The Doomsday Equation and 50 years beyond: new perspectives on the human-water system

In 1960, von Foerster et al. humorously predicted an abrupt transition in human population growth to occur in the mid-21st century. Their so-called ‘Dooms-day’ emerged from either progressive degradation of a finite resource or faster-than-exponential growth of an increasingly resource-use efficient popula-tion, though what constitutes this resource was not made explicit. At present, few dispute the claim that water is the most fundamental resource to sustainable human population growth. Multiple lines of evidence demonstrate that the global water system exhibits nontrivial dynamics linked to similar patterns in population growth. Projections of the global water system range from a finite carrying capacity regulated by accessible freshwater, or ‘peak renewable water,’ to punctuated evolution with new supplies and improved efficiency gained from technological and social innovation. These projections can be captured, to first order, by a single delay differential equation with human–water interactions parameterized as a delay kernel that links present water supply to the population history and its impacts on water resources. This kernel is a macroscopic representation of social, environ-mental, and technological factors operating in the human-water system; however, the mathematical form remains unconstrained by available data. A related model of log-periodic, power-law growth confirms that global water use evolves through repeated periods of rapid growth and stagnation, a pattern remarkably consistent with historical anecdotes. Together, these models suggest a possible regime shift leading to a new phase of water innovation in the mid-21st century that arises from delayed feedback between population growth and development of water resource.