Population Collapse in Elite-Dominated Societies: A Differential Equations Model without Differential Equations

Many civilizations have risen and then collapsed. There can be many causes. A major influence can be how Elite (wealthy or ruling) populations interact with the Commoners (workers) and with the environment. Each population's size can fluctuate. We say a model is "Elite-dominated" when the Elites' per capita population change rate is always at least as large as the Commoners'. First we present a class of ordinary differential equation models in which the Commoner population C(t) as a function of time t always crashes (i.e., C(t) -> 0 as t -> infinity), sometimes only after undergoing many oscillations. Our main tool is a new Lyapunov function theorem. Next, we discard the equations entirely, replacing them with qualitative conditions, and we prove these conditions imply population collapse must occur.