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

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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 ‘Doomsday’ emerged from either progressive degradation of a finite resource or faster‐than‐exponential growth of an increasingly resource‐use efficient population, 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, environmental, 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 resources. WIREs Water 2015, 2:407–414. doi: 10.1002/wat2.1080 This article is categorized under: Engineering Water > Planning Water
Power‐law growth in global human population. The solid line is a log‐periodic fit, described in Equation , with parameters a = 1.98 · 108, b = 1.66 · 1012, tc = 2062.7, c = 1.82 · 1011, β = − 1.37, ω = 5.82, and ϕ = 6.17. The dashed line is a power‐law fit, N(t) = k(tct)− 1/δ with k = 4.2 · 1011, tc = 2062.7, and δ = 0.957. The dot‐dashed line is the original power‐law fit by von Foerster et al. with k = 1.8 · 1011, tc = 2026.9, and δ = 1.01.
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Historical predictions of global water withdrawals (a) and per capita water demand (b) from the log‐periodic model. The inset in (a) shows the observed 20th century expansion and the predicted mid‐21st century expansion. In (b), population is obtained from the log‐periodic fit in Figure .
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The 20th century global water withdrawals (a) and per capita water demand (b) with corresponding log‐periodic, delay, and logistic model fits. Parameters for the log‐periodic fit are a = 0, b = 5.65 · 107, tc = 2092.5, c = 1.58 · 107, β = − 2.13, ω = 6.23, and ϕ = 2.02. Parameters for the delay model are Aw = 1072, B = 1.77, τ = 50, r/w = 4.59 · 10− 5, W(t = 0) = 394, w = 6.58 · 10− 7. Parameters for the logistic model are Aw = 2902, B = 0.45, r/w = 1.41 · 10− 5, W(t = 0) = 273, w = 6.58 · 10− 7. In (b), w(t) for the log‐periodic model is calculated with population obtained from the log‐periodic fit in Figure and the horizontal line corresponds to the constant w assumed for the delay and logistic models. Other references are: Refs Rockstrom et al. 2009 (R09), Shiklomanov 2003 (S03), Gleick 2013 (G13).
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Conceptual models of renewable (a) and nonrenewable (b) water use. In these examples, the delay kernel from Yukalov et al. is adopted, which can be written as B(t) = B δ (tτ), where δ(0) = 1 and 0 otherwise. In (a) B = 0.65 and in (b) B = − 0.8. Common parameters are A = 3.5 · 106, τ = 15, N0 = 103, and r = 1. (a) Punctuated evolution to peak renewable water: Coupled human‐water systems with a carrying capacity that grows slower than the population (i.e., 0 < B < 1) evolve in a punctuated manner. N and K grow through successive logistic phases that reach incrementally larger plateaus, eventually reaching an asymptote N = K. At the carrying capacity, water is produced at the maximum sustainable rate, which may be governed by natural renewal rates or ecological considerations. (b) Overshoot of nonrenewable water: Coupled human‐water systems with a carrying capacity that is initially large (i.e., A > |B|N0) and decays slower than the population (i.e., − 1 < B < 0) evolves in a punctuated manner between oscillating levels. These dynamics are characteristic of nonrenewable water extraction, such as groundwater use, where the production rate initially exceeds the natural recharge rate. As aquifers are depleted or contaminated, production costs increase and supply decreases back toward the sustainable extraction rate. Such oscillatory dynamics are common in consumer‐resource systems.
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