Klausmeier Model — Turing Instability in Semi-Arid Ecosystems
Vegetation density (green=high)
Water density (blue=high)
Klausmeier Parameters
Simulation
Step: 0
Pattern: —
Max veg: —
The Klausmeier model (1999) describes vegetation-water feedbacks in dryland ecosystems. Two species interact: plant biomass W and soil water h.
∂_t w = a − w − w·n² + d_W Δw (water) ∂_t n = w·n² − b·n + d_N Δn (plants)
Plant uptake (~wn²) is facilitated: plants concentrate water, making denser patches more productive — a positive feedback. Combined with water diffusion (much faster than plant spread), this creates a Turing instability: the homogeneous state becomes unstable to spatial perturbations, spontaneously forming vegetation patterns — stripes on slopes, spots on flat terrain. These patterns are real — visible from satellite in Sahel, Somalia, Australia. Near the desertification threshold, patterns collapse suddenly (catastrophic bifurcation).