Dynamic equilibrium, stability and sensitivity analysis of anthropogenic metal cycles with an application to copper

Anthropogenic metal cycles shape resource availability, recycling performance and long-term sustainability: understanding their dynamic behaviour, including long-term stability and sensitivity to change, is essential for sustainable resource management and supply-chain resilience. Here, a system of ordinary differential equations for the global copper cycle linking extraction, processing, manufacturing, use, waste management and recycling is developed to analyse the dynamic equilibrium, stability and sensitivity of metal cycles. Results demonstrate that dissipative losses are necessary for asymptotic stability of the stocks, underscoring the importance of quantifying dissipation and recycling for long-term forecasting. Multi-decadal copper inventory oscillations can be simulated by integrating a second-order investment accelerator to endogenise mining dynamics, while also creating conditions for endogenous growth. Sensitivity analysis shows that new and old scrap rates exert the largest influence on the system's dynamics, whereas refining and manufacturing flows provide the strongest stabilising effect. This study extends material flow analysis with tools from dynamical systems theory by linking physical flows with economic feedbacks and offers a general framework for assessing stability, resilience and policy levers in metal cycles and other resource systems.