Abrupt transitions between large-scale grassland and desert in arid and semiarid regions have been observed in nature and reproduced by modeling studies. Observations also show the existence of nonuniform fine-scale vegetation patterns along the transition zone. This paper attempts to better understand these observations from two very different spatial scales. By explicitly introducing horizontal interaction terms into our previous dynamical grassland model, vegetation patterns with high diversities are found in the transition zone, and the system possesses an infinite number of equilibrium states in response to a given climatic forcing. The transition can be elucidated in two ways. In terms of the vegetation formations, the ecosystem undergoes the transition from uniform grassland to regular and irregular vegetation patterns, and then to pure desert as the moisture index (i.e., the ratio of precipitation over potential evaporation) decreases. In terms of biomass, the transition from grassland to desert goes through a narrow range of moisture index under which grassland is most fragile, as indicated by erratic vegetation patterns and large variation of average biomass. The existence of this range, however, has not been reported in previous modeling studies, and still needs to be validated using observational data.
 Dramatic changes in the terrestrial ecosystems, e.g., desertification or deforestation, have significant impacts on human activities and hence are of great concern [iLEAPS, 2005]. Land ecosystems are primarily determined by climatic and environmental conditions, and they also feed back to the atmosphere [e.g., Cox et al., 2000]. The gradual changes in the ecosystem formation as well as the shift of the boundaries of large ecosystems are observed to be driven by climate change at long timescales (e.g., from hundreds to thousands of years). However, abrupt transitions of ecosystems can also happen at a much shorter timescale even in a region where the climate change is smooth and slow, and this, in the long run, can lead to significant changes in climate through atmosphere-vegetation interactions. This phenomenon is called “catastrophic shifts” in the literature [Scheffer et al., 2001; Scheffer and Carpenter, 2003]. How to explain and further predict such irreversible changes is an important task in the study of global change.
 The changes of distributions and formations of vegetation as well as their impacts on climate have been studied extensively. With the help of multidisciplinary theoretical studies, field measurements, remote sensing, and empirical evidence, numerous models have been developed to reasonably simulate the land-atmosphere system at different spatial and temporal scales. At the global level, general circulation models (GCMs) with the prescribed plant ecosystems can reasonably simulate the current climate regimes as well as the major energy and hydrological cycles of the climate system [Bonan et al., 2002; Zeng et al., 2002]. When coupled with a dynamical global vegetation model (DGVM), GCMs or land surface models (LSMs) can reproduce the regimes of major plant functional types [Kucharik et al., 2000; Cramer et al., 2001; Bonan et al., 2003; Sitch et al., 2003], and even predict a remarkable collapse of the Amazon forest in response to climate changes driven by an assumed doubling of CO2 concentration in the next 50 years [Cox et al., 2000; Huntingford et al., 2000].
 While such comprehensive models are very powerful in providing the general distributions of the climate and ecosystem components in the whole system, the conceptual and intermediate level models as well as the analytically tractable simple models are more flexible in capturing the mechanisms of specific events at the regional scale with clear physical and ecological meanings. Results from simpler models might also be used to better understand and further improve comprehensive models. Many such simpler models indeed focus on the coexistences and transitions between plant ecosystems [Svirezhev and von Bloh, 1997, 1998; Oyama and Nobre, 2003; Rodriguez-Iturbe et al., 1999a, 1999b], especially the sharp transitions of grassland to pure desert as driven by the variation of precipitation in the arid and semiarid regions [Brovkin et al., 1998; Claussen et al., 1999; Zeng and Neelin, 2000; Wang, 2004; Wang and Eltahir, 2000a, 2000b; Zeng and Zeng, 1996; Zeng et al., 2004, 2005b]. The essential concept in these models is that multiple stable equilibrium states coexist in the system as a result of the self-organization of the positive and negative interactions within the system. Under certain conditions, the effect of small changes in climate condition can be amplified by vegetation-environment interactions (in the land only system model), or the subtle variations of climate could be strongly amplified by atmosphere-vegetation feedbacks (in the land-atmosphere coupled system), and then trigger an abrupt switch of ecosystem from one state to another.
 The above models assume a large-scale homogenous ecosystems (i.e., well-mixed from 1 km to 100 km), and can be called single column models. Observations also show that spatially nonuniform vegetation patterns at much smaller scales (from decimeters to 10 m) exist along the transition zone in the arid and semiarid regions. The vegetation patterns vary from irregular spots and dashes to bands and labyrinths, depending on the precipitation. The spatial heterogeneity is partly considered in some land models in an aggregated way [e.g., Zeng et al., 2005a]. The small-scale horizontal heterogeneity has also been explicitly investigated in ecological models by introducing spatial horizontal interactions, and the occurrence of the pattern and the diversity of its formation have been reproduced [von Hardenberg et al., 2001; Rietkerk et al., 2002, 2004; Lejeune et al., 2004; Meron et al., 2004; van de Koppel and Rietkerk, 2004] by using the concepts of dissipative structure [Prigogine, 1969] and hysteresis. A major conclusion of these models is that, over arid and semiarid regions, water is more concentrated into patches of vegetation due to spatial interactions so that patterned vegetation can extend to arid regimes where the homogeneous vegetation state cannot be observed.
 Some questions arise in relating the fine-scale patterns to the large-scale behaviors: (1) what is the variability (and diversity) of pattern formation? (2) how is the fine-scale heterogeneity relevant to the mean variables in a large enough area (like a global model grid box)? and (3) how does the fine-scale heterogeneity influence the system behaviors such as transition and resilience? The purpose of this study is to address these questions by introducing horizontal interaction terms into our previous single column model of grassland [Zeng et al., 2004, 2005b]. Section 2 describes the model formulation processes and parameters. Section 3 discusses the condition under which the spatial homogenous solution (uniform grassland) becomes unstable, and section 4 provides the mathematical analyses of the spatially periodic patterns. Based on sections 3 and 4, sections 5 and 6 investigate the more general vegetation pattern – the spatially aperiodic solution, and show the existence of erratic vegetation patterns with high diversity of averaged biomass. These patterns are most fragile to perturbations but have not been reported in the literature with similar models. Section 7 provides summary and further discussions.
2. Model Description
 Consider a grassland ecosystem model of three state variables, i.e., the mass density of living leaves x (kg m−2), the available soil wetness y (in the rooting zone) (kg m−2, or mm), and the mass density of wilted and dead leaves z (kg m−2). The governing equations of the system can be written as
where terms G, D, and C are the growth (photosynthesis subtracts plant maintenance and growth respiration), wilting, and consumption (grazing) of the living leaves, Gz, Dz, and Cz are the accumulation, decomposition, and consumption of the wilted leaves, Ev is soil evaporation, Et is vegetation transpiration, and R is runoff or drainage. These terms describe the vertical processes within the system. Their formulations can be found in the work of Zeng et al. [2004, 2005b]. For simplicity, here the consumption terms C and Cz are assumed to be zero, i.e., the undisturbed situation. Horizontal interaction terms (Hx, Hy, and Hz) will be discussed later.
 The formulations of these terms are closer to those in the widely used climate system models for global change study [e.g., Dickinson et al., 1993, 1998; Sellers et al., 1986, 1992; Dai et al., 2003; Oleson et al., 2004]. The major differences consist of: (a) in our model, the dry wilted leaves are not considered as the litters mixed with soil but rather those accumulated on the ground surface; (b) we separately derive the wilting term D independently from the photosynthesis (growth) term G; and (c) we treat the soil wetness as a single variable for simplicity rather than a layered one.
 This ecosystem model is driven by climate conditions. In arid and semiarid regions, the sparse precipitation is the most important climatic forcing in influencing the growth of vegetation. Hence, precipitation (denoted as P) is taken as the only input term in our model, while the effects of other conditions (such as air temperature, humidity and sunlight) are implicitly considered as some adjustable parameters in the vertical process terms. Here all parameters including P are assumed to be uniformly distributed in the horizontal direction, and also independent of the state variables (i.e., the feedback from vegetation to the climate is neglected).
 For the convenience of mathematical analysis, the state variables are then scaled by proper characteristic values x*, y*, and z* so that the dimensionless variables x′ = x/x*, y′ = y/y*, and z′ = z/z* are usually within the range of 0 to 2. Also, terms G, D, Gz, and Dz are scaled by the maximum growth rate of the living grass, α*, and terms Ev, Et, and R are scaled by the potential evaporation e*. Accordingly, precipitation P is replaced by a dimensionless index μ = P/e* which is called the “moisture index”.
 Neglecting the horizontal interaction terms Hx, Hy, and Hz, the spatially uniform solutions of equations (1)–(3) have been thoroughly investigated in our previous studies [Zeng et al., 2004, 2005b] and validated with the observation over well-mixed grassland in Inner Mongolia [Zeng et al., 2005b]. Note that horizontal heterogeneity is implicitly included in the model by dividing the soil surface into vegetated (the part covered by the living leaves) and nonvegetated areas, and the fraction of vegetation coverage is a function of the living biomass. This reflects the fact that there are always gaps within the foliage and spaces between individual plants. Processes (e.g., evaporation, transpiration, and runoff) over the vegetated and nonvegetated areas are calculated separately, and the use of a single value of soil water over the column implies an instantaneous horizontal water exchange between these two areas. In the arid and semiarid regions, water supply is limited, and observations show that water is more concentrated into vegetated areas due to spatial interactions. This effect can be largely represented in our single column model by the changes of some model parameters, e.g., the increase of the coefficient in the biomass growth dependence on soil water [Zeng et al., 2005a, 2005b].
 To explicitly investigate the spatial heterogeneity within the vegetated areas and how the vegetation pattern formations influence the abrupt transition, spatial horizontal interactions should be explicitly included in the model. Usually, the spatial interactions are expressed by the dissipation, facilitation and competition [Prigogine, 1969; Rietkerk et al., 2002; Lejeune et al., 2004]. Similar to von Hardenberg et al. , here we take
where ∇2 = ∂2/∂ξ2 + ∂2/∂ζ2 is the two-dimensional Laplacian (with ξ and ζ denoting the horizontal coordinates), δx describes the ability of spatial spreading of living grasses, δy reflects the horizontal conductivity of soil water, δz represents the horizontal mixing of wilted grass due to environmental factors (e.g., wind) or rearrangement by activities of human beings or herbivores, and η describes the effect of soil water suction by vegetation roots. Here, we assume a uniform soil surface, i.e., with no variation in elevation or soil properties.
 Similar to the vertical process terms, a characteristic scale of the horizontal process, Lξ, is introduced, and the parameters of horizontal interactions are rewritten in dimensionless forms as,
The value of Lξ is chosen so that either δ′x or δ′y is fixed.
 Without causing confusion, we omit the symbol of apostrophe on the variables and parameters, and all terms without asterisk are dimensionless hereafter. The parameters of vertical interactions are kept as in the single column model [Zeng et al., 2005b]. The dimensionless horizontal interaction parameters are taken as: δx = 1, δy = 50, δz = 4, and η = 0.7. These parameters, except for δz, are chosen according to the values from von Hardenberg et al.  along with the consideration of the different ratios of equilibrium values of x and y between the two models as well as some realistic constraints. For instance, the choice of parameters δz > δx reflects that wilted grass is easier to move horizontally due to environmental factors (e.g., wind) or human activities. Our sensitivity tests show that the essential characteristics of the system (e.g., the formation and diversity of vegetation pattern, the existence of erratic versus regular patterns) are robust with respect to different choices of parameters, although the exact parameter regimes where different patterns persist depend on the values of these parameters.
3. Transition and Instability of the Spatially Uniform Solutions
 Our previous single column model corresponds to the special case of Hx = Hy = Hz = 0 and represents the spatially uniform solutions. Multiple equilibrium states are found in this model [Zeng et al., 2004, 2005b]. The state of bare soil (with no biomass) is always a solution, but it is stable only when μ is smaller than a threshold, μ2. Stable grassland can be formed when μ is larger than another threshold μ1 < μ2 (see Figure 1, curves UIa and UIb). There also exists an unstable grassland state when μ1 < μ < μ2 (Figure 1, curve UII), but this state is not expected in nature and is shown here just for reference. The hysteresis diagram in Figure 1 implies that abrupt transition of ecosystem can occur in two ways, i.e., grassland collapses into desert as μ decreases across μ1, and desert converts to grassland as μ increases across μ2.
 For the general situation with horizontal interactions, the stability of the above multiple equilibrium states might change when the system is disturbed by spatially inhomogeneous perturbations. The general method in stability analysis is to solve the eigenvalue problem of the relevant linearized equations. Suppose there is a spatial harmonic perturbation (with wave number m) superimposed on the uniform solution,
where = (ξ, ζ) is the spatial vector, = (mξ, mζ) is the wave vector and m = ∣∣, and (Ex, Ey, Ez) is the amplitude (eigenvector). For equations (1)–(3), there are three eigenvalues for a given m, and the maximum value of real parts of these eigenvalues can be denoted as λm. If λm is positive, the uniform state is unstable because the perturbation grows with time based on (7), and eventually a spatial pattern is formed. A larger λm implies that the system will diverge more quickly from the spatially uniform equilibrium state. Figure 2 shows that, the uniform grassland solution is unstable when disturbed by perturbations with wave number m1 < m < m2, where m1 and m2 depend on μ and are calculated as the m values when the λm curve crosses the line of λm = 0. At the bifurcation point of μ = μ1, the equilibrium state has λm = 0 with the homogenous perturbation, hence m1 = 0 (in Figure 2b), i.e., this state is unstable for any perturbations with an arbitrarily small wave number. As μ increases, m1 increases and m2 − m1 decreases (Figure 2b). The critical value of μ, below which the uniform grassland is unstable, is determined when m1 = m2 and is denoted as μ3 (Figures 1 and 2b) hereafter. On the other hand, the most unstable wave number (denoted as mmax) under the given μ, i.e., the wave number corresponding to the maximum λm, is almost a constant, approximately equal to 0.19 (Figure 2a).
 The unstable uniform solutions will eventually diverge to spatially periodic or aperiodic patterns, which will be discussed in sections 4 and 5, respectively. The periodic solutions are easier to obtain mathematically; for instance, the stability of the solution in the parameter space can be easily analyzed. In contrast, the aperiodic solutions represent the more general and realistic cases, but are more complicated. Although the characteristics of aperiodic patterns are difficult to analyze directly, it could roughly be inferred from the analysis of the periodic solutions.
4. Spatially Periodic Stationary Patterns
 As a general method in mathematical analysis, the spatially periodic patterns can easily be obtained by disturbing the unstable uniform solution with spatial harmonic perturbations, or by integrating equations (1)–(3) with periodic initial conditions. Note that adding random perturbations to the spatially homogeneous state or integrating equations (1)–(3) with random initial conditions would lead to the more general aperiodic patterns (to be discussed in section 5) rather than the special cases of periodic patterns in this section. Because of the nonlinearity of the system, the profile shape and even the wave number of the stable periodic solution is usually different from the initial perturbations. Furthermore, as will be demonstrated later, the range of μ where a stable periodic solution exists is different from the range of μ1 < μ < μ3 where the uniform solutions are unstable (Figures 1 and 2).
 The typical two-dimensional (2-D) periodic patterns are found to be parallel straight stripes and hexagonal lattices of spots or gaps (Figure 3), consistent with Lejeune et al. . We first analyze the simplest case in this system, i.e., the parallel straight stripes. The profiles along the transect perpendicular to the stripe can be described by a 1-D periodic solution. Here, it is more convenient to use the wavelength (i.e., 2π/m, corresponding to the distance between adjacent vegetation patches) to describe the periodic behavior. Our numerical simulations show that, in the region of μ1 < μ < μ3, a stable periodic solution occurs only with wavelength L′1 > L > L′2, where L′1 and L′2 depend on μ, consistent with the analysis in section 3. Hence, there are an infinite number of, possibly continuous, stable equilibrium states under a given μ (e.g., Figure 4). In the wetter region with a relatively high μ (e.g., μ = 0.32 in Figure 4a), quasi-harmonic solutions with a single peak within a period are found only around L ∼ 2π/mmax. As L increases, the solution will first have multiple peaks within a period, and then becomes unstable (Figure 4a). The spatially averaged living biomass under different wave length L is then calculated and denoted as 〈x〉. It varies little in comparison with that of the uniform solution (Figure 5, curve 1, calculated from Figure 4a). Stable stripe patterns can also occur in a narrow region of μ > μ3 (with μ3 shown in Figure 1), and their characteristics are very similar to those in the region of μ ∼ μ3.
 As μ decreases toward the drier region of μ ∼ μ1, the range of (L′2 − L′1) for a stable periodic solution increases. While the minimum value of 〈x〉 is still comparable with that of the uniform solution, the maximum value of 〈x〉 becomes larger. As μ further decreases into the region of μ < μ1 where no uniform grassland solution exists, stable periodic solution still occurs with wavelength L′2 < L < ∞ (i.e., it can be arbitrarily large), and always has one peak within a spatial period (Figure 4b). The averaged biomass 〈x〉 reaches its maximum value near the wavelength 2π/mmax, and then monotonously decreases to 0 as L approaches infinity (Figure 5, curves 3 and 4). As μ further decreases, no stable periodic stripe solution exists for any wavelength.
 The above results indicate that in a nonlinear system, multiple equilibrium solutions can exist beyond the region of linear instability (i.e., μ1 < μ < μ3). The maximum and minimum values of 〈x〉 and 〈y〉 as a function of μ are then calculated and summarized in Figure 6 as curves Smax and Smin.
 The hexagonal spot lattices can be considered as a composition of three sets of parallel stripes with the angle of any two sets to be 2π/3. Curves Dmax and Dmin in Figure 6 show the maximum and minimum values of 〈x〉 and 〈y〉 of the hexagonal spot lattices as a function of μ. The relations between the profile and 〈x〉 with the wavelength are also found to be similar to the periodic stripe solutions in Figures 4 and 5 except that: (1) the stable hexagonal lattice solutions exist in a relatively drier region than the stripe solutions, with the upper bound smaller than μ3, and the lower bound (denoted as μ0) extends to more arid areas (see the Dmax curve in Figure 6), and (2) the critical value of μ for having stable solution of L → ∞ is smaller than μ1, and this critical value is then denoted as μ1s (see the Dmin curve in Figure 6). This is because the vegetation in the spots can uptake water from the surrounding bare areas in all directions, while the vegetation in the stripes can do so only in the direction perpendicular to the stripe, hence stable spots can extend to more arid regions than stripes [von Hardenberg et al., 2001].
 The hexagonal gap lattices can be considered as the opposite case of spots, and appear in the wetter regions. Different from the spot solutions, however, 〈x〉 (or 〈y〉) changes little with the wavelength, i.e., the difference between 〈x〉max and 〈x〉min (or between 〈y〉max and 〈y〉min) is very small (Figure 6, curves Gmax and Gmin).
5. Spatially Aperiodic Solutions: Diversity of Vegetation Patterns
 The general vegetation patterns are aperiodic. They can be obtained by integrating equations (1)–(3) with random initial conditions, or by adding large enough random perturbations to the state variables of the periodic solutions. When the moisture index (μ) is time invariant, our numerical simulations show that the system always approaches to a stationary state, and no temporal chaotic or oscillatory behaviors are observed.
 Mathematically, the periodic solutions from section 4 form the base functions of the solution space, and the general patterns can be considered as a composition (i.e., Fourier integral) of these bases. Because different basis has different stability (i.e., the strength of perturbation it can resist, and the speed of recovery from perturbation), its possibility to become a real pattern is also different. Roughly speaking, the range of μ where spots, stripes, and gaps can occur in the general patterns, as well as the minimum and maximum distances between adjacent vegetation patches, should be within the ranges determined by the corresponding periodic solutions (e.g., see Figure 6).
 In the region of μ > μ1s, different pattern types (i.e., spots, stripes, and gaps) can coexist, and the fraction of areas where a pattern type occurs in a solution can be different. For example, Figure 7b shows the coexistence of spots and stripes [patterns (b3) and (b4)], and stripes and gaps [patterns (b5) and (b6)]. However, the distances between neighboring vegetation patches also change slightly so that 〈x〉 under a given μ (but with different patterns) are within a narrow range (Figure 7a), consistent with the results from the periodic solutions (i.e., a smaller range of 〈x〉 for μ > μ1s than for μ < μ1s in Figure 6a).
 In the dry region of μ < μ1s, the pattern mostly consists of irregular and erratic distributions of spots that are very sensitive to the initial conditions, and the spacing can be very large [Figure 7, patterns (b1) and (b2)]. As a result, 〈x〉 under a given μ can be very different, varying from 0 to near Dmax (Figure 7a). This can be understood from the perspective of soil water availability. Once a grass patch is established, the maximum soil water available is μπr2 in the patch's neighborhood with the radius r also representing the average length of its roots. Roughly speaking, r is as large as, or several times larger than, the vegetation patch size, and in the model a larger r can be represented by a larger η in equation (5). Since the average distance between two adjacent patches is larger than r by an order of magnitude [e.g., see Figure 7, patterns (b1) and (b2)], the two patches are almost independent of each other in the competition for limited water resources due to small horizontal conductivities. Therefore, the irregular patterns of grass patches are randomly distributed and determined by the initial condition.
6. Impact of Pattern Diversity on Grassland Maintenance
 In our previous study using a single column model, we have addressed the practical issue of how the grassland maintenance in the arid and semiarid regions is influenced by the coexistence of the two stable equilibrium states, i.e., a uniform grassland and a pure desert. In the region of μ1 < μ < μ2, an equilibrium state can recover from a small perturbation, but will be converted to another state under a large enough one [Zeng et al., 2005b].
 Similarly, we may ask the question of how vegetation pattern responds to large perturbations (e.g., significant decrease or even total removal of biomass in an area due to fire, incidental disease or grazing)? Our numerical results suggest that, in the region of μ < μ1s (slightly less than 0.26 from Figure 6), the ecosystem is more sensitive to perturbations. Figure 8a shows an example of μ = 0.25. Some patches are removed (the light gray areas) from a stable spots pattern. The remaining spots can expand and shift to the vacant space, and the total biomass begins to recover from the perturbation [Figure 8, pattern (a2)]. Some partially removed spots can have the chance to regrow [e.g., the three spots near the bottom-left corner in panels (a1) and (a2)], but the totally removed spots cannot grow back, and the total number of spots remains the same as that right after the perturbation (i.e., after the vegetation patches are removed). The averaged biomass of the new pattern is 0.205, higher than the value right after the perturbation (0.189), but significantly smaller than the value before perturbation (0.248) (see Table 1). Our computations also show that further removal will eventually damage the ecosystem and lead to the desert state.
Table 1. Values of Spatially Averaged x, y, and z of the Original Stable Solution (T0), Right After Perturbation (T1), and Recovered Solution (T2) for μ = 0.25 and 0.26 (as in Figure 8) as Well as 0.30
μ = 0.25
μ = 0.26
μ = 0.30
 In the region of μ > μ1s, however, the response of vegetation patterns to perturbations is quite different. For comparison, a stable pattern of μ = 0.26 is generated by using the undisturbed stable pattern of μ = 0.25 as the initial condition, and undergoes the same perturbation [Figure 8, pattern (b1)]. Figure 8, pattern (b2) shows that the remaining spots can also expand bigger and shift as in Figure 8, pattern (a2), and the spots near large vacant space can split and send newborn spots to the space. The total number of spots is larger than the number right after the perturbation, although it is still smaller than the original one. The values of averaged biomass are 0.345 before perturbation, 0.262 right after perturbation, and 0.320 for the new pattern (Table 1).
 For the wet region with a relatively higher μ, the system can more easily recover from perturbations, and the averaged biomass in the original and recovered solutions is similar (Table 1).
 The above experiments show how the ecosystem responds to the perturbation of partial removal of vegetation. Another type of perturbation can be the small random increase in biomass, especially over the bare area (e.g., through seed dispersal). However, because the bare ground state is stable, a small amount of biomass surrounded by a large bare area cannot survive in general, as confirmed by our modeling tests. These results imply that the recolonization of grass over the bare area in such an arid region is mostly through expansion or tillering from existing plants [e.g., Figure 8, pattern (b2)], or implanting new plants, rather than seed dispersal.
 Hence, the maintenance of grassland in the region of μ > μ1s is more or less similar to the result in the single column model. The system can recover if biomass in part of the region is removed, although the spatial pattern may change slightly. Desertification happens only if the grassland is under a large perturbation to the entire area under this condition. On the other hand, grassland in the region of μ < μ1s is more fragile, the loss of a whole vegetation patch (spot) cannot recover, and over-grazing can result in the gradual decrease of vegetated patch numbers and eventually lead to the stable state of desert.
7. Conclusions and Discussion
 Our previous dynamical grassland model [Zeng et al., 2004, 2005b] investigates the vegetation-soil water interaction within a single vertical column and is able to simulate the abrupt transition of ecosystem between uniform grassland and desert. In this paper, the fine-scale horizontal interaction terms are added into the model to address the three questions in section 1 in relating the fine-scale patterns to the large-scale behaviors. Results show that vegetation patterns occur in the transition zone and exhibit high diversities in terms of both pattern formation and the spatially averaged biomass. (1) In terms of spatial vegetation formation (microscale), the ecosystem undergoes the transition from uniform grassland to regular and irregular vegetation patterns, and then to pure desert, as the moisture index μ decreases. An infinite number of equilibrium states coexist in the transition zone. (2) In terms of spatially averaged biomass (macroscale), the transition zone from grassland to desert consists of a narrow region (μ0 < μ < μ1s, see Figure 6) with erratic vegetation patterns and high diversity of averaged biomass, in contrast to just a single critical point (μ = μ1) in representing the transition in the single column model. (3) The behavior (e.g., transition, resilience) of grassland with spatial patterns in the region of μ > μ1s is more or less similar to that of the uniform grassland, but vegetation patches in the region of μ < μ1s are more fragile, and over-grazing can result in the gradual decrease of vegetated patch numbers and eventually lead to the stable state of desert. The existence of a narrow region with erratic vegetation patterns and high diversity of averaged biomass has not been reported before using this type of models. On the contrary, the previous studies always show a single-value function (or with small variations) of averaged biomass versus precipitation [von Hardenberg et al., 2001; Rietkerk et al., 2004]. Our results show that ecosystem behavior (e.g., resilience) in this region is most fragile compared with other vegetation pattern zones as well as the uniform grassland state. Just as the occurrence of vegetation patches from uniform grassland is a precursor of desertification [Rietkerk et al., 2004], such erratic vegetation patterns are an indicator that the ecosystem is at the very edge of catastrophic shift. These modeling results, however, still need to be validated using observational data.
 The essential mechanism for the persistence of vegetation pattern in our and other similar models [e.g., von Hardenberg et al., 2001; Rietkerk et al., 2002; Meron et al., 2004] is that water is more concentrated into patches of vegetation due to spatial interactions. This implies that the horizontal spread speed of soil water should be much higher than that of living biomass [e.g., δy ≫ δx in equations (4) and (5)]. In contrast, some models [e.g., Klausmeier, 1999] consider the horizontal water transportation over slopes only and hence cannot reproduce vegetation patterns on flat ground without prescription of slight topographic variation.
 Our model and many other conceptual models on vegetation patterns use a prescribed atmospheric forcing (e.g., precipitation) so that the land feedback to the atmosphere cannot be addressed. Other conceptual models considering the atmosphere-land coupling (but without considering the fine-scale vegetation pattern) often introduce an additional precipitation term which is mainly determined by the total living biomass (or leaf area index) [e.g., Wang, 2004; Wei et al., 2006]. When this kind of coupling is added to the models possessing vegetation patterns, an issue related to the question 3 in section 1 is: what is the impact of vegetation pattern on the atmosphere-land coupling? Qualitatively, the impact of vegetation patterns would be larger in the drier zone (e.g., μ < μ1s in Figure 7) where the average biomass is sensitive to initial conditions and precipitation inhomogeneity. Furthermore, vegetation patterns could affect the radiative transfer within and below the canopy (e.g., through shading), which has not been considered yet in modeling studies.
 All these models discussed so far consider one vegetation type only and hence cannot simulate the competition and facilitation between vegetation types (e.g., grass versus shrub or tree). A few models have investigated the coexistence and transition between plant ecosystems [Rodriguez-Iturbe et al., 1999a, 1999b; Breshears and Barnes, 1999], but they did not explicitly consider the spatial vegetation patterns. It is a major challenge to combine these studies to form a comprehensive model that can efficiently predict spatial patterns, interaction between vegetation types, and abrupt transitions in biomass over arid and semiarid regions.
 This work was supported by NASA (NNG06GA24G). Michael Barlage, the Associate Editor, and two anonymous reviewers are thanked for helpful comments.