Booming and Crashing Populations and Easter Island
Booming and Crashing Populations and Easter Island
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DOI：
10.1137/s0036139903426952
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
Bill Basener;D. Ross
中科院分区：
文献类型：
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作者：
Bill Basener;D. Ross
复活节岛的人口在一段时间内稳定增长，然后突然急剧下降。这不是孤立种群的通常逻辑微分方程模型或使用资源的种群的捕食者-猎物模型所预测的那种行为。我们提出了一个数学模型，当食物和树木等资源的增长率低于资源收获的速率时，可以预测这种类型的行为。我们的模型在数学上表达为两个一阶微分方程组，这两个方程都是广义逻辑方程。使用实际参数对方程进行数值解，预测值非常接近复活节岛的考古观测值。我们通过使用坐标变换来炸毁原点处的奇点来分析模型。我们的分析揭示了令人惊讶的丰富动态，包括简并的 Hopf 分岔。
The population of Easter Island grew steadily for some time and then suddenly decreased dramatically. This is not the sort of behavior predicted by the usual logistic differential equation model of an isolated population or by the predator-prey model for a population using resources. We present a mathematical model that predicts this type of behavior when the growth rate of the resources, such as food and trees, is less than the rate at which resources are harvested. Our model is expressed mathematically as a system of two first-order differential equations, both of which are generalized logistic equations. Numerical solution of the equations, using realistic parameters, predicts values very close to archaeological observations of Easter Island. We analyze the model by using a coordinate transformation to blow up a singularity at the origin. Our analysis reveals surprisingly rich dynamics including a degenerate Hopf bifurcation.