Overview
Predicting the future of human population growth is a problem that cuts across multiple disciplines: mathematics, physics, biology, economics, and environmental science. The challenge lies not in a lack of population data, but in understanding which mathematical models best capture the dynamics of human population change. In 2017–2018, I collaborated with fellow physics researcher Chris Fickess to investigate nonlinear models of population growth, applying techniques from chemical kinetics and differential equations to historical population data spanning from 1 AD to the present day.
Methodology
Rather than assume a single model form, we derived a power-law equation from first principles using chemical kinetics theory—specifically the law of mass action and nuclear reaction models. We then compiled extensive historical population data from multiple sources (U.S. Census Bureau, World Bank, academic databases) spanning nearly 2,000 years. Using Excel's Solver optimization tool, we fit the power-law model and several alternative models (exponential, logistic, and other nonlinear forms) to different data windows. Each model's parameters were systematically optimized, and we calculated error estimates and R² goodness-of-fit values for every model-data pair combination.
Key Findings and Insights
Our analysis revealed a striking result: different models produce dramatically different long-term forecasts despite fitting the historical data reasonably well. The exponential model predicts explosive growth, the logistic model predicts stabilization, and nonlinear models derived from chemical kinetics produce more complex behavior including potential overshoot and decline scenarios. Critically, the models remained nearly identical in their predictions until reaching a transition point—after which they diverged sharply. This parameter sensitivity shows that forecasting population futures requires extreme caution: small variations in model structure or parameter values can yield dramatically different outcomes. Some models projected unsustainable growth, others predicted stabilization or decline, and still others indicated potential collapse scenarios by mid-century.
Contribution and Limitations
The project built on prior research at the University of Central Oklahoma while applying new optimization techniques and data sources. We created comprehensive comparison tables analyzing multiple models' parameters, uncertainties, and forecasts. The work demonstrates that rigorous mathematical modeling of population dynamics can generate multiple self-consistent projections—but cannot predict which future will actually occur. Population growth depends not only on mathematical structure but on technological change, resource availability, policy decisions, and human behavior—factors external to any mathematical model. The value of this work lies not in predicting the future, but in illuminating the range of plausible futures and the sensitivities embedded in different modeling assumptions.
