Integrated mathematical modeling of cross-scale dynamics and equity constraints in water–energy–food systems

Abstract Global sustainability challenges manifest as complex system dynamics operating across multiple scales, where water, energy, and food systems interact through amplified stochastic shocks and equity constraints. Prevailing methodologies overlook cross-scale feedbacks, represent resilience as fixed, and neglect random disturbances and equity concerns, restricting their capacity to guide adaptive and just sustainability pathways. This study addresses the existing gap by introducing the Sustainability Nexus Transition Model (SNTM). As a unified mathematical framework, the SNTM synthesizes seven theoretical traditions including resilience theory, panarchy, complex adaptive systems, cross-scale systemic resilience, adaptive cycles, source-access-flow models, and stochastic viability theory, transforming these theories into a single operational system. The methodology integrates hybrid differential equations, stochastic processes, equity-constrained optimization, participatory calibration, agent-based governance, and transformative adaptive pathways to capture nonlinear dynamics, feedback amplification, and multi-actor adaptation across water–energy–food sectors. Simulation experiments across 50 independent runs reveal that local water shocks below 40% of carrying capacity trigger cascading failures, with energy flux amplifying over 170% within 20 time steps. Incorporating equity weights above 0.7 sustains system stability 89% longer than equity-agnostic models, while decentralized agent-based governance decreases recovery time by 52% compared to centralized approaches. Adaptive pathway switching lowers maladaptation risk by 58%. Sensitivity analysis confirms robustness under ± 30% parameter variations, and empirical validation against Murray–Darling Basin sectoral fluxes (2000–2020) demonstrates real-world applicability. The SNTM provides a platform for scenario analysis, early warning detection, and participatory policy design under deep uncertainty, contributing to sustainability science applied to water–energy–food systems by translating multi-actor processes into formal models that support equitable adaptive governance. Graphical abstract