Modeling co–opetition and forecasting of courier, cold chain, and instant delivery services using an integrated Grey Lotka–Volterra model

Driven by the rapid expansion of global e-commerce and escalating consumer demands for logistics timeliness and service quality, the modern logistics sector has evolved into a tightly interconnected and co-evolving ecosystem comprising traditional courier services, cold chain logistics, and instant delivery. Accurately disentangling their dynamic co-opetitive relationships and forecasting their evolution is crucial for both strategic and policy decision-making. However, existing approaches struggle to simultaneously capture complex interactions and deliver reliable predictions under small-sample conditions. To address these challenges, we originate an integrated Grey Lotka–Volterra (IGL-V) model. Specifically, the proposed model incorporates a structurally optimized grey framework by introducing a background value coefficient and a grey derivative coefficient to mitigate discretization errors. In addition, Robust Least Squares (RLS) is combined with the Whale Optimization Algorithm (WOA) to enhance parameter estimation robustness under data fluctuations. The model is validated using annual business volume data from three major logistics service modes in China: traditional courier, cold chain logistics, and instant delivery. Empirical results demonstrate that the IGL-V model outperforms six benchmark models in terms of both forecasting accuracy and robustness. Furthermore, the estimated parameters provide direct empirical evidence of the underlying co-opetitive dynamics, revealing bidirectional strategic complementarity between traditional courier services and the other two modes, alongside mutual competition between cold chain logistics and instant delivery. These findings offer a valuable decision-support framework for stakeholders navigating the increasingly complex logistics ecosystem.