For thousands of years, human population growth occurred so slowly that there wasn’t even a noticeable curvature in the graph of humanity’s civilization. Villages became towns. A harvest fed another generation. Empires grew and collapsed while the total number of people on Earth crept upward by degrees.
This has changed dramatically with the onset of the modern age, as industry, medicine, energy production, agriculture, and technology drove our population curve into one of the most spectacular population explosions in human history. This growth, however, has also defied mathematical explanations, challenging some of the best models used to explain life on our planet.
Now, a new mathematical model suggests that hidden within that rise is a deeper pattern, one that may also point to how quickly things could change if humanity abruptly runs into the planet’s limits.
A Worst-Case ‘Crisis’ Scenario
Published in Chaos, Solitons and Fractals, the study was authored by University of Milan physicist Dr. Alessio Zaccone and the late Dr. Kostya Trachenko of Queen Mary University of London. Their work used more commonly used mathematical methods to describe disordered materials, where scientists study how complex systems evolve, relax, and respond over time.
By applying this new model to our population growth, Dr. Zaccone and Dr. Trachenko have discovered that their simple equation appears to embrace a wide range of growth regimes observed over the last 12,000 years, from long periods of relative stability to rapid acceleration of our growth after the onset of the industrial age.
They also demonstrated just how rapidly our growth curve could shift if we lost the underlying assumptions for rapid human growth.
In a deliberately conservative worst-case scenario in which carrying-capacity constraints became abruptly active today, the researchers found that the global population could be cut in half as early as 2064.
Instead of trying to predict the future by looking at factors such as migration, fertility rates, technological development, economic changes, climate policy, and others, Dr. Zaccone and Dr. Trachenko sought to address a simpler, much more profound issue. Namely, can a general nonlinear model be used to describe the population growth curve in the history of humanity?
The answer is yes, though with important caveats.
“We show that a simple nonlinear differential equation (originally studied in the physics of disordered systems) mathematically describes key regimes of global population growth over the past 12000 years,” researchers write. “The proposed framework provides a compact analytical setting to explore future scenarios, including a deliberately conservative, worst-case illustration in which the global population could halve as early as 2064 if carrying-capacity constraints became abruptly active today.”
Why Population Models Are Hard to Build
Historically, modeling of population growth has been a controversial issue. As far back as 1798, English cleric Thomas Malthus proposed a simple exponential growth curve. According to his framework, the growth rate is determined by the difference between birth and death rates. If birth exceeds deaths, the population grows exponentially. If the opposite happens, it declines.
The problem with that approach is that the population of any species, including humans, doesn’t grow indefinitely. The carrying capacity, i.e., how many individuals of the species can be sustained, is limited.
It was Dr. Pierre François Verhulst who, in the 19th century, added this factor to our population growth models. He showed that population growth occurs, though it is progressively slowed by resource limitations and eventually comes to a stop.
Later, in 1960, Dr. Heinz von Foerster and colleagues famously proposed a hyperbolic model suggesting that human population growth was accelerating toward a mathematical “doomsday” singularity in 2026.
Obviously, Dr. von Foerster’s prediction did not come to fruition. However, his model raised a further crucial issue in population dynamics—namely, that any mathematical population framework can be fitted to describe certain historical events. The problem is that when applied to a much wider timeframe, they can completely break down.
According to Dr. Zaccone and Dr. Trachenko, the problem isn’t that those models were useless per se. On the contrary, most of them are very useful and supply valuable information about various aspects of population dynamics. However, none of them can be universally used, as they are typically local estimates valid for a specific timeframe.
A Single Mathematical Model To Capture It All
In their new study, Dr. Zaccone and Dr. Trachenko developed a nonlinear differential “rate-feedback” equation. In essence, it implies that the population growth rate depends on the population size, and a single parameter K determines whether the dependence is positive or negative.
If K = 0, the model yields a simple exponential growth curve. For negative values of K, the behavior approaches logistic dynamics, with population growth being increasingly slowed by resource restrictions. If K is positive, the model shows a rapidly accelerating growth curve.
Importantly, according to researchers, the classic models aren’t equivalent to theirs. Rather, these behaviors appear as local approximations within the proposed framework. It means the researchers do not claim to have developed a magical equation that will solve all problems. Instead, what they propose is a mathematical tool to bring a few key models under a single umbrella.
“Different growth regimes since the early Neolithic until the present can be interpreted within a single nonlinear rate-feedback equation in appropriate limits,” researchers write. “These include the well-known Malthus (exponential) and Verhulst (logistic) growth laws, as well as von Foerster-type hyperbolic growth as a controlled low-order truncation.”
Humanity’s Population Growth Regimes Keep Changing
Based on empirical estimates of the global population over the last 12,000 years, researchers discovered that our species has experienced multiple regimes throughout its history. While some of these periods were defined by relative population stability, others featured exponential growth, and others featured compression or stretching of the growth curve.
While there were shorter periods of population decline, for instance, during the Black Death in Europe, researchers focused on broader trends in population growth. These regimes, they say, were clearly distinct from each other.
The era of early agricultural societies was relatively stable. Later periods featured increasing acceleration in our population growth. Since the 1970s, the authors argue, our population dynamics can be best approximated by a stretched exponential regime, suggesting that population growth has slowed significantly compared to earlier stages.
Within this mathematical model, the current stretched-exponential regime implies K < 0. In other words, humanity’s growth doesn’t approach a critical threshold, and the possibility of catastrophic runaway growth can be ruled out.
However, the paper’s most attention-grabbing scenario explores what could happen if that trend were disrupted by a major crisis. Researchers say that in a sudden global catastrophe, carrying capacity constraints could suddenly become active.
When the mathematical Model Runs Into Earth’s Limits
Researchers suggest that if there were a serious shock to our planet, such as a global war, rapid climate change, or a massive pandemic, we could potentially see a collapse of our growth regime due to a drastic reduction in the exploitation efficiency of available resources.
To illustrate this, researchers introduced an additional term in their equation. Specifically, they accounted for the carrying capacity of our planet. Using an extremely conservative estimate of the carrying capacity of 2 billion individuals, they found that under these assumptions, our population would halve by 2064.
However, it’s important to note that this estimate is highly speculative. It cannot be viewed as an exact prediction of our future for several reasons. First, researchers explicitly state that their model is purely illustrative and not intended for prediction.
Secondly, the choice of a carrying capacity of 2 billion is highly debatable. The carrying capacity of Earth itself, rather than per person, depends on many parameters and is not a constant. Technological progress, energy efficiency, agricultural productivity, climatic stability, and international cooperation determine, to a great extent, how many people our planet can sustain at any given time.
Still, the study’s warning is clear. Mathematical population trends can look stable until the assumptions behind them suddenly change. A world that continues along its current stretched-exponential trajectory may avoid doomsday-style runaway growth. But a world that abruptly runs into hard limits could experience a very different future.
The Real Warning Is in the Curve, Not the Date
The researchers acknowledge that the model’s empirical fits vary in strength. The 1970–2023 regime shows a stronger fit than the earlier compressed-exponential periods analyzed in the study, as indicated by the goodness-of-fit metrics reported for each historical window.
However, the significance of their analysis lies not in the exact numbers but in what they imply. According to researchers, their results show that human population growth is not governed by a single law throughout its entire history.
Ultimately, the model’s value may lie less in its specific dates than in its wider message. Human population growth is not governed by a single permanent law. It is formed by feedback, constraints, and changing historical conditions.
The future, in this mathematical model, depends not only on how many people exist, but also on whether the systems supporting them continue to function efficiently enough to avoid sudden encounters with carrying-capacity limits.
“While the current global population growth trend corresponds to 𝐾 < 0 and does not lead to a doomsday criticality, reverting to an effectively 𝐾 > 0 regime would reintroduce a finite-time divergence in the uncontrolled dynamics,” researchers conclude. “In a separate conservative scenario where carrying-capacity constraints become abruptly active, [it] predicts a rapid population decline.”
Tim McMillan is a retired law enforcement executive, investigative reporter and co-founder of The Debrief. His writing typically focuses on defense, national security, the Intelligence Community and topics related to psychology. You can follow Tim on Twitter: @LtTimMcMillan. Tim can be reached by email: tim@thedebrief.org or through encrypted email: LtTimMcMillan@protonmail.com
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