe such as a hurricane. There appears to be a gradual decline in the number of trees available, which can be detected by looking at the construction methods the Maya used in their later years. This would suggest a progressing environmental decline as opposed to a catastrophe.
Archeology is very good at determining what happened. Mathematics, and differential equations in particular, is very good at determining cause and effect. So, archeology determines that a collapse happened in a particular society, and we can use the mathematics to determine if a suggested set of causes could lead to the collapse.
Its worth mentioning that some researchers are using what are called multi-agent models, such as Robert Axtell who is at The Brookings Institution. These models have many more factors built in, modeling each individual person, and thus in some ways are more realistic. The drawback is that it becomes more difficult to understand the cause-and-effect because there are so many factors involved. In the terminology of mathematicians, the phase space is very high, which often leads to instabilities and chaos, some of which may not be present in the actual civilization. It’s a lot like modeling a gas by describing every molecule individually or using the ideal gas law PV=nRT.
Q: Can your work be applied to modern civilizations? Perhaps for the use of oil?
A: We believe so. We are collaborating with some economists on applications to the present. Long-term sustainability is a well-studied topic, although much of the work is still being done. We believe we can contribute to the understanding of the nonlinear differential equations involved, but its just part of a much larger discussion.
The important thing here is for researchers from a variety of fields to collaborate together. It is difficult to communicate between mathematics, economics, ecology, sociology, archeology, etc., but very important. We all have our own way of looking at things and our own jargon.
Present civilizations are more difficult to model. Ancient civilizations such as the population on Easter Island or the Maya were relatively isolated, which makes the easier to model, making the phase space 2 or 3 dimensional. Modern civilizations have more interactions and the phase space for such models could easily have thousands of dimensions, making the model more difficult to analyze. (Each dimension corresponds to a factor, say war, trade, corn, rice, weather, each tribe or nation, etc. that you build into the model.)
Concerning oil, the replenishing rate (growth rate) of oil is very low (practically nil) so this puts oil in the “collapse zone” as a resource. The real question is whether we will make the change to other sources of energy early enough to avoid repercussions of a shortage. Solar energy and nuclear energy are both good sources of energy from the point of view that they consume relatively small amounts of resources.
Philosophically, our model does say something important in general which I mentioned earlier. Societies can be near a collapse even when they are growing steadily. The rate of resource use is key. For example, improving oil drilling may in fact be detrimental to the world; it could allow us to continue using oil as a main resource longer, leading to a more drastic collapse.
Modeling, particularly modeling human civilizations, is a many step process. Someone has to study the civilization. Someone has to develop a mathematical model. Someone has to analyze the model. Someone has to compare the analysis to history. The hard part is to get the specialists in each step to work together. We are mathematicians, and hence work mostly on refining models that others have developed and analyzing the models, although we can compare our results with observations of archeologists and economists.
Q: Is your approach to blending archeology, economics, biology, and mathematics new?
A: Some of it is and some of it isn’t. Economics in general uses mathematics and observations about a society to determine possible outcomes. As I mentioned before, Brander and Taylor used differential equations to study the Polynesians on Easter Island, and their work was published in the American Economic Review. They certainly incorporated all of these areas. Peter Turchin, whom I mentioned earlier, discusses the role of differential equations in modeling human populations, and entire civilizations in particular. I would say that we are part of an emerging trend.
We did have two new ideas. First, we developed a more mathematically sophisticated model. Most population models are built on neoclassical economics. There is an implicit assumption in this approach that excludes extinction as a possibility; every solution tends to either a periodic cycle or equilibrium. Yet, the trees on Easter Island were completely used up. Reality doesn’t match the model, so we rebuilt to model from a large-scale perspective taking more of the mathematics into account. As mathematicians, we think of models in terms of their long-term behavior: equilibrium, periodic cycles, chaos, and singularities. Our model has a singularity that enables collapse and extinction.
The singularity can be described in common sense terms. What we did is assume that the growth rate of the population depends on the amount of food available per person. (This is what economists call a ratio-dependent model.) We also assume that even as resources become scarce, people still harvest enough food to eat. The is an appropriate assumption for a population on an island whose primary resource (for construction and food) was trees. Trees are easy to find and cut down on an island even when they are scarce. To put it another way, when fish become scarce, their natural predators starve. People, on the other hand, build bigger nets.
The second original contribution we made was to take advantage of the power of the mathematics. Complete mathematical analysis of a model determines every possible outcome for every possible situation that the model describes. Archeology is about answering “what if” questions. People suppose a “what if” consider the implications, and compare the implications to empirical observations. Mathematics answers the “what if” question absolutely. It establishes the implications of a given set of “what ifs” absolutely. Moreover, proper use of the mathematics determines the implications for all possible civilizations for a given model simultaneously. In other words, properly exploited mathematics determines the outcomes of infinitely many different “what if” societies at once. We try our best to comment on the comparison to empirical data, but that is done much more effectively by archeologists and applied scientists.
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