The model consists of a system of four coupled differential equations describing the mass balances of soil moisture and three SOC pools. The dynamics of soil moisture and the carbon pools evolve deterministically with perturbations introduced by stochastic rainfall. To address the issue of spatial heterogeneity induced by vegetation structure, we have adapted the approach of Caylor et al.  and introduce three variables describing the vegetation structure at a point: nC, the number of shrub canopies overlapping the point; nR the number of shrub root systems overlapping the point; and G, which is equal to 1 if the point is covered by a perennial grass patch and equal to 0 otherwise. We assume that grass patches do not overlap, and the “shrubs” in the model could also be trees. The model computes unique results for each combination of nR, nC, and G.We take a weighted average of the model results to be indicative of landscape SOC stocks. We assume steady-state conditions to make possible analyses of the effect of patch-scale vegetation structure on SOC and the relative sensitivity of SOC in shrub- and grass-dominated landscapes to vegetation, climate, and soil parameters.Hibbard et al.  modeled the transient dynamics of SOC during shrub encroachment and found only monotonic changes. Though many landscapes affected by woody encroachment may not be in steady state, we would expect this to influence the accuracy of the absolute SOC abundance predicted by the model, but not the pattern of relative differences that we are trying to explain. We describe our temporal and spatial averaging methods in sections (2.4) and (2.6).
2.1. SOC Dynamics
 Following Porporato et al. [2003a], we model SOC as a three-pool system, with coupled differential equations characterizing the dynamics of carbon in fast-decaying litter (Cf), slow-decaying organic matter (Cs), and microbial biomass (Cb) in units of gC m−3, such that
where rr is the fraction of decomposed organic matter that is lost as CO2 (i.e. 1 − rr is the microbial efficiency), rsis the fraction of fast-decaying litter that is stabilized by physical and chemical factors that make it less available to decomposers, andkb is the rate constant for death of microbial biomass (m−3 d−1). ADD is the rate of inputs from vegetation (gC m−2 d−1), which we discuss in section (2.3), and Zr is the active soil depth (m).
 The decomposition rates of organic matter (DECf and DECs; gC m−3 d−1) represent the stochastic component of the equation. We model decomposition of the two substrate pools using first-order rate kinetics with constantskf and ks (m−3 d−1; kf ≫ ks) that represent the decomposition rates in the absence of water stress. This is a simple approach that works well in cases where a physical factor, soil moisture in this system, is the dominant limitation on decomposition [Manzoni and Porporato, 2009]. We calculate the DEC terms as
where W[s(t)] is a dimensionless parameter describing soil moisture limitation of decomposition as a function of relative soil moisture content, s(t) (m3H2O m−3void space). This term represents the stochastic forcing in the SOC system. We incorporate short-term variability in soil moisture into the decomposition term while assuming a constant temperature. Temperature-based respiration models do not perform well in water-limited ecosystems [Reichstein et al., 2003; Bahn et al., 2010], and soil moisture controls the variability of respiration at short time scales [Xu and Baldocchi, 2004] due to the necessity of water for both microbial activity and substrate diffusion [Davidson et al., 2006]. We calculate W[s(t)] using the formulation presented by Cabon et al.  and Gusman and Marino  in which the parameter increases linearly from 0 at a microbial wilting point (sb) to 1 at field capacity (sfc) and decreases hyperbolically up to soil saturation (s = 1), such that
 We determine s with a model of soil moisture dynamics forced by stochastic rainfall, as described in the following section.
2.2. Soil Moisture Dynamics
 Rodriguez-Iturbe et al.  and Laio et al. [2001a] model the daily soil moisture balance of the plant rooting zone with the stochastic differential equation
where n is the soil porosity, Zr is the depth of the active root zone, and s(t) is the relative soil moisture content. Inputs of water from stochastic rainfall [R(t)] are modeled as a marked Poisson process of storms with arrival rate λr (d−1) and exponentially distributed depth with mean α (mm). Inputs are balanced by losses due to canopy interception [I(t)], runoff (Q[s(t); t]), evapotranspiration (ET[s(t)]), and drainage (L[s(t)]).
 Caylor et al.  extend the probabilistic soil moisture modeling framework to heterogeneous landscapes by parameterizing equation (4) for each combination of nC and nR to account for the local effects of shrub canopies and root systems on the soil moisture balance. We modify this approach to include grass. We characterize the influence of the structural differences between grasses and shrubs on aboveground processes but are unable to account for potential differences in rooting depth between shrubs and grasses within the modeling framework. When averaged globally, shrublands have deeper roots than grasslands [Jackson et al., 1996]. However, there is mixed evidence on whether shrubs and grasses growing in the same conditions have different rooting depths [Schenk and Jackson, 2002], especially in the case of fine roots [Bai et al., 2009], which produce most below ground contributions to labile carbon pools. Additionally, the majority of water uptake by woody plants is from surface layers [Kulmatiski et al., 2010]. Since soil moisture and SOC input are the processes of interest in this study, we assume equal active soil depths for grasses and shrubs.
 At points covered by grass and shrub canopies, some rainfall will be intercepted by and evaporated directly from plant canopies without reaching the soil. The depth of rainfall reaching the soil in a given storm is the storm depth minus a characteristic interception value, Δ(nC, G) (mm), or 0 if the storm depth is less than Δ(nC, G). We estimate Δ empirically as
where LAIs and LAIg are the leaf area index (LAI; m2 m−2) beneath a single shrub canopy and a typical grass patch, respectively. The parameter h is the characteristic amount of interception per unit leaf area, which we take to be 1 mm [Aston, 1979]. If the depth of non-intercepted rainfall exceeds available storage in the soil, the excess is converted to surface run-off (Q[s(t); t]).
 Once water has infiltrated the soil, it leaves via drainage and evapotranspiration, which constitute the soil moisture-dependent loss function of the system. FollowingLaio et al. [2001a], we model the rate of gravity-driven drainage as an exponential decay from the saturated hydraulic conductivity of the soil (Ks; mm d−1) at s = 1 to 0 at soil field capacity, independent of nC, nR, and G. Plants reduce bare soil evaporation by shading and determine the spatial distribution of water loss via root uptake, so we model the evaporation and transpiration components separately, following Caylor et al. . The interception of energy and loss of water through leaves, which we will refer to as transpiration, may be spatially decoupled from the absorption of soil moisture by roots, which we will refer to as uptake. In the original model formulation, potential evapotranspiration (PET; mm d−1) provided an upper bound on the rate of soil moisture loss via evapotranspiration. To provide upper bounds on the processes of shrub transpiration, grass transpiration, and bare soil evaporation we multiply PET, which we take to be a constant, long-term value, by the fractions of incoming energy absorbed by shrub canopies, grass canopies, and the soil surface. We model canopy energy interception using Beer's law and assume shrub canopies are taller than grass canopies. Thus, the fractions of incident radiation at a point contributing to potential shrub transpiration (Φs), grass transpiration (Φg), and evaporation (ΦE) are given by
where k is the extinction coefficient of evaporative demand, which we take to be 0.35 [Brutsaert, 1982]. We calculate these terms recursively to represent the vertical structure of the ecosystem.
 Evaporation and uptake by grasses, which we assume do not have roots extending beyond their canopies, extract water from areas of soil directly below the surfaces absorbing the energy that drives these processes. Therefore, the maximum rates of soil moisture loss from evaporation and grass uptake (φe and φg; mm d−1) at a point are a function of the energy interception pattern at that point:
The root system of a shrub may extend well beyond its canopy edge, so shrubs can extract water from the soil at points not overlapped by their canopies. By doing this, shrubs spatially decouple transpiration and uptake. In some landscapes grasses also have laterally extensive root systems [Casper et al., 2003], in which case the formulations for the belowground structure and function of shrub roots could be applied to grasses. To estimate the maximum rate of uptake by shrubs for a point with nR overlapping shrub root systems, it is necessary to estimate the energy absorbed by the canopy associated with each root system. Some canopies will absorb less energy per unit area because they overlap with others, but it would be computationally intensive to model them individually. We simplify the system by calculating the fraction of incident energy absorbed by a shrub canopy with an average amount of overlap with other canopies, , following [Caylor et al., 2006]. This is given by the summation of the fraction of energy absorbed by a canopy at a point with each nCvalue, weighted by the fraction of the canopy-covered landscape (nC ≥ 1) with that nC value, such that
where is the probability that a point on the landscape is overlapped by nC canopies. Its derivation is described in section (2.5). We update an earlier form of the model [Caylor et al., 2006] by multiplying by the ratio of the average canopy area (AC, m2) to the average root system area (AR, m2), because energy absorbed by the canopy drives uptake through the entire root system. This results in a lower estimate of potential uptake at a point. The potential shrub uptake at a point (mm d−1) is given by
 Evaporation and uptake may also be limited by soil moisture. The bare soil evaporation rate ranges from zero at the soil hygroscopic point to its maximum rate (φe) at the soil field capacity and above. We model the rate with a linear increase between these two points (following Caylor et al. ). We model uptake with a similar linear model ranging from zero at the plant wilting point, sw, to a maximum at the point of incipient stomatal closure, s∗ [following Laio et al., 2001a]. This leads to a unique soil moisture loss function for each combination of nC, nR, and G.
2.3. Productivity and SOC Input
 The SOC model also requires that vegetation productivity be estimated so the addition term, ADD (gC m−2 d−1), can be determined. Water is the primary limitation on photosynthesis in the ecosystems considered in this study, so we model carbon assimilation as the product of transpiration (mm) occurring over the course of a day and measured water use efficiency values (gC mm−1 H2O; measured values from Scholes and Walker ). In doing this, we assume that water use efficiency does not decrease when plants are under water stress. However, this simplifying assumption could easily be removed if information on the relationship between water use efficiency and soil moisture were available. We calculate grass and shrub net primary productivity (NPP) separately, such that
where WUEg and WUEs are the water use efficiency values for grasses and shrubs. Measured values of NPPg and NPPs could be substituted if available. Growth and mortality create a time lag between transpiration and SOC input. We assume that these processes are in equilibrium, so SOC input rate can be modeled using the equations in (10). This assumption allows us to derive a steady-state solution to the SOC model, which we describe insection (2.4). We present a comparison of our simple empirical representation of productivity with more complex, mechanistic physiology models in Appendix B. The simple model captures the trend of increasing productivity with increasing rainfall and vegetation density, while allowing us to develop a tractable model that accounts for patch-scale vegetation heterogeneity.
 One of our goals is to account for the local effects of vegetation patches on both inputs to and decomposition of SOC. Therefore, it is necessary to calculate ADD, the SOC addition rate defined in equation (1), as a function of nC, nR, and G. Grasses and shrubs affect the SOC input rate differently, so we introduce separate addition terms that sum to ADD. The aboveground and belowground biomass of grasses contributes to SOC over the same area, but shrub root systems and canopies, which respectively determine the spatial distribution of inputs from belowground and aboveground biomass cover different areas. Therefore, it is necessary to divide NPPs into aboveground and belowground components. We make the simplifying assumption that aboveground/belowground plant allocation is proportional to the ratio of average canopy area to average root system area, so the fraction of biomass allocated aboveground is equal to AC/AR.
 We then determine the addition term at a point by multiplying NPPg and NPPs by vegetation structure weighting terms, which are a function of nC, nR, and G:
The vegetation structure weighting terms are determined by dividing nC, nR, and G by their expected values, such that
for grasses and
for shrubs. Expected values are denoted by the operator E[⋅] and are derived from the probability distributions of nC, nR, and G, which we define in section (2.6).
 Abiotic processes, primarily photodegradation, have been identified as having significant effects on surface litter dynamics in dryland ecosystems [Austin and Vivanco, 2006; Rutledge et al., 2010]. A term could be added to reduce the addition rate to account for this. We did not do so, because reported rates of mass loss from photodegradation are less than 16 g m−2 yr−1 [Austin, 2011], which is small compared to NPP and thus unlikely to significantly reduce soil carbon inputs.
2.4. Temporal Averaging
 The SOC model given by equation (1) represents a system that exhibits deterministic behavior with perturbations driven by stochastic rainfall, as described in section 2.1. Previous stochastic models of hydrologically-driven soil biogeochemistry have relied on a rainfall simulation to force the model [Porporato et al., 2003a; Wang et al., 2009]. The simulation approach, which models soil moisture dynamics with a daily time step, is useful in drylands, where decomposition rates respond quickly to rainfall events [Xu and Baldocchi, 2004; Jarvis et al., 2007]. Such a model is impractical in our case, because unique simulations would need to be run for each combination of nC, nR, and G.To address this, we introduce a steady-state version of the SOC model. For simplicity, we describe the three state variables in the SOC system, fast-decaying litter (Cf), stabilized organic matter (Cs), and microbial biomass (Cb), as a vector:
The temporal dynamics of the SOC pools, driven by stochastic soil moisture, for a given combination of nC, nR, and G can be described as a function of xt. It is derived from equation (1) by substituting the expressions relating decomposition rates (DECf and DECs) and SOC input rate (ADD) to soil moisture, which are described by equations (3) and (11), respectively, giving
The terms relating transpiration (Tg[s(t)] and Ts[s(t)]) and decomposition rate (W[s(t)]) to soil moisture control the variability of xt in time.
 We simplify equation (15) by dividing it into two components: (1) a deterministic expression, given by H(xt), describing the behavior of the system under average conditions, and (2) an expression, given by h(xt), describing the variability introduced to the system by the stochastic nature of soil moisture. The deterministic component is given by evaluating dxt/dt for temporal averages of Tg[s(t)], Ts[s(t)], and W[s(t)]:
where 〈 ⋅ 〉 indicates a temporal average. To determine the temporal averages in equation (16), we follow the approach presented by Rodriguez-Iturbe et al.  and Laio et al. [2001a] and modified by Caylor et al. . We solve the stochastic differential equation describing the soil moisture balance (equation (4)) for the steady-state probability density function of soil moisture in time,pS(s), for each combination of nC, nR, and G. The temporal averages of transpiration rate and the decomposition limitation term are given by their expected values:
 The deterministic system described by equation (16) is an adequate representation of soil carbon additions, because growth and mortality decouple this process from the temporal variability of soil moisture. However, it is important to account for the influence of soil moisture variability on decomposition rate through the limitation term W[s(t)]. To do so, we first define the system h(xt) as all components of equation (15) that multiply W[s(t)]:
We then approximate W[s(t)] as the sum of its temporal average and zero-mean, normally distributed perturbations (white noise), such that
where ξ is a normally distributed random variable with mean 0 and variance 1, and σW2 is the variance of W[s(t)]. This approximation of the variability of decomposition conditions as white noise assumes no temporal autocorrelation in the values of W[s(t)], which is physically unrealistic. However, other applications of multiplicative noise models in environmental systems have made the same assumption about temporally autocorrelated variables such as wind speed [Brubaker and Entekhabi, 1996] and precipitation [Rodriguez-Iturbe et al., 1991] without introducing significant model error. We present a comparison of our model results with a simulation-based form that does not make this assumption inAppendix A.
 The variance of W[s(t)] in equation (19) is given by
The expected value of W[s(t)] is its temporal average, given by equation (17). Following the approach used in equation (17), we derive the expected value of W[s(t)]2 using the probability density function of s, so that the full expression for the variance of W[s(t)] is
 Combining the deterministic and stochastic components leads to a single expression describing the dynamics of the system:
where dwt is the infinitesimal of a Wiener process. We then solve equation (22)numerically to determine the steady-state sizes of the SOC pools for a point on the landscape covered by a given combination ofnC, nR, and G. In our analysis of the interactions between vegetation structure, soil moisture, and soil carbon, it is useful to determine temporal averages of all components of the soil moisture balance as a function of nC, nR, and G. The temporal means (mm d−1) of the soil moisture-dependant processes of leakage, evaporation, and transpiration are calculated in the same way as mean soil moisture, as described inequation (17). We calculate the average daily canopy interception following Laio, et al. [2001a]:
In the following sections, we describe the spatial averaging methods used to determine landscape averages of the SOC pools and components of the soil moisture balance.
2.5. Vegetation Structure
 Following the approach presented by Caylor et al. , we use a probabilistic description of vegetation structure based on a two-dimensional, marked Poisson point process. Using a random point process neglects any clustering or inhibition in the location of shrubs but enables us to determine analytically the fraction of the landscape covered by each combination ofnC, nR, and G. We model shrub locations as a Poisson process of rate λs (individuals m−2) with circular canopies having radii drawn from an exponential distribution of mean μs (m). We assign the ratio of root radius to canopy radius of an individual, at, a constant value of 2, following Caylor et al. . Because we assume this description of shrub location and size, the probability that a point on the landscape is covered by a number of root systems, nR, is given by a Poisson distribution
where 0! is, by definition, equal to 1. The probability that a point is overlapped by a number of canopies, nC, is given by
Because canopies and root systems are connected, nC and nR are not independent. We address this by calculating the conditional probability that a point overlapped by nR root systems is overlapped by a number of canopies, nC, given by
Equations (24) and (26) can be used to find the joint distribution of nR and nC and the expected fraction of the landscape covered by any combination of nR and nC values such that nR ≥ nC.
 We consider the shading effects of shrub canopies when calculating the probability that Gis equal to one for a point, because grass productivity has been shown to be inversely related to shrub LAI in mixed shrub-grass systems [Caylor et al., 2004]. As with our calculation of canopy energy interception, we estimate shading effects using Beer's law. G is determined so that its expected value equals measured grass fractional cover, cg. We calculate the probability that G = 1 for a point overlapped by nC canopies as
Note that the term in brackets is the expected value of on the landscape.
2.6. Spatial Averaging
 As outlined in section (2.4), we calculate unique, steady state values of the SOC pools (〈Cf(nC, nR, G)〉, 〈Cs(nC, nR, G)〉, and 〈Cb(nC, nR, G)〉) and components of the soil moisture balance (〈I(nC, G)〉, 〈E(nC, nR, G)〉, 〈Tg(nC, nR, G)〉, 〈Ts(nC, nR, G)〉, and 〈L(nC, nR, G)〉) for each combination of nC, nR, and G. To find a landscape average of any of these values, we take an average over the combinations of nC, nR, and G values weighted by the fraction of the landscape covered by each combination. Utilizing the conditional probability distributions presented in the previous section, this is given by
where the overbar indicates a spatial average.
 The turnover time for the stabilized organic matter pool is usually larger than the timescales on which changes in vegetation structure occur. Therefore, if we wish to calculate the total SOC abundance for a point on the landscape, we add point-specific values forCf and Cb to the landscape average of Cs abundance, such that
where we multiply by the active soil depth, Zr, to express in units of gC m−2. As with the individual pools, we determine a weighted landscape average of SOC abundance for intersite comparisons, which is given by