In a new open-access study that I published with my late colleague Kostya Trachenko from Queen Mary University of London, I propose a surprisingly simple nonlinear mathematical equation that unifies 12,000 years of human population growth and points to stark possible futures if global environmental crises intensify.

The research, published in the journal Chaos, Solitons & Fractals, introduces a nonlinear "rate-feedback" model for global population growth that I originally worked out with Trachenko in a different context, that is, the physics of disordered materials such as glasses and amorphous solids.

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We now show that the same mathematics can reproduce many of the major population growth patterns observed throughout the last 12,000 years, from the Neolithic era to the modern age.

Unlike classical demographic models that treat growth as either exponential or logistic, our new framework can switch naturally between multiple historical regimes using a single parameter. In some periods, human population expanded slowly and steadily; in others, growth accelerated explosively. According to our derivation, these shifts can all emerge from the same underlying nonlinear dynamics.

Our model also revisits one of the most famous predictions in population science: the "doomsday" scenario proposed in 1960 by Heinz von Foerster and colleagues, who mathematically extrapolated that the world population would diverge toward infinity around 2026.

Humanity avoided that trajectory as fertility rates declined globally, but our new study argues that the underlying mathematics of runaway growth can still reappear under certain conditions.

To test the theory, we compared our equation (sometimes also referred to as the Trachenko-Zaccone equation) against empirical population data across several historical eras. We found that the model successfully reproduces both "compressed exponential" growth phases, such as the rapid industrial-era expansion, and the slower "stretched exponential" regime that has characterized global population growth since about 1970.

The most provocative part of our paper explores hypothetical future scenarios. In our baseline analysis, the current global trend does not produce a catastrophic singularity like the one predicted by von Foerster and co-workers, because the governing parameter remains in a stabilizing regime.

However, we also modeled what could happen if major environmental crises abruptly imposed severe carrying-capacity limits on Earth, through climate collapse, pandemics, conflict, or resource shortages.

Under a deliberately conservative worst-case assumption that Earth's sustainable carrying capacity suddenly dropped to around 2 billion people, our model predicts a rapid global population decline, with humanity potentially halving by around the year 2064.

In the article we stress that this is not a forecast, but rather an illustrative mathematical scenario intended to show how sensitive population dynamics may be to abrupt environmental or societal changes. We emphasize that the current trajectory remains relatively stable and does not imply imminent collapse.

Beyond demography, we believe that this work could be interesting for importing ideas from condensed matter physics into population science. The same mathematical structures used to describe how the atomic dynamics in glasses relax over time appear capable of describing how human societies grow, stabilize, and potentially destabilize over centuries.