Booming and Crashing Populations and Easter Island

Abstract

The population of Easter Island grew steadily for some time and then suddenly decreased dramatically; humans almost disappeared from the island. 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 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 archeological 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.

Growth on a finite planet: resources, technology and population in the long run

“…The interaction between population growth and biological laws generates rich dynamics, including feast-famine equilibrium paths and/or envi-ronmental crises that can eventually drive the human society to extinction (Taylor, 2009). Bio-economic models have been calibrated to replicate the collapse of Easter Island and similar historical episodes (Brander and Taylor, 1998;Basener and Ross, 2005;Good and Reuveny, 2009). Several authors have argued that the Easter Island economy is a metaphor of resource-based closed systems like Planet Earth and extended the model to include manufactured goods (Reuveny and Decker, 2000), intentional capital bequests (Harford, 2000) and endogenous technological change (Dalton et al, 2005).…”

Noether Symmetries Quantization and Superintegrability of Biological Models

“…In [10], a mathematical model that describes the dynamics of growth and sudden decrease in the population of Easter Island was proposed. The model is a nonlinear system of two first-order ordinary differential equations, i.e.,ẇ…”

“…More details on the construction of this model can be found in [10] where numerical solutions of the system (1)- (2) were given and a qualitative analysis of the general behavior of solutions both in finite and infinite time was presented.…”

“…Indeed, one may quantize and also detect superintegrability of biological models by means of Noether symmetries. We exemplify the method by using a mathematical model that was proposed by Basener and Ross (2005), and that describes the dynamics of growth and sudden decrease in the population of Easter Island. …”

The slow demise of Easter Island: insights from a modeling investigation

“…In contrast to most previous modeling studies on Easter Island (Brander and Taylor, 1998;Anderies, 2000;Dalton and Coats, 2000;Reuveny and Decker, 2000;Basener and Ross, 2004;D'Alessandro, 2007;Basener et al, 2008;Sprott, 2011;Motesharrei et al, 2014), we evaluate the simulations by comparing the model results against two compilations of radiocarbon dates (Hunt and Lipo, 2006;Mulrooney, 2013), because the frequency distribution of these radiocarbon dates provide a temporally resolved estimate of the forest clearance intensity, which can be directly compared to the forest clearance rate predicted by the model. Unlike other studies (Collard et al, 2010;Mulrooney, 2013;Shennan et al, 2013), therefore, we do not consider the radiocarbon frequencies as an indication of the size of the human population, but rather of the islanders' clearance activities.…”