Assessing the robustness of spatial pattern sequences in a dryland vegetation model

Abstract

A particular sequence of patterns, 'gaps→labyrinth→spots', occurs with decreasing precipitation in previously reported numerical simulations of partial differential equation dryland vegetation models. These observations have led to the suggestion that this sequence of patterns can serve as an early indicator of desertification in some ecosystems. Because parameter values in the vegetation models can take on a range of plausible values, it is important to investigate whether the pattern sequence prediction is robust to variation. For a particular model, we find that a quantity calculated via bifurcation-theoretic analysis appears to serve as a proxy for the pattern sequences that occur in numerical simulations across a range of parameter values. We find in further analysis that the quantity takes on values consistent with the standard sequence in an ecologically relevant limit of the model parameter values. This suggests that the standard sequence is a robust prediction of the model, and we conclude by proposing a methodology for assessing the robustness of the standard sequence in other models and formulations.

Keywords: desertification; dryland ecosystems; early-warning signs; models of vegetation pattern formation; pattern formation.

Figure 1.

Example of the standard ‘gaps → labyrinth → spots’ sequence in the vegetation model by Rietkerk et al. [14] Qualitatively, different patterns occur at successively smaller values of a precipitation parameter. Darker shading denotes higher levels of vegetation biomass. (Online version in colour.)

Figure 2.

Schematic diagram depicts the uniform steady-state solutions of R02 (2.1), with insets show examples of patterned states occurring at different values of p. The uniform desert state is stable on the interval p∈[0,p₀), and the uniform vegetated state is stable to spatially uniform perturbations for p>p₀. The vegetated state is unstable to spatially periodic perturbations at a range of wavelengths on the interval p∈(p_ℓ,p_u). We refer to the endpoints p_ℓ and p_u as the lower and upper Turing points, respectively. (Online version in colour.)

Figure 3.

Examples of hexagon (H⁻ and H⁺) and stripe (S) patterns on a two-dimensional hexagonal lattice. We idealize gaps as H⁻ patterns, labyrinths as S, and spots as H⁺. (Online version in colour.)

Figure 4.

Diagram of numerical simulation procedure. Numerical simulations are run at discrete values of p marked by dots. The procedure is initialized with p just below the upper Turing point p_u (star) and run forward in time until a steady state is reached. Then, p is stepped upward by a small increment and the simulation is once again allowed to reach steady state. This is repeated until patterns lose stability to a uniform vegetated state at p=p_u+ (right circle). Using the previous patterned steady state (right square) as an initial condition, p is then stepped downward in the same way until patterns lose stability to a uniform state at p=p_ℓ− (left circle). Then, p is incremented upward a final time, and the procedure terminates when patterns once again lose stability (octagon). Note that p_u+ and p_ℓ− do not necessarily coincide with the Turing points p_u and p_ℓ, because patterns may persist outside of the Turing instability interval. (Online version in colour.)

Figure 5.

Summary of upper Turing point amplitude equation calculations over α−f and D_h−f parameter spaces, along with schematic bifurcation diagrams. The coefficients of the amplitude equations (2.4) are computed, and the curves a=0, b−c=0, c+2b=0 and b=0 separate the parameter spaces into regions labelled A–E. Qualitatively similar bifurcation structures occur within each region. In the white region, no Turing points occur on the uniform vegetated steady state of R02 and no calculations are performed. (Online version in colour.)

Figure 6.

Summary of lower Turing point amplitude equation calculations over α−f and D_h−f parameter spaces. The coefficients of the amplitude equations (2.4) are computed, and the curves c−b=0, b+2c=0 and b=0 separate the parameter space into regions labelled F–I, each of which exhibits a qualitatively distinct bifurcation structure. Bifurcation diagrams applicable to regions F–I resemble diagrams B–E, respectively, in figure 5, with the roles of gaps and spots exchanged and the solutions reflected, so that supercritical branches bifurcate in the direction of increasing precipitation. (Online version in colour.)

Figure 7.

Summary of pattern transitions observed numerical simulations over α−f and D_h−f parameter spaces in R02, along with representative examples of transitions from numerical simulations at f=0.2 and ${\log }_{10}(D_h)=0.6-4.0$. Number lines plot the relative locations of the upper and lower Turing points (p_u and p_ℓ, respectively), the transcritical point (p₀) and upper and lower pattern stability boundaries (p_u+ and p_ℓ−) for the example simulations shown. The parameter values corresponding to the example simulations are circled. Although p_ℓ and p₀ are nearly coincident, the distance between these points is exaggerated to illustrate that p_ℓ>p₀. (Online version in colour.)

Figure 8.

Example profiles of individual spot patterns taken from numerical simulations at f=0.2 and ${\log }_{10}(D_h)=0.6$ (sinusoidal), ${\log }_{10}(D_h)=2.0$ (sharply peaked) and ${\log }_{10}(D_h)=4.0$ (plateau-like). The example sinusoidal profile comes from a spot-patterned state near the upper Turing point. The sharply peaked and plateau-like profiles come from spot-patterned states well below the lower Turing points in their respective simulations.