We follow Lefever and Lejeune  and model spatial interactions as the combined effect of facilitation-competition mechanisms: near-neighbor interactions facilitate vegetation survival and growth, due to the favorable environment existing in the subcanopy soils, e.g., higher moisture contents due to lower evaporation losses and/or higher infiltration capacity [Walker et al., 1981; Scholes and Archer, 1997; Zeng et al., 2004; Greene, 1992; Greene et al., 1994]. Long-range interactions are dominated by competitions between lateral root systems, which expand beyond the vertical projection of the canopy [e.g., Casper et al., 2003; Caylor et al., 2006]. At any point, x(x, y), in a two-dimensional domain, Ω, the effect of interactions with vegetation at another point, x′(x′, y′), is assumed to be proportional to the biomass of the neighboring vegetation and to a weight function, w(r), of the distance, r = ∣r∣ = ∣x − x′∣, between x′ and x. This weight function (Figure A1a) is positive for relatively small values of r (facilitation) and negative at larger distances (competition). In the absence of relief, the spatial interactions (hence, w(r)) depend on the length but not on the direction of the displacement vector, r, i.e., the process is isotropic. An integral formulation is used to account for the effect of the interactions with vegetation existing at all points, x′(x′, y′), in Ω [e.g., Murray, 1989]. For each of the two climate-controlled states (i.e., stressed and unstressed conditions) vegetation dynamics (1) can be expressed as
where the subscripts 1 and 2 denote the local dynamics of stressed and unstressed vegetation, respectively, while V represents vegetation biomass, which is normalized with respect to the ecosystem carrying capacity in state 2 (i.e., 0 ≤ V ≤ 1). The local dynamics are expressed by f1,2(V) as in equation (2) and (3). The weighting (or “kernel”) function, w(r), is here modeled as the difference between two Gaussian functions [e.g., Murray, 1989]
where b1 and b2 express the relative importance of facilitation and competition processes, while d1 and d2 are related to the radii of canopy and root footprints, respectively. The kernel function, w(r), has qualitatively the shape shown in Figure A1a when d1 < d2 and b1 > b2.
Figure A1. (a) Example of kernel function (equation (6)) used to model the spatial dynamics of facilitation and competition. (b) Relation between growth factor, γ, and wave number, k, in solutions of (A4) expressed as linear perturbations of V0 (ζ = 0.4; χ = 2.0; ɛ = 0.25; η = 1.454).
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 When dichotomous Markov noise is used to model the random switching, vegetation dynamics can be expressed by the stochastic integral-differential equation
with ξDM being a dichotomous Markov noise assuming values −1 and 1 with probability P and (1 − P), respectively, and f± (V(x, t)) = (f1 (V(x, t)) ± f2 (V(x, t)))/2.
 Buceta et al.  have shown how systems with fast switching between two processes having the same Swift-Hohenberg spatial coupling but different local dynamics may be able to exhibit pattern formation even though – separately - neither of the two processes is capable of generating patterns. We apply the same approach to the case in which spatial interactions are expressed by the integral term in (A1).
 If the random switching is fast with respect to the relaxation times of the deterministic dynamics, pattern emergence can be investigated replacing the (stochastic) local term in (A3) with its average (equation (4)). In dimensionless form equation (A3) becomes
where ζ and ɛ < 1 represent the relative importance of local/spatial dynamics (see equation (1)), and of competition/facilitation processes, respectively; χ > 1 depends on the ratio between the radii of root system and canopy footprints, while η expresses the relative importance of logistic growth and stress-induced mortality. To simplify the notation, in what follows we drop the header “∼” and indicate with x(x, y) the dimensionless coordinate vector.
 The homogeneous steady states, are obtained as solutions of (A4) for V = V0 = const.
 In particular, we study the linear stability of the homogeneous stable state
by seeking for solutions of (A4) in the form of a sum of V0,with a perturbation term, δV,
where V is the amplitude of the perturbation, γ is its growth factor, k the wave-number vector, i = the imaginary unit, and “·” the scalar product operator. V = V0 is unstable when γ > 0, because any disturbance, δV, of V0 would indefinitely grow with time. To determine the relation between growth factor and wave number in (A7), we insert equation (A7) in (A4), and obtain (after a Taylor expansion for small values of V)
with k = ∣k∣ and W(k) = w(∣x∣)eik · xdx. Figure A1b shows that, for given values of ζ, ɛ, and η, γ is negative both for P = 0 and for P = 1, indicating that the corresponding homogeneous solutions (i.e., V0 = 0 and V0 = 1, respectively) are stable. Thus, no patterns emerge when, in the absence of climate fluctuations (i.e., with no switching), the system evolves towards homogeneous unvegetated (P = 0; Figure A1b, dotted line) or vegetated (P = 1; Figure A1b, dashed line) states. γ may be positive for intermediate values of P, indicating how climate-driven random switching between deterministic dynamics may indeed trigger instability. The instability of the state, V0, is associated with the emergence of spatial patterns when the most unstable mode, kmax, is different from zero (uniform solution), i.e., when W(0) < 0 [see Murray, 1989, p. 488]. kmax can be determined from equation (A8), kmax = . The condition of marginal stability (γmax = γ(kmax) = 0) becomes
with V0 in (A9) being a function of P (equation 10). Because in Figures 2 and A1 the most unstable mode, kmax, is greater than zero (i.e., χ > 1/), the instability (i.e., ζ > ζ*) of the homogeneous stable state, V0, is associated with the emergence of spatial patterns.
 Numerical simulations were carried out to simulate these patterns and to investigate the effects of the bound V = 0 on the stability of the state V0 = 0. To this end, equation (A4) was solved by finite differences (in time) over a square lattice (128 pixels × 128 pixels, with pixels having about the same size as an individual plant canopy), using a simulation length that allowed for 10,000 switches between the stressed and unstressed deterministic dynamics. In these simulations the deterministic vegetation decay was faster than vegetation encroachment and the ratio between the dynamical timescale and the characteristic time of the noise was equal to 25. The numerical results (Figure 1) were –qualitatively - in good agreement with the analytical model (equation (A8)).