The World4 model

The simple model, called World4, consists of four stocks and four flows, forming two interacting binary subsystems. There are two stocks quantifying states of the global environment (humansphere versus ecosphere) and two stocks quantifying states of technological development (knowledge versus ignorance). Both are closed systems, which means there are no outside sources or sinks. The only source of domesticated land (humansphere) is wild land (ecosphere), and life-saving technologies (knowledge) can only save more lives to the extent that there is a non-zero death rate (ignorance). In this model, knowledge draws down the death rate, while ignorance is the death rate itself. Knowledge does double duty by also increasing the carrying capacity. Other forms of knowledge and technology are ignored. We’ll get back to technology, but first let’s discuss humanity.

The environment subsystem

Humanity is represented by its ecological footprint within the Environment subsystem. The footprint is the amount of Earth’s biocapacity appropriated for human use [13]. The Earth has a maximum total biocapacity (E₀) estimated to be around 1.12e10 global hectares (gha) which is shared between wild and domesticated land and sea. Humansphere is equal to population times the average consumption per capita, times a term derived from the state of technology. In terms of the popular "I = PAT" [24] for ecological impact, humansphere is "I", population is "P", and the carrying capacity equation (CC) expresses the "A" (affluence) and "T" (technology) terms. In this model, population is viewed as a consequence rather than a cause of consumption. Therefore the expression becomes P = I/(AT), or (1) where CC is the reciprocal encoding of the informal concepts A and T. Humansphere grows intrinsically in the model, via the flow (2) because human need for more land grows with, and is proportional to, the population. The rate constant (I₀ + ln(2)/τ) (1-exp(u ecosphere/E₀) approaches zero as ecosphere approaches zero because ecosphere cannot be negative. u is a negative number that models the strength of this feedback. I₀ + ln(2)/τ is the birth rate, which is the replacement of deaths (I₀) plus the net growth, where the latter is reciprocally related to the intrinsic doubling time τ, in years. The approach of basing population on carrying capacity is contrary to most population models, including World3 [19], that express population in terms of births and deaths. Hopfenberg [8] and others have argued that population growth should be viewed as food-supply driven rather than the result of births and deaths. This view has been met with skepticism [25] but it is consistent with the treatment of humans as a K-selected species. To treat humans differently from all other K-selected species would be a form of "human exceptionalism" [26], which is not scientific. Moreover, ecosphere cannot be expressed in terms of a population, so gha units make sense for these stocks.

The technology subsystem

In this model, technology is a driver of growth through the suppression of mortality and is a factor in amplifying the human carrying capacity. In a sense, technology (or knowledge of technology) is an independent living entity since it can exist in written form outside of humanity itself and since it has its own catalytic effect on the development of new technology. Thus it is not wrong to treat technology as a living thing that can grow exponentially. The expression for intrinsic flow into knowledge is (3) where κ is a constant. Also, like a living thing, technology can "die" by obsolescence. Even though the knowledge may still remain, its usefulness towards human survival can whither to zero (buggy whip technology, for instance). Thus in the model we lump together unlearned or unknown technology with obsolescent or ineffective technology under the term ignorance, a quantity that is rate constant for mortality.

Feedback between environment and technology subsystems

The two subsystems, Environment and Technology, interact to form a feedback loop. Specifically, since humanity is expressed in terms of its ecological footprint in gha units, and because death represents the return of human appropriated biocapacity to the ecosphere, therefore ignorance is the rate constant for that flow from humansphere to ecosphere, called rewilding.

(4)

It is postulated that the loss of the undomesticated environment will cause technological challenges to human survival due to the loss of ecosystem services that are required for food production. To model this, ecosphere feeds back to obsolescence as follows. (5) where v is a negative number. Eq 5 expresses the idea that as the wild environment disappears (depleting fresh water aquifers and fossil fuels, for instance), old technologies lose their usefulness (center-pivot groundwater irrigation systems, oil-fired electric generators), and humanity must develop new technologies to survive (ocean desalination, photovoltaic panels). Thus it makes sense that ecosphere depletion leads to obsolescence. This completes a negative feedback loop (Fig 2A). To play out a scenario on this loop, we can imagine that ignorance goes to zero, therefore rewilding decreases. This leads to a decrease in ecosphere by domestication. Depletion of ecosphere causes an increase in ignorance, therefore an increase in rewilding, reversing the effect. In the context of the exponentially depleting quantities ignorance and ecosphere, the system reaches a switching point and thereafter equilibrates (Fig 2B). Interestingly, this is the same feedback cycle that generates Lotke-Volterra oscillation (inset in Fig 2B) Interestingly, this feedback cycle generates Lotka-Volterra oscillation [27] (inset in Fig 2B) under different parameter setting. In the current setting, both ecosphere and ignorance are decreasing exponentially, causing the oscillation to be severely damped, producing just a single oscillation.

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Fig 2. Feedback.

(a) Ignorance feeds back in a positive way to rewilding. Rewilding increases ecosphere. Ecosphere feeds back negatively to obsolescence. Obsolescence increases ignorance. (b) Exponential decrease of both ecosphere (pE) by domestication and ignorance (pI) by learning, results in a switch, first in ignorance then in ecosphere. Inset: undamped Lotke-Volterra oscillation.

https://doi.org/10.1371/journal.pone.0247214.g002

Fitting the parameters of the model to population data

Parameter settings for the model were determined by non-linear least squares fitting of population output to historical population data, as described in greater detail in Methods. The intent of the project was to explain the hyper-exponential growth that was observed in the 19th and 20th centuries, to explain why recent population growth has been slowing, and to predict the future (Fig 3D). Fitting data before 1970 to a hyper-exponential model asks the question “How did humanity grow in the absence of limits?” Whereas, fitting the late 20^th century slowing asks the question "How do planetary limitations slow the growth of humanity?" The result of the fitting is a model that projects into the future, providing a range of different population projections. Details of the process are found in Methods.

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Fig 3. Least-squares fits to years 1970–2010 are non-linearly correlated in the space of the four variables (E₀, a, u and v) that effect only recent population data, as shown using hyperfit.

For example, as seen in (a), the best-fit setting for v (ecosystem-dependent obsolescence of technology) goes down as we increase the setting for a (ecosystem fragility). Each image is a projection of minimum values of residuals from the 4D space to 2D spaces (a) a, v, (b) a, E₀, (c) E_0, v. (d) A plot of five trajectories using optimal and suboptimal values, demonstrating the effect of choice of E₀ (total ecosystem size) on growth rate (1960–2000) and on the position of the population peak, ignoring other parameters. A hypothesized infinite ecosystem, E₀ = ∞ (black), leads to massive overestimate of growth rate. E₀ = 0.800e10 (green) or E₀ = 0.750e10 (orange) overestimates growth rate and predicts a later peak. E₀ = 0.695e10 (cyan) is optimal and predicts a 2020 peak. E₀ = 0.600e10 (magenta) underestimates growth and predicts that we are past the peak. Thick blue line is population data up to 2010. 2020 population numbers are not available.

https://doi.org/10.1371/journal.pone.0247214.g003

Output of the model

After fitting, the output of this model matches all historical population data from years 1500 to 2010 within ±10% with the exception of the 1950 census, which is overestimated by 14%. It should be noted that 1950 census number was revised 17 times from 1951 to 1996, mostly upward [28]. Population growth has been sub-exponential over the last 50 years, suggesting that humanity is passing through an inflection point of a curve that is the product of two steep trends, one upward, the other downward. The upward curve is the combined exponential expansion of humanity and the intrinsically exponential increase in technological innovation, and the downward curve is the accelerating depletion of non-renewable resources and the loss of food security. The model predicts that the results of the 2020 census (not yet available) will be in the range 7.2 to 7.6 billion (80% confidence) instead of the projected 7.8 billion [11]. The model predicts that the population will peak or has peaked, with the peak in the range 2018 to 2023 (80% confidence) and will decline to between 2.1 and 6.6 billion by 2060. The nearness of the peak is supported by accelerating increases in adult mortality and decreases in birth rates since 2016 [29].

Significance of the model

The system dynamics simulation assumes fixed set of parameters and equations throughout the simulated time period. The model has no outside sources or sinks. There are no built-in switches and no settings are changed at any point in the simulation. Nonetheless, the simulation matches very closely to 2010 years of population history spanning periods of slow growth, rapid growth, and recent deceleration. The results of the study are a set of meaningful parameters that attach numbers to well-established but theoretical human/environment feedback mechanisms. Parametric solutions from the multi-variable least squares fit span only a narrow range in most variables, varying for the most part in the parameters that define the feedback between human population and the environment. Multiple solutions fit and explain past population data equally well but project very differently into the future as shown by the narrow 80% confidence ranges for the population peak and the side confidence range for the population in 2060 (see Fig 1B).

Limitations of the model

The theoretical appearance of a population decline in the near future is a foregone conclusion of the design of the model itself. The model encodes the business-as-usual (BAU) assumption that humanity will not react to change and will continue to degrade the environment. It also assumes that the carrying capacity depends critically on resources that are not under human control nor are regenerated by human activity, and which will not come under human control within the timeframe of the simulation. These model design choices are not to be considered incontrovertible, but they are consistent with the dominant theory in human ecology and are intended to be free of human exceptionalism.

Modeling philosophy and motive

Human population dynamics has its root causes in the environment and in human behavior. An understanding of the essence of our interactions with Nature and with each other can be best gained by expressing human population in the simplest possible dynamic model. The approach taken here has been to define the components and equations of a systems model and to tune the parameters of these components to fit population data. Prior to the current model we explored a number of alternative models or equations and rejected them, either because they could not be tuned to fit the data, or because they did not make real world sense, or because they were too complex. Because accurate prediction of future population was the motive from the outset, the simplest possible model was designed, giving the best chance of being robust to errors in the data and mistakes in the functional forms.

Exploring parameter space

The behavior of the model with respect to its parameters is complex and surprising. For example, if E₀ is set to 7.50e9, it places the peak population year at 2038, but then the model fits poorly to the data, showing an upwardly curving trajectory through the 1980’s and 90’s (see Fig 3D), though we know that did not happen. Setting E₀ to a low value of 5.5e9 places the peak at 2016 and shows a downward curving population trajectory from 1990 to 2010, which again we know did not happen. But if all other parameters are allowed to float and are fit to the data by exhaustive multidimensional search, then both high and low settings of E₀ match very well to the 1970–2010 data (see Fig 1B). However, the results are reversed. The peak for low E₀ is now at 2040, and the peak for high E₀ is now at 2016. Surprisingly, in the context of the full complement of model parameters, the effect of E₀ has become completely counter-intuitive. With a little effort we can rationalize this strange behavior, as follows. If E₀ is high (8.0e9 gha) and yet population data is well fit, the parameter a adopts a high value (a = 0.48), which means the ecosystem is fragile and fails well before it is completely depleted. This leads to a steeper downturn in carrying capacity, which leads to the observed prediction, a counterintuitive earlier peak. On the other hand, if E₀ is set to a low value (E₀ = 4.2e9) and yet population data is well fit, then the parameter a adopts a low value (a = 0.10) and a later population peak is predicted. Low a is interpretted to mean that the ecosystem is robust and the ecosystem services upon which we all depend are not entirely provided by the wild ecosphere but may be partially provided by the humansphere. To be clear, these very different predicted futures are not scenarios that can be affected by policy change, rather they are the results of uncertainty and a lack of understanding of the fragility or robustness of the global ecosystem with respect to human carrying capacity. Upcoming results from the 2020 census will greatly resolve this uncertainty.

Analysis of the median business-as-usual projection

The World4 model is a BAU projection in the sense that human growth behavior and behavior towards the environment are treated as parameters that are determined by fitting past population data, not parameters that in any way seek an outcome. A range of parametric solutions has been found by the process described here. Each solution fits past population data the same but projects differently into the future. The median projection (Fig 1) peaks at or around 2022, then falls steeply, leveling off at around 3 billion by 2060. The cause of the decline within the model is a decrease in the food supply caused in turn by degradation of the environment and the concomitant atenuation of essential ecosystem services. The model reflects the current thinking on climate change and its repercussions. Climate change leads to weather uncertainty, increased severe storms, draught and floods, and sea level rise affecting low-lying areas—each a factor in decreasing agricultural output. Increased hunger in turn fuels conflict [30, 31]. Conflict leads to further decreases in food production and to mass migration, as we have seen recently from the rapidly heating Sahel region of Africa [32, 33]. In the BAU projection we see an increase in human mortality, followed by a decrease in carrying capacity. The recent worldwide spread of Covid-19 is an example of an emerging source of mortality for which human technology was not ready. Societal stressors such as hunger or a pandemic can drive violent behavior [34]. In the medan projection, following the peak, population drops quickly, accelerating to 100 million net lives lost per year through the years 2030 to 2040, which is faster than the fastest growth during the 20th century. In this model, we clearly see the cause of the rapid decline—the exponential growth of the consumption of finite vital natural resources.

Real world examples of current ecosphere collapse

We are beginning to see the effects of the decline of natural resources and ecosystem services. Fossil fuels, fresh water aquifers, and greenhouse gas sequestration by plants are all regarded in this model as part of the ecosphere since they are not generated by human activity. The decline of one or more natural resources is cited as a cause of, for example, the ongoing deadly conflicts in Syria starting since 2011, the conflicts and famine in Yemen beginning in 2015, and the economic collapse in Venezuela that began in 2014, to mention a few. Draught and desertification were blamed for conflicts and mass migration out of the Sahel region of Africa, where Lake Chad has all but disappeared. Conflicts and famine have produced millions of refugees. Innumerable lives have been lost crossing the Sahara or crossing the Mediterranean, or in primitive camps along the southern borders of Europe. The 2017 documentary film "Human Flow" [35] reveals the massive scale of the refugee issue. Meanwhile, the global north has responded to the aggregate changes of the last 50 years with a dramatic decrease in the birth rate. An increasingly technological workforce has meant women spend more time in school and marry later. Rising oil prices have steadily ramped up the cost of raising a child, leading to smaller families by choice. Total fertility rate (TFR) globally is projected to reach replacement level (2.11) this year, 2020.

Real world examples of the predicted decline of technology

Along with ecosystem decline, the model predicts changes in technology. In the projection, knowledge will be lost or made obsolescent during a population collapse. Much of our cultural technology is composed of laws, governance and economics. In recent economic history, consumerism has become engrained in our culture [36, 37]. Stability and prosperity in the context of the current economic system relies on population growth, according to economists. A technology shift in economics is likely when population begins to decline, since growth-based economic systems and the associated body of knowledge will become obsolescent in the sense that they will not produce stability. In effect, economics will have to be re-learned. Obsolescence of growth-based economics may manifest itself in real-world breakdown in economic systems leading to decreased efficiency in manufacture and trade, in turn leading to a decrease in the effective food supply, which in turn will cause an increase in malnutrition and a decreased birth rate. Already, increased adult mortality and decreased birth rate are both current trends in global vital statistics [38]. In other areas, medical technology is partially responsible for a historic low death rate worldwide, but successful treatment of disease requires instruments and drugs that depend on a complex supply chain and high level engineering skills. In the event of an economic shift, supply chains will be disrupted unless a new system of economic motivation is quickly invented to replace the growth motive. In agriculture, technology to increase crop yields will become obsolescent as climate challenges, biodiversity losses (especially the loss of pollinating insects), and depletion of freshwater aquifers, combine with economic changes to reduce the efficiency of food production and distribution.

Hopeful what-if scenarios as add-ons to World4

The future could easily be different than what is predicted by World4. World4 does not model changes in attitude and policy. Humanity could adapt in ways that may be good or bad. Modeling adaptation mechanisms opens a non-BAU modeling space that is too large to thoroughly explore. Taking inspiration from E. O. Wilson’s book "Half Earth" [39], parameters for policies to preserve wild nature were implemented. Four new variables were added, w, y, py and sy, as defined in Table 1, for the target amount of ecosphere to save, the level of policy enforcement, the phase-in period and the date on which the policy begins, respectively. These variables do not affect populations prior to and including 2010. Preserving wild land wild allows humans to thrive. The optimal result (coindidentally it is w = 0.5, half earth!) gives, as Wilson predicted, a stable and high human population (Fig 4). This makes sense mathematically, because the carrying capacity equation contains a term of the form x(1-x), which has a maximum at x = 0.5 where x is the fraction of the Earth dedicated to the ecosphere. But it also makes sense ecologically, because maximum sustainable food production is a trade off between maximizing arable land and maximizing climate stability, the latter embodied in wild forests and arctic ice, and other buffers to change. A global climate awareness campaign might lead to such a balance.

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Fig 4. Half-Earth scenarios.

Population projections for conservation efforts with various % enforcement, length of phase-in period and the % of ecosphere to be preserved, as compared to a BAU scenario. Dotted line is the present year, 2020.

https://doi.org/10.1371/journal.pone.0247214.g004

Any attempt to halt growth has to address the population. Cultural taboos currently prevent discussing, much less solving, this problem [4]. But in a what-if scenario, we can imagine ways that population growth can be halted or even reversed while preserving peace and prosperity. To build a mental picture of a society that has achieved balance with nature, imagine a people with a strong religious prohibition against growth, so engrained that no policing is required. A woman of child-bearing age in the Half-Earth world are permitted to have another child only if she is "blessed" by an elderly person, who, on his deathbed, bequeaths to her his one and only "blessing"—the right to procreate. The one-to-one matching of deaths to births would guarantee population stability.

The hyper-exponential growth model explains most of human history

The hyper-exponential growth of human population from 1000 CE to the present is revealed by simple curve fitting against historical population data. (6) where t is the year since 0 CE and P_t is population at t. This formula fits all population numbers from 1700–1987 within 10%, and all numbers from the present back to 1000CE within 17%, except 2010. Note that this equation differs from an earlier published one (Eq 12 in [5]) with the addition of a log-linear term (7.78e-4 t). This term improves the fit to numbers in the Renaissance period. Eq 6 may be interpreted as exponential growth in which the growth rate grows exponentially due to an exponential decline in the death rate. Other functional forms of the growth equation were considered and discarded, either because they did not fit the data or because they were not realistic. For example, normal exponential growth fits poorly to the data after 1500. On the other hand, a hyperbolic function fits all the data but lacks a physical rationale. Only the hyper-exponential equation achieves the steepness of human population growth in the 20th century without implying an unrealistic asymptote or arbitrary timepoints of discontinuity.

Hyper-exponential growth equation implies a 2 subsystem model

The hyper-exponential equation has two intrinsic growth rates. Therefore it implies two interacting subsystems, one containing the human population and the other containing a quantity that affects the rate of human growth. For this quantity we use the term "knowledge", meaning knowledge of technology. Technology generally improves life expectancy and efficiency in the use of natural resources and thus accelerates growth [6]. Knowledge of technology grows intrinsically, while humanity grows both intrinsically and extrinsically, depending on knowledge. The hyper-exponential growth model therefore breaks down into two growth models, as described below, one for the Humansphere and one for Knowledge.

Exponential growth of the Humansphere, domesticated land

Drawing inspiration from the "ecological footprint" literature, the stock quantity that includes humanity is modeled as the total human-dominated portion of the global ecosystem, called the humansphere [12]. The remaining global biocapacity is the ecosphere, representing the portion of the global environment that is not under human domination. Humansphere and ecosphere are measured in global hectares (gha) and sum to a constant, which is the maximum total biocapacity, E₀. Humans are assumed to be a K-selected species as opposed to r-selected [8]. The population of a K-selected species is determined by the carrying capacity, therefore multiplying humansphere by the carrying capacity per gha gives the population.

Growth of humanity is modeled as a flow of gha from ecosphere to humansphere, with value domestication = g • humansphere. This flow goes to zero as the ecosphere, the space into which humansphere must expand, goes to zero. (7) where g is the domestication rate constant measured in reciprocal years (y^-1), p_E = ecosphere/E_0, τ is the initial doubling time of the population in years, and I₀ is the initial value of ignorance. The negative-valued variable u models the aggressiveness of human growth as ecosphere approaches zero, discussed below. A large negative u means aggressive domestication (u is negative so that exp(u p_E) is bounded between zero and one.). Flow of humansphere back to ecosphere is called rewilding (see Eq 4). Note that ignorance, a number that reflects the mortality rate, initially declines exponentially as knowledge takes its place.

Ecosystem model versus birth/death model

Expressing human population using the total ecological footprint of humanity is functionally equivalent to the more traditional birth/death model. Rewilding is equivalent to death, since, upon death, a human’s resources are returned to the ecosphere. Food that is not eaten counts as carbon sequestered, and waste that is not produced counts as waste assimilated. Therefore, death converts humansphere to ecosphere. By the same token, domestication is functionally equivalent to birth since it converts ecosphere to humansphere to support an increase in our numbers.

Exponential growth of Knowledge of technology

Knowledge (of technology) flows from ignorance (of technology), at the intrinsic rate called learning = κ *knowledge. The optimal value κ = 9.6e-3 y^-1 was determined from the data. The amount of learning is the degree to which life expectancy has been increased by science and technology in a given year. Theoretically, maximum knowledge implies zero ignorance, which unrealistically implies zero death. However, simulations never come close to this value. On the other hand, zero knowledge implies a mortality rate of equal to the base value, estimated as I₀ = 0.11 y^-1. Unfortunately I₀ could not be precisely fit because it affects the simulations only after the present time. This estimate is derived from the estimated current overshoot of the global biocapacity [40].

Eventual decline of Knowledge

Knowledge can and does flow back to ignorance in the sense that technology becomes lost or obsolescent. Obsolescence happens when new forms of mortality emerge from advancing technology, such as cancer arising from chemical synthesis of pesticides, draught arising from the efficient depletion of aquifers, and disease transmission arising from the increased ease of long-distance travel. In these cases, innovations that initially increased lifespan later uncovered a cryptic ignorance. To model cryptic ignorance, knowledge flows back to ignorance with value obsolescence = r * knowledge, where (8) where p_E = ecosphere/E₀. Obsolescence is increased by environmental degradation. The negative-valued variable v was fit to late 20th century population data. A larger negative value for v leads to less obsolescence and steeper population growth.

Carrying capacity formulation

In this model, the human population is calculated as the amount of Humansphere times the carrying capacity for humans, which is the situation for all K-selected species. The equations for carrying capacity were worked out such that they reproduced population growth. To achieve the observed rapid population growth in the 20th century, the exponentially-growing quantity knowledge was applied to both population growth and the growth of the carrying capacity (CC). In preliminary studies, applying knowledge to only the mortality rate (rewilding) or to only the CC did not reproduce the observed steepness of 20th century population growth. CC is defined as the number of people that can be supported on one global hectare (gha) of humansphere. The global biocapacity is treated as an unknown constant value E₀, an amount of the world’s biocapacity which is split between ecosphere and humansphere [13]. Both ecosphere and humansphere contribute resources needed for human life, therefore CC = cc_E + cc_H, but only the carrying capacity contributed by the humansphere (cc_H) is augmented by knowledge, which makes logical sense.

(9)

In this view, the carrying capacity contributed by the humansphere is wholly dependent on the ecosphere carrying capacity (cc_E, Eq 10 below), because food production approaches zero as ecosystem services approach zero. The loss of ecosystem services that maintain climate stability would leave us with unpredictable temperature, precipitation, and storms. Season-to-season instability and unpredictability limit the efficiency of agriculture. Thus environmental degradation (loss of ecosphere) leads to a decreased carrying capacity and therefore a loss of human population. Consider for example that the ecosphere includes fresh water aquifers and fossil fuels, factors in food production which are not likely to be replaced by any amount of human invention. Ecosphere also encompasses the wild Earth systems and bodies (oceans, atmosphere) that provide ecosystem services and thus define the climate. Loss of ecosphere is interpretted to mean climate change, consistent with the role of the ecosystem in maintaining climate. Human food production is also limited by the total area of agricultural lands (humansphere) times the maximum carrying capacity of those lands under intense cultivation (c) [40]. The term 1-exp(d • knowledge) expresses the rise in food production efficiency per gha as knowledge increases [6]. Following the "law of diminishing returns" [41], food production rises more slowly with each additional unit of knowledge. The optimal value for the degree of diminishing returns was d = -110.

The optimized model (Table 1) was found to closely reproduce a 65% increase in the total caloric output of agriculture observed during the Green Revolution from 1960–2010 [42, 43]. In the model, the carrying capacity cc_H goes from 0.80 gha per capita in 1960, to 1.33 gha per capita in 2010, a 66% increase. The model also roughly reproduces the total human ecological footprint increase of 225% from 0.75 Earths in 1960 to 1.7 Earths in 2010 [44]. In the World4 simulations, humansphere grows 187% over the same period.

Ecosystem fragility, the parameter a

Although carrying capacity is wholly dependent on ecosystem services, those services are related to the biocapacity of the ecosphere in a way that cannot be easily assumed. We may hypothesize that damage to the ecosystem is not felt until a certain threshold in degradation is reached. Thereafter the damage to ecosystem services may be rapid and go asymptotically to zero, as described allegorically in Ehrlich’s "The Population Explosion" [45] with the "rivet popper" story. Indeed, in many ways, the global ecosystem, like the airplane wing in the story, can take serious damage before it suddenly fails in flight. On the other hand, the environment may be more robust than we know and the ecosphere may pass a percentage of its ecosystem services to the humansphere, never really collapsing to zero. An asymmetric sigmoid function (Eq 10) is used to express a continuum of unknown non-linear relationships between ecosphere and ecosystem services (cc_E). The asymmetric sigmoid curve allows us to optimize a single parameter (a) to express the degree of robustness or fragility of the ecosystem. If a is low, the environment responds robustly to depletion of ecosphere by retaining ecosystem services within the humansphere, whereas if a is high then the ecosystem is fragile and critical services are lost suddenly, as described in Ehrlich’s story. This function allows to explore ecosystem fragility by fitting proposed non-linear ecosystem behavior to population. (10) where p_E = ecosphere/E₀. Fig 5 shows the shape of the cc_E function and how it changes with a.

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Fig 5. Plot of Eq 10, the ecosphere component of the carrying capacity for humans.

Increasing values of a move the curve to the right, meaning ecosystem services are more fragile with respect to the ecosphere.

https://doi.org/10.1371/journal.pone.0247214.g005

Variable a was initially set to a value that places the global ecosystem today at about one-half humansphere and one-half ecosphere in 2005, the year of "peak oil" [46], albeit that peak date should perhaps be pushed forward by new discoveries of natural gas and new mining technologies. Fossil fuel is a dominant resource in raising the carrying capacity in the 20th century and its numbers are well studied and readily available, however it should be recognized that other limiting natural resources and ecosystem services may "run out" before fossil fuel does. Therefore, the variable a was set, not by a priori, but by fitting to late 20th century population data as described below. This variable, along with E₀, is one of the most critical to the accurate fitting of late 20th century, post-hyperexponential growth. The optimal value of a was found to be in a range from 0.27 to 0.48 depending other variables. Fig 3A–3C shows the shape of the range of optimality in dimensions a, E₀, and v.

Model fitting to population data

Variable setting were determined by least-squares fitting using the program Hyperfit. All variables in bold italics in Table 1 were fit to data. The most accurate and reliable data relevant to human population is the population itself. Global population numbers have been estimated for as far back as 10,000 years ago, and recent numbers are likely to be correct within 3% [28]. Exploratory curve fitting was used to test functional forms using Microsoft Excel, using the Solver plug-in [47]. Other global numbers, such as the ecological footprint, world forest cover, atmospheric carbon, global economic output, etc. were used in supporting roles only.

Variables were solved in three stages. First, intrinsic growth parameters (H₀, K₀, τ, and κ) were fit to years 1–1970. Second, the three "footprint" variables that affect the balance between normal exponential growth and hyper-exponential growth (b, c and d) were fit to years 1900–2010. Third, the late 20th century discrepancy (1970–2010) was reconciled using the variables (E₀, a, u, v) that determine the total global biocapacity, the fragility of ecosystem services, and the strength of feedback from the environment on the carrying capacity for humans. One variable, the baseline mortality rate in Year 0 (I₀) does not have an effect on population until after the present date, and therefore could not be fit. I₀ may be arbitratilly set to 0.11 y^-1 to reproduce previously estimated overshoot values (1.7 earths [44]). With this setting, the current population is 1.7 times the equilibrium value, which would be around 4 billion in 2100 using I₀ = 0.11. Also, four variables (w, p, s_y, p_y) were created for the hypothetical implementation of environmental conservation policy in the 21st century.

Hyperfit, a multivariable sampling algorithm

Finally, a range of least-squares minima was identified by exhaustive sampling of variables within the ranges defined by the heirachical fit method, using Hyperfit. Hyperfit is a program in Fortran90 that runs the World4 model for any number of ranges of variables, either randomly or exhaustively. In exhaustive sampling mode, Hyperfit reads an input file containing the ranges to be sampled and parameters for the range of years to be fit and the sampling density. The output is a matrix of mean square residuals summed over the range of years sampled. Up to 4 variables can be sampled exhaustively. The program outputs to the plotting program gnuplot [50]. In random sampling mode, Hyperfit samples all variables within ranges defined in the input file using a flat probability distribution, then runs a simulation and saves the parameters if the residual is below a cutoff (e.g. 0.5e8 for the years 1970–2010). The set of best fit parameters were passed back into Hyperfit to generate trajectories for plotting, using Microsoft Excel.