Existence of localised radial patterns in a model for dryland vegetation

Abstract

Localised radial patterns have been observed in the vegetation of semi-arid ecosystems, often as localised patches of vegetation or in the form of `fairy circles'. We consider stationary localised radial solutions to a reduced model for dryland vegetation on flat terrain. By considering certain prototypical pattern-forming systems, we prove the existence of three classes of localised radial patterns bifurcating from a Turing instability. We also present evidence for the existence of localised gap solutions close to a homogeneous instability. Additionally, we numerically solve the vegetation model and use continuation methods to study the bifurcation structure and radial stability of localised radial spots and gaps. We conclude by investigating the effect of varying certain parameter values on the existence and stability of these localised radial patterns.

Figures (10)

• Figure 1: a) Due to evaporation of the soil water (illustrated by the blue arrows) and insufficient resources to sustain uniform plant growth, semi-arid ecosystems can exhibit localised patches of vegetation. The soil water is also localised, subsisting in the shadow of the vegetation. b) For sufficiently large domains, these patches can also form part of a larger pattern (such as spots, stripes or hexagons, for example). c) By measuring the vegetation and soil water densities, vegetation patterns can be described by continuum models. Here, we present a schematic density plot of the vegetation in panel b), where the darker green indicates a higher intensity of vegetation.
• Figure 2: a) For localised spikes, the top picture shows a pair of real eigenvalues bifurcating from the origin (blue circle with cross) onto the real line (red circles). This results in a localised solution with monotonic tails, as shown in the bottom picture. b) For localised spots, the top picture shows four purely imaginary eigenvalues bifurcating from $\pm \textnormal{i} k$ (blue circle with cross), for some $k\in\mathbb{R}$, into the complex plane (red circles). This results in a localised solution with spatially-oscillatory tails with wave number $k$, as shown in the bottom picture.
• Figure 3: A summary of our results regarding the existence of different localised radial patterns. For the five classes of patterns, we present their relative vegetation density plot and radial profile, as well as their bifurcation type and whether or not they exist in the vegetation model.
• Figure 4: Diagram of the bifurcation structure for a) the full model \ref{['model:wn']}, where the vegetation density $n$ is plotted against the precipitation $p$, and b) the weakly nonlinear reduction \ref{['model:NW']}, where the scaled vegetation density $N$ is plotted against the scaled precipitation $P$. Stable solutions are indicated by solid lines and unstable solutions by dashed lines; points at which there is a change in stability are highlighted by red circles. In both equations, the uniform vegetated state $\mathcal{V}$ (green) bifurcates from the bare state $\mathcal{B}$ (blue) and undergoes at least one change of stability.
• Figure 5: In the weakly reduced model \ref{['model:NW']}, we expect there to be three possible cases in which localised radial solutions can emerge. In case 1, solutions bifurcate from the bare state $\mathcal{B}$ (blue) sub-critically at $P=0$. In case 2, solutions bifurcate from the uniform vegetated state $\mathcal{V}$ (green) super-critically at $P=0$. In case 3, solutions bifurcate from the uniform vegetated state $\mathcal{V}$ (green) sub-critically at $P=P_{0}$.

Theorems & Definitions (9)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Theorem 1: Existence of Spot A
• Theorem 2: Existence of Rings
• Theorem 3: Existence of Spot B
• Remark 3.6