Mathematical Modelling for Sustainability: Making Trade-offs Explicit

Sustainability is usually defined by slogans such as “reduce emissions,” “save water,” “go green.” Real sustainability, however, is a system problem. Energy, food, land, cities, finance, and human behaviour interact, with feedback loops and unintended consequences.

Mathematical modelling disciplines that complexity by forcing clear questions: What is the objective? What are the non-negotiable constraints? Who gains and who pays over what time horizon?

This multi-objective reality is made explicit in the UN SDGs adopted in 2015: social and economic progress must be pursued along with environmental protection. So, trade-offs are unavoidable—and quantifying them is the honest starting point.

Key modelling approaches used in sustainability

1. System dynamics: stocks, flows, feedback.

System dynamics models track accumulations—like population, industrial assets, or pollutants—and the feedback loops that connect them. A well-known example is the World3 model, popularized by The Limits to Growth, which shows how delays and reinforcing loops can lead to overshooting long before the warning signs become obvious.

2. Threshold thinking and the “safe operating space.”

Not all environmental risks follow a gradual pattern—some behave like tipping points. The “planetary boundaries” concept outlines physical limits to define a safe operating space. Even if the exact numbers are debated, the math behind it is vital: real sustainability often means staying away from dangerous thresholds, not just improving average performance.

3. Optimization under constraints.

Many sustainability challenges are optimization problems—how to allocate limited water between users, plan electric grids with renewables and storage, or craft emissions policies that balance costs. Good models don’t just give a single “best” answer; they reveal the landscape of trade-offs (cost, carbon, reliability, equity), giving leaders the insight to make informed choices—not guesses.

4. Scenario and integrated assessment modelling.

To address climate change, the IPCC uses scenarios and integrated assessment models (IAMs) that link energy, land, the economy, and emissions. These models check whether action plans are consistent under real-world limits—like available technology, political choices, and human behaviour. But they also call for humility: results depend on assumptions, so transparency and sensitivity testing are essential.

A complementary tool is life-cycle assessment (LCA). Many “green” choices simply shift impacts to other stages of a product’s life. ISO 14040 formalizes LCA to assess the full cycle—from goal setting and inventory to impact analysis and interpretation—so that decisions are grounded in evidence, not just marketing claims.

What makes models accountable when they aren't crystal balls?

A sustainability model is not an oracle; rather, it is a tool for making decisions. It shouldn't influence policy if it can't be audited. Calibration to observed data when feasible, uncertainty and sensitivity analysis (report ranges rather than single numbers), and explicit value choices (what qualifies as "cost," whose wellbeing is included, which affects matter) are all examples of responsible practice. differing stakeholders accept differing trade-offs in many "technical" issues, which are partially ethical. It is the model's responsibility to explicitly state such trade-offs, including distributional effects in addition to averages.

References

• UN Sustainable Development Goals (SDGs) overview. (Sustainable Development Goals)
• Meadows et al., The Limits to Growth (World3/system dynamics). (The Academy for Systems Change)
• Herrington (Club of Rome) reflection on World3 and long-run trajectories.
• Rockström et al. (2009), “A safe operating space for humanity” / planetary boundaries. (Nature)
• IPCC AR6 WGIII Mitigation report and modeling annex (scenarios & methods). (IPCC)
• ISO 14040:2006 Life Cycle Assessment—principles and framework.