Spatial pattern formation in a vegetation–water model with uptake-diffusion feedback and nonlocal delay

Vegetation patterns in arid and semi-arid regions exhibit diverse spatial structures that reflect ecosystem dynamics and functioning. Although vegetation–water interaction models have been widely studied, the effect of delayed nonlocal uptake has been explored mainly in specific model settings. Its role within the reduced Zelnik-type uptake-diffusion framework has received less systematic attention. In this study, we extend the reduced vegetation–water model proposed by Zelnik et al. by incorporating a phenomenological delayed nonlocal water-availability term. Based on this extended model, we perform a theoretical analysis to derive the conditions for Turing instability and apply multiple-scale analysis to obtain the corresponding amplitude equations near the bifurcation threshold. The stability of these amplitude equations is further investigated to characterize the emergence of stripe, spot, and mixed patterns. Numerical simulations are carried out to verify the analytical results and illustrate the evolution of vegetation patterns under different parameter regimes. The results demonstrate that, within appropriate parameter ranges, uniform vegetation states can undergo transitions to gap patterns, thereby providing a model-based perspective on degradation-related pattern transitions within this framework.