We derive and analyse a model with unusual features characterizing human activities over the long-run. First, human population dynamics draw heavily on consumer-resource modelling in ecology in that humans must consume biological resources to produce new humans. Second, the model also draws heavily from economic growth theory in that humans do not simply consume biological resources; they also produce the resources they consume. Finally, humans use two types of technology. Consumption technology affects the rate at which humans can extract resources. Production technology controls how effectively humans convert labour into new resources. The dynamics of both types of technology are subject to cumulative cultural evolutionary processes that allow both technological progress and regress. The resulting model exhibits a wide range of dynamical regimes. That said, the system is routinely sensitive to initial conditions, with wildly different outcomes given the same parameter values. Moreover, the system exhibits a basic fragility in the sense that human activities often lead to the endogenous extinction of the human species. This can happen gently, or it can follow periods of explosive human activity with super-exponential growth that ends in collapse.This article is part of the theme issue 'Evolution and sustainability: gathering the strands for an Anthropocene synthesis'.