SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1736-1757, January 2020.
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., $C(t)\to 0$ as $t\to\infty$), 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.

中文翻译：

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精英统治社会中的人口崩溃：没有微分方程的微分方程模型

SIAM应用动力系统杂志，第19卷第3期，第1736-1757页，2020年1月。
许多文明崛起然后崩溃。可能有很多原因。主要影响因素可能是精英（富裕或统治）人群与平民（工人）以及环境之间的相互作用。每个人口的规模都可能波动。当精英的人均人口变化率始终至少等于平民百姓时，我们说模型是“精英主导”的模型。首先，我们提出一类常微分方程模型，其中平民百姓$ C（t） $作为时间的函数$ t $总是崩溃（即$ C（t）\ to 0 $作为$ t \ to \ infty $），有时只有在经历多次振荡之后，我们的主要工具是一个新的Lyapunov函数定理。接下来，我们将等式完全抛弃，将其替换为定性条件，然后证明这些条件暗示必须发生人口崩溃。