A nonlinear discrete-time compartmental model for rare earth e-waste recycling

Electronic waste (e-waste) contains valuable rare earth elements (REEs) essential for modern technologies, yet current recycling systems recover only a fraction of these materials. This study develops a discrete-time compartmental model to track REE flows within e-waste systems, explicitly capturing e-waste stocks, recovered REEs, and cumulative dissipation, including nonlinear feedback between past losses and future recovery. Analysis identifies two steady states: an idealized dissipation-free equilibrium and a dissipation-included equilibrium with persistent losses. Local and numerical stability analyses show that the dissipation-free state is dynamically unstable under realistic recycling and recovery parameters, while the dissipation-included equilibrium is globally attracting. Results indicate that dissipation is an inherent structural feature that stabilizes the system, highlighting the bounded nature of circularity for critical materials. Policies that increase recycling alone are insufficient; effective strategies must also target loss pathways and inefficiencies to improve long-term sustainability of REE recovery from e-waste.