Booming and Crashing Populations and Easter Island

Abstract

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.