Localized vegetation patterns, fairy circles, and localized patches in arid landscapes

We investigate the formation of localized structures with varying widths in one- and two-dimensional systems. The mechanism of stabilization is attributed to strongly nonlocal coupling mediated by a Lorentzian type of kernel. We show that, in addition to stable dips found recently [see, e.g. Fernandez-Oto et al., Phys. Rev. Lett. 110, 174101 (2013)], there are stable localized peaks which appear as a result of strongly nonlocal coupling. We applied this mechanism to arid ecosystems by considering a prototype model of a Nagumo type. In one dimension, we study the front connecting the stable uniformly vegetated state to the bare one under the effect of strongly nonlocal coupling. We show that strongly nonlocal coupling stabilizes both—dip and peak—localized structures. We show analytically and numerically that the width of a localized structure, which we interpret as a fairy circle, increases strongly with the aridity parameter. This prediction is in agreement with published observations. In addition, we predict that the width of localized patch decreases with the degree of aridity. Numerical results are in close agreement with analytical predictions.

Figure 1

Localized vegetation patterns: (a) An aerial photo taken on 05 March 2010 shows fairy circles in the Namibrand region, Namibia. Image courtesy of Professor Norbert Juergens. (b) An aerial photograph showing LVPs, Zambia (taken from the review [29]).

Figure 2

Front propagation obtained from numerical simulations of Eq. (1). (a) Biomass front profile. (b) Spatiotemporal diagram that shows the front movement with a constant speed. The parameters are $\alpha =0.51$, $\sigma =0.7$, $n=2$, and $\epsilon =1.0$.

Figure 3

Numerical solutions of model (1) with the Lorentzian kernel (2). (a) Front profile at the Maxwell point $\alpha =0.5$; other parameters are $\sigma =2$, $n=2$, and $\epsilon =0.35$. (b) Decay of front to the uniformly vegetated state for $\alpha =0.5$, $\sigma =2$, $n=2$, and $\epsilon =0.1$. In (b), the top panel shows the comparison between the numerical calculation (dotted line) and the analytical estimation (10) (continuous line), while the bottom panel displays the same comparison in a log-log plot.

Figure 4

Stationary FC obtained from numerical simulations of Eq. (1). (a) One dimension with $\alpha =0.492$, $\sigma =0.8$, $n=2$, and $\epsilon =0.5$. (b) Two dimensions with $\alpha =0.46$, $\sigma =0.1$, $n=2.5$, and $\epsilon =10$.

Figure 5

Width of a FC as a function of aridity $\alpha$ in one dimension. The solid line represents the analytical prediction from Eq. (16). The squares are obtained by numerical simulations of Eq. (1). The inset is the spatial profile of biomass. Parameters are $\sigma =0.8$, $n=2$, and $\epsilon =0.5$.

Figure 6

Stationary localized patch obtained from numerical simulations of Eq. (1). (a) One dimension with $\alpha =0.52$, $\sigma =0.8$, $n=2$, and $\epsilon =0.5$. (b) Two dimensions with $\alpha =0.56$, $\sigma =0.1$, $n=2.5$, and $\epsilon =10$.

Figure 7

Width of a LVP, as a function of aridity $\alpha$ in one dimension. The solid line represents the analytical prediction from Eq. (18). The dots are obtained by numerical simulations of Eq. (1). The inset is the spatial profile of biomass. Parameters are $\sigma =0.8$, $n=2$, and $\epsilon =0.5$.

Figure 8

Bifurcation diagram of LSs for model Eq. (1), in parameter plane $(\alpha ,\epsilon )$, with $\sigma =0.8$ and $n=2$.

Figure 9

Two-dimensional localized vegetation patterns obtained from numerical simulations of Eq. (1). (a) Multi-FCs for $\alpha =0.44$; (b) multi-LVPs for $\alpha =0.57$. Other parameters are $\sigma =0.1$, $\epsilon =10$, and $n=2.5$.