Stability of banded vegetation patterns under seasonal rainfall and limited soil moisture storage capacity

Stability of banded vegetation patterns under seasonal rainfall and limited soil moisture storage capacity

Advances in Water Resources 29 (2006) 1556–1564 Stability of banded vegetation patterns under seasonal rainfall and...

436KB Sizes 0 Downloads 16 Views

Advances in Water Resources 29 (2006) 1556–1564

Stability of banded vegetation patterns under seasonal rainfall and limited soil moisture storage capacity Nadia Ursino *, Samuel Contarini Dipartimento di Ingegneria Idraulica, Marittima e Geotecnica, Universita´ di Padova, via Loredan 20, 35100 Padova, Italy Received 1 April 2005; received in revised form 23 September 2005; accepted 7 November 2005 Available online 4 January 2006

Abstract The delicate equilibrium of soil moisture and biomass may become unstable under water scarcity conditions causing banded vegetation patterns to form on hillsides of semi-arid catchments. Soil related processes that induce instability (namely: soil moisture advection and diffusion), have been evaluated numerically for different rainfall regimes. This study addresses the combined influence of some relevant soil characteristics, and the effect of seasonal precipitation on vegetation patterns, advancing the comprehension of those mechanisms that cause shifts toward banded vegetation patterns or bare states. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Semi-arid grasslands; Vegetation banding; Stability analysis; Seasonality of rainfall; Soil moisture storage

1. Introduction Regular patterns of vegetation have been observed in many arid and semi-arid regions of the world (e.g. [2,21,27–29]). The formation of regular vegetation bands on hillsides of semi-arid catchments is often attributed to a low scale process of water redistribution by runoff [1,24,5]. One way to better understand the organization of these vegetation patterns is to recognize that local feedbacks along with dispersion cause regular vegetation patterns to develop as a result of Turing-like instability [22,14]. This phenomenon motivated many recent studies concerning the stability of the soil moisture and plant biomass balance equations (e.g. [15,9, 26,10]). The stability analysis demonstrates that, when the soil moisture and biomass balance is unstable, an even slightly heterogeneous initial distribution of the vegetation may turn into a reorganized one over time


Corresponding author. Fax: +39 49 8275446. E-mail address: [email protected] (N. Ursino).

0309-1708/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2005.11.006

without being ordered by external forces. Resource scarcity leads to spatial reorganization of consumers (plants) and resources (e.g. water), and as a consequence, the structures observed in reality emerge from fine-scale interactions. Rietkerk et al. [16] revise recent studies on vegetation patterns formation and the mechanisms involved. The effect of inter-annual variability of rainfall (decade to century time scale) on the dynamics of soil moisture and plant physiology has recently been investigated by hydro-ecologists, whose main achievements are (i) the characterization of the soil moisture statistical variability as a function of the random rainfall distribution (e.g. [18,3,11]), and (ii) the linkage between the soil moisture time variability and the spatial vegetation organization in water limited ecosystems (e.g. [7]). Very often stability models account for rainfall within the moisture balance in terms of a uniform supply, neglecting its seasonality and the frequency of showers. Rainfall fluctuations at daily time scale determine water loss due to runoff generation, as soon as the soil moisture storage capacity is exceeded. Their effect on the stability of soil

N. Ursino, S. Contarini / Advances in Water Resources 29 (2006) 1556–1564

moisture and biomass balance has not been evaluated yet. Soil texture has been proven to be of crucial importance with regard to climate, soil and vegetation dynamics (e.g. [6,12]). Despite the fact that soil characteristics may have a relevant role for water and nutrient transport and thus for vegetation growth, the appearance of banded vegetation patterns is rarely related to effective soil parameters. According to the model of Klausmeier [10], the instability is driven by the lateral loss of resources (soil moisture). Lateral advection and dispersion depend on soil parameters, and have been expressed as a function of hydraulic conductivity and mean capillary rise by Ursino [23] who went on to study whether the extended model could fit experimental data from literature. She found that even if the conductivity and the capillary rise may be ‘‘tuned’’ in order to obtain a good fit, the values to be attributed to these parameters may be far beyond their usual range of variability. Soil properties that are defined at the continuum scale (the scale of the model discretization) may be obtained by upscaling effective processes that occur at smaller scales (e.g. the scale of the pores, fractures or roots). Thus, Ursino [23] concluded that: either relevant processes occurring at a smaller scale must be upscaled to justify the out-of-scale values of the soil parameters, or relevant processes influencing the dynamics of soil moisture and biomass at the larger scale must be taken into account within the model. The following processes have already been proven to induce instability and thus vegetation patterns: the feedback between vegetation density and infiltration (see e.g. [9]); the short range facilitation and the long range competition for resources [13]; the transport of limited nutrients [17]. These processes have not been modeled here according to Klausmeier [10] and von Hardenberg et al. [26]. Ecosystem engineers [8] are ignored and plant physiology is simplistically modeled in accordance with previous stability models (e.g. [10,15]), even though a more accurate representation of plant response to prolonged soil moisture deficit is likely to be an important factor in climate change scenarios [4]. We will not address the question of to what extent the above mentioned processes have an impact on the bands formation, but rather investigate, through a sensitivity analysis, the effect of different rainfall distributions on soils with limited moisture storage capacity. We solved numerically the Klausmeier [10] model further expanded upon by Ursino [23], believing that it may be suited for the aim of this study. We must nevertheless acknowledge that this simplistic model might not be reliable for quantitative prediction purposes and thus, our study represents only one step towards the comprehension of those mechanisms that might be responsible for ecosystem shifts to vegetation mosaic or bare state. The model (Section 2) is solved numerically for different soil prop-


erties and rainfall regimes (Section 3) and the results are compared with the analytical solution obtained in the case of uniform rainfall supply (Section 4).

2. Theory The model for soil moisture (W) and plant biomass (N) balance is defined on an infinite one-dimensional domain indexed by X as a function of time T, as follows: oW oW d2 W ¼ A  LW  rwn2 þ V w þ Dw ; oT oX dX 2 oN o2 N ¼ RJWN 2  MN þ D 2 . oT oX


It does not take explicitly into account the surface water balance, the transport of nutrients and the action of ecosystem engineers that have been recognized to have an influence on stability [16]. Neither does it take into account fire and grazing, whose influence on the stability condition (as far as we know) has not yet been evaluated. According to Klausmeier [10] and von Hardenberg et al. [26] the net inflow A somehow resembles the rainfall regimes and does not account for enhanced infiltration in the vegetated areas. This approach differs from the one of Gilad et al. [8] and HilleRisLambers et al. [9]. LW is the water loss due to evaporation and leakage, that may be approximately considered to be a linear function of the soil moisture under stress conditions [18,19]. Close to saturation, the evapotranspiration reaches an upper limit that is ignored here. Leakage out of the root zone could be otherwise expressed as a function of the soil conductivity. Indeed, the deeper soil layers may block the vertical flow or enhance the vertical loss that is crucial for the band formation (not considered here). MN is the plant biomass loss due to mortality; Dw and Vw represent the soil moisture diffusion coefficient and transverse velocity; J is the yield of plant biomass per unit water consumed; D is the diffusion coefficient of plant dispersal, and rwn2 is the plants’ uptake. The balance of soil moisture and biomass (1), may be conveniently expressed in dimensionless form ow ow o2 w ¼ a  w  rwn2 þ v þd 2 ; ot ox ox on o2 n ¼ rwn2  mn þ 2 ; ot ox


where t = LT, x = D1/2L1/2X, m = L1M, W ¼ W 1 s W, N = (WsJ)1N, a = (LWs)1A, and r ¼ W 2s J 2 RL1 . Ws represents the soil moisture storage capacity that is approximately Ws = hsH. H is the soil depth where the root uptake occurs and hs is the saturated soil moisture content.


N. Ursino, S. Contarini / Advances in Water Resources 29 (2006) 1556–1564

The dimensionless numbers in the first of (2) might be expressed as functions of the saturated conductivity Ks and the inverse of the capillary rise a: d = Ks(hs Æ a)1 1=2 D1 and v ¼ i  K s h1 [23]. When the capillary s ðLDÞ forces are negligible and a becomes very large, d tends to 0 and (2) reduces to the model proposed by Klausmeier [10]. The stability of the steady state solution (w0, n0) of (2) is tested by linearizing the problem about (w0, n0). For a small perturbation   w  w0 u¼ ð3Þ n  n0 in the form X ck ekt U k ðxÞ. u¼



The eigenvalue k represents the growth rate of the disturbance and the constants ck are determined by a Fourier expansion of the perturbation in terms of Uk(x). Substituting (4) into (2) leads to the following relation: ut ¼ Au þ V ru  Dr2 u; where  lw A¼ hw

 ln ; hn

 V ¼

ð5Þ v 0

 0 ; 0


d 0

0 1


with l = a  w  rwn2, and h = rwn2  mn; lw ¼ ð1þ rn20 Þ, ln = 2m, hw ¼ rn20 and hn = m are the derivatives of l and h with respect to w or n, evaluated in (w0, n0). The steady state (w0, n0) is stable if the real part of the growth rate k is Re(k) < 0 and unstable when Re(k) > 0. The equation Re(k) = 0 identifies the boundary between stable and unstable conditions, is called critical stability condition and is associated with the critical wave number k0. The critical stability condition of problem (2) can be obtained analytically 1 1=2 1=2 lw þ hn  ðd þ 1Þk 20 þ pffiffiffi ½ðq2 þ p2 Þ þ q ¼ 0; 2

3. Numerical solution of the stability problem under seasonal rainfall The analytical critical stability condition (7), holds for continuous net inflow A and unlimited W, therefore, the amount of moisture that the soil can really store may be overestimated by applying (7). Indeed W cannot exceed the soil storage capacity Ws and thus 0 6 w 6 1. Physically, moisture is stored for plants in a shallow upper soil layer and as soon as this layer is saturated during a storm event, the excess rainfall runs off. Whether or not a limitation on the net extractable soil moisture produces an effect on the vegetation organization can be verified by solving numerically (2). We investigated seasonal rainfall showers characterized by a deterministic frequency instead of more realistic shower distributions with the same average characteristics. By doing so we obtain general results that do not depend on a particular realization of a stochastic rainfall process. The seasonal net rainfall is assumed to be concentrated over a time-limited wet season (5 months). A daily frequency F is attributed to the rain events and the net inflow during the rainy days is set at A 0 = A Æ 12 Æ 51 Æ F1 (namely: F = 1, 0.1 and 0.2 day1) (see Fig. 1). Reasonable values for the cumulative annual rainfall in semi-arid regions are less than 800 mm year1 [24]. A was set accordingly. Cases with A < 240 mm year1 are not considered since the steady state solution (w0, n0) whose stability is investigated here, exists only for A > 240 mm year1. The other parameters were M = 1.8 year1; J = 0.003 kg m2 mm1; D = 1 m2 year1, L = 4 year1 and R = 100 m4 kg2 year1 [10]. Thus, m = 0.45 and r = 0.36. Ks and a have been set according to Ursino [23] as specified in Table 1.



where p ¼ 2k 0 v½lw  hn  ðd  1Þk 20 , and q ¼ ½lw  hn  ðd  1Þk 20 2 þ 4ln hw  k 20 v2 . Eq. (7) may be regarded as a relation between any set of model parameters. It will be represented hereafter as a relation between the net inflow A and the gradient slope i, for different v and d. We chose to represent the critical stability condition on the plane A–i, since many authors observed that a slope gradient threshold below which no banded patterns develop does exist and that it tends to increase with the mean annual rainfall (e.g. [20,25,30], cited by Valentin et al. [24]). By representing the stability condition on the A–i plane, the consistency of the model results with the experimental evidence of a threshold i, may be easily verified [23].



0 0




Fig. 1. Simulated precipitation regimes. Mean uniform water supply (dashed line) and seasonal rainfall (continuous line). Cases of daily showers frequency F = 1 day1 (thick line) and F = 0.1 day1.

N. Ursino, S. Contarini / Advances in Water Resources 29 (2006) 1556–1564 Table 1 Characteristic soil effective parameters investigated by numerical simulation i = 0.02

Case 1 Case 2 Case 3

Ks [m year1]

a [m1]

v [–]

d [–]

v/d [–]

4000 4000 400

100 1000 10

100 100 10

100 10 100

1 10 101

Fig. 2. Critical stability condition evaluated by linear stability analysis (continuous line) and cases to be investigated by numerical simulation (bold circles): (i = 0.02, A = 300 mm) (unstable according to the linear stability analysis), (i = 0.02, A = 410 mm) (critical stability condition), (i = 0.02, A = 500) and (i = 0.02, A = 600 mm) (stable cases).


With reference to i = 0.02, v/d ranges from 0.1 to 10. Indeed, the instability may theoretically be driven either by the advective or by the diffusive subsurface flow or by both of them. Other models (e.g. [10]) do not account for soil moisture diffusion, or conversely describe the soil water movement just by diffusion (e.g. [9]). At time T0 = 0 we assumed that the soil moisture and the biomass distribution were (w0 + , n0 +  0 ), where  and  0 are small heterogeneous disturbances of order O(w0 Æ 103) and O(n0 Æ 103) respectively. By numerical integration we obtained the spatial soil moisture and biomass distribution at time (T20 = 20 year), and discriminated between stable and unstable conditions on the basis of the long term vegetation distribution n(x, T20). The linear stability analysis summarized in Section 2 instead, provides the instantaneous conditions for a single mode of a small perturbation to remain stationary: Re(k) = 0 (that corresponds to Eq. (7)). To clarify this point, in Fig. 2 the critical stability condition provided by the linear stability analysis has been plotted in the A–i plane (the parameters are those relative to case 1 and A is uniform). Fig. 3 shows the soil moisture and vegetation distribution after 20 years obtained by numerical simulation for the stable and unstable cases indicated in Fig. 2 with solid circles. Fig. 3 demonstrates that, according to our numerical results, there is no abrupt transition between mosaic and homogeneous vegetation, but rather the bands become less pronounced as A increases. The critical stability condition

Fig. 3. Soil moisture (w) and vegetation distribution (n) evaluated at time T = 20 years of numerical simulation for different mean annual precipitation (A = 300, 410, 500 and 600 mm) and for gradient slope i = 0.02. As shown in Fig. 2 the linear stability analysis predicts for the four cases considered here: instability, critical stability and twice stability.


N. Ursino, S. Contarini / Advances in Water Resources 29 (2006) 1556–1564

predicted by linear stability analysis is A = 410 mm year1. The inter-bands soil is progressively colonized by plants for A > 410 mm year1. Indeed, we found heterogeneous vegetation with n > 0 everywhere for A = 500 mm year1. For A = 600  410 mm year1, at time T, the soil moisture and the vegetation demonstrated uniform distribution. When a dry season and a wet season were simulated, the soil moisture distribution at the end of the wet season was similar to the one shown in Fig. 3, whereas at the end of the dry season the soil was almost completely dry. The bands of vegetation as well as the uniform vegetation cover, persisted over the entire year reappearing strengthened at the end of every wet season. We discriminated between stable and unstable conditions on the basis of the deviation of the vegetation distribution from its spatial average  n (evaluated at the end of the wet season for seasonal rainfall). We set a threshold value Th = 0.8, and assumed that only vegetation distributions characterized by a deviation from the mean above Th had to be considered as organized in bands. This means that the critical stability condition was conventionally sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ‘ 1 1  n ðnðx; T 20 Þ   nÞ2 dx ¼ Th; ð8Þ ‘ 0 R ‘ where  n ¼ ‘1 0 nðx; T 20 Þ dx, and ‘ is the dimensionless size of our 1D domain with wrap-around boundaries (‘D1/2L1/2 = 200 m). 4. Results Fig. 4 shows the critical stability condition evaluated by numerical simulation in case of uniform mean annual

rainfall (triangles) along with the critical stability condition estimated by linear stability analysis (continuous line). Open symbols are used for the cases of unlimited w and bold symbols for those with w 6 1. As expected, the critical stability condition evaluated by numerical simulation (Eq. (8)) and the critical stability condition estimated analytically by linear stability analysis (Eq. (7)) came close but did not coincide. The limitation imposed upon w did not significantly affect the evaluated critical stability condition. Indeed for uniform rainfall, the soil moisture seldom exceeds its natural limit in the cases analyzed here. The flat ground (i = 0.0) was found to be unstable in cases 1 and 3. Conversely, in case 2 (instability driven by advection) the flat ground was always stable, in accordance with the previous findings of Klausmeier [10], since the destabilizing effect of advection vanishes on flat ground. Furthermore, in case 3 (instability driven by diffusion), the instability field was restricted to values of the net inflow between A = 240 mm year1 and A  260 mm year1 for 0.0 < i < 0.04, demonstrating the major influence of advection on the band formation. Of course, it does not make sense to discuss the 1D analysis results on flat ground, as well as those cases of instability caused by diffusion since, in these cases, no preferential flow direction may be identified. Nevertheless, we do report the 1D results for the sake of comparison between cases where instability is caused by advection and similar cases where it is caused by diffusion. When the advective term may be neglected, the extension of the 1D results to the 2D case may be achieved by replacing the squared mode of the disturbance with the summation of the squared unstable mode in the two main directions, leaving the results unchanged.

Fig. 4. Critical slope i versus mean annual precipitation A for different soil properties. Continuous line: marginal stability condition evaluated by linear stability analysis. Triangles: stability condition estimated by numerical simulation and uniformly distributed rainfall supply A. Open symbols: unlimited soil moisture. Solid symbols: limited soil moisture storage (w 6 1).

N. Ursino, S. Contarini / Advances in Water Resources 29 (2006) 1556–1564


Fig. 5. Critical slope i versus mean annual precipitation A for different soil properties and different daily shower frequency F. Circles: F = 1 day1. Diamonds: F = 0.2 day1. Squares: F = 0.1 day1. Open symbols: unlimited soil moisture. Solid symbols: limited soil moisture storage (w 6 1).

In Fig. 5 the critical stability condition is plotted for seasonal rainfall. The frequency of showers is F = 1 (circles), F = 0.2 (diamonds), and F = 0.1 (squares). For limited soil moisture storage w 6 1 (bold symbols), and given gradient slope, a lower limit narrowed consistently the instability field. Below the lower boundary of the stability field, the long term solution of (2) was bare soil; beside the upper boundary, similar to the case with unlimited soil moisture storage (open symbols), the soil resulted uniformly covered by vegetation. Within the instability field, regular bands of vegetation were simulated. The lower limit of the instability field was visible also in case 1 and unlimited w, in the gentle slope domain (i 6 0.01), whereas it was not detected in case of uniform A (Fig. 4). The instability field resulted narrowed and shifted towards the higher A and i for decreasing F. This result is not visible in case 3 because the instability field, already for F = 1, was found above the gradient slope i = 0.04 (results obtained for higher gradient slopes have not been shown here). In case 3 (instability driven by diffusion) for i < 0.04 the evaluated vegetation distribution at time T20 was either bare soil or uniform cover. Thus, the lateral advection sustains the bands over a wide range of mean annual precipitation and when it vanishes the model predicts a sudden shift from uniform vegetation to bare soil as the mean annual rainfall decreases (e.g. case 3). This result was not predicted in case of uniform A (Fig. 4). In order to explain the discrepancies between results obtained for limited and unlimited w, one may check when the soil moisture tends to rise above its upper limit. The maximum and minimum point values of w and n are plotted, as a function of time in Fig. 6 (unlimited w), and in Fig. 7 (w 6 1). As shown in Figs. 6 and 7 for case 1, i = 0.02 and F = 1, the limitation imposed upon w had a major effect at very low A

(A = 300 mm year1). In this case w tends to rise above 1, leading to a positive growth rate of the perturbation that causes the formation of the vegetation bands. Differently, when the soil moisture is forced to remain w 6 1, the initial heterogeneous perturbation of the soil moisture and vegetation distribution decay (maximum and minimum w and n coincide). The resulting soil moisture and vegetation distribution is homogeneous but the limited rainfall supply cannot sustain the vegetation growth and its density decays to zero. The final result of the simulation in this case is bare soil. For A = 400 mm year1 the soil moisture seldom reaches its upper limit until the initial perturbation amplifies and the bands form. Only after the band formation, when the vegetation is already strongly heterogeneous, the soil moisture tends to rise above the limit, but the limitation imposed upon w does not hamper the band developments. For A = 500 mm year1 the problem is stable and its solution is uniform vegetation cover. In the unlimited case, w results slightly above 1 and the solution obtained by limiting w does not differ substantially from the former. The vegetation density is higher locally when the bands R ‘ develop but the spatial average of plants uptake: ‘1 0 rwn2 dx increases with a, balancing the major inflow (not shown here). We verified that limiting the evapotranspiration to a more realistic maximum value LW* as W reaches a critical value W* has a minor impact on the overall results. In Table 2 the limits of the instability field in terms of A have been evaluated for uniform precipitation and for seasonal rainfall (F = 1), in case 1. We set i = 0.04 and W* = 0.4Ws. In brackets, for the sake of comparison, we reported the results obtained for W* = Ws (no evapotranspiration limit). They are shown in Fig. 5. The instability field resulted just slightly shifted toward the lower A values for W* < Ws due to the reduced vertical loss.


N. Ursino, S. Contarini / Advances in Water Resources 29 (2006) 1556–1564

Fig. 6. Maximum and minimum point values of the soil moisture and biomass distribution as a function of time. Case of unlimited soil moisture storage (case 1; i = 0.02 seasonal rainfall with F = 1).

Fig. 7. Maximum and minimum point values of the soil moisture and biomass distribution as a function of time (w 6 1) (case 1; i = 0.02 seasonal rainfall with F = 1).

N. Ursino, S. Contarini / Advances in Water Resources 29 (2006) 1556–1564


Table 2 Limits of the instability field in terms of A, evaluated for continuous (left) and seasonal rainfall with F = 1 (right) and i = 0.04 Continuous rainfall

Seasonal rainfall (F = 1) 1

Case 1 Case 2 Case 3


A lower limit [mm year ]

A upper limit [mm year ]

A lower limit [mm year1]

A upper limit [mm year1]

– (–) – (–) – (–)

646 (646) 545 (660) 276 (283)

351 (367) 346 (362) 371 (389)

605 (610) 623 (627) 374 (390)

W* = 0.4Ws. In brackets, corresponding values obtained for unlimited evapotranspiration W* = Ws.

Finally, we found that the model is particularly sensitive to changes in the form of the plant uptake term (not shown here). Thus, the definition of the biological aspect of the model that we inherited from Klausmeier [10], should be handled with care since it could be crucial for the set up of a predictive tool, that was beyond the aim of this investigation.

5. Discussion and conclusions The critical stability condition of the problem of soil moisture and biomass balance (2) has been investigated by numerical simulation for different soil properties and rainfall regimes. In particular, the impact of seasonal rainfall has been demonstrated. Our main finding confirmed the intuitive thought that intense showers falling on soils with a limited soil moisture capacity generate relevant losses of resources due to runoff and, as a consequence, the vegetation may either disappear—leaving the soil bare—or may organize in patterns, depending on the intensity and frequency of showers. The runoff generation due to soil moisture excess strikingly affects the transition from vegetated to bare soil (lower limit of the instability field) as the net rainfall supply decreases. Previous studies could not prove the importance of the limited soil moisture storage capacity since they were based on the assumption of uniform rainfall distribution over time. In particular, neither the linear stability analysis, nor the numerical solution of the problem could predict the transition to bare soil, overestimating the extension of the instability field, under the unrealistic assumption of uniform rainfall. In cases 1 and 2 (advection induced instability), the evaluated soil moisture seldom exceeded its upper limit, unless the soil was bare, indicating that the plants could optimize the use of the scarce resource by organizing in patterns and varying their spatial density. Advection, was crucial for the gradual transition from uniform cover—to mosaic—to bare soil. Indeed, for small gradient slope and negligible v/d ratio (case 3), the instability field disappeared and the solution of the soil moisture and biomass balance shifted from uniform cover to bare soil as the net rainfall decreased. Our results suggest that acting on the soil storage capacity and on the soil connectivity (that enhances

the lateral advection) may be one way to avoid environmental shifts. Certainly, plants roots contribute to loosening and increase in porosity and depth of the active upper soil layers, and thus root architecture and development might deserve special attention when designing predictive models for ecosystem shifts or ecosystem management support tools. Finally, though this point has not been addressed here, for further advances toward a really predictive tool, one should model biology with care and consider how prolonged inter-rainfall dry intervals might affect plant physiology. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.advwatres.2005.11.006. References [1] Boaler SB, Hodge CAH. Vegetation stripes in Somaliland. J Ecol 1962;50:465–74. [2] Boaler SB, Hodge CAH. Observations on vegetation arcs in the northern region Somali Republic. J Ecol 1964;52:511–44. [3] D’Odorico P, Ridolfi L, Porporato A, Rodriguez-Iturbe I. Preferential states of seasonal soil moisture: the impact of climate fluctuations. Water Resour Res 2000;36:2209–19. [4] Fay PA, Carlisle JD, Knapp AK, Blair JM, Collins SL. Productivity responses to altered rainfall patterns in a C4dominated grassland. Oecologia 2003;137:245–51. [5] Fernando TM, Cortina J. Spatial patterns of surface soil properties and vegetation in a Mediterrean semi-arid steppe. Plant and Soil 2002;241:279–91. [6] Fernandez-Illescas CP, Porporato A, Laio F, Rodriguez-Iturbe I. The ecohydrological role of soil texture in a water-limited ecosystem. Adv Water Resour 2001;12:2863–72. [7] Fernandez-Illescas CP, Rodriguez-Iturbe I. The impact of interannual rainfall variability on the spatial and temporal patterns of vegetation in a water-limited ecosystem. Adv Water Resour 2004;27:83–95. [8] Gilad E, von Hardenberg J, Provenzale A, Shachak M, Meron E. Ecosystem engineers: from pattern formation to habitat creation. Phys Rev Lett 2004;93:098105. [9] HilleRisLambers R, Rietkerk M, van den Bosch F, Prins HHT, de Kroon H. Vegetation patterns formation in semi-arid grazing systems. Ecology 2001;82(1):50–61. [10] Klausmeier CA. Regular and irregular patterns in semiarid vegetation. Science 1999;284:1826–8. [11] Laio F, Porporato A, Ridolfi L, Rodriguez-Iturbe I. Plants in water controlled ecosystems: active role in hydrological processes




[14] [15]





N. Ursino, S. Contarini / Advances in Water Resources 29 (2006) 1556–1564

and response to water stress. II. Probabilistic soil moisture dynamics. Adv Water Res 2001;24:707–23. Laio F, Porporato A, Fernandez-Illescas CP, Rodriguez-Iturbe I. Plants in water controlled ecosystems: active role in hydrological processes and response to water stress. IV. Discussion of real cases. Adv Water Res 2001;24:745–62. Lejeune O, Tlidi M, Couteron P. Localized vegetation patches: a self-organized response to resource scarcity. Phys Rev E 2002;66: 010901. Murray JD. Mathematical Biology. Berlin: Springer-Verlag; 1989. Rietkerk M, Boerlijst MC, van Langevelde F, HilleRisLambers R, van de Koppel J, Kumar L, et al. Self-organization of vegetation in arid ecosystems. Am Nat 2002;160:524–30. Rietkerk M, Dekker S, de Ruiter PC, van de Koppel J. Selforganized patchiness and catastrophic shifts in ecosystem. Science 2004;305:1926–9. Rietkerk M, Dekker SC, Wassen MJ, Verkroost AWM, Bierkens MFP. A putative mechanism for bog pattering. Am Nat 2004;163:699–708. Rodriguez-Iturbe I, Porporato A, Ridolfi L, Isham V, Cox DR. Probabilistic modelling of water balance at a point: the role of climate, soil and vegetation. Proc R Soc Lond A 1999;455: 3789–805. Salvucci GD. Estimating the moisture dependence of root zone water loss using conditional averaging precipitation. Water Resour Res 2001;37(5):1357–65.

[20] Slatyer RO. Methodology of water balance study conducted on a desert woodland community in central Australia. In: Plant–water relationships in arid and semi-arid conditions. In: Proceedings of the Madrid symposium, UNESCO Arid Zone Research, vol. 16, 1961, p. 15–25. [21] Tongway D, Ludwig JA. Vegetation and soil patterning in semiarid mulga lands of Eastern Australia. Austral J Ecol 1990;15: 23–34. [22] Turing AM. Philos Trans R Soc London, Ser B 1952;237:37. [23] Ursino N. The influence of soil properties on the formation of unstable vegetation patterns on hillsides of semiarid catchments. Adv Water Resour 2005;28:956–63. [24] Valentin C, d’Herbes JM, Poesen J. Soil and water components of banded vegetation patterns. Catena 1999;37:1–24. [25] Valentin C, d’Herbes JM. Niger tiger bush as a natural water harvesting system. Catena 1999;37:231–56. [26] Von Hardenberg J, Meron E, Shachak M, Zarmi Y. Diversity of vegetation patterns and desertification. Phys Rev Lett 2001;87: 198101. [27] White LP. Brousse tiger patterns in southern Niger. J Ecol 1970; 58:549–53. [28] White LP. Vegetation stripes on sheet wash surfaces. J Ecol 1971; 59:615–22. [29] Worral GA. The Butana grass patterns. J Soil Sci 1959;10:34–53. [30] Worral GA. Patchiness in vegetation in the northern Sudan. J Ecol 1960;48:107–17.