Development of a Novel, Continuous-Time Dynamical Modelling Paradigm That Integrates the Cycles of Natural Resources with Economics

Natural resource cycles operate under both physical constraints and socio-economic drivers. To represent these coupled dynamics, a mineral commodity model is developed by integrating material flow analysis with market behaviour using a system of ordinary and delay differential equations \cite{Gambaro_2025} . The formulation is grounded in mass conservation and the cobweb economic theorem, enabling simulation of anthropogenic metal stocks, flows and market variables, including price dynamics. An application to the global copper cycle demonstrates the model's ability to reproduce long-term trends in both material flows and market determinants. Numerical integration of the differential equations shows that longer mining lead times amplify and extend price fluctuations, whereas shorter delays lead to more stable cycles. Stability and eigenvalue sensitivity analyses \cite{Phillips_1982} reveal that new and old scrap flows dominate overall dynamics, while refining and manufacturing activity provides the strongest stabilising influence. Dissipative flows and alternative recycling (tailings, slag and landfills) affect the system's speed of convergence towards equilibrium. This work illustrates the feasibility of unifying physical and economic theories in the simulation of resource systems. The framework also accommodates other economic mechanisms, such as the second-order accelerator of investment \cite{Hillinger_1989} , offering further avenues for advancing resource cycle modelling. Introducing this economic formalism to the equation set generates endogenous oscillations consistent with observed multi-decadal fluctuations in copper inventories \cite{Crowson_2023} , offering a mechanistic explanation for cyclical behaviour in metal supply. The results highlight the importance of integrating theories from social sciences and economics into the modelling of resource cycles to improve forecasts of material flows and optimize exploitation and future use.