Delay‐Driven Spatial Patterns in a Network‐Organized Vegetation Model With the Finite Soil Carrying Capacity in Semiarid Areas

ABSTRACT Traditional vegetation models are typically limited to simulating specific localized regions or discrete points, failing to capture the spatial heterogeneity of vegetation–water interactions in complex terrains. This study explores a novel network‐organized vegetation model for semiarid areas, which extends the Klausmeier framework to network topology through Laplacian matrix quantification of interpatch diffusion, overcoming the uniform terrain limitations of conventional continuum models. The model integrates finite soil carrying capacity with delayed water absorption mechanisms, unveiling a new pathway through which time delay drives spatial pattern transitions via the Hopf bifurcation. Furthermore, it establishes stability criteria for delay systems and develops a center manifold‐based approach for analyzing periodic solutions. Theoretical analysis demonstrates that the equilibrium remains stable without delay but loses stability through the Hopf bifurcation when exceeds critical thresholds. Numerical simulation not only supports the theoretical results but also further demonstrates the evolution process of the system under various parameter conditions, especially exploring the impact of key parameter changes such as soil bearing capacity and time delay on system stability. This work provides a novel framework for modeling semiarid ecosystems and explains the mechanisms by which landscape fragmentation influences vegetation patterns under delay‐driven dynamics.