Nearly twenty years ago, the mathematical ecologist Joel E. Cohen published a landmark book scientifically evaluating the question, “How many people can the Earth support?” In a companion article in the journal *Science*, Cohen introduced and explained what he called a mathematical cartoon of human population change. He made two very simple suggestions about how we should alter our assumptions concerning the relationship between our population size and the ecosystem resources that feed its growth. First, Cohen said, don’t worry first and foremost about absolute physical limits to key resources–of fresh water or arable land, say. Second, worry more about the social relationship between the number of people and the efficiency of economic production.

Mathematically, this involves a minor change in notation. We’ll get to that in a moment, but it is worth noting that it is almost absurdly simple.

Let’s consider the traditional model that Cohen wanted to update. The classic logistic growth model represents how finite resources will limit a population’s growth–specifically when there is a constant population level, denoted by the letter *K*, at which that population’s aggregate extraction, consumption, and impact on resources is in equilibrium with the wider ecosystemic renewal of those resources. *K*–or the population’s ecological carrying capacity–is known as the largest size the population can reach while leaving just enough material, nutrients, and energy to sustain demographic replacement in the next generation. In nature, a population at this ecological limit would not literally be in perfect harmony from one generation to the next. But it is an equilibrium level to which slight fluctuations in population growth or decline will return. What did Cohen change? He suggested that we should no longer see *K* as a constant. Rather, we should see carrying capacity as a factor that *changes over time. *Thus, he suggested designating carrying capacity as a mathematical function of time, *K(t)*, which would vary in a mutually causal relationship with population change.

More specifically, Cohen suggested that–at least up to a point–having more people around will have a positive impact on the efficiency of nutrient, energy, and material extraction, without impacting the ecosystemic renewal of those resources. Thus, he suggested an equation in which change in carrying capacity at a given time, *dK(t)/dt*, would be determined by the balance between two factors: (1) the potential for the current population level, *P(t)*, to yield efficiencies in resource extraction and utilization … and (2) the marginal impact on resources due to demographic growth–that is, adding more mouths to feed. Cohen’s differential equation for change in *K(t)* may be written as follows: *dK(t)/dt =* *rL*[*K(t)-P(t)*].* *Here, r is a constant that represents the population’s average biological capacity to transform ecosystem resources into births net of deaths. *L** *is a constant that marks the population limit on achieving economies of scale in supporting survival and reproduction. The systemically linked change in actual population may be represented by the following differential equation: *dP(t)/dt = rP(t)*[*K(t)-P(t)*]. As long as *P(t) < L*, the population will grow AND raise the prevailing limits on growth. As *P(t)* surpasses *L*, growth in carrying capacity, *dK(t)/dt*, will decline.

The resulting pattern is that, as long as the potential for economies of scale–*L*–has already become large, the population can continuously, gradually build up carrying capacity, leading to a surprisingly rapid population explosion. Yet, that explosive, super-exponential growth has neither come out of nowhere, nor will it continue forever. The figure above shows the surprisingly close fit between the Cohen (1995) model’s interpolation between 1 CE and 2012 and independently derived historical estimates and modern UN census values. Although human societies seem to be eminently capable of depleting resources that sustain the populations constituting them, cultural factors shaping our ability to use those resources efficiently has been–and is likely to continue to be–more important.

In fact, Cohen’s analysis suggests that cultural structures favoring efficiencies at larger population sizes were in place centuries or even millennia before industrial, demographic / epidemiological, ideological, and political modernity.

In other words, the modern population explosion may have been proximately caused by industrialization, vaccinations, and improved sanitation, but those are technologies, and as such, their development and adoption was made possible by much earlier change in the effective scalable structure of hierarchical organizational institutions; the cooperation among diverse institutions; and competition among individual and institutional actors adding dynamism and an element of biocultural evolutionary selection.

Remember that Cohen described his model of coupled change in population and carrying capacity as a “mathematical cartoon.” A key cartoonish aspect of the model is the notion that *L*–the potential for economies of scale–is really constant. The surprising fit of the Cohen-model interpolation and the independently derived historical estimates hints that some very important cultural features of economic scalability were in place quite early. But the question for us now is whether our economic systems, their political institutional actors and network connections, and the ideologies that structure challenges and resolutions of conflicts of interest are best fit for a world in which population growth yields economic growth. If indeed we’re reaching the limits of cultural support for further growth in carrying capacity, then the very ideological tools for mobilizing humanity’s recent population success may turn out to be counterproductive and lead to more rapid economic–and population–decline than we expect.

This is certainly an instance in which mathematical formalization–kept simple–helps us to focus on the assumptions surrounding that formalization. Just beyond the limits of the solution to Cohen’s model is the fact that economies of scale both structure and are structured by symbolically constituted ideologies of order, change, or competition. One surprising lesson to draw from Cohen’s work is that we need to look much more carefully at the relationship between ideologies of economic production, our awareness of them, and actual growth or decline in economies of scale.

### References

Cohen, J. E. (1995). *How Many People Can the Earth Support?* W. W. Norton & Company.

Cohen, J. E. (1995). Population growth and earth’s human carrying capacity. *Science*, *269*(5222), 341–346. doi:10.1126/science.7618100