Fifty years to prove Malthus right.

A major question confronting sustainability research today is to what extent our planet, with a finite environmental resource base, can accommodate the faster than exponentially growing human population. Although these concerns are generally attributed to Malthus (1766–1834), early attempts to estimate the maximum sustainable population (ergo, the carrying capacity K) were reported by van Leeuwenhoek (1632–1723) to be at 13 billion people (1). Since then, the concept of carrying capacity has evolved to accommodate many resource limitations originating from available water, energy, and other ecosystem goods and services (1, 2). In PNAS, Suweis et al. (3) apply the concept of carrying capacity using fresh water availability on a national scale as the limiting resource to infer the global K. They estimate a decline in global human population by the middle of this century. We ask to what extent models that are premised on a constant global K as in Suweis et al. (3) agree with an alternative class of carrying capacity models on predicting the timing for unsustainable global population growth. In particular, factors that symbolize innovation and adaptation lead to K representations that are not constant (4, 5) butmaydependonadynamicpopulation size. The workhorse model for describing population growth remains the logistic equation (6–8). Attributed to Pierre-François Verhulst (9) and popularized by Pearl and Reed (10), the logistic equation is dN(t)/dt = rN(t)(1 − N(t)/K), where N(t) is the population, t is time, and r is the growth rate. For a constant K and r, the solution is an S-shaped function saturating to a constant population size N(∞) = K. Suweis et al. (3) determine the r and K for each country, using a population time series from 1970 to 2010, water availability, and water use per capita. If a country has a water surplus, it can export water to water-poor countries as virtual water (VW) in the form of food. The momentary K of a nation is based on a combination of local and traded VW that provides water-poor countries the possibility to extend their population size above the local carrying capacity. The global K was assumed to be constant in time, reflecting the finite fresh water supply of the planet. A point of departure from this assumption is provided by two dynamic-K models allowing for innovative solutions. The one assumes K changes instantaneously with N(t) (4, 5), whereas the other considers large-scale social changes introduced with delays in the dependence between N and K (8, 11). In 1960, von Foerster et al. (5) humorously developed the first model type to predict a doomsday associated with a mathematical singularity in population dynamics on November 13, A.D. 2026. They concluded that “[. . .] our great-great-grandchildren will not starve to death. They will be squeezed to death” (5, 12). To illustrate occurrences of singularities in finite times, the logistic equation can be recasted as dN(t)/ dt = r′N(t)(K − N(t)). Von Foerster et al. (5) made the assumption that K(t) ∼ N(t) with δ > 1. For a large N, this yields a solution as N(t) ∼ (tc − t). Here, tc is known as the doomsday (4, 5). The singularity originates from N(t) and K(t) becoming infinite as t approaches a finite tc. However, given the finite resources of the planet, the singularity only signifies a “regime shift” to qualitatively different dynamics (4), such as population decline. Fitting this model to available population records in 1958, von Foerster et al. (5) determined a tc ≈ 2027. Their ansatz reveals that even in this optimistic view of an infinite K(t), a decrease in the population is inevitable. Von Foerster et al. (5) labeled the constant-K model capping N(t) as a pessimistic view, resonating with Suweis et al. (3). A refinement to the first model type was proposed by Johansen and Sornette (4) to explain nonrandom oscillatory behavior in N(t) around the previous power-law solution (Fig. 1). They added an imaginary component to the exponent δ without providing physical justification but noted that numerous complex systems exhibit log-periodic oscillations before a collapse (4, 13). This amendment increased the model parameters to seven but reproduced the observed N(t) as shown in Fig. 1. Because of the large number of parameters, the robustness of the calculated tc remains to be determined. To address this, we analyzed the available data by terminating them at different dates t ∈ [0, 2010]. An ending date of 1985 then signifies fitting the log-periodic function for