Socio-economic and environmental changes are well known causes of demographic collapse of agrarian cultures. The collapse of human societies is a complex phenomenon where historical and cultural dimensions play a key role, and they may interact with the environmental context. However, the importance of the interaction between socio-economic and climatic factors in explaining possible breakdowns in Native American societies has been poorly explored. The aim of this study is to test the role of socio-economic causes and rainfall variability in the collapse suffered by the Aymara people of the semiarid Andean region of Tarapacá during the period 1820-1970. Our motivation is to demonstrate that simple population dynamic models can be helpful in understanding the causes and relative importance of population changes in Andean agro-pastoral societies in responses to socio-environmental variability. Simple logistic models that combine the effects of external socio-economic causes and past rainfall variability (inferred from Gross Domestic Product [GDP] and tree-rings, respectively) were quite accurate in predicting the sustained population decline of the Aymara people. Our results suggest that the depopulation in the semiarid Tarapacá province was caused by the interaction among external socio-economic pressures given by the economic growth of the lowlands and demands for labor coupled with a persistent decline in rainfall. This study constitutes an example of how applied ecological knowledge, in particular the application of the logistic equation and theories pertaining to nonlinear population dynamics and exogenous perturbations, can be used to better understand major demographic changes in human societies.