Compartmental Modeling of Woodland Ecosystem Services for Evaluating Urban Natural Capital in Sustainable Cities

ABSTRACT Ecosystem services (ES) from natural assets such as woodlands, freshwater, and air are vital for human well‐being and economic sustainability. In the UK, ES contribute approximately £1.5 trillion annually, with woodland ecosystem services (WES) alone valued at £382 billion. Key WES include carbon capture, air and water purification, recreation, and urban health services. Despite their significance, urban ES management often lacks comprehensive computational models for effective accounting and resource planning. When developed, sustainable smart city governance technologies such as the Internet of Things (IoT), artificial intelligence (AI), and predictive analytics can be integrated to enhance the forecasting and valuation of ES for natural capital accounting. In addition, other frameworks such as game theory, Leontief Input–Output analysis, gene–environment networks, and homogenization theory can be explored. This study models WES components as compartments using ordinary differential equations (ODEs) and incorporates management practices, such as forest conservation (e.g., reforestation), recreational infrastructure conservation (e.g., site hardening, visitor management), and water conservation (e.g., habitat protection, species management), as control variables. Since the resulting model is nonlinear and analytical solutions do not exist, we apply the widely used and complementary numerical methods, Non‐Standard Finite Difference (NFSD) and Runge–Kutta 4th Order (RK4), to approximate solutions of such ODE problems, thereby preserving accuracy, stability, and nonnegativity. The NSFD and RK4 are applied to assess three critical WES: carbon sequestration, recreational infrastructure, and water filtration. The NSFD method is suited for global, long‐term analyses and achieved up to 55.68% effectiveness, while RK4 is more appropriate for short‐term, detailed dynamics and attained 48.66% effectiveness. These results highlight that compartmental modeling with either method can accurately capture ecosystem services for smart city planning and valuation. Both approaches are robust, with control variables such as reforestation enhancing carbon capture, improving water quality, and promoting urban health. This study emphasizes the importance of integrated governance frameworks that combine reforestation, conservation, and smart infrastructure policies to support long‐term urban sustainability.