The application of mathematics to the prediction of population dynamics has challenged demographers for at least two hundred years. Various proponents have developed formulae for both the calculation of population growth as well as the potential limits to such growth. While these formulae on their own have not always been able to accurately predict human carrying capacity limits, in many cases they have contributed to the development of more complex carrying capacity models.
[i] As such, they have often been theoretic in nature, rather than having direct applicability to a particular landscape.
One of the earliest known equations relating to population dynamics was Thomas Malthus’ exponential growth theory of 1798. According to Malthus,[ii] “[p]opulation, when unchecked, increases in a geometric ratio,” while its means of subsistence, namely its food supply, increases only in a linear or arithmetic manner. The exponential growth formula is relatively simple and can be given as;
P(t) = P_{o} e^{rt},
where P(t) is the population at a point in time, P_{o}_{ }is the initial population, e is the base of natural logarithms (2.718...), r is the growth rate and t is time. This formula generates a j-shaped curve with population reaching to infinity (figure 1a). However, according to Malthus, this infinite growth is inevitably halted by the inability of food production to keep up with the population’s exponential expansion (figure 1b).
Figure 1a. (left): Malthusian exponential growth curve showing how the population increases infinitely. Figure 1b. (right): Malthus’ exponential population growth curve limited by the linearly increasing food supply. The assumed carrying capacity is the point at which the population projection intersects with the food supply projection. The carrying capacity is assumed in this instance because Malthus did not refer to it as carrying capacity.
Malthus’ theories have been largely derided for over 200 years, mainly due to the fact that his most dire predictions have yet come to pass. However, many authors such as physicist Albert Bartlett,
[iii] still warn of exponential population growth, stating, “[t]he greatest shortcoming of the human race is our inability to understand the exponential function.” The main objection that Malthus’ detractors level against his population theory involves the implausible nature of both linear agricultural growth and never-ending exponential population growth. Ginzburg & Colyvan
[iv] explain that the exponential formula, while theoretically correct, rarely reflects actual circumstances, describing the approach as the, “default situation for populations - how they behave in the absence of any disturbing factors,” even though most environmental conditions are replete with disturbing factors. Despite ample evidence suggesting that human populations have sometimes experienced periods of increasing exponential growth, the smooth curve that Malthus’ equation generates, rarely reflects the broader historic outcome. For instance, even though the 1,800 year period leading up to Malthus’ era displays a strong correlation to his exponential growth curve, there is still a degree of fluctuation in the growth rate (figure 1a). In an analogy with financial accounting, Coutts
[v] points out that the population curve generally follows a variable rate of compound growth rather than a smooth fixed compound rate. This variability of the growth rate, visible in an examination of the last 200 year period (figure 1b), can be attributed not only to environmental irregularity but also, as Hussen
[vi] argues, to institutional and technological intervention in the population dynamic. He states, “There are social and economic factors that induce humans to check their own population growth under adverse conditions”, which “make the Malthusian margin a moving target.”
[vii] Ultimately, criticism of Malthus on the impossibility of uninterrupted exponential population growth is most likely unfounded because Malthus
[viii] himself agrees that this formula is to be considered more as a theoretic foundation than a practical reality, stating that, “in no state that we have yet known has the power of population been left to exert itself with perfect freedom.”
Figure 1a (left): Global population numbers from 1AD to 1800[ix] showing occasional falls, but overall growth. Figure 1b (right): Global annual population growth rate since 1800,[x] showing a variable growth trend.
The second criticism of Malthus’ work is the suggestion that food production need not grow in a linear fashion. Lomborg,[xi] for example, points out that, “the quantity of food seldom grows linearly. In actual fact, the world’s agricultural production has more than doubled since 1961.” Coutts[xii] supports such criticism, stating that Malthus’ position of first proposing a universal law of exponential population growth but then effectively arguing against it by suggesting, “that food (which grows in populations!) grows arithmetically is logically contradictory.” Kendall and Pimentel,[xiii] on the other hand, provide evidence to support Malthus’ assertions, stating that between 1950 and 1984, “[w]orld grain output expanded by a factor of 2.6… increasing linearly, within the fluctuations.” Invoking a Malthusian disaster, Kendall and Pimentel continue,“[r]ising growth of population,… and a linearly increasing food production have persisted over the recent 40 years,” thus potentially leading to “great human suffering.” Comparing these authors’ views, Coutts is at least clear about Malthus’ theorem, while it is clear that Lomborg is not. According to Malthus, the difference between population and food production was not their potential for growth, but the rate at which that growth might potentially occur. He argued that the rate of growth may increase in populations but remains relatively steady with food. Hence, Lomborg’s assertion that a doubling of food production since 1961 disproves linear growth is clearly unfounded. This statement merely proves that food production grew, but does not explain whether the growth was constant (linear) or increasing (exponential). On the other hand, Kendall and Pimentel’s observations of one instance of linear food growth and Coutts’ assertion that the growth rate of food production may be variable (sometimes constant, sometimes increasing dependant on timeframes and external circumstances) rather than fixed doesn’t actually prove or disprove whether population growth is likely to ever outstrip food production.
Thus, it seems that Malthus’ theories for exponential population growth and linear food production growth are both highly conditional on the timeframe, societal influences and location to which they are applied. Malthusian predictions of premature death visiting the human race[xiv] have opened his theories to much criticism but his warnings that infinite population growth will ultimately be limited by the finite nature of its means of subsistence, or in other words, its carrying capacity, seem as pertinent today as they were in 1798.
Even though Malthus wrote of food constraints to population growth, his exponential population equation failed to actually incorporate it. However, four years after Malthus’ death, in 1838, Belgian mathematician, Pierre-Francois Verhulst developed a theorem that began to incorporate these limits, in the form of the logistic curve, stated as;[xv]
dN/dt = rN (K - N)/K,
where N is the population size, r is the rate of population growth, K is the carrying capacity and dN/dt is the rate of population increase. This formula, when graphed (figure 2), takes on a characteristic S-shaped sigmoid curve, beginning with exponential growth at low densities, but transitioning to a tapering off at higher densities, “as resources become insufficient to sustain continued population growth.”[xvi]
Figure 2 (left): Verhulstian logistic growth curve illustrating eventual levelling-off of populations at a carrying capacity limit.
Even though the logistic equation may be instructive of theoretical population dynamics, in criticism analogous to that of Malthus, Fearnside[xvii] suggests that, applying it to human carrying capacity assessment oversimplifies the complexity inherent in societal interactions. Price[xviii] also doubts the usefulness of the logistic equation in predictions of human population dynamics, finding fault in the premise that environmental conditions might exert unchanging constraints on a population as well as the assumption that populations grow until automatically stabilising at a carrying capacity limit. He points out that even in non-human populations, these aspects rarely hold true, stating, “seldom if ever does a natural population rise sharply and then stabilize in the form of sigmoid curve.”[xix] There is some evidence however, to suggest that population growth over the last hundred years has followed a sigmoid curve pattern (figure 3), with the growth rate initially rising gradually, accelerating after about 1950 and then starting to decline over the last twenty years. Whether this pattern will continue to follow the logistic curve remains to be seen, but the United Nations[xx] does make predictions to this effect.
Figure 3 (left): The United Nations[xxi] record of global population growth over the last 100 years and their estimate (median fertility variant) for the next 90 years suggests a logistic growth pattern.
The aspect most clearly lacking from nineteenth century authors Malthus and Verhulst’s formulas is the effect of societal behaviour on population dynamics. Subsequent dramatic changes in science and technology in the intervening 200 years has only served to exaggerate this omission.
A more recent population equation which attempts to address the societal influences of population dynamics was devised by Ehrlich and Holdren in the early 1970s. In 1971 they initially proposed the equation;
I = PF,
where I is impact, P is population and F is a function measuring per capita impact.[xxii] In order to realign this formula to carrying capacity imperatives, it could also be given as a population projection;
P = I / F,
where population is equal to its total environmental impacts divided by the impacts per person. Ehrlich and Holdren subsequently expanded on the F in this equation to also include affluence (A) and technology (T) in order to highlight that environmental impacts are not only influenced by the population’s size but also by the consumption patterns (represented by A and T) of its participants shown as;[xxiii]
I=PAT (or simply referred to as IPAT).
In this formula, affluence is defined by economic activity per person and technology as the environmental impact per unit of economic activity. It is perhaps not immediately obvious why technology is an apt description of environmental impact per economic activity, but Dietz and Rosa[xxiv] suggest that it roughly represents the efficient utilisation of resources available to the population. In other words, the T component is “determined by the technology used for the production of goods and services and by the social organization and culture that determine how the technology is mobilized.”
An alternate population-focused form of the IPAT formula would become;
P = I / (AT),
suggesting that population size has a direct correlation with impacts and an inverse relationship to affluence and technology. In this alternate equation, if P is considered to be the maximum allowable population then it could also be thought of as the carrying capacity. As such, the equation shows that as acceptable impacts grow, so does the carrying capacity, but as personal consumption grows, the carrying capacity falls. In other words, if a population wishes to grow, then it needs to accept either greater environmental impacts and/or a reduction in its per capita consumption.[xxv]
While this rearrangement of Ehrlich and Holdren’s equation serves to illustrate its potential in calculating carrying capacities, in reality, a method for assigning numeric values to the impacts, affluence and technology components would have to first be derived. Schulze[xxvi] points out that, “[t]he equation is not intended as a formal mathematical model, but rather as a conceptual framework.” In order to transform this conceptual equation into a comprehensive quantitative one, modes of impact such as habitat destruction, pollution levels, climate change and other measures of environmental damage would need to be pursued; affluence would need to be further defined by elements such as economic performance and consumption of goods and services; and various facets of technological usage would also need to be validated and quantified.
While a comprehensive approach to the IPAT equation is yet to be developed, there is evidence of some progress. For example, Dietz and Rosa[xxvii] developed a method of assessing societal carbon dioxide impacts based on Ehrlich and Holdren’s work. They state, “[a]lthough there have been attempts to assess the validity of the [IPAT] model, they have typically relied on qualitative assessments, field study demonstrations, or projections rather than on an assessment of the model’s overall fit to an appropriate data base. This was our main task.”[xxviii] Dietz and Rosa redefined the components of the model to suit their own focus, with environmental impacts reflecting only industrial CO2 emissions, GDP representing affluence, and population data utilised on a national scale. In a further alternate version of the IPAT formula, Dietz and Rosa rearranged it in order to derive the technology index. So their formula reads,
T = I / (PA).
The application of the IPAT formula proved useful for Dietz and Rosa in determining correlations between populations, economic growth and environmental impacts. They found that a population’s size is roughly proportional to its impacts but that “when affluence approaches about $10,000 in GDP, CO2 emissions tend to fall below a strict proportionality.”[xxix] However, given that the authors didn’t expect economic growth to rise to this level in most nations for two or three decades, they deduced that “[e]conomic growth in itself does not offer a solution to environmental problems.”[xxx]
While theoretically instructive, the exponential, logistic and IPAT formulae have not yet facilitated the accurate assessment of human carrying capacity. However, more quantitative approaches do exist and according to Sayre,[xxxi] the earliest known carrying capacity assessment performed under that name, was conducted in Africa by William Allan[xxxii] in 1949. Although he didn’t pioneer the particular food-based approach employed, he was amongst the first to clearly articulate the methodology. Firstly, Allan estimated the agricultural yield of regionally grown staple crops (Y) and this was then divided by the average amount of food required per person (F). Then, drawing on existing ecological survey data of regional soil and vegetation types, he calculated the amount of land available for growing staple crops (L) and divided this total land by the amount of land required per person. So, in summary, the formula reads, carrying capacity is equal to the area of land available for food production; divided by the food required per person, divided by the area required per food, or:
K = L / (F / Y).
In its simplest form, the equation is merely the total area of land (L) divided by the area of land required per person (A):
K = L / A.
In this form, the carrying capacity equation mirrors Ehrlich and Holdren’s original formula (P = I / F)[xxxiii], except that land area is substituted for impacts. Consequently, Allen’s formula can be seen as a resource-based approach focussing only on the constraint of food production and consumption, while the IPAT formula is an environmental impacts-based methodology.
While the formula developed by Ehrlich and Holdren predominantly serves to highlight societal trends, Allan’s resource-based approach actually generates a quantitative carrying capacity result. Allen’s simple methodology only makes estimates of basic food production and consumption requirements for a small population. However, his methodology has subsequently been refined and developed by other carrying capacity proponents[xxxiv] who added further detail to the equation relating to production techniques, resource demands beyond just food, land use variables and consumption choices. This additional level of complexity allows more recent carrying capacity assessment approaches to be categorised as models rather than just formulae.[xxxv]
[i] This chapter only looks at population equations rather than carrying capacity models such as the Carrying Capacity Dashboard.
[ii] MALTHUS, T. R. (1959) Population: The First Essay, Michigan, University of Michigan Press.
[iii] BARTLETT, A. (2012) Al Bartlett, Professor Emeritus Physics. Boulder.
[iv] GINZBURG, L. R. & COLYVAN, M. (2004) Ecological orbits: how planets move and populations grow, New York, Oxford University Press.
[v] COUTTS, D. A. (2009) Reverend Thomas Robert Malthus - An Exponentialist View. Melbourne.
[vi] HUSSEN, A. M. (2004) Principles of environmental economics, London, Routledge.
[viii] MALTHUS, T. R. (1959) Population: The First Essay, Michigan, University of Michigan Press.
[ix] Graph derived from Cohen’s COHEN, J. (1995) How Many People Can the Earth Support?, New York, W. W. Norton. summary of global population estimates.
[x] WIKIPEDIA (2012) Malthusian catastrophe. Wikipedia.
[xi] LOMBORG, B. (2001) The Skeptical Environmentalist: Measuring the Real State of the World, Cambridge, Cambridge University Press.
[xii] COUTTS, D. A. (2012) Couttsian Growth Model. Academic Publishing Wiki.
[xiii] KENDALL, H. W. & PIMENTEL, D. (1994) Constraints on the Expansion of the Global Food Supply. Ambio, 23, 198-205.
[xiv] MALTHUS, T. R. (1959) Population: The First Essay, Michigan, University of Michigan Press.
[xv] PRICE, D. (1999) Carrying capacity reconsidered. Population and Environment. 21.
[xvi] FEARNSIDE, P. (1986) Human carrying capacity of the Brazilian rainforest, New York, Columbia University Press.
[xviii] PRICE, D. (1999) Carrying capacity reconsidered. Population and Environment. 21.
[xx] UNITED NATIONS (2011) World Population Prospects, the 2010 Revision. New York, United Nations.
[xxi] The United Nations (Ibid.) have used t
[xxii] EHRLICH, P. R. & HOLDREN, J. P. (1971) Impact of Population Growth. Science, 171, 1212-1217.
[xxiii] EHRLICH, P. R. & HOLDREN, J. P. (1974) Human Population and the Global Environment: Population growth, rising per capita material consumption, and disruptive technologies have made civilization a global ecological force. American Scientist, 62, 282-292. and DIETZ, T. & ROSA, E. (1997) Effects of population and affluence on CO2 emissions. Proceedings of the National Academy of Sciences, 94, 175-179.
[xxiv] DIETZ, T. & ROSA, E. (1997) Effects of population and affluence on CO2 emissions. Proceedings of the National Academy of Sciences, 94, 175-179.
[xxv] To further illustrate this relationship; if A is 10, I is 1 and T is 0.1, then the carrying capacity (P) equals 100; but if per capita consumption is decreased so that A is 10, I is 0.1 and T is 0.01, then the carrying capacity would increase to 10,000; or if impacts are to decrease so that A is 1, I is 1 and T is 0.1, then the carrying capacity falls to only 10 people.
[xxvi] SCHULZE, P. C. (2002) I=PBAT. Ecological Economics, 40, 149-150.
[xxvii] DIETZ, T. & ROSA, E. (1997) Effects of population and affluence on CO2 emissions. Proceedings of the National Academy of Sciences, 94, 175-179.
[xxxi] SAYRE, N. F. (2008) The Genesis, History, and Limits of Carrying Capacity. Annals of the Association of American Geographers, 98, 120-134.
[xxxii] ALLAN, W. (1965) The African husbandman, Munster, Lit Verlag.
[xxxiii] EHRLICH, P. R. & HOLDREN, J. P. (1971) Impact of Population Growth. Science, 171, 1212-1217.. Where P is population, I is impacts and F is the impact per person.
[xxxiv] More recent proponents include Fairlie FAIRLIE, S. (2007) Can Britain Feed Itself? The Land, 4, 18-26. and Peter, Wilkins and Fick PETERS, C. J., WILKINS, J. L. & FICK, G. W. (2007) Testing a complete-diet model for estimating the land resource requirements of food consumption and agricultural carrying capacity: The New York State example. Renewable Agriculture & Food Systems, 22, 145-153.
[xxxv] More on this later.
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