Figure 1

Biomass $b$ vs precipitation $p$ for plane topography. Solid lines denote stable solutions while dashed (dotted) lines denote solutions which are unstable to uniform (nonuniform) perturbations. The branch $\mathcal{B}$ represents a uniform bare-soil solution; The two branches marked $\mathcal{A}$ and $\mathcal{E}$ represent uniform vegetation states for two different parameter choices, corresponding to ($\mathcal{A}$) low biomass density, high growth rate, and short root lengths and ($\mathcal{E}$) higher biomass density, low growth rate, and long root lengths. The insets show typical biomass patterns of the $\mathcal{E}$ species along the precipitation gradient. Dark shades of gray represent high biomass. Parameter values used are $\mu =0.3,\alpha =10,f=0.1,{\delta }_b={10}^{-4},{\delta }_w={10}^{-2}$, ${\sigma }_0=0.1$, and ($\mathcal{A}$): $\eta =0.7,\beta =1,\rho =0.6,q=0.25$; ($\mathcal{E}$): $\eta =3.5,\beta =5,\rho =3,q=0.05$. These values reflect realistic orders of magnitudes for shrubs in water limited systems [17, 19]. Nondimensional units have been converted to dimensional quantities using the following scalings (obtained for a particular choice of scaling parameters): precipitation $P=(500\mathrm{m}\mathrm{m}/\mathrm{y}\mathrm{r})p$, biomass $B=(1\mathrm{K}\mathrm{g}/{\mathrm{m}}^2)b$.Reuse & Permissions

Figure 2

Spatial profiles of $b$, $w$, and $h$ obtained by taking cross sections through two-dimensional numerical solutions of (1). The profiles represent three different species (different $\eta$) at two levels of crust coverage, with dimensional precipitation $P=75\mathrm{m}\mathrm{m}/\mathrm{y}\mathrm{r}$ ($p=0.15$): (a) $\eta =12,f=0.1$; (b) $\eta =3.5,f=0.1$; (c) $\eta =2,f=0.1$; (d) $\eta =12,f=0.9$; (e) $\eta =3.5,f=0.9$; (f) $\eta =2,f=0.9$. Other parameters are the same as those of the $\mathcal{E}$ branch in Fig. 1. Panels (a) and (d) span an horizontal range corresponding to 14 m, all others span 3.5 m.Reuse & Permissions

Figure 3

Habitat change induced by a transition from a banded pattern (a) to a spotted pattern (b), on a slope, in response to a precipitation drop from $P=175\mathrm{m}\mathrm{m}/\mathrm{y}\mathrm{r}$ (a) to $P=150\mathrm{m}\mathrm{m}/\mathrm{y}\mathrm{r}$ (b). The upper frames show two-dimensional biomass patterns (darker gray shades correspond to higher biomass density) obtained by solving numerically Eqs. (1). The lower panels show soil-water profiles along the transects shown by the dashed lines in the upper frames. The soil-water density under spots is higher than the density under bands. Spots may therefore provide habitats to species that bands cannot accommodate. The slope is $10°$, $f=0.1$, and all other parameters are those of the $\mathcal{E}$ branch in Fig. 1. Precipitation has been scaled to dimensional units using $P=(500\mathrm{m}\mathrm{m}/\mathrm{y}\mathrm{r})p$.Reuse & Permissions