From single-scale to multi-scale Turing patterns: A two-layer vegetation model

Dryland ecosystems often exhibit striking vegetation patterns across multiple spatial scales, reflecting strong feedbacks between plants, soil water, and climate. We develop a two-layer vegetation model that explicitly incorporates vertical soil heterogeneity and phenotypic plasticity in rooting depth to investigate the ecological origins of multiscale pattern formation. Shallow-rooted plants primarily exploit transient topsoil moisture and generate large-scale spatial patterns, whereas deep-rooted plants access more stable subsoil water and persist as fine-scale vegetation clusters. The interaction of these contrasting strategies produces nested spatial structures that remain robust across a wide range of precipitation regimes. By comparing idealized and biologically realistic inter-layer coupling, we show that multiscale organization emerges from ecologically meaningful processes such as water infiltration, evaporation, and irreversible investment in deeper roots. The model provides a mechanistic realization of Walter's two-layer hypothesis and demonstrates how variations in spatial scale separation can give rise to complex vegetation structures. Our findings suggest that multiscale vegetation patterns may emerge as adaptive responses to water limitation in dryland ecosystems.