Fifty years to prove Malthus right

Abstract

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 (1766Malthus ( -1834, early attempts to estimate the maximum sustainable population (ergo, the carrying capacity K) were reported by van Leeuwenhoek (1632Leeuwenhoek ( -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) but may depend on a dynamic population 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)...

Resilience and reactivity of global food security

“…The emergence of global food scarcity is confirmed by a number of recent studies suggesting that the limited resources of the planet [primarily, land (18) and water (19)] would soon become insufficient to meet the escalating demand for food, fibers, and biofuels by the increasingly numerous and affluent human population (10,(20)(21)(22)(23). Such conclusions indicate that we may be at the verge of a global-scale Malthusian catastrophe (24,25). One then wonders how sensitive global food security is to perturbations arising from drought occurrences, changes in energy and trade policies, or food price spikes.…”

Power Law Growth and Delayed Feedbacks in Socio‐Hydrological Systems

“…Von Foerster et al () assumed that K N ( t )∼ N ( t ) σ with σ >1 and provided the general solution N ∼( t − t c ) −1/ σ . This model suggests faster‐than‐exponential population growth and a finite‐time singularity at time t c =2,026 (i.e., the “doomsday”) when a regime shift in global population dynamics is predicted to occur (Kaack & Katul, ). Hence, from equation , it follows that growth in human water use can also be described by a power law, , where t c is the critical time at which the solution diverges and β * =− β / σ is the scaling exponent.…”

Cultural Collapse and System Survival Due to Environmental Modification

“…In either case, the model has no stable solutions: either collapse eliminates the population, or the population grows by achieving a continuous increase in carrying capacity N s . This case would give exponential growth, slower than the supraexponential growth exhibited by the human population [3,4,6].…”

“…The application of mathematical models to human societies has a long history [1][2][3][4][5][6]. Such models typically enable one to cast verbal arguments into a quantitative form and see how they play out over various scenarios.…”