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The influence of soil properties on the formation of unstable vegetation patterns on hillsides of semiarid catchments

Nadia Ursino *

Universitá di Padova, Dept. IMAGe, via Loredan 20, I-35131 Padova, Italy

Received 17 June 2004; received in revised form 14 February 2005; accepted 22 February 2005

Available online 10 May 2005

Abstract

On hillsides of semiarid catchments regular bands of vegetation have been observed to form under low rainfall conditions. Many

authors have observed the existence of a slope gradient threshold below which no banded patterns were observed; this increases with

the mean annual rainfall. A simple model for soil moisture balance and vegetation growth, explicitly accounting for basic soil phys-

ics, is demonstrated and discussed. The influence of relevant soil characteristics on vegetation patterns and their patchiness is

addressed by linear stability analysis. The results confirm the experimental evidence of a threshold slope gradient depending on

the mean annual net water supply, and demonstrate the influence of the soil properties (saturated conductivity and capillary rise)

on the stability condition and on the threshold slope. Nevertheless, the concluding remark concerns the oversimplified models for

vegetation patterns, such as the one discussed here, that require properly defined effective soil parameters in order to become pre-

dictive tools.

2005 Elsevier Ltd. All rights reserved.

Keywords: Ecohydrology; Semiarid catchments; Vegetation banding; Stability analysis; Patterns initiation

1. Introduction

Regular bands of vegetation have been observed to

form on hillsides of semiarid catchments under low rain-

fall conditions. Numerous experimental studies have

been conducted in different parts of the world, mainly

in Africa and in Australia, in order to collect data and

classify patterns of vegetation organized in a two-phase

mosaic as well as to define their connection with geo-

morphic and climatic conditions. Valentin et al. [24]present a summary of the experimental evidence col-

lected in the last 30 years circa. There is a continuous

interest in trying to understand and model vegetation

patterns formation and evolution. Lefever and Lejeune

[8] explain the formation of tiger bush through a mech-

anism of resources competition. Gilad et al. [5] attribute

the appearance of mosaic vegetation to the cooperation

of plant and microorganism. Ares et al. [1] recognize

that grazing and thus anthropic use may modify the spa-

tial patterns of vegetation. The formation of vegetation

bands alternating with bare soil has been linked by

many authors to a low scale process of redistribution

of water by runoff [22,8,9,7,24,3,6,25,14,2]. The redistri-

bution of surface water is considered as an important

factor explaining the display of mosaic vegetation pat-

terns, according to the following self-organizing mecha-nism. Rain falling on bare patches of soil fails to

infiltrate and runs off [19]. This run-off water accumu-

lates in the vegetated patches, where it can infiltrate

more easily; the spatial vegetation distribution then

influences the spatial distribution of soil particles, or-

ganic matter and nutrients [14]. HilleRisLambers et al.

[6] demonstrate the effect of bare soil infiltration

capacity on patterns formation. Rietkerk et al. [15] re-

cently summarized model results and perspectives on

Advances in Water Resources 28 (2005) 956–963

www.elsevier.com/locate/advwatres

0309-1708/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2005.02.009

* Fax: +39 49 8275446.

E-mail address: [email protected]

mailto:[email protected]:[email protected]


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echosystems bistability and catastrophic shifts that may

be linked to vegetation patchiness. Despite the many di-

verse recent contributions to the comprehension of the

genesis and evolution of vegetation mosaic, there remain

questions concerning the origin of these vegetation and

soil moisture paths that still remain unanswered, such

as: do the averaged soil properties have an impact onvegetation patchiness? And, eventually, to what extent?

Notwithstanding the complexity of the interplay be-

tween vegetation, soil, moisture and nutrients, a simple

model for soil moisture and plant biomass [7] shows that

local interactions (such as the feedback between water

and biomass), coupled with dispersion, can cause regu-

lar vegetation patterns to develop in the absence of soil

heterogeneity as the result of Turing-like instability

[23,10]. The results previously obtained with this model

demonstrate that the emergent behavior of large, com-

plex landscapes in a prolonged period of disequilibrium

may simplify into a form that can be observed without

an explicit consideration of soil effective parameters. In

other words, although soil may have a crucial role in

water and nutrient transport and thus in vegetation

growth, soil properties are usually not explicitly consid-

ered within this model. The following questions arise:

how may relevant soil properties be taken into account

in hydro-biological models? Are the parameters and

concepts commonly adopted in soil physics adequate

to describe the role that the soils play in creating vegeta-

tion patterns?

The model of Klausmeier [7] (revised in Section 2.1),

is reinterpreted here in order to account for relevant soil

properties. The new modified version of this model isdemonstrated in Section 2.2. Factors such as surface

water, nutrient transport as well as animals and micro-

organism action are neglected, in the belief that the im-

pact of soil properties on the stability condition could be

captured anyway. Many authors have observed that a

slope gradient threshold, below which no banded pat-

terns were observed, exists and that it tends to increase

with mean annual rainfall (e.g. [26,21]). The new model

allows derivation of the marginal stability condition as a

function of the hill slope, the saturated conductivity,

and the capillary rise. The results presented here were

obtained analytically and confronted with experimental

data taken from literature (Section 3). In Section 4, the

different processes that appear to have an influence on

the interaction between water, soil and vegetation are

discussed. It has been demonstrated that the relationship

between critical annual rainfall and critical slope is

strongly affected by the averaged over depth saturated

conductivity and by the mean capillary rise. Neverthe-

less, the parameterization of the model results is partic-

ularly delicate. It may also be said that properly defined

effective parameters (which may resemble the subscale

soil processes that determine the macroscopic evolution

of the unstable ecosystem) are needed.

2. Theory

Turing [23] provided a hypothesis to explain the gen-

eration of patterns through mechanisms of reaction and

diffusion. By superimposing an initial pattern of given

wave-number upon the steady state homogeneous solu-

tion and allowing reaction and diffusion to act overtime, the initial disturbance is expected to either grow

or decline. Obviously, the patterns which grow faster

in time will be most visible. The mathematical frame-

work for this stability analysis is the Fourier analysis.

The relations between the attributes of the unstable pat-

terns (growth rate and characteristic length) and those

of the environment (mean rainfall, hill slope and soil

properties) are investigated below.

The characteristic length of the bands of vegetation

considered here is much larger in the direction perpen-

dicular to the slope than in the parallel one. The soil

moisture follows a spatial distribution similar to that

of the vegetation. Furthermore, the water content vari-

ability in the direction perpendicular to the soil surface

may be neglected since the soil depth that influences veg-

etation growth is shallow as compared to the surface

extension of the domain (a similar hypothesis is com-

monly adopted in remote sensing studies, see e.g. [20]).

Thus, based on the above simplified hypothesis, the

theoretical analysis of the soil moisture and of the vege-

tation balance equation is limited here to the one-dimen-

sional case only. On flat ground the extension of the

model and the related results to the two-dimensional

case is straightforward, by replacing the squared mode

considered here with the summation of the squaredunstable modes in two directions.

2.1. Linear stability analysis of soil moisture

and biomass balances

The model of Klausmeier [7] is a pair of partial differ-

ential equations for soil moisture (W ) and plant biomass

(N ) that are defined on an infinite one dimensional do-

main indexed by X as a function of time T .

oW

oT ¼ A LW RWN 2 þ V oW

o X

o N oT

¼ RJWN 2 MN þ D o2 N o X 2

ð1Þ

where A is the uniform water supply (according to

Klausmeier [7] A does not account for the feedback be-

tween infiltration and biomass [6] even if this mechanism

has been proved to induce instability); LW is the water

loss due to evaporation and leakage (this may be

approximately considered to be a linear function of

the soil moisture under stress conditions, e.g. [16,18]);

MN is the plant biomass lost due to mortality; V is

the downhill water speed; J is the yield of plant biomass

per unit water consumed; D is the diffusion coefficient of

N. Ursino / Advances in Water Resources 28 (2005) 956–963 957


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plant dispersal, and RWN 2 is the plants uptake. Accord-

ing to Klausmeier [7] plants take up water at rate

RG (W )F (N )N –where G (W ) is the response of plants to

water and F (N ) is a function that describes how plants

increase infiltration. Klausmeier [7] linearizes the pro-

blem assuming G (W ) = W and F (N ) = N , and arguing

that the results are not sensitive to the exact form of these functions. The lateral loss V oW

o X , that is responsible

for the instability of the model, will be disclosed in the

next section.

Equation (1) can be reduced in dimensionless form

assuming T = L1t, X = D1/2L1/2x, V = (LD)1/2v, M =

Lm, W = W sw, N = W sJn and A = LW sa, and obtaining:

ow

ot ¼ a w rwn2 þ v ow

o x

on

ot ¼ rwn2 mn þ o

2n

o x2

ð2Þ

where r

¼W 2s J

2 RL1.The spatially uniform steady state solution (w0, n0) of

the coupled soil moisture and vegetation problem (1) is

the solution of

lðw; nÞ ¼ a w rwn2 ¼ 0hðw; nÞ ¼ rwn2 mn ¼ 0 ð3Þ

namely: w0 ¼ 0.5ða ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 4m2r 1p

Þ and n0 ¼ mw10 r 1.Adding a small perturbation to the steady state

(w0, n0),

u ¼ w w0n

n0

ð4Þ

in the form

u ¼X

k

ck ekt U k ð xÞ; ð5Þ

where the growth rate k is the eigenvalue, the constants

ck are determined by a Fourier expansion of the pertur-

bation in terms of U k (x), and k is the wave number.

Substituting (4) into (2) and linearizing yields to the fol-

lowing relation

ut ¼ Au þ V ru Dr2u; ð6Þwhere

A ¼ lw hnhw ln

; V ¼ v 0

0 0

; D ¼ 0 0

0 1

and l w, hw, hn and l n are the derivative of l and h with re-

spect to w or n, evaluated in (w0, n0). Substituting (5) into

(6), the following expression for the eigenvalue k may be

found:

k ¼ A þ jkV k 2 D; ð7Þwhere j is the imaginary unit. The steady state (w0, n0) is

stable in the absence of any spatial effect (when k = 0)

and unstable for some k 50. The first condition gives

jAj > 0 and l w + hn < 0. The second condition is fulfilledwhen the real part of k is negative [10].

2.2. A linear stability model for subsurface soil

moisture dynamics on slopes accounting for effective

soil parameters

The model of Klausmeier [7] includes within the

water balance equation the term V oW o X

, that accounts

for the flow parallel to the soil surface and induces insta-

bility [10]. It may be interpreted as the flow of soil mois-

ture within the root depth and may be expressed as a

function of the slope gradient and of the soil properties

as will be shown below. Further, by analyzing the sub-

surface flow by the continuum approach, an additional

diffusive term appears. The diffusive term neglected by

Klausmeier [7] is found in other instability models (e.g.

[14]) and will also be demonstrated to lead to instability.

According to the Richards equation [13] that governs

the vadose-zone water flow, the net variation of soil

moisture due to the horizontal soil water flux q (x-direc-

tion) may be expressed as follows

dqd x

¼ dd x

K ðhÞ dWd x

i

ð8Þ

where W 6 0 is the negative pressure head in the unsat-

urated soil, h is the soil moisture content, K (h) is a mean

value of the soil moisture dependent conductivity in the

unsaturated root zone i ¼ d z d x

is the slope gradient (the

vertical coordinate z is taken as positive upward). Set

h = hs Æ eaW and K = K s Æ e

aW = h Æ K s/hs [4], where a is

the inverse of the mean capillary rise and depends on

the soil characteristics [11,12]. a represents the ratio be-

tween the gravity and the capillary forces. Substituting

K = K (W) and h = h(W) into (8) yields to

dqd x

¼ Dw d2h

d x2þ V w dh

d x; ð9Þ

where Dw = K s/(hs Æ a) and V w = i Æ K s/hs.

By averaging q and h we obtain the horizontal flux Q

through the root zone depth H : Q ¼ R H 0

q d z , and the

averaged soil moisture content W ¼R

H

0 hd z the upper

limit of which W s is the extractable soil moisture and

may be estimated from the difference between the wet-

test and the dryest profile in a time series of profile soil

moisture data. W s depends on the soil type and on the

vegetation type since deeper rooted vegetation has ac-

cess to a larger depth H . Here the extractable soil mois-

ture is set approximately W s = hsH . Taking into account

the vertical flux of water through a plane element, due to

the net rainfall supply, evapotranspiration and leakage,

we find an expression similar to the first of Eq. (1),

where V w = V , and the additional term Dwd2W d x2

appears

on the right side accounting for soil moisture diffusion

due to tension gradients.

958 N. Ursino / Advances in Water Resources 28 (2005) 956–963


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The destabilizing advective term that appears in the

model of [7] Eq. (2) has been demonstrated to be the

component of the subsurface flow in the direction paral-

lel to the hill slope. It depends on the nature of the lower

boundary of the root zone which is investigated here. An

impermeable boundary deviates the whole infiltrating

soil moisture, whereas, if the rain infiltrates into a slopedsoil of infinite extension, the subsurface flow parallel to

the slope reduces to the projection of the vertical flux in

the direction of the slope. In order to properly account

for the transverse loss of moisture, the leakage must

be carefully estimated and expressed as a function of

the soil properties.

The dimensionless form of the new pair of water and

vegetation balance equations is the following:

ow

ot ¼ a w rwn2 þ v ow

o x þ d o

2w

o x2

on

ot ¼ rwn2

mn þo

2n

o x2

ð10Þ

where d = DwD1. In the new water balance equation

(the first of (10)) both dimensionless numbers v and d

are linked to the soil properties K s and a, since

d = K s(hs Æ a)1D1 and v ¼ i K sh1s ð LDÞ1=2. When the

capillary forces are negligible and a becomes very

large, d tends to 0 and the proposed model (10) for-

mally reduces to the model of Klausmeier [7] (2). The

diffusive term allows instability to appear also on flat

ground.

The eigenvalues k as a function of the wave number k

are still given by (6), where now

D ¼ d 00 1

; ð11Þ

whereas A and V (Eq. (3)) and the steady state solution

(w0, n0) remain unchanged.

The critical stability condition upon the real part of

the growth rate Rek = 0, identifies the boundary be-

tween stable and unstable conditions. It may be ex-

pressed in terms of the correspondence between two or

more soil and/or vegetation parameters, such as i andA in the cases presented here. It allows the evaluation

of the critical wave number k that has maximum growth

rate Rek = 0, and is derived below (according to Rovin-

sky and Menzinger [17]).

Rek ¼ lw þ hn ðd þ 1Þk 2 þ 1 ffiffiffi2

p q2 þ p 2 1=2 þ qh i1=2

ð12Þwhere Rek is the real part of the growth rate k,

p ¼ 2kv lw hn d 1ð Þk 2

;

and

q ¼ lw hn ðd 1Þk 2 2 þ 4lnhw k 2v2.

Based on (3): lw ¼ ð1 þ rn20Þ, l n = 2m, hw ¼ rn20 andhn = m.

Fig. 1 illustrates the significance of the critical stabil-

ity condition Rek = 0. The linear stability analysis dem-

onstrated in this section distinguishes between stable

and unstable scenarios on the base of the growth rate

of a very small initial disturbance, but says nothing

about the resulting finite size pattern. Fig. 1 shows the

growth rate curves (12) at the critical stability condition

(A = 374 mm, and i = 0.02), below (A > 374 mm, and

i < 0.02), and beyond (A < 374 mm, and i > 0.02). Belowthe critical stability condition all the perturbation modes

are stable since Rek is always negative. Beyond, a

Fig. 1. Growth coefficient Rek as a function of wavelength k . Depending on A and i no wavelength may lead to instability (Rek < 0), or a continuous

range of unstable modes may do this. At the critical stability condition only one k has Rek = 0; the other modes are stable. In the case shown here, the

parameters that characterize the marginal stability condition are A = 374 mm, and i = 0.02. The other relevant parameters are M = 1.8 y1,

J = 0.003 kg m2 mm1 y1; R = 100 m4 kg2 y1, D = 1 m2 y1, L = 4 y1, hs = 0.4, H = 0.1 m, K s = 3000 m y1 and a = 100 m1.



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continuous range of unstable modes may arise if stimu-

lated. In the next section, the focus is on the pair A – i

leading to the marginal stability condition for different

biomass and soil parameters.

3. Results

A sensitivity analysis of the critical stability condition

(12) for soil parameters was developed. All variables

were set according to literature data or reasonable

assumptions.

Rainfall in semi-arid regions is between50 and 750 mm

y1 [24]. A is set in this range. Klausmeier [7] assumes for

grass M = 1.8 y1; J = 0.003 kgm2mm1y1; R =

100 m4kg2y1 (obtained from the expression for equi-

librium plant biomass); D = 1 m2y1, and L = 4 y1.

Thus, m = 0.45 and r = 0.36. The maximum stored soil

moisture W s

hsH depends on the active soil depth H .

If hs = 0.4, and H = 0.1 m, then W s = 40 mm. In the depth

averaged water balance only the transverse conductivity

comes into play. It was set K s = 3 · 102 and 3 ·

103 m y1 (for more clayey and more sandy soils respec-

tively). Rather low wettability of the bare zones and lat-

eral diffusion due to capillary forces are expected to

support the so called ‘‘runoff runon’’ mechanism of water

redistribution, which is responsible for the formation of

vegetation patterns. Thus, very high values of the ratio be-

tween gravity and capillary forcesa have been considered:

a = 10, 100, 1000 m1. Klausmeier [7] does not take into

account soil properties and assumes v = 182.5 and d = 0.

Here, for i = 4%, L = 4 y1

, and a = 100 m1

,v ¼ iK sh1s ð LDÞ1=2 ¼ 16, 160 and d = K s(hsaD)1 = 7.5,75 depending on K s (case 1). Setting L = 2 y

1, v = 22,

220 and d = 11, 110 (case 2). Assuming that a = 1000,

which indicates negligible capillary forces acting on soil

moisture, the proposed model reproduces the results pre-

viously obtained by Klausmeier [7] neglecting soil mois-

ture diffusion.

The critical slope i is plotted in Fig. 2 as a function of

the net precipitation A for different K s and a. It is worth-

while repeating here that on sites characterized by a hill-slopes below the critical i no banded vegetation patterns

are expected to appear, and that the threshold gradient

slope has been observed to increase with the mean rain-

fall intensity. Fig. 2 demonstrates that the results of the

proposed model are in agreement with this experimental

evidence. Furthermore, the range of precipitation that

induces the formation of banded vegetation extends to-

wards the higher A with increasing saturated conductiv-

ity K s (curves on the left side of Fig. 2 versus

corresponding curves on the right side). This result

could be expected since K s influences both: the advective

and the diffusive terms in the water balance Eq. (10). In-

deed advection and diffusion contribute toward the

reduction of the already scarce soil moisture.

The influence of the newly introduced diffusive term

may be inferred confronting case a = 1000 m1 (circles)

with the other two: a = 10 and 100 m1 (squares and tri-

angles, respectively). For gentle hill slopes, soils charac-

terized by lower a values reach the critical stability

condition at higher A (e.g. triangles and squares for

i < 0.04). Conversely, for steeper slopes, the critical sta-

bility condition is reached at higher net precipitation by

the soils characterized by larger a. It may otherwise be

stated that for very low A the critical stability is reached

at gentle slope i when a is low; conversely, for higher Athe soil with larger a shows an unstable behavior at steep

slopes. Fig. 2 demonstrates that the lower a values

(a = 10) are associated with unstable behavior also for

Fig. 2. Case 1. Grass critical stability condition for different slope gradients i and net inflow A. () Experimental data taken from literature. Opensymbols: analytical solutions of the marginal stability condition problem Rek = 0 (12). Water loss due to evaporation and leakage per unit soil

moisture L = 4 y1 and coefficient of plant uptake R = 100 m4 kg2 year 1. Soil parameters: saturated conductivity K s = 3000 m y1 (left) and

K s = 300 m y1 (right); a = 1000 m1 (n); a = 100 m1 (

) a = 10 m1 (h).



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i = 0 in the lower range of precipitation (between a min-

imum Amin and the intersection of the marginal stability

condition with the axis i = 0). In this case, instability is

impelled by capillary diffusion only (d is maximum when

a is minimum). In general, the influence of K s, a and i on

the critical stability condition is more evident for large

K s (left side of Fig. 2).In Fig. 2 the analytical model results are plotted with

the critical stability conditions observed by different

authors at different sites in Africa and Australia (bold

circles in Fig. 2). All the experimental data are referred

by Valentin et al. [24] in a review paper. Unfortunately

the sparse details concerning the soil characteristics re-

ported in the literature references do not allow us to

accurately parameterize our model. At any rate, the

hypothesis of homogeneous soil adopted here for the

sake of simplicity sounds rather unrealistic, and thus it

does not justify a very accurate description of the sub-

soil, other than in terms of homogeneous, effective

parameters. The effective parameters are meant here to

be those values that somehow match the experimental

evidence and thus make the model soil behavior similar

to the observed one. Indeed, only the processes in action

and the relevant parameters are discussed, whereas the

exact comparison of the experimental data with the the-

oretical prediction is retained to be a complementary

target of this study. Further, the data shown are taken

from diverse experimental sites, requiring, in principle,

separate interpolations. Many of the experimental data

shown in Fig. 2 match poorly with the theoretical pre-

diction, since the experimental precipitation rate A is

below the theoretical minimum meaningful precipita-tion: Amin = 2 mr

1/2 (see the constraint for w0 and n0to be real). In order to overcome this limitation, a sec-

ond case was tackled. R and L have been adjusted main-

taining the same ratio LR1 in order to leave the plant

biomass unchanged [see Klausmeier [7] for more details].

The critical condition, obtained for L = 2 y1 and

R = 50 m4 kg2 y1 is shown in Fig. 3. A better corre-

spondence between model results and field data is

achieved. The influence of K s and a on the stability con-

dition remains the same as in the first case (Fig. 2) and

similar comments apply to Figs. 2 and 3.On slopes steeper than 0.2% in arid regions, typical

vegetation bands with a width in the range of a few doz-

ens of meters are observed (see [24] and [15] for a re-

view). Lefever and Lejeune [8] report on the evidence

of an inverse correlation between the band wavelength

and the ground slope, and of a direct correlation be-

tween band length and aridity. Figs. 4 and 5 show the

unstable vegetation mode k versus A corresponding to

the critical stability condition plotted in Figs. 2 and 3.

k in the range 0.32–0.48 corresponds to bands of width

2pk 1D1/2L1/2 in the range 26–39 m for L = 4 y1 (Fig.

4); and 20–28 m for L = 2 y1 (Fig. 5). The evaluated

critical mode appears to be reasonably in agreement

with the observed band characteristic length. There is

no unique trend in the vegetation critical mode k as a

function of the net rainfall amount A. Indeed, depend-

ing on K s and a, k increases with A, decreases, presents

a maximum or a minimum. Conversely, R has minor

influence on k , in fact Figs. 4 and 5 show substantially

similar results.

4. Discussion and conclusions

A new model for linear stability analysis of vegetationpatterns was derived and solved analytically in order to

obtain the critical stability condition of the soil moisture

and biomass balance equations. The model explicitly

takes into account two commonly referred soil proper-

Fig. 3. Case 2. Grass critical stability condition for different slope gradients i and net inflow A. () Experimental data taken from literature. Opensymbols: analytical solutions of the marginal stability condition problem Rek = 0 (12). L = 2 y1; R = 50 m4 kg2 year1. K s = 3000 m y

1 (left) and

K s = 300 m y1 (right); a = 1000 m1 (n); a = 100 m1 (

) a = 10 m1 (h).



7/8

ties: the soil saturated conductivity and the ratio be-

tween the gravity and the capillary forces. The model

is based on the Klausmeiers [7]. All terms and parame-

ters were set accordingly, except those related to subsur-

face flow. The model results were compared with

literature data. In order to fit the experimental data,

the soil parameters had to be stretched beyond the usual

range. The result suggests that the subsurface processes

that have a relevance for banded vegetation formation

may not have been captured and other relevant pro-

cesses must be taken into account. At least three terms

within the soil moisture balance should be parameter-

ized with care: net vertical inflow, leaching, and lateral

destabilizing flow. These depend on the upper and lower

boundary conditions that do not come into play within

the horizontal soil moisture balance. In this case, the

effective parameters are the upscale keys to the scale of

the soil moisture balances relevant processes taking

place at smaller scales.

Despite the limitations discussed above, the results

support some relevant phenomenological evidence: (i)

the threshold slope increases with effective rainfall; (ii)

lateral soil moisture diffusion (that may be impeded by

the low wettability of the bare zones) has a destabilizing

effect (leading to a lower threshold gradient) in case of

low precipitation, whereas, conversely, the effect is stabi-

lizing in the case of higher precipitation (leading to a lar-

ger threshold gradient); (iii) the soils characterized by

larger saturated conductivity are more prone to the

appearance of unstable plant development (the instabil-

ity field is broader); (iv) different ratios a between the

gravity and the capillary forces have a minor impact

on the behavior of the less conductive soils. Only point

(i) has already been proven experimentally.

Due to the many simplifying hypotheses that have

been postulated, it seems rather ambitious to pretend

that the model prediction fully matches the manifold

experimental data taken from literature. Indeed they

Fig. 5. Case 2. Grass critical wave length for different slope gradients i and net inflow A. L = 2 y1; R = 50 m4 kg2 year1. Soil parameters:

saturated conductivity K s = 3000 m y1 (left) and K s = 300 m y

1 (right); a = 1000 m1 (n); a = 100 m1 () a = 10 m1 (h).

Fig. 4. Case 1. Grass critical wave length for different slope gradients i and net inflow A. L = 4 y1 and R = 100 m4 kg2 year1. Soil parameters:

saturated conductivity K s = 3000 m y1 (left) and K s = 300 m y

1 (right); a = 1000 m1 (n); a = 100 m1 () a = 10 m1 (h).



8/8

relate to different heterogeneous scenarios. Nevertheless,

the comparison between theoretical and experimental

data suggests that effective soil parameters may be found

resulting in a good fit. A major question concerning the

effective parameters is whether they could be physically

supported. The early results presented here induce the

belief that such a physical basis may exist.The model introduced and discussed here was used to

speculate on the impact of soil properties on unstable

environments. Thus, it has been kept as simple as possi-

ble. Nevertheless, the impact that soil parameters may

have on the stability condition, has been clearly demon-

strated. The model may be led closer to reality by adding

new degrees of complexity, such as the consideration of

soil heterogeneity or non-averaged over depth soil

parameters. Nevertheless, in its present form, it evi-

dences the crucial role of soil physics in plant develop-

ment. It may seem rather obvious, but this point is

often neglected. The lack of relevant elements makes

the predictive power of the model less quantitative than

it might be. In order to achieve a predictive power, it

should be coupled, at least, with the surface water and

nutrients balance equations.

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