Periodic versus scale-free patterns in dryland vegetation

Abstract

Two major forms of vegetation patterns have been observed in drylands: nearly periodic patterns with characteristic length scales, and amorphous, scale-free patterns with wide patch-size distributions. The emergence of scale-free patterns has been attributed to global competition over a limiting resource, but the physical and ecological origin of this phenomenon is not understood. Using a spatially explicit mathematical model for vegetation dynamics in water-limited systems, we unravel a general mechanism for global competition: fast spatial distribution of the water resource relative to processes that exploit or absorb it. We study two possible realizations of this mechanism and identify physical and ecological conditions for scale-free patterns. We conclude by discussing the implications of this study for interpreting signals of imminent desertification.

Figure 1.

Single-patch dynamics as affected by the infiltration and root-augmentation feedbacks. (a,b) No infiltration and root-augmentation feedbacks (f = 1, E = 0): an initial spot-like patch either (a) shrinks to zero, when the precipitation rate is low enough (P = 75 mm y⁻¹), or (b) expands indefinitely when the precipitation rate is high (P = 225 mm y⁻¹). (c) No root-augmentation feedback (f = 0.1, E = 0): when the precipitation rate is high enough (P = 105 mm y⁻¹), patches grow but their areas are limited by central dieback processes that lead to ring-shape patches. (d,e) No infiltration feedback and moderate root-augmentation feedback (f = 1, E = 1 m² kg⁻¹): (d) when the precipitation rate is sufficiently high (P = 165 mm y⁻¹), patches grow but form ring shapes owing to central dieback processes; (e) at lower precipitation rates (P = 140 mm y⁻¹) growing patches approach a fixed size. (f) No infiltration feedback and strong root-augmentation feedback (f = 1, E = 4 m² kg⁻¹): when the precipitation rate is sufficiently high (P = 195 mm y⁻¹), patches initially grow but then split owing to peripheral dieback processes. Other parameters are S₀ = 0.125 m, K = 1 kg m⁻², Q = 0.05 kg m⁻², M = 1.2 y⁻¹, A = 400 y⁻¹, N = 4 y⁻¹, Λ = 0.032 m² (kg y)⁻¹, Γ = 20 m² (kg y)⁻¹, D_B = 0.000625 m² y⁻¹, D_W = 0.0625 m² y⁻¹, D_H = 0.2 m⁴ (kg y)⁻¹, R = 10.

Figure 2.

(a) A typical scale-free pattern, obtained for global competition induced by fast surface-water flow relative to infiltration and (b,c) the appearance of characteristic patch sizes upon decreasing the competition range, either by increasing the root-to-shoot allocation parameter E, or (c) by increasing the time scale of surface-water flow. Global competition induced by fast soil-water diffusion relative to water uptake also gives rise to scale-free patterns. (e–h) show patch-size distributions, determined by censuses of connected areas above 0.05 kg m⁻² in density. (i–l) show average power spectra. Grey bands indicate the range in which 94 per cent of 32 independent realizations fall. The domain size shown is 64 × 64 m² for (a), 16 × 16 m² for (b) and 32 × 32 m² for (c,d). In all simulations, the integration time is 166 y. Parameter values: (a,l,i): E = 0, D_H = ∞, f = 0.1; (b,f,j): E = 4 m² kg⁻¹, D_H = ∞, f = 0.1; (c,g,k) E = 0, D_H = 1 m⁴ (kg y)⁻¹, f = 0.1; (d,h,l) E = 2 m² kg⁻¹, D_W = ∞, Γ = 7.2 m² (kg y)⁻¹, f = 1. All panels: P = 120 mm y⁻¹, A = 40 y⁻¹. Other parameters are as in figure 1.