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# Mathematically Modeling Population Collapse in “Elite-Dominated” Societies

Throughout history and prehistoric times, many civilizations have risen and collapsed. Examples range from the Mayan people and isolated island populations to regions of ancient Egypt, India, and China. However, some countries and civilizations, such as the Pacific island of Tikopia, are able to avoid collapse. Tikopia maintained an average population change rate of zero and a sustainable rate of resource use by preventing overpopulation with “virtual suicide” — during which citizens embarked on dangerous overseas voyages that resulted in population control. Primogeniture, wherein the oldest male child inherits the family’s wealth, limits the size of the wealthy or ruling communities and can also prevent collapses. However, societies like Tikopia are rare exceptions. So the question that remains is, Why did so many civilizations collapse?

In a paper that published today in SIAM Journal on Applied Dynamical Systems, Naghmeh Akhavan and James A. Yorke present a class of differential equation models that they use to determine the cause of population collapse. The authors compare the wealthy/ruling community, referred to as the “Elite” population, to the workers, referred to as the “Commoner” population. Their work is supported by the notion that an elite-commoner economic stratification can sometimes collapse for good. Akhavan and Yorke employ the new Lyapunov function theorem tool to demonstrate an equilibrium point's stability, then replace the equations with qualitative conditions to prove that population collapse is inevitable.

Many civilizations throughout history have risen and collapsed. Public domain image.

The four main causes of civilization collapse are environmental damage, climate change, hostile neighbors, and trade partners. The interaction between Elite populations, Commoners, and the environment is the biggest contributor to populations’ rise and collapse. Since population size continually fluctuates, one can consider a mathematical model “Elite-dominated" when the Elites' per capita population change rate is always at least as large as the Commoners’ rate.

The Commoners hunt and gather food for the community while the Elites organize and control food distribution. Akhavan and Yorke offer two criteria for Elite-dominated models; in these models, (i) an increase in food consumption never decreases the per capita population change rate, and (ii) Elite individuals always consume more food than Commoners. A model experiences population collapse when the Commoner population (the food suppliers) dies out.

The authors investigate the HANDY (Human and Nature Dynamics) model, which serves as an example of Elite dominance. This model helps to explain the causes of per capita population change rates, also known simply as change rates. The Elite-dominated HANDY model assumes the change rates to be equal and positive for plentiful food. When food is scare, the change rate is negative. This model accounts for various factors, including the amount of food that is distributed to the people, the amount of stored food, and the climate. It excludes scenarios in which one population receives more food per capita than the other while maintaining a lower per capita change rate.

Akhavan and Yorke’s study concludes that when Elites and Commoners assume their respective roles, populations will collapse due to overpopulation among the Commoners and the fact that the Elites consume more food. When food is plentiful and the normalized food supply is greater than or equal to 1, Commoners consume just enough to maintain their maximal change rate. When the normalized food supply is less than 1, every Commoner gets less food. Only the amount of food is allocated in this scenario; this amount is distributed on a per capita basis, with each Elite receiving more than a Commoner. The Commoner change rate thus decreases and can even become negative.

This study provides a means of assessing the issues and causes within collapsed societies. When the ruling Elite community has excessive control of the food source's quantity and allocation, the community that works to hunt, gather, and provide food suffers. If the Elites receive more food, the workers or commoners lack the sustenance or stability necessary to continue providing for the civilization as a whole. This imbalance results in the indefinite collapse of Elite-dominated civilizations.