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MATHEMATICAL MODELS OF VEGETATION

PATTERN FORMATION IN ECOHYDROLOGY

F. Borgogno,1 P. D’Odorico,2 F. Laio,1 and L. Ridolfi1

Received 13 November 2007; accepted 24 October 2008; published 18 March 2009.

[1] Highly organized vegetation patterns can be found in anumber of landscapes around the world. In recent years,several authors have investigated the processes underlyingvegetation pattern formation. Patterns that are inducedneither by heterogeneity in soil properties nor by the localtopography are generally explained as the result of spatialself-organization resulting from ‘‘symmetry-breakinginstability’’ in nonlinear systems. In this case, the spatialdynamics are able to destabilize the homogeneous state of

the system, leading to the emergence of stableheterogeneous configurations. Both deterministic andstochastic mechanisms may explain the self-organizedvegetation patterns observed in nature. After an extensiveanalysis of deterministic theories, we review noise-inducedmechanisms of pattern formation and provide someexamples of applications relevant to the environmentalsciences.

Citation: Borgogno, F., P. D’Odorico, F. Laio, and L. Ridolfi (2009), Mathematical models of vegetation pattern formation in

ecohydrology, Rev. Geophys., 47, RG1005, doi:10.1029/2007RG000256.

Ecohydrology is the science, which seeks to describe the hydrologic

mechanisms that underlie ecologic patterns and processes.

[Rodriguez-Iturbe, 2000, p. 3]

1. INTRODUCTION

[2] In many landscapes around the world the vegetation

cover is sparse and exhibits spectacular organized spatial

features [e.g.,Macfadyen, 1950b] that can be either spatially

periodic or random. Commonly denoted as ‘‘vegetation

patterns’’ [e.g., Greig-Smith, 1979; Lejeune et al., 1999],

these features can be found in many regions around the

world, including Somalia [Macfadyen, 1950b; Boaler and

Hodge, 1964], Burkina Faso and Sudan [Worrall, 1959,

1960; Wickens and Collier, 1971], South Africa [van der

Meulen and Morris, 1979], Niger [White, 1970; Adejuwon

and Adesina, 1988], Australia [Slatyer, 1961; Mabbutt and

Fanning, 1987; Burgman, 1988; Tongway and Ludwig,

1990; Ludwig and Tongway, 1995], Mexico [Cornet et al.,

1988; Montana et al., 1990; Acosta et al., 1992; Mauchamp

et al., 1993], United States [Fuentes et al., 1986], Argentina

[Soriano et al., 1994], Chile [Fuentes et al., 1986], Japan

[Sato and Iwasa, 1993], and Jordan [White, 1969]. Vegeta-

tion patterns are often undetectable on the ground but

became visible with the advent of aerial photography

[e.g., Macfadyen, 1950b]. Figures 1–4 show some exam-

ples of spectacular spatially periodic vegetation patterns that

can be found especially in arid and semiarid landscapes

around the world. These patterns exhibit amazing regular

configurations of vegetation stripes or spots separated by

bare ground areas. In some cases, patterns may spread over

relatively large areas (up to several square kilometers)

[White, 1971; Eddy et al., 1999; Valentin et al., 1999;

Esteban and Fairen, 2006] and can be found on different

soils and with a broad variety of vegetation species and life

forms (i.e., grasses, shrubs, or trees) [Worrall, 1959, 1960;

White, 1969, 1971; Bernd, 1978; Mabbutt and Fanning,

1987;Montana, 1992; Lefever and Lejeune, 1997; Bergkamp

et al., 1999; Dunkerley and Brown, 1999; Eddy et al., 1999;

Valentin et al., 1999].

[3] The study of vegetation patterns is motivated by their

widespread occurrence in dryland landscapes and by the

possibility to infer from their presence and features useful

information on the underlying processes, including the

susceptibility of the system to abrupt shifts to a desert

(i.e., unvegetated) state as a result of climate change or

anthropogenic disturbances [e.g., van de Koppel et al.,

2002; D’Odorico et al., 2006c]. However, there is no doubt

that the beauty of some natural patterns of vegetation

contributed to draw the attention of a number of scientists,

who remained fascinated by their breathtaking natural

features and, thus, engaged themselves in the observation,

understanding, and modeling of these spatially organized

distributions of vegetation.

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1Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnicodi Torino, Turin, Italy.

2Department of Environmental Sciences, University of Virginia,Charlottesville, Virginia, USA.

Copyright 2009 by the American Geophysical Union.

8755-1209/06/2007RG000256$15.00

Reviews of Geophysics, 47, RG1005 / 2009

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Paper number 2007RG000256

RG1005

http://dx.doi.org/10.1029/2007RG000256

[4] Early studies on vegetation patterns began to appear

in the 1950s and 1960s [e.g., Macfadyen, 1950a, 1950b;

Worrall, 1959, 1960; Boaler and Hodge, 1962, 1964;

Greig-Smith and Chadwick, 1965] and became increasingly

popular in recent years [e.g., Lefever and Lejeune, 1997;

Klausmeier, 1999; Lejeune and Tlidi, 1999; Couteron and

Lejeune, 2001; Buceta and Lindenberg, 2002; D’Odorico et

al., 2007b] (see also section 4 for more references). Two

major approaches have been followed in the study of

vegetation patterns, depending on whether the focus was

on their qualitative empirical description and characteriza-

tion or on the mechanistic understanding of the key pro-

cesses determining pattern formation.

[5] The first group of studies concentrated on the

qualitative analysis of vegetation patterns [e.g., Worrall,

1959, 1960; Boaler and Hodge, 1962, 1964; Greig-Smith

and Chadwick, 1965; Greig-Smith, 1979; Adejuwon and

Adesina, 1988; Burgman, 1988; Acosta et al., 1992; Aguiar

and Sala, 1999] and recognized the recurrence of some

main types of spatial configurations, exhibiting organized

distributions of either stripes, spots, or gaps. Stripes consist

of an alternation of fairly regular vegetated bands with stripes

Figure 1. Example of aerial photographs showing vegetation patterns (tiger bush). (a) Somalia (9�200N,48�460E), (b) Niger (13�210N, 2�50E), (c) Somalia (9�320N, 49�190E), (d) Somalia (9�430N, 49�170E),(e) Niger (13�240N, 1�570E), (f) Somalia (7�410N, 48�00E), (g) Senegal (15�60N, 15�160W), and(h) Argentina (54�510S, 65�170W). Google Earth imagery # Google Inc. Used with permission.

RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION

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of bare soil (Figure 1). Depending on the topography and

other external conditions (wind direction, light exposure, or

land degradation) the stripes can be perpendicular or parallel

to the slope [Maestre et al., 2006]. When stripes emerge on

flat terrains, they become less regular and exhibit Y-shaped,

arc-shaped, or labyrinthine patterns (Figure 2). In particular,

arc-shaped stripes have been named ‘‘brousse tigree’’ [Clos-

Arceduc, 1956] or ‘‘tiger bush’’ [Bromley et al., 1997;

Hiernaux and Gerard, 1999; Couteron et al., 2000] because

of their resemblance to the tiger’s coat [Macfadyen, 1950a,

1950b] (see Figures 1a, 1b, 1e, and 1f). It has been argued that

stripes emerging on hillslopes tend tomigrate uphill [Worrall,

1959;Hemming, 1965;Montana, 1992; Valentin et al., 1999;

Sherratt, 2005], though their slow migration rate may limit

our ability to provide conclusive experimental evidence in

support of this mechanism. Spots and gaps can be viewed as

two complementary configurations. In fact, spots are little

round-shaped aggregations of vegetation interspersed within

a bare soil background (Figure 3), while gaps are round-

shaped bare soil islands surrounded by relatively homoge-

neous vegetation (Figure 4). Both spots and gaps can be

arranged in randomly distributed or more regular configu-

rations.

[6] In addition to a qualitative description, a number of

authors have provided a quantitative characterization of

vegetation patterns, often based on a variety of indices

and parameters as descriptors of the geometry of vegetated

soil patches and of their spatial arrangement [e.g., Dale,

1999]. These indicators are often used to classify the

different types of configurations and relate them to land-

scape or climate variables [e.g., Couteron, 2002; Barbier et

al., 2006; Caylor and Shugart, 2006; Okin et al., 2006]. For

example, several authors have used some of these geomet-

rical parameters (i.e., wavelength of stripes, bandwidths,

and periodicity of spots) to investigate the association

between pattern shape and mean annual rainfall, tempera-

ture, ground slope, wind, or other topographic variables

[Gunaratne and Jones, 1995; Perry, 1998; Giles and Trani,

1999; Okin and Gillette, 2001; Augustine, 2003; Webster

and Maestre, 2004]. This empirical approach is useful to

shed light on the relation between pattern geometry and the

‘‘external’’ environmental conditions. For example, it

allows one to predict the type of pattern that is more likely

to emerge under given climate, soil, and topographic con-

ditions or to understand how vegetation patterns are

expected to change in response to changes in the external

drivers. Moreover, these empirical studies provide some

criteria to test mathematical models of pattern formation

through the comparison of their results with ‘‘real-world’’

observations.Figure 2. Example of aerial photographs showing vegeta-tion patterns (labyrinths). (a) Senegal (15�220N, 15�210W)and (b) Senegal (15�140N, 15�70W). Google Earth imagery# Google Inc. Used with permission. (c) Aerial obliquephotograph of vegetation patterns from SW Niger (courtesyof Nicolas Barbier, Oxford University).

Figure 3. Example of aerial photographs showing vegeta-tion patterns (spots). (a) Zambia (15�380S, 22�460E),(b) Australia (15�430S, 133�100E), and (c) Australia(16�140S, 133�100E). Google Earth imagery # GoogleInc. Used with permission.


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[7] The second group of studies investigated the physical

mechanisms of pattern formation in vegetation and their

response to changes in environmental conditions and dis-

turbance regime. These studies related vegetation patterns to

underlying ecohydrological processes, mechanisms of spa-

tial redistribution of resources [e.g., Klausmeier, 1999;

Barbier et al., 2008; Ridolfi et al., 2008], the nature of

the spatial interactions existing among plant individuals

[e.g., Lefever and Lejeune, 1997; Zeng and Zeng, 2007;

Barbier et al., 2008], the stability and resilience of dryland

ecosystems [Rietkerk et al., 2002; van de Koppel and

Rietkerk, 2004], and the landscape’s susceptibility to de-

sertification under different climate drivers and management

conditions [e.g., von Hardenberg et al., 2001; D’Odorico et

al., 2006c]. Because vegetation patterns are observed even

when topography and soils do not exhibit any heterogeneity,

their formation represents an intriguing case of self-orga-

nized biological systems, which results from completely

intrinsic vegetation dynamics [Lejeune et al., 1999]. This

fact is particularly manifest in the case of periodic patterns

emerging in systems that do not display periodicity in

topography, landforms, or the spatial distribution of other

environmental drivers.

[8] To analyze these intrinsic processes and their role in

the emergence of vegetation patterns, it is important to

capitalize on the understanding of pattern-forming mech-

anisms gained in other fields, such as biology and physics.

In fact, the understanding of mechanisms frequently in-

voked to explain the formation of self-organized patterns

in vegetation originated from studies in other fields,

including fluid dynamics (e.g., the Rayleigh-Bernard con-

vection [Chandrasekhar, 1961; Cross and Hohenberg,

1993], convection in fluid mixtures [Platten and Legros,

1984], or the Taylor-Couette flow [DiPrima and Swinney,

1981]), electrodynamics (e.g., instabilities in nematic

liquid crystals [Dubois-Violette et al., 1978]), chemistry

(morphogenesis in chemical reactions [Turing, 1952]),

and biology (morphogenesis, patterns on animals’ coats

or skin, pigment patterns on shells, and hallucination

patterns [Murray, 2002]). This broad body of literature

inspired a number of studies proposing a variety of

ecological models to explain the fundamental mecha-

nisms conducive to vegetation pattern formation. One of

the early examples of process-based analyses is given by

Watt [1947], who invoked, among others, mechanisms of

reallocation of nutrients and water to explain the emer-

gence of patchy vegetation covers: this model suggested

that as nutrient and water availability decrease, plant

individuals tend to grow in clumps. The emergence of

these aggregated structures is motivated by the need to

concentrate the scarce resources (e.g., soil moisture and

soil nutrients) in smaller areas, thereby increasing the

likelihood of vegetation survival within vegetated

patches that are richer in resources. In the subsequent

years the idea that the mechanisms underlying vegetation

pattern formation are intrinsically dynamic and originate

from interactions among plant individuals was better ar-

ticulated and formalized [e.g., Greig-Smith, 1979; Wilson

and Agnew, 1992; Thiery et al., 1995]. These studies

paved the way to a new generation of models explaining

vegetation patterns as the result of self-organization

emerging from symmetry-breaking instability, i.e., as a

process in which the existence of both cooperative and

inhibitory interactions at two slightly different spatial

ranges may induce the appearance of heterogeneous dis-

tributions of vegetation with wavelengths determined by

the interactions between the two spatial scales [Lefever

and Lejeune, 1997; Lejeune and Tlidi, 1999; Lejeune et

al., 1999; Barbier et al., 2006; Rietkerk and van de

Koppel, 2008]. Three major classes of deterministic

models explain pattern formation as the result of self-

organized dynamics conducive to symmetry-breaking

instability: models based on (1) Turing-like instability

(hereafter named ‘‘Turingmodels’’ [Turing, 1952]), (2) short-

range cooperative and long-range inhibitory interactions

among individuals (hereafter named ‘‘kernel-based mod-

els’’ because short- and long-range spatial interactions are

expressed through a kernel function (see section 2)), and

(3) instability induced by differential flow rates between

Figure 4. Example of aerial photographs showing vegeta-tion patterns (gaps). (a) Senegal (15�90N, 14�360W) and(b) Senegal (15�120N, 14�540W). Google Earth imagery #

Google Inc. Used with permission.


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two interacting species in an activator-inhibitor system

[Klausmeier, 1999]. We stress that these models differ only

in the mathematical description of the dynamics, while they

exhibit essentially the same mechanism of pattern forma-

tion. In fact, in all of them, patterns are induced by

symmetry-breaking instability in activation-inhibition sys-

tems. To further stress this point, in this paper we will

demonstrate that the first two types of models lead to

patterns that are qualitatively the same; that is, patterns

emerging from Turing models can be expressed as a

particular class of those generated by neural models.

[9] More recently, some stochastic models have been

developed, which explain vegetation patterns as noise-

induced effects. In this case patterns emerge as a result

of the randomness inherent to environmental fluctuations

and disturbance regime. We will review two major mech-

anisms of noise-induced pattern formation based either on

random switching between alternative dynamics or on

phase transitions with breaking of ergodicity in systems

driven by different types of noise. We will then discuss the

feedback mechanisms and spatial interactions commonly

invoked to explain the emergence of vegetation patterns.

The rest of the review provides a synopsis and a critical

discussion of the broad literature on the theory of process-

based pattern formation in landscape ecohydrology.

2. MECHANISMS OF PATTERN FORMATION

2.1. Deterministic Mechanisms of Pattern Formation

[10] In this section we provide a mathematical descrip-

tion of the three major deterministic models of self-

organized pattern formation that are commonly invoked

to explain the spatial organization of vegetation. These

three models invoke the same physical mechanism, i.e.,

symmetry-breaking instability. Spatial interactions induce

this instability, while the resulting patterns are stabilized

by suitable nonlinear terms. In Turing and kernel-based

models, symmetry-breaking is the result of the interactions

between short-range activation and long-range inhibition,

i.e., of positive and negative feedbacks acting at different

spatial scales [e.g., Rietkerk and van de Koppel, 2008]. In

the third class of models (i.e., differential flow models),

symmetry-breaking emerges as a result of the differential

flow rate between two (or more) species.

[11] In Turing and differential flow models the nonlinear-

ities are local (i.e., they do not appear in the terms express-

ing spatial interactions), while in kernel-based models the

nonlinearities can be, in general, nonlocal; that is, they can

appear as multiplicative functions of the term accounting for

spatial interactions [e.g., Lefever and Lejeune, 1997]. In a

particular class of kernel-based models, known as ‘‘neural

models’’ [e.g., Murray and Maini, 1989], the nonlinearities

are only local and do not affect the spatial interactions. In

these models the nonlinear terms appear as additive func-

tions of the spatial interaction term. Here we will describe

the Turing and kernel-based models separately because they

use a different mathematical representation of the spatial

dynamics. However, to stress that Turing and neural models

invoke similar mechanisms of morphogenesis (namely,

symmetry-breaking instability induced by spatial interac-

tions in activation-inhibition systems and stabilization by

local nonlinearities) in section 2.1.3 we will show the

relation existing between the analytical frameworks used

by these two models.

2.1.1. Turing-Like Instability[12] In the study of nonlinear chemical systems, Turing

[1952] found that the diffusion of two species (reagents)

may lead to pattern formation when they have different

diffusivities. In the absence of diffusion both species reach

a stable and spatially uniform steady state, while diffusion

may be able to destabilize this state (‘‘diffusion-driven

instability’’) leading to the formation of spatial patterns.

Known as ‘‘Turing’s instability,’’ this mechanism seems to

be counterintuitive. In fact, diffusion is usually believed to

act as a homogenizing process, leading to the dissipation

of concentration gradients of the diffusing species. Con-

versely, Turing’s [1952] model shows that diffusion may

lead to the emergence of spatial heterogeneity in the

coupled nonlinear dynamics of two diffusing species. In

the literature on symmetry-breaking instability in chemis-

try and biology the two diffusive species are often named

‘‘activator’’ and ‘‘inhibitor,’’ and pattern emergence

requires (1) nonlinear local dynamics and (2) the inhibitor

to diffuse faster than the activator [e.g., Meinhardt, 1982;

Murray and Maini, 1989].

[13] In this section we will present Turing’s [1952] model

and determine the conditions leading to Turing’s instability.

We will also stress that patterns emerging from this insta-

bility are self-organized, in that they originate from the

internal dynamics of the system and are not imposed by

heterogeneities in the external drivers. Thus, this mecha-

nism has been often invoked to explain the emergence of

self-organized patterns also in fields other than chemistry,

such as physics and biology, in systems with two or more

diffusing species. Notable examples include convection in

fluid mixtures [Platten and Legros, 1984], the formation of

shell patterns from pigment diffusion [Murray, 2002], and

vegetation pattern formation from diffusion-induced insta-

bility in arid landscapes [e.g., HilleRisLambers et al., 2001].

The emergence of natural patterns from Turing’s instability

has been experimentally demonstrated in a chemical system

[Castets et al., 1990] and in nonlinear optics [e.g., Staliunas

and Sanchez-Morcillo, 2000]. We are not aware of any

similar experiment for the case of vegetation patterns. Thus,

although models based on Turing’s instability are capable of

generating vegetation patterns which resemble those ob-

served in nature, there is no conclusive experimental evi-

dence suggesting that the organized spatial configurations

of vegetation observed in nature do emerge from Turing’s

dynamics (see also the discussion in section 5). One of the

major challenges in the application of Turing’s activator-

inhibition model to the field of landscape ecohydrology

arises from the need to recognize two or more leading state

variables and to assess whether they do diffuse in space.

The diffusive character of the spatial dynamics of both

activator and inhibitor is fundamental to the development of


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a sound Turing-like model of pattern formation, in that

diffusion is crucially important to the emergence of sym-

metry-breaking instability in a Turing’s system.

[14] The mathematical description of Turing’s instability

will be presented here for the case of two species, u and v,

diffusing across a two-dimensional infinite domain {x, y}.

The dynamics of u and v are modeled by two differential

equations involving both diffusive terms and functions of

the local values of the state variables [e.g., Murray and

Maini, 1989; Murray, 2002; Henderson et al., 2004]:

@u

@t¼ f u; vð Þ þ r2u; ð1aÞ

@v

@t¼ g u; vð Þ þ dr2v; ð1bÞ

where t is time and f and g are the local reaction kinetics,

while d is the ratio, d = d2/d1, between the two diffusivities,

d1 and d2, of u and v, respectively, and r2() is the Laplace

operator (@2/@x2 + @2/@y2). All variables are in dimension-

less units.

[15] Turing [1952] demonstrated that this diffusive sys-

tem exhibits diffusion-driven instability if (1) in the absence

of diffusion the homogeneous steady state is linearly

stable (i.e., stable with respect to small perturbations)

and (2) when diffusion is present the homogeneous steady

state is linearly unstable. Thus, we first need to determine

the homogeneous steady state (u0, v0) as the solution of

equations (1) with r2u = r2v = 0 (homogeneous state)

and @u/@t = @v/@t = 0 (steady state); therefore f(u0, v0) = 0

and g(u0, v0) = 0. Then, we need to impose the condition

that this solution is stable in the absence of diffusion. To

this end, we can study the stability of (u0, v0) with respect

to small perturbations,

w ¼ u� u0v� v0

� �; ð2Þ

around the steady state. For small perturbations of the

steady homogeneous state (i.e., for jwj ! 0) the system (1)

can be linearized around (u0, v0). Using a linear Taylor’s

expansion we have

@w

@t¼ Jw; J ¼

@f@u

@f@v

@g@u

@g@v

0@

1A

u0 ;v0

; ð3Þ

where J is the Jacobian of the dynamical system (1).

[16] The solutions of this system are in the form w(x, y, t)/ est and express the temporal evolution of the perturbation

of the homogeneous steady state, where s is an eigenvalue

of the system (1), i.e., a solution of the secular polynomial

jJ � sI j ¼ 0; ð4Þ

with I being the identity matrix.

[17] When the real part of s, Re[s], is negative,

jwj tends to zero for t ! 1, and the steady homogeneous

state, (u0, v0), is linearly stable with respect to small

perturbations. From the analysis of equation (4) we obtain

that this condition is met when

@f

@uþ @g

@v< 0 and

@f

@u

@g

@v� @f

@v

@g

@u> 0; ð5Þ

with all derivatives being calculated in (u0,v0) [Murray,

2002].

[18] To study the effect of diffusion on the stability of (u0,

v0), we consider the full system (1) and use a Taylor’s

expansion to linearize this set of equations around the

homogeneous steady state, (u0, v0):

@w

@t¼ Jwþ Dr2w; D ¼ 1 0

0 d

� �: ð6Þ

[19] The solution of the system (6) can be written in the

form of a sum of Fourier modes,

w r; tð Þ ¼ Wkestþik�r; ð7Þ

with k = (kx, ky) being the wave number vector, r = (x, y)

being the coordinate vector, and Wk being the Fourier

coefficients [Murray, 2002].

[20] The relation between eigenvalues and wave numbers

(known as ‘‘dispersion relation,’’ see Figure 5) can be

obtained inserting equation (7) into (6) and searching for

nontrivial solutions. Setting k =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k2y

q, one obtains

jsI � J þ Dk2j ¼ 0: ð8Þ

[21] For the state (u0, v0) to be unstable with respect to

small perturbations, the solution of the dispersion relation

(8) should exhibit positive values of Re[s(k)], for some

wave number k 6¼ 0. Using equation (8), it is possible to

demonstrate that this condition is met when

d@f

@uþ @g

@v> 0 and

d@f

@uþ @g

@v

� �2

�4d@f

@u

@g

@v� @f

@v

@g

@u

� �> 0: ð9Þ

[22] The first of equations (5) combined with the first of

equations (9) implies that d 6¼ 1, indicating that the system

cannot be unstable with respect to small perturbations if

both species have the same diffusivity. If all four conditions

(5) and (9) hold, at least one eigenfunction is unstable with

respect to small perturbations and grows exponentially with

time as a consequence of the destabilizing effect of diffu-

sion. The dispersion relation (8) imposes a specific link

between eigenvalues, s, and wave numbers, k. The wave

number, kmax, corresponding to a maximum positive value

of Re[s] represents the most unstable mode of the system.

This implies that if Re[s(kmax)] > 0, this mode grows faster

than the others, and the state of the system for t ! 1 is

dominated by kmax, in the sense that as t ! 1, only kmax

dictates the length scale of the spatial pattern.


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[23] Because this linear stability analysis is developed in

the limit jwj ! 0 (i.e., under the assumption of small

perturbations of the homogeneous steady state), it cannot

provide any information on the state of the system when the

perturbation grows in amplitude. In the absence of non-

linearities in f(u, v) and g(u, v) the solutions (7) of the

linearized model would coincide with the exact solutions of

system (1) even away from the state (u0, v0). In this case,

equation (7) clearly shows that if the state (u0, v0) is

unstable the perturbations, w, grow indefinitely. Thus,

suitable nonlinear terms are needed to stabilize the pattern

through higher-order terms in the Taylor’s expansion, which

become important when the amplitude of the perturbation is

finite. In other words, the system reaches a steady config-

uration when the exponential growth of the eigenfunction is

limited by second-order terms (or higher) that come into

play once the perturbation has finite amplitude. In these

conditions the (nonlinear) stability of the system can be

partly studied through a more complex mathematical frame-

work based on the so-called amplitude equations, which

investigate the dynamics of the system in the neighborhood

of the most unstable mode. This review will not present

these nonlinear methods, and we refer the interested reader

to specific literature on this topic for further details [Cross

and Hohenberg, 1993; Leppanen, 2005].

[24] The above theory refers to the case of unconfined

spatial domains. In the case of confined (i.e., spatially

limited) systems the set of unstable eigenvalues is no longer

infinite, and only a discrete set of nonnull wave numbers

k 6¼ 0 may correspond to solutions of equation (6) with

Re[s (k)] > 0 (i.e., unstable perturbations of the homogeneous

steady state). Similarly to the case of unconfined domains,

the dynamics lead to pattern emergence when there are

unstable modes, and, again, the pattern geometry is deter-

mined by the wave number of the most unstable mode.

However, the effect of the finite size of the domain is

seldom accounted for in deterministic models of vegetation

self-organization, as these patterns typically stretch across

relatively large areas (several square kilometers) [White,

1971; Eddy et al., 1999; Valentin et al., 1999; Esteban and

Fairen, 2006] compared to the size of vegetation patches.

[25] We present here a simple example of an ecological

Turing model able to generate spatial patterns in a system

with two species, u (activator) and v (inhibitor). To this end,

we use equations (1) with local kinetic functions

f u; vð Þ ¼ u avu� eð Þ; ð10aÞ

g u; vð Þ ¼ v b� cu2v

; ð10bÞ

where a, b, c, and e are dimensionless positive constants.

[26] Equation (10a) describes the growth or the decay of

the activator and accounts for a positive interaction between

u and v. In fact, as v increases, the growth rate of species u

increases. Moreover, the growth rate of u increases with

increasing values of u. Equation (10b) is a generalized

logistic growth [e.g., Murray, 2002] with carrying capacity

b and a strong negative influence (inhibition) of species u

on the growth rate of v: in fact, as u increases, the second

term of the function g decreases as a second-order power

law.

[27] The homogeneous steady state of this system is u0 =

ab/ce and v0 = ce2/ba2. The derivatives of the two functions

(10) calculated in (u0, v0) are @f/@u = e, @f/@v = a3b2/c2e2,

@g/@u = �2g2e3/ba3, and @g/@v = �b, while the four

conditions (5) and (9) leading to diffusion driven instability

become e � b < 0, eb > 0, de � b > 0, and d2e2 � 6bde +

b2 > 0, respectively.

[28] Figure 6 shows an example in which these condi-

tions are met and patterns emerge from diffusion-driven

instability as a hexagonal arrangement of spots with wave-

length l ’ 2p/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1= 1� dð Þ½ � eþ b� 1þ dð Þ=d½ �

ffiffiffiffiffiffiffiffiffiffi2bde

p� �qin agreement with the wavelength of the most unstable

mode obtained through the dispersion relation (8).

2.1.2. Kernel-Based Models of Short-RangeCooperative and Long-Range Inhibitory Interactions[29] We classify as kernel-based models those modeling

frameworks in which spatial interactions are expressed

through a kernel function accounting both for short-range

and long-range coupling. In most models of self-organized

vegetation, patterns arise as a result of short-range cooper-

ation (or ‘‘activation’’) and long-range inhibition. In these

models, stable patterns emerge when spatial interactions cause

symmetry-breaking instability and the system converges to

an asymmetric state, which exhibits patterns. The conver-

gence to this state is due to suitable nonlinear terms, which

prevent the initial (linear) instability to grow indefinitely.

[30] A kernel-based model with multiplicative nonlinear-

ities (i.e., with nonlinearities embedded also in the spatial

interaction term) was developed by Lefever and Lejeune

[1997] to explain the formation of patterns in dryland

vegetation. This model will be discussed at the end of this

section. We first consider a particular type of kernel-based

models, whereby the nonlinearity is not in the spatial

coupling but in an additive term. These models are often

Figure 5. Generic dispersion relation for a two diffusivespecies monodimensional system. The extremes of therange of unstable Fourier modes are represented by k1 andk2, while kmax represents the most unstable Fourier mode.


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known as ‘‘neural models’’ because of their applications to

neural systems.

[31] Some of the most fascinating and complex pattern-

forming processes existing in nature are associated with

neural systems. Typical examples include the process of

pattern recognition, the transmission of visual information

to the brain, and stripe formation in the visual cortex

[Murray, 2002]. The framework of a neural model is

often used to represent other systems, including the case

of vegetation dynamics in spatially extended systems

[D’Odorico et al., 2006b].

[32] Neural models can, in general, be developed for

systems with more than one state variable. However, unlike

Turing models, pattern-forming symmetry-breaking insta-

bility can emerge even when the dynamics have only one

state variable. Thus we concentrate on the case of neural

models that are mathematically described usually by only

one state variable, say u, representing, for example, the

population density in a two-dimensional domain (x, y). At

any point, r = (x, y), of the domain the population density,

u(r), undergoes local dynamics (i.e., independent of spatial

interactions) expressed by a function, h(u), with a steady

state at u = u0 (i.e., h(u0) = 0). For the sake of simplicity we

will assume that the local dynamics exhibit only one steady

state. To express the effect of spatial interactions on the

dynamics of u, we account for the impact that individuals at

other points, r0 = (x0, y0), of the domain have on the

population density, u(r, t), at the location r. It is sensible

to assume that this impact depends on the relative position

of the two points r and r0. Because the strength of the

interactions with other individuals is likely to decrease with

the distance, a weighting function w(r, r0) is introduced to

describe how the effect of spatial interactions depends on r0

and r. We integrate r0 over the whole domain, W, to account

for the interactions of u(r, t) with individuals at any point r0

in W

@u

@t¼ h uð Þ þ

ZWw r; r0ð Þ u r0; tð Þ � u0½ �dr0: ð11Þ

[33] The right-hand side of (11) consists of two terms: the

first term, h(u), describes the local dynamics, i.e., the

dynamics of u that would take place in the absence of

spatial interactions with other points of the domain. The

second term expresses the spatial interactions and depends

both on the shape of the weighting function (or ‘‘kernel’’) and

on the values of u in the rest of the domain W. If w(r, r0) > 0,

the spatial interactions affect the dynamics of u(r) positively

or negatively depending on whether u(r0) is smaller or greater

than u0, respectively. The opposite happens when w(r, r0) < 0.

Notice how the dynamics expressed by (11) are not neces-

sarily bounded at u = 0, and a bound may need to be imposed

to ensure that u � 0, if in the model u represents population

density or vegetation biomass.

[34] When the processes underlying the spatial interac-

tions are homogeneous (i.e., they do not change from point

to point) and isotropic (i.e., they are independent of the

direction), the kernel function is independent of r and

exhibits axial symmetry. In this case, w is a function only

of the distance, z = jr0 � rj, between the two interacting

points (w(z) = w(jr0 � rj)). It will be shown that even thoughthe underlying mechanisms are homogeneous, they can lead

to pattern formation, i.e., to nonhomogeneous distributions

of the state variable.

[35] In neural models of pattern formation the interac-

tions between cells are typically represented by short-range

activation and long-range inhibition [Oster and Murray,

1989]. In this case the kernel is positive at small distances,

z, and becomes negative at greater distances (Figure 7). This

type of framework has been proposed as a model for spatial

interactions within plant communities [e.g., Lefever and

Lejeune, 1997; Yokozawa et al., 1999; Couteron and

Lejeune, 2001] in other kernel-based models. A kernel with

the shape illustrated in Figure 7 can be obtained, for

example, as the difference between two exponential func-

tions of the form

w zð Þ ¼ b1 exp � z

q1

� �2" #

� b2 exp � z

q2

� �2" #

; ð12Þ

with 0 < q1 < q2, while b1 and b2 are two coefficients

expressing the relative importance of the facilitation and

competition components of the kernel.

Figure 6. Spatial pattern emerging for the variable u in theTuring system in equations (10a) and (10b). The parametersare a = 22, b = 84, c = 113.33, e = 18, and d = 27.2. Theparameters a and c do not influence the emergence of spatialpatterns (see the end of section 2.1.1), while they influenceonly the shape of spatial patterns. The simulation is carriedout over a domain of 256 � 256 cells, each cell representinga spatial step Dx = Dy = 0.2. Equations are solvednumerically by means of finite difference method.


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[36] To intuitively understand how equation (11) with a

kernel w(z) shaped as in Figure 7 can lead to the emergence

of spatial patterns, we show how the spatial dynamics can

render unstable the spatially uniform steady state, u0,

similarly to the case of Turing’s instability discussed in

section 2.1.1. In fact, starting from a small heterogeneous

perturbation of the state, u0, each point, r, positively

interacts with the nearby points, r0, that are located at a

distance, z, such that w(z) > 0. Thus, small perturbations

with u > u0 tend to further increase u, while those with u <

u0 tend to decrease the value of u in the surrounding points,

thereby enhancing the heterogeneity. The integrated impact

of the interaction with all individuals in the neighborhood of

r may be able to induce pattern formation. While short-

range positive interactions activate the formation of patterns

through the instability of the uniform steady state, u0,

mechanisms of long-range inhibition represented by the

negative part of the kernel (Figure 7) prevent the perturba-

tion of the uniform state from growing indefinitely in space.

Thus, inhibition (along with suitable nonlinearities) is

needed to stabilize the pattern in a way that the perturbed

state can reach a steady configuration [Murray, 2002].

[37] To apply this framework to the case of vegetation

patterns we need to justify the use of a kernel with the shape

shown in Figure 7 and determine its parameters on the basis

of what is known about mutual interactions between plant

individuals. There is no doubt that these are challenging

tasks [Barbier et al., 2008]. The existing models of vege-

tation pattern formation invoking kernel-based activation-

inhibition frameworks [Lefever and Lejeune, 1997] recognize

that spatial interactions typical of dryland plant communities

exhibit short-range cooperative effects (facilitation) which

concentrate the resources in a relatively small area, thereby

providing more favorable conditions for plant establishment

and growth in the surroundings of existing plant individuals

[Rietkerk and van de Koppel, 2008] (Figure 8). At the same

time, as noted, long-range negative interactions are needed to

stabilize the pattern [Rietkerk and van de Koppel, 2008].

Thus, these models invoke root competition for water and

nutrients as the main mechanism of long-range inhibition.

Section 3 will provide more details on the ecosystem pro-

cesses determining short-range cooperation and long-range

competition.

[38] The dynamics expressed by equation (11) may lead

to pattern formation through mechanisms that resemble

those of Turing’s instability. In fact, patterns emerge as a

result of the spatial interactions, which destabilize the

uniform stable state, u0, of the local dynamics. To study

the stability of the state u = u0 with respect to infinitesimal

perturbations, we linearize equation (11) around the steady

state u = u0. Indicating with u = u � u0 the amplitude of the

(‘‘small’’) perturbation, we obtain

@u

@t¼ uh0 u0ð Þ þ

ZWw jr0 � rjð Þu r0; tð Þdr0; ð13Þ

where h0(u0) is the derivative of the function h(u), calculated

for u = u0.

[39] Solutions of equation (13) can be expressed in the

form of integral sums of the harmonics u(r, t) / exp[st +ik � r], where each harmonic is a solution of (13), k = (kx,

ky) is the wave number vector, and the growth factor, s, isan eigenvalue of equation (13). Substituting this solution

in equation (13), setting z = jr0 � rj, and canceling out the

exponential function, we obtain the dispersion relation,

that is, the relation between k and s in solutions of

equation (11) obtained as small perturbations of the state

u = u0,

s kð Þ ¼ h0 u0ð Þ þZWw zð Þ exp ik � z½ �dz ¼ h0 u0ð Þ þW kð Þ; ð14Þ

with k = jkj. If W is infinitely extended both in the x and y

directions, W(k) is the Fourier transform of the kernel

function. The dispersion relation obtained with the kernel

(12) is shown in Figure 9. Notice how the shape of the

dispersion relation is entirely determined by W(k), i.e., by

the effect of the kernel function on the spatial dynamics,

while the local dynamics affect equation (14) only through

the constant h0(u0). In fact, changes in this constant

Figure 7. Typical kernel which exhibits local activationand long-range inhibition.

Figure 8. Visualization of the positive and negativeinteractions typical of a tree.


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determine a vertical shift of the curves in Figure 9 without

modifying their shape. This vertical shift affects the sign

of s(k), thereby determining the stability/instability of

the system and the range of unstable modes. All modes

with s < 0 are linearly stable because they vanish as time,

t, passes. Conversely, all modes with s > 0 are linearly

unstable and tend to grow with time. However, even in

this case, when the amplitude of the unstable modes

becomes finite, the assumptions underlying this linear

stability analysis (i.e., that perturbations are ‘‘small’’/

infinitesimal) are no longer valid. Thus, the linear stability

analysis does not shed light on the state approached by the

system as an effect of the unstable modes. However, as

noted for Turing’s instability, the dominant wavelength of

patterns emerging from this instability is dictated by the

most unstable mode, kmax (which grows faster than the

other unstable modes, thereby determining some key

aspects of pattern geometry). This wavelength depends

only on the shape of the kernel function and is not affected

by the term h0(n0) (see equation (14)), even though h0(n0)

determines the stability of the system and the emergence

of spatial patterns: for relatively low values of h0(u0), s < 0

for all wave numbers k (see Figure 9), while as h0(u0)

increases above a critical value, s(kmax) becomes positive,

and the mode, kmax, is unstable. Larger values of h0(u0)

correspond to broader ranges of unstable wave numbers.

[40] Spatial interactions lead to pattern formation in

equation (11) when the following conditions are met:

[41] 1. In the absence of spatial interactions the uniform

steady state, u = u0, of the local dynamics is stable. The

linear stability analysis demonstrates that the stability of u =

u0 requires h0(u0) to be negative, as shown by equation (13)

when the integral term is set equal to zero.

[42] 2. In the presence of spatial interactions there should

be at least one wave number (kmax) associated (through the

dispersion relation) with a positive value of s.[43] 3. Because the mode k = 0 corresponds to a spatially

uniform perturbation of u = u0, instability does not lead to

the emergence of any spatial pattern if the most unstable

mode, kmax, is zero.

[44] Thus, in a neural model, patterns emerge from spatial

interactions when

h0 u0ð Þ < 0; W kmaxð Þ þ h0 u0ð Þ > 0; kmax > 0; ð15Þ

where kmax is the solution of W 0(kmax) = 0 with W 00(kmax) <

0. We also notice that if h(u) is a linear function of u,

equation (11) is also linear. Thus, solutions of equation (13)

are exact expressions (rather than approximations) of the

perturbed state of the system (u). In this case, because of the

linearity of (13), the perturbed state remains an exponential

function of s even when the amplitude of the perturbation is

no longer infinitesimal. In other words, if h(u) is linear and

the conditions (15) are met, the steady homogeneous state is

unstable, and any perturbation of u = u0 grows indefinitely

without ever reaching a steady configuration. Thus, a

suitable nonlinear function, h(u), is needed for the neural

model to have a steady state in which patterns emerge from

symmetry-breaking instability. In this case, as soon as the

initial perturbation of the steady homogeneous state grows

in amplitude, suitable nonlinear terms can come into play

and prevent the indefinite growth of the perturbation.

[45] In the particular case of the kernel function

expressed by (12) the dispersion relation becomes

s ¼ h0 u0ð Þ þW kð Þ

¼ h0 u0ð Þ þ pb1q21 exp � q21k2

2

� �� pb2q22 exp � q22k

2

2

� �; ð16Þ

while the most unstable mode is

kmax ¼ q1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln �c4ð Þc2 � 1

s; ð17Þ

with � = b2/b1 and c = q2/q1. The last two conditions (15)

can be rewritten as

h0 u0ð Þb1q

21

>p�c2 c2 � 1ð Þ

�c2ð Þc2

c2�1

and �c4 > 1: ð18Þ

[46] As noted, one of the first models of vegetation self-

organization [Lefever and Lejeune, 1997] used a kernel-

based framework that resembles that of equation (11), with

spatial interactions involving both short-range activation

and long-range inhibition. The model by Lefever and

Lejeune [1997] differs from a neural model in that the

nonlinearities are not strictly local but modulate the spatial

interactions.

[47] In some cases, the spatial interactions modulated by

the kernel function have only a limited effect (i.e., w(z)! 0)

at relatively large distances, z. Thus, depending on the

shape of w(z), conditions leading to pattern formation in

neural models can be formalized through a Taylor’s expansion

(for small values of z) of the integral term of equation (11)

to the fourth order. This approach leads to the so-called

Figure 9. Dispersion relation s(k) as a function of thewave number k for various values of the bifurcationparameter h0(u0). The critical value for h0(u0) that dis-criminates the situations of stability and instability isrepresented by ac.


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long-range diffusion (or biarmonic) approximation of the

neural model [Murray, 2002],

@u

@t� h uð Þ þ w0 u� u0ð Þ þ w2r2uþ w4r4u; ð19Þ

where r4 is the biarmonic operator (@4/@x4 + 2@4/@x2@y2 +@4/@y4), while wm are the mth-order moments of the kernel

function

wm ¼ 1

m!

ZWzmw zð Þdz; m ¼ 0; 2; 4; . . . ð20Þ

[48] In equation (19) we have assumed that the dynamics

are isotropic, i.e., that the kernel function has axial sym-

metry. Thus, because in this case the odd-order moments of

w(z) are zero, we have not included the odd-order terms in

the Taylor’s expansion. Table 1 reports the moments of w(z)

for the case of the kernel function (12) in one- and two-

dimensional domains.

[49] Because the moment w2 multiplies the Laplacian of

u, it modulates the effect of ‘‘short-range diffusion,’’ while

the moment w4 multiplies the biarmonic term, which

accounts for long-range interactions (‘‘long-range diffu-

sion’’). It can be shown that the diffusion term alone is

unable to lead to persistent patterns [e.g.,Murray, 2002] and

that the biarmonic term is needed in the series expansion to

obtain (with equation (19)) patterns that do not vanish with

time. In fact, the linear stability analysis of the state u = u0with respect to a perturbation J(r, t) / est+ik�r leads to the

dispersion relation

s ¼ h0 u0ð Þ þ w0 � 2w2k2 þ 4w4k

4: ð21Þ

[50] In the absence of the long-range diffusion (biar-

monic) term (i.e., when w4 = 0), the most unstable mode,

kmax, is zero, and no patterns emerge. In the case of the

biarmonic equation (19) (i.e., when w4 6¼ 0), the most

unstable mode can be easily obtained from equation (21)

as kmax =12

ffiffiffiffiffiffiffiffiffiffiffiffiffiw2=w4

p. Patterns emerge when kmax is real and

different from zero (i.e., w2 and w4 need to have the same

sign), and s(kmax) > 0,

s kmaxð Þ ¼ h0 u0ð Þ þ w0 �w22

4w4

> 0: ð22Þ

[51] In addition, the stability of u = u0 in the absence of

spatial dynamics requires h0(u0) to be negative as in the first

of equations (15). Moreover, in most ecohydrological appli-

cations u is always nonnegative. This condition is met when

w0 < 0. Because, in this case, w0 and h0(u0) are both

negative, equation (22) combined with the requirement that

w2 and w4 have the same sign imply that pattern formation

occurs only if w2 and w4 are also negative. However, the

condition that w0, w2, and w4 are negative is only necessary

and not sufficient for pattern formation as the condition (22)

would still need to be met for the instability to emerge.

[52] An ecohydrologic neural model of vegetation pat-

tern formation is given by D’Odorico et al. [2006b], where

a typical kernel accounting for short-range cooperation and

long-range inhibition (Figure 7) is used to describe the

spatial interactions. Here we want to show how spatial

patterns may also emerge when the kernel is ‘‘upside

down’’ with respect to the case of Figure 7, i.e., in the

presence of short-range inhibition and a long-range coop-

eration. We develop a numerical simulation of a simple

neural model, using equation (11) with local dynamics ex-

pressed by a generalized logistic function, h(u) = a(u0� u)u2,

where a is a positive constant. Because h0(u0) = �au02 < 0 for

any u0, in the absence of spatial interactions the homoge-

neous state u = u0 is linearly stable. The results of the

numerical simulation of equations (11) and (12) are shown

in Figure 10. In this case the nonlinearity of h(u) is capable of

limiting the growth of the perturbations of the homogeneous

state. However, we recall that only some suitable nonlinear

functions, h(u), can prevent the indefinite growth of these

perturbations. For example, when h(u) = a(u0 � u)u, the

TABLE 1. Moments of the Kernel Function in Equation (12)

in One- and Two-Dimensional Systems

Moment 1-D 2-D

w0

ffiffiffip

p(b1q1 � b2q2) p(b1q1

2 � b2q22)

w2

ffiffiffip

p=4(b1q13 � b2q2

3) p/2(b1q14 � b2q2

4)

w4

ffiffiffip

p=32(b1q15 � b2q2

5) p/12(b1q16 � b2q2

6)

Figure 10. Spatial pattern emerging for variable u in theneural system in equation (11). The parameters of the kernel(see equation (12)) are q1 = 1, q2 = 0.8317, b1 = 242.45, b2 =1046.2, and a = 0.01. The simulation is carried out over adomain of 256 � 256 cells, each cell representing a spatialstep Dx = Dy = 0.2. Equations are solved numerically bymeans of finite difference method.


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nonlinear terms are not able to constrain the growth of u,

which tends to ±1.

2.1.3. Relation Between Patterns Generated by Turingand Neural Models[53] The relation between neural models and Turing’s

systems is mentioned in few studies commenting on sim-

ilarities existing between the variety of patterns generated

by these two classes of models. For example, Dormann

et al. [2001] demonstrated that simple cellular automata

models of activator-inhibitor systems resembling simplified

neural models can lead to the emergence of patterns that

are very similar to those obtained with a reaction-diffusion

model (i.e., Turing models). Moreover, von Hardenberg

et al. [2001] pointed out that the same patterns emerging

from Turing-like instability can be obtained with neural

models, which account for only one state variable and one

dynamic equation. Thus, the relation between these two

classes of models has been described mostly qualitatively.

In this section we develop a mathematical framework to

show the link between Turing’s model and the biarmonic

approximation (19) of the neural model. To this end, we

first notice that in both models the spatial means, �u or �v, inthe asymptotic states reached by the system for t ! 1 are

the same, in the linear approximation, as the homogeneous

steady states, (u0, v0) (Turing) or u0 (neural model). In

fact, at t ! 1 the terms @/@t are zero. Expanding the

local functions (i.e., f, g, and h) on the right-hand sides of

equations (1) and (19) in Taylor’s series around (�u, �v) and(�u), respectively, and taking only the linear terms, we find

that these equations reduce to f(�u, �v) = g(�u, �v) = 0 and h(�u)= 0.

[54] Combining the same linear approximations of equa-

tions (1) at steady state, we obtain

w04r4uþ w0

2r2uþ w00 u� uoð Þ ¼ 0; ð23Þ

w04r4vþ w0

2r2vþ w00 v� voð Þ ¼ 0; ð24Þ

where

w00 ¼ fvgu � gvfuð Þ; w0

2 ¼ � gv þ dfuð Þ; w04 ¼ �d; ð25Þ

with fu = @f/@u, gu = @g/@u, etc. (calculated for (u0, v0)).

Equations (23) and (24) are the same as equation (19) at

t ! 1, with h(u) linearized around u = u0 and with w0 =

w00 � h0(u0), w2 = w2

0, and w4 = w40. Thus, at t ! 1 the

two equations of Turing’s model (i.e., (1a) and (1b))

reduce to the same equation as (23) or (24) with the same

coefficients. This equation is also the same as (19) at

steady state. In the case of equation (19), spatial dynamics

associated with short- and long-range diffusion induce the

formation of patterns when the condition (22) is met and

w4 and w2 have the same sign (see section 2.1.2). Because

w4 < 0 (see equation (25)), w2 needs to be negative. Using

equations (25) it is easy to show that these conditions lead

to the same relations (9) determined for the emergence of

diffusion-induced instability in Turing’s model. Thus, in

both classes of models the conditions determining the

formation of patterns as a result of spatial interactions are

the same. Moreover, using equations (25), it can be shown

that the most unstable mode, kmax = 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiw2=w4

p, of the

biarmonic model is the same as the one obtained from the

dispersion relation for Turing’s model in conditions of

marginal stability (i.e., when s = 0), indicating important

commonalities in the steady state geometry of the patterns

generated by these two models. Thus, Turing’s model can

be viewed as a particular case of the neural model. In fact,

in a neural model the dynamics of only one species are

explicitly described, while Turing’s model describes the

dynamics of at least two species. This means that pattern

formation in a neural model imposes constraints only for

one species, while in a Turing model the constraints are

required for at least two species.

[55] A biarmonic approximation of the example of neu-

ral model presented in Figure 10 can be obtained from

equation (19) with moments calculated (equation (20) and

Table 1) using the same parameters b1,2 and q1,2 as in

Figure 10. Patterns generated by this biarmonic model are

shown in Figure 11. Using equations (25) it can be shown

that the linearization of the Turing model in Figure 6 leads

to the biarmonic model (19) with the same coefficients as

the example in Figure 11.

[56] It is possible to observe that the patterns generated

by these three models (Figures 6, 10, and 11) exhibit the

same wavelengths. As noted, the same wavelength in

Figures 6 and 11 is found because in this case the dispersion

Figure 11. Spatial pattern emerging for the variable u inthe long-range approximation of the kernel-based model inequation (11). The parameters are w0 = �1512, w2 =�405.6, w4 = �27.2, a = 1, and u0 = �0.906. Thesimulation is carried out over a domain of 256 � 256 cells,each cell representing a spatial step Dx = Dy = 0.2.Equations are solved numerically by means of finitedifference method.


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relation (equation (8)) of the Turing model is tangent to the

x axis.

2.1.4. Patterns Emerging From Differential FlowInstability[57] The third major deterministic mechanism of self-

organized pattern formation associated with symmetry-

breaking instability is due to differential flow. This mech-

anism resembles Turing’s dynamics, in that it involves two

diffusing species, u and v (‘‘activator’’ and ‘‘inhibitor,’’

respectively). However, unlike Turing’s model, diffusion is

not important to the destabilization of the homogeneous

state. In this case, one or both species are subjected to

advective flow (or ‘‘drift’’), and instability emerges as a

result of the differential flow rate of the two species

[Rovinsky and Menzinger, 1992]. While diffusion is not

fundamental to the emergence of differential flow instabil-

ity, it plays a crucial role in imposing an upper bound to

the range of unstable modes, k, and determines the

wavelength of the most unstable mode [Rovinsky and

Menzinger, 1992]. As a result of the drift, patterns gener-

ated by this process are not time-independent as those

associated with Turing’s instability. Rather, they exhibit

traveling waves in the flow direction. Self-organized

patterns of this type have been observed in nature mainly

in chemical systems (the ‘‘Belousov-Zhabotinsky reaction’’

[Rovinsky and Zhabotinsky, 1984]). The same mechanism

has been also invoked to explain ecological patterns

subject to drift, including banded vegetation [Klausmeier,

1999; Okayasu and Aizawa, 2001; von Hardenberg et al.,

2001; Shnerb et al., 2003; Sherratt, 2005]. We note that

this mechanism of pattern formation induced by differen-

tial flow is often classified as a Turing model in that in

both models the dynamics can be expressed by the same

set of reaction-advection-diffusion equations. In the case of

Turing models, instability is induced by the Laplacian

term, while in the case of differential flow instability it

is the gradient term that causes instability. For sake of

clarity, here we discuss the case of differential flow

instability separately.

[58] We introduce the mathematical model of differential

flow instability [e.g., Rovinsky and Menzinger, 1992] as-

suming that only one of the two species undergoes a drift,

and we orient the x axis in the direction of the advective

flow. The activator-inhibitor dynamics can be expressed as

@u

@t¼ f u; vð Þ þ p

@u

@xþ d1r2u; ð26aÞ

@v

@t¼ g u; vð Þ þ d2r2v; ð26bÞ

where p is the drift velocity and with d1 and d2 being

the diffusivities of u and v, respectively. Notice that when

p = 0, equations (26) can be written in the same form as

equations (1).

[59] When p 6¼ 0, the conditions on d1 and d2 for the

emergence of patterns from equation (26) are less restrictive

than those for Turing’s instability. To stress the fact that

patterns emerge from the differential flow rates of u and v

we first consider the conditions leading to instability in the

absence of diffusion and set d1 = d2 = 0. The homogeneous

steady state, (u0, v0), obtained as solution of the equation set

f(u0, v0) = g(u0, v0) = 0 is stable when the conditions (5) are

met. To determine the conditions in which the differential

flow destabilizes the state (u0, v0), we linearize f(u, v) and

g(u, v) around (u0, v0) and seek for solutions of the

linearized equations in the form of

u ¼ uþ u0; ð27aÞ

v ¼ vþ v0: ð27bÞ

We obtain

@u

@t¼ fuuþ fvvþ p

@u

@x; ð28aÞ

@v

@t¼ guuþ gvv: ð28bÞ

[60] The solution of system (28) can be expressed as a sum

(or integral sum in spatially infinite domains) of Fourier

modes, uk =Ukexp(st + ik � r) and vk =Vkexp(st + ik � r), withUk and Vk being the Fourier coefficients of the kth mode.

Because equations (28) need to be satisfied for each mode, k,

we have

sUk ¼ fuUk þ fvVk þ ipUkkx; ð29aÞ

sVk ¼ guUk þ gvVk : ð29bÞ

[61] Nontrivial solutions of system (29) exist when its

determinant is zero:

s2 � fu þ gv þ ipkxð Þs þ fugv � fvgu þ ipkxgv ¼ 0: ð30Þ

[62] Notice how in this case, s is a complex number. The

emergence of instability requires the real part of s to be

positive. Traveling wave patterns require that the imaginary

part of s is different from zero. It has been noticed

[Rovinsky and Menzinger, 1992] that equation (30) does

not lead to the selection of any finite value for the most

unstable wave number in that s is a monotonically increas-

ing function of k, and the wave number interval of the

unstable modes has no upper bound. However, the addition

to equation (29) of a diffusion term to either the first or the

second equation (or to both, as in equation (26)) imposes an

upper bound to the range of unstable modes. In this case the

most unstable mode corresponds to a finite value of the

wave number.

[63] We present, as an example of differential flow

instability, a model developed to study the formation of

patterns in young mussel beds [van de Koppel et al., 2005].

The model can be adopted also to describe a system

involving trees or grasses. Two (dimensionless) state vari-


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ables, representing nutrient concentration, u, and vegetation

density, v, are used. The dynamics of the two variables are

expressed as

@u

@t¼ 8 1� uð Þ � uvþ pruþ d1r2u; ð31aÞ

@v

@t¼ huv� d

v

1þ vþ d2r2v: ð31bÞ

[64] The first term on the right-hand side of the first

equation represents the rate of increase in nutrient concen-

tration, the second term accounts for the consumption of

nutrient by biomass, while the third term is the loss of

nutrients by advection; the fourth term models the spreading

of u by diffusion. The first term on the right-hand side of the

second equation represents the nutrient-dependent rate of

biomass growth, the second term represents the state-

dependent mortality rate, and the third term accounts for

the diffusion-like spatial spreading of biomass. The steady

homogeneous state (u0 = (h8 � d)/[h(8 � 1)], v0 = [8(d �h)]/(h8 � d)) is stable in the absence of drift and diffusion

when the conditions (5) are met. Drift-induced instability

occurs if the drift term is able to destabilize the homogeneous

state (u0, v0) even when the Laplacian terms are set equal to

zero. In this case the dispersion relation (30) provides the

range of Fourier modes that are destabilized by drift (see

Figure 12). As noted by Rovinsky and Menzinger [1992], in

the absence of a diffusion term the interval of the unstable

modes has no upper bound (equation (30)). When a diffusion

term is added to the first equation (i.e., d1 6¼ 0), the dispersion

relation becomes

s2 þ d1k2 � fu � gv � ipkx

s

þ fugv � fvgu þ ipkxgv � d1k2gv

¼ 0: ð32Þ

[65] The plot of this relation (see Figure 12) shows that in

this case the interval of the unstable wave numbers has an

upper bound and the most unstable mode has finite wave

number. When a diffusive term is added also to the second

equation (i.e., d1 6¼ 0, d2 6¼ 0) as in equation (29), the

dispersion relation becomes

s2 þ d1k2 þ d2k

2 � fu � gv � ipkx

s

þ fugv � fvgu þ ipkxgv � id2pkxk2 � d1k

2gv � d2k2fu þ d1d2k

4

¼ 0; ð33Þ

with no substantial differences in the amplitude of the

interval of unstable modes (see Figure 12).

[66] An example of spatial patterns emerging with this

model is shown in Figure 13.

2.2. Stochastic Models

[67] Pattern formation in ecology has been often associ-

ated with the deterministic mechanisms of symmetry-break-

ing instability described in sections 2.1.1, 2.1.2, and 2.1.4,

while random environmental drivers have been usually

considered to be only able to introduce noise in the ordered

states of the system. Thus, random environmental fluctua-

tions are usually believed to disturb the states of the system

and to destroy the patterns formed by deterministic dynam-

ics [e.g., Rohani et al., 1997]. However, it has been shown

that random fluctuations are able to also play a ‘‘construc-

tive’’ role in the dynamics of nonlinear systems, in that they

can induce new dynamical behaviors that did not exist in the

deterministic counterpart of the system [e.g., Horsthemke

and Lefever, 1984]. In particular, stochastic fluctuations

have been associated with the emergence of new ordered

states in dynamical systems, in both time [e.g., Horsthemke

Figure 12. Different dispersion relations for the case ofdifferential flow instability. The parameters for the boldcontinuous line are 8 = 0.72, h = 6.10, d = 5.14, p = �1.315,d1 = 1, and d2 = 2.

Figure 13. Spatial pattern emerging for the variable uusing the model (26). The parameters are h = 6.10, d = 5.14,8 = 0.72, p = �1.315, d1 = 0, and d2 = 1. The simulation iscarried out over a domain of 256 � 256 cells, each cellrepresenting a spatial step Dx = Dy = 0.8. Equations aresolved numerically by means of finite difference method.


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and Lefever, 1984] and space [Garcia-Ojalvo and Sancho,

1999]. Known as ‘‘noise-induced phase transitions,’’ these

‘‘constructive’’ effects of noise may occur in systems forced

by multiplicative noise (i.e., when there is a state depen-

dency in the impact of random fluctuations on the system).

[68] Thus, random environmental drivers are not neces-

sarily in contraposition to pattern formation. Indeed, it has

been shown that noise may induce pattern formation [van

den Broeck et al., 1994; Garcia-Ojalvo and Sancho, 1999;

Loescher et al., 2003; Sagues et al., 2007]. Although these

noise-induced mechanisms of pattern formation have been

investigated by the physics community for over a decade,

they have found only limited applications in ecohydrology.

This fact is quite surprising, in that environmental dynamics

are undoubtedly affected by random fluctuations, which

might have the potential of playing a fundamental role on

the composition and structure of plant ecosystems.

[69] We will present two major mechanisms of noise-

induced pattern formation, based either on nonequilibrium

phase transitions or on the random switching between

dynamics. We will also discuss the few existing examples

of ecohydrological models of noise-induced pattern forma-

tion (section 4).

2.2.1. Nonequilibrium Phase Transition Models[70] Recently, it has been found that patterns may also

emerge as ordered symmetry-breaking states induced by

noise in nonlinear, spatially extended systems [van den

Broeck et al., 1994, 1997; Parrondo et al., 1996]. These

ordered states result from phase transitions, which break the

ergodicity of the system. In the thermodynamics literature

these transitions are often referred to as ‘‘nonequilibrium

phase transitions’’ to stress the fundamental difference in the

role of noise (i.e., its ability to generate order) with respect

to the case of equilibrium phase transitions [van den Broeck

et al., 1994]. In these (nonlinear) systems, multiplicative

noise destabilizes a homogeneous steady state of the under-

lying deterministic dynamics thereby leading to an ordered

state that is stabilized by the spatial dynamics [Sagues et al.,

2007]. For noise to be able to induce phase transition with

breaking of ergodicity, it has to be ‘‘multiplicative’’; that is,

its effect on the dynamics needs to be modulated by a

(multiplicative) term, which depends on the state of the

system. However, it has been recently found that order can

also be induced by additive noise acting in concert with

multiplicative noise in spatially extended systems [Sagues

et al., 2007]. These symmetry-breaking states are purely

noise induced; that is, they are induced by local fluctuations

and do not occur in the deterministic counterpart of the

system. In fact, they vanish as the noise intensity (i.e., the

variance) drops below a critical value, suggesting that a

threshold needs to be exceeded by the noise intensity for

noise-induced patterns to emerge. At the same time, these

nonequilibrium phase transitions have been found to be

reentrant, in that the ordered phase is destroyed when the

noise intensity exceeds another threshold value. In other

words, the multiplicative noise has a ‘‘constructive’’ effect

only when the variance is within a certain interval of values.

Smaller or larger values of the variance correspond to

conditions in which noise is either too weak or too strong

to induce ordered states. van den Broeck et al. [1994, 1997]

used an approximated analytical framework to investigate

conditions leading to nonequilibrium phase transitions with

breaking of ergodicity. This framework, which is based on

mean field analysis, was first developed for the case of

Gaussian noise [van den Broeck et al., 1994, 1997; Parrondo

et al., 1996] and then applied to systems forced by Poisson

[Porporato and D’Odorico, 2004] and dichotomous [Bena,

2006] noise. The only applications of this mechanism to

landscape ecology we are aware of are based on a model of

fire-vegetation interaction in which random fire occurrences

are represented as Poisson noise [D’Odorico et al., 2007b].

[71] When the state of the system is determined by only

one state variable, u, its temporal dynamics can be, in

general, modeled by a differential equation expressing the

temporal variability of u at any point, (x, y), as the sum of

three terms: a function of local conditions (i.e., of the value

of u at (x, y)), a term representing a state-dependent noise,

and a term accounting for the spatial interactions with the

other points of the domain. These spatial interactions are

modeled as a diffusion process

@u

@t¼ f uð Þ þ g uð Þx tð Þ þ dr2u; ð34Þ

where d is a diffusivity coefficient, f(u) and g(u) are two

functions of u(x, y), and x(t) is the noise term. Equation (34)

is a stochastic partial differential equation (a stochastic

reaction-diffusion equation) where the multiplicative term is

interpreted in the Stratonovich sense [van Kampen, 1981].

The solution of this equation would provide the probability

distribution of u as a function of (x, y) and t. However, there

are no known exact methods for the integration of (34).

Thus, the analytical evidence for the existence of a noise-

induced transition comes from an approximated approach

based on the mean field (Weiss) approximation [van den

Broeck et al., 1994, 1997; Buceta and Lindenberg, 2003].

These approximated analytical results have been supported

by numerical simulations [van den Broeck et al., 1994,

1997; Buceta and Lindenberg, 2003; Porporato and

D’Odorico, 2004].

[72] First, a finite difference representation of the diffu-

sion term is used and equation (34) is rewritten for the

generic site, i, in a square lattice domain

@ui@t

¼ f uið Þ þ xig uið Þ þ d

4

Xj�n ið Þ

ui � uj

; ð35Þ

where ui and xi are the values of u and x at site i,

respectively; n(i) is the set of the 4 nearest neighbors, j, of

site i. The solution of equation (35) is impeded by the fact

that the dynamics of ui are coupled to those of the

neighboring points. In fact, the spatial interaction term in

(35) depends on the mean value, E[u]i, of u in the

neighborhood of i,P

j�n(i)(ui � uj)/4 = ui � E[u]i. Van

den Broeck et al. [1994] solved equation (35) using the

mean field approximation. To this end, they assumed that


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the local mean can be approximated by the spatial mean,

E[u], of u across the whole domain, i.e., E[u]i � E[u]. The

effectiveness of this approximation can be improved by

taking E[u]i ’ 12[ui + E[u]] to account for the dependence

of E[u]i on the local conditions, as explained by Sagues et

al. [2007]. It can be shown that this last approximation

corresponds to weakening the spatial coupling by using an

effective diffusivity deff = d/2 [Sagues et al., 2007].

Because under this assumption the dynamics of ui do not

depend on those of the neighboring points, it is possible to

determine exact expressions for the steady state probability

distributions, pst(u; E[u]), of u obtained from equation (35)

for the cases of Gaussian, dichotomous, or Poisson noise.

Exact expressions for these distributions are given by van

den Broeck [1983]. The distribution pst(s; E[u]) will

necessarily depend on a number of parameters of the

dynamics and on E[u], which remains unknown. To

determine E[u], a self-consistency condition is used:

E u½ � ¼ hui ¼Z þ1

�1u pst u;E u½ �ð Þdu ¼ F E u½ �ð Þ: ð36Þ

[73] Multiple solutions of this equation correspond to

the existence of multiple possible average values of u in

statistically steady state conditions, i.e., of multiple possi-

ble steady state probability distributions of u. Thus, the

emergence of multiple solutions of the self-consistency

equation indicates the occurrence of ergodicity- and sym-

metry-breaking nonequilibrium phase transitions. Numeri-

cal simulations are then used to confirm the validity of the

approximated results obtained from equation (36) and to

show the actual emergence of nonequilibrium pattern

formation. Alternatively, the occurrence of nonequilibrium

phase transitions can be studied through an approximated

analytical solution of equation (36) obtained using the

Taylor’s expansion of its right-hand side term [Sagues et

al., 2007].

[74] Zhonghuai et al. [1998] used the framework by van

den Broeck et al. [1994] to study noise-induced phase

transitions in generic two-variable systems exhibiting

Turing instability. When a control parameter of the kinetics

is perturbed by noise, new kinds of patterns arise (transi-

tion from single spiral to double spiral waves). A similar

study was developed by Carrillo et al. [2004] for the

analysis of pattern formation in chemical reactions and

fluid convection.

2.2.2. Patterns Induced by Random Switchingof Dynamics[75] In the mechanism of noise-induced pattern formation

described in section 2.2.1, ordered states of the system

emerge as a result of local, random fluctuations of the state

variable, which can be either spatially correlated or uncor-

related [Sagues et al., 2007]. In the other major class of

stochastic models, noise-induced patterns result from the

random switching between dynamics that simultaneously

occurs at all points of the spatial domain [Buceta and

Lindenberg, 2002; Buceta et al., 2002a, 2002b, 2002c].

This mechanism is based on the idea that if the random

switching between dynamics is global (i.e., it simulta-

neously occurs across the domain) and ‘‘rapid,’’ the

system behaves as if it was undergoing the average

dynamics obtained as a weighted mean of the two states.

In this context with ‘‘rapid’’ switching we mean that the

average residence time of the dynamics in either one of

the two dynamical states is much shorter than the time

needed by the system to reach the equilibrium state in

each dynamics.

[76] Thus, if we take as an example two Turing models

and we randomly and rapidly switch between them the

system experiences only the average Turing dynamics. We

can envision cases in which, separately, neither of the two

dynamics are able to lead to pattern formation, while their

average exhibits diffusion-induced symmetry-breaking in-

stability. In these conditions, patterns emerge from the

nonequilibrium random (global) alternation between dy-

namics. Similar models can be constructed using two

suitable biarmonic (or neural) models with the same spatial

dynamics term but different local dynamics [Buceta et al.,

2002a]. Separately, these models are unable to exhibit

symmetry-breaking instability. The random switching be-

tween them may lead to mean dynamics that are capable of

generating patterns.

[77] We consider as an example a system in which

Turing’s instability is induced by noise. To this end, we

consider two reaction-diffusion systems (1):

@u

@t¼ f1;2 u; vð Þ þ r2u; ð37aÞ

@v

@t¼ g1;2 u; vð Þ þ d1;2r2v; ð37bÞ

with f1,2(u, v) and g1,2(u, v) being two pairs functions

describing the dynamics of states 1 and 2. The system

switches between state 1, where the local kinetics are

expressed by f1 and g1 and the diffusion ratio is d1 (these three

quantities are hereinafter simply indicated as A1), and state 2,

where the kinetics are modeled by the functions f2 and g2 and

the diffusion ratio is d2 (indicated as A2). Neither one of the

equations (37) for state 1 or 2 meets all the conditions (5) and

(9). Thus, neither one of the two dynamics can separately lead

to pattern formation. Each control parameter (f, g, or d)

alternates dichotomously in a way that the temporal

evolution, A(t), of each parameter can be expressed as

A tð Þ ¼ A1L tð Þ þ A2 1� L tð Þ½ �; ð38Þ

with L(t) being a dichotomous variable assuming values 0

and 1. When the switching is fast in the sense discussed

before, L(t) can be replaced by its average value, L(t) ’hL(t)i, and in this case

A tð Þ ’ �A ¼ A1P1 þ A2P2; ð39Þ

with P1 = hLi and P2 = 1 � hLi.


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[78] The dynamics resulting from the fast switching

between the two states are then

@u

@t¼ �f u; vð Þ þ r2u; ð40aÞ

@v

@t¼ �g u; vð Þ þ �dr2v; ð40bÞ

where �f = f1P1 + f2P2, �g = g1P1 + g2P2, and �d = d1P1 +

d2P2. Patterns emerge if the average dynamics meet the

conditions (5) and (9).

[79] The emergence of switching-induced patterns

depends on the velocity of the alternation between the

two dynamics. Over a relatively long time, both dynamics

would lead to spatially homogeneous configurations; how-

ever, if the switching is sufficiently fast the system can

experience the average dynamics (40). In fact, in these

conditions, the homogeneous steady state can never be

reached, and the system always remains in a nonequilibrium

configuration, which can be described by the mean of the

two states. The separation between slow and fast switching

can be defined using a control parameter, r, representing the

ratio between an ‘‘external’’ time scale, text, associated with

the random switching (i.e., average time the system spends

in each configuration), and an ‘‘internal’’ time scale, tint,

associated with the time needed by the system to reach

equilibrium in each of the two states. If r = text/tint ! 1, no

switching-induced instability emerges. On the contrary,

when r < 1 the dynamics can be described in average

terms as in equations (40), and patterns may emerge if

equations (5) and (9) are met. The following patterns have

been noted:

[80] 1. Patterns may emerge from random alternation of

dynamics even when both of them have the same homoge-

neous steady state. In this case, it has been found that the

random switching leads to spotted structures, while when

the two dynamics have different steady states the random

alternation may lead to labyrinthine-striped configurations

[Buceta et al., 2002a].

[81] 2. A similar model of pattern formation can be

developed using a random switching between two biar-

monic [Buceta and Lindenberg, 2002] or two neural models

[D’Odorico et al., 2006a], as discussed in section 4.2.

[82] 3. Although the random switching can play a crucial

role in this process of pattern formation, similar patterns

emerge when the switching mechanism is deterministic. In

fact, Buceta and Lindenberg [2002] showed that periodic

alternation of dynamics can also lead to pattern formation.

Thus, unlike the mechanism described in section 2.2.1,

patterns emerging from nonequilibrium dynamics associ-

ated with global alternation of dynamics are not noise

induced sensu strictu. We refer the interested reader to Bena

[2006] for a more detailed discussion of ordered states

induced by periodic and random drivers.

[83] In the specific case of ecohydrology the random

switching can be caused, for example, by interannual

rainfall variability and the consequent alternation of stressed

and unstressed conditions in vegetation [D’Odorico et al.,

2005, 2006b]. An example is discussed in section 4.

2.2.3. Case Study: Turing Instability Inducedby Random Switching[84] As an example of patterns induced by random

switching between two Turing models we consider the

case of a system that switches between the two states:

(1) state 1 expressed by equations (37) with f1(u, v) = avu2

and g1(u, v) = bv (with a = 27.5, b = 105, and d1 = 41.25)

and (2) state 2 with f2(u, v) = �eu and g2(u, v) = �cu2v2

(with e = 90, c = 566.67, and d2 = 20). The system is in

state 1 with probability P1 = 0.8 and in state 2 with

probability P2 = 0.2; using equations (40) we then have�f = u(P1avu � P2e), �g = v(P1b � P2cu

2v), and �d = P1d1 +

P2d2 (as in equations (10)). It is possible to show that unlike

the dynamics in states 1 and 2, the average dynamics

associated with the process of fast switching can induce

pattern formation through Turing instability. In fact, in this

case the average dynamics are the same as those of the

deterministic model presented at the end of section 2.1.1,

and the spatial configuration arising with this last model has

the same features as those shown in Figure 6.

2.3. Other Mechanisms of Noise-Induced PatternFormation

[85] Other authors have investigated the role of noise in the

process of pattern formation in ecohydrology [e.g.,Ruxton and

Rohani, 1996; Satake et al., 1998; Durrett, 1999; Spagnolo et

al., 2004]. For example, Spagnolo et al. [2004] proposed

multivariate models of noise-induced pattern formation in

generalized Lotka-Volterra systems with multiplicative

(Gaussian) noise and diffusion-like spatial interactions. Con-

sistently with van den Broeck et al. [1994], they found that

multiplicative noise can play a constructive role in the non-

linear dynamics in that it can lead to a variety of dynamical

behaviors including pattern formation. However, apart from

these few examples, there have been only limited ecohydro-

logical applications of some recent theories of noise-induced

order.

[86] Analyzing the geometrical features of patchy land-

scapes, it has been observed that in a number of natural

systems patterns may exhibit order at all scales; that is, the

spatial organization has no characteristic scale, and features

of all sizes exist [Morse et al., 1985; Krummel et al., 1987;

Sole and Manrubia, 1995; Malamud et al., 1998; Guichard

et al., 2003; Caylor and Shugart, 2006; Kefi et al., 2007b,

2007a]. Known as ‘‘fractals,’’ objects exhibiting (statistically)

the same geometrical properties at all scales are ubiquitous

in nature, as evidenced by the recurrence of power laws in

the probability distributions of geometrical features of

natural systems [Mandelbrot, 1984; Lam and de Cola,

1993; Rodriguez-Iturbe and Rinaldo, 2001]. The presence

of fractality is often associated with criticality [e.g., Bak,

1996], i.e., the state of systems close to a phase transition

[e.g., Herbut, 2007]. A few theoretical frameworks have

been developed to relate these patterns to the underlying

stochastic processes. In particular, in this review we briefly


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mention the theory of the self-organized criticality [Bak et

al., 1988].

[87] The concept of self-organized criticality provides a

rather general framework explaining some general features

of complex systems in which scale invariance has been

observed to emerge both in time and in space [Bak, 1996].

These systems are generally open, nonlinear, dissipative,

and with many degrees of freedom. They consist of many

components, with each component interacting with its

neighbors (internal dynamics). The overall dynamics result

from the combined effect of local interaction and external

stochastic drivers: at any given time the external driver may

activate some components of the system if some threshold

conditions are exceeded. Through a network of local inter-

actions the activation can spread to other components,

thereby propagating the signal until the system relaxes in

a marginally stable state in which all components are again

below threshold conditions. The system remains in this state

until the external driver activates some other components

thereby spreading through other pathways. Thus, an impor-

tant feature of these systems is that external drivers acting in

conjunction with local interactions lead to nonlocal changes

that propagate throughout the system (‘‘avalanche’’ effect).

Through a number of simple cellular automata models [Bak

et al., 1988; Bak and Tang, 1989; Bak et al., 1989, 1990;

Bak, 1996], P. Bak and coworkers demonstrated that these

dynamics are typical of systems that self-organize them-

selves in critical states (i.e., in states close to a phase

transition) which do not exhibit any dominant scale either

in space or time. Self-organized criticality (SOC) occurs

when the critical state is attained with no parameter tuning.

[88] Examples relevant to the study of vegetation pattern

formation are based on discrete (cellular automata) models.

In these systems the emergence of self-organized criticality

has been investigated mostly through numerical simula-

tions, though theoretical frameworks based on mean field

theory (section 2.2.1), group theory, and Langevin equa-

tions have been also developed and applied [e.g., Jensen,

1998]. The forest fire model [Bak et al., 1990; Drossel and

Schwabl, 1992] is a classical example of dynamics sugges-

tive of self-organized criticality, which could be relevant to

the study of vegetation patterns observed in nature [e.g.,

Malamud et al., 1998]. Developed as a ‘‘toy model’’ to

study SOC mechanisms, this model did not claim to

reproduce forest dynamics that are ecologically realistic.

Moreover, it requires parameter tuning [e.g., Jensen, 1998].

[89] Sole and Manrubia [1995] found scale invariance

(i.e., power laws) in the distribution of gap sizes in the

rainforests of Barro Colorado island. These authors devel-

oped a simple spatially extended cellular automata model to

show how scale-invariant order may spontaneously emerge

in these systems from SOC dynamics, i.e., as a result of the

tendency of the system to relax in a critical state. A few

other models have been developed to explain scale invari-

ance in vegetation patterns as a sign of self-organized

criticality [Sprott et al., 2002; Bolliger, 2005; Zeng and

Malanson, 2006; Kefi et al., 2007a; Manor and Shnerb,

2008].

[90] More recently, Scanlon et al. [2007] explained the

emergence of scale-invariant order in savanna vegetation

[Caylor and Shugart, 2006] without invoking the occur-

rence of criticality. These authors showed how power law

scaling found in the cluster size distribution of vegetation

[Caylor and Shugart, 2006] can result from the interplay of

positive feedbacks with local interactions and global con-

straints in an Ising-like model [e.g., Brush, 1967].

[91] In addition to the mechanisms reported in the

previous paragraphs, there is a relatively rich body of lit-

erature on pattern formation based on spatiotemporal

stochastic resonance [Benzi et al., 1985; Loescher et al.,

2003; Spagnolo et al., 2004], coherence resonance [e.g.,

Sagues et al., 2007], noise-induced phenomena in excit-

able systems [Sagues et al., 2007], and front dynamics

in the presence of external noise [Garcia-Ojalvo and

Sancho, 1999]. However, these more recent theories of

noise-induced order in spatially extended systems have had

only limited applications to the environmental sciences.

3. BIOECOLOGICAL MECHANISMS LEADING TOVEGETATION PATTERNS

[92] A common feature of the three deterministic models

of symmetry-breaking instability presented in section 2.1 is

that the emergence of periodic vegetation patterns arises

from the balance between positive (activation) and negative

(inhibition) interactions [e.g., Shnerb et al., 2003; Rietkerk

and van de Koppel, 2008]. For example, in the neural model

both pattern emergence and pattern geometry are deter-

mined by the interplay between short-range facilitation (or

cooperation) and long-range competition (or inhibition), as

reflected by the kernel shape shown in Figure 7 [Lejeune et

al., 2004; Rietkerk and van de Koppel, 2008]. The dominant

wavelength of the resulting pattern is typically larger than

the range of either facilitative or competitive interactions

between individuals because of the ability of spatial insta-

bility to amplify local processes and to induce pattern

formation at the patch scale [Couteron and Lejeune, 2001].

[93] Cooperative and synergistic short-range effects are

usually associated with positive feedbacks resulting from

the ability of some species or functional types to create

environmental conditions that favor plant establishment,

growth, and survival. These feedbacks typically operate

within a short range. For example, cooperation among

neighboring individuals may lead to the concentration of

resources in vegetated areas where plant individuals find

more favorable conditions for establishment and survival

[Charley and West, 1975; Schlesinger et al., 1990; Greene,

1992; Wilson and Agnew, 1992; Bhark and Small, 2003;

D’Odorico et al., 2007a]. The aerial parts of plant individ-

uals that have already established in a certain patch may

favor the growth of other plants in the same area by limiting

soil moisture losses associated with evapotranspiration

[Vetaas, 1992; Thiery et al., 1995; Lejeune et al., 2004;

Zeng et al., 2004; D’Odorico et al., 2007b] either through a

mulching effect (i.e., soil evaporation limited by wilted

leaves and litter) or shading (i.e., when the foliage shades


18 of 36

RG1005

the ground surface thereby limiting evaporation) [Zeng and

Zeng, 1996; Scholes and Archer, 1997; Zeng et al., 2004;

Caylor et al., 2006; Borgogno et al., 2007]. Moreover, the

formation of physical and biological crusts on bare soil

may further reduce the infiltration of surface water [e.g.,

Fearnehough et al., 1998]. Physical crusts, typically 1–

3 mm thick, are generated by rain splashing [Issa et al.,

1999; Esteban and Fairen, 2006], while biological crusts

are formed by micro-organisms such as cyanobacteria,

which exude mucilaginous secretions that bind together soil

grains and organic fractions [Issa et al., 1999; Meron et al.,

2004; Belnap et al., 2005]. These crusts greatly reduce the

soil infiltration capacity thereby decreasing the soil moisture

available in the underlying soil layers, with consequent

limitations on the establishment and growth of perennial

vegetation [Fearnehough et al., 1998]. Because soil crusts,

on the other hand, seldom develop beneath vegetation

canopies owing to the reduced raindrop impact [Boeken

and Orenstein, 2001; Meron et al., 2004; Borgogno et al.,

2007] and the limited light available to the photosynthetic

activity of biological crusts [Walker et al., 1981; Greene,

1992; Joffre and Rambal, 1993; Greene et al., 1994, 2001],

a positive feedback exists between presence of vegetation

and absence of crusts. Conversely, in some arid and semi-

arid areas the absence of biological crusts leads to increased

erosion and loss of organic matter, fine soil particles, and

nutrients [Schimel et al., 1985; Harper and Marble, 1988].

In fact, cyanobacterial-lichen soil crusts have been shown to

be the dominant source of nitrogen in a cold desert and a

grassland ecosystem in southern Utah [Evans and Ehlringer,

1993]. Moreover, there is some discussion as to whether

biological crusts cause reduced infiltration [West, 1990] or

whether, as a result of the formation of microtopography,

they limit runoff and surface erosion [Belnap et al., 2005].

[94] In vegetated areas the protection against evapotrans-

piration and soil crust formation enhances surface water

infiltration which, in turn, favors vegetation growth. The

associated increase in root density, in turn, enhances the soil

infiltration capacity [Walker et al., 1981; Greene, 1992;

Joffre and Rambal, 1993; Greene et al., 1994, 2001;

HilleRisLambers et al., 2001; Okayasu and Aizawa,

2001; Gilad et al., 2004; Yizhaq et al., 2005; Borgogno

et al., 2007]. Moreover, a dense canopy of established

plants provides protection against herbivores (e.g., birds),

thereby favoring plant reproduction and growth [Lejeune

et al., 2002] in densely vegetated areas (propagation by

reproduction effect) where higher rates of seed production

and germination occur [e.g., Lefever and Lejeune, 1997;

Lejeune and Tlidi, 1999; Lejeune et al., 1999; Lefever et al.,

2000; Couteron and Lejeune, 2001].

[95] Species able to modify the abiotic environment, re-

distribute resources, and facilitate the growth of other species

as well as their own are known as ‘‘ecosystem engineers’’

[Jones et al., 1994; Gilad et al., 2007; Bonanomi et al.,

2008]. For example, the improvement of conditions exist-

ing in the microenvironment underneath the canopy of so-

called ‘‘nurse plants’’ [Neiring et al., 1963; Kefi et al.,

2007b] favors the establishment and growth of other plants

[e.g., Garcia-Moya and McKell, 1970; Burke et al., 1998;

Aguiar and Sala, 1999]. Vegetation cover may decrease the

amplitude of temperature fluctuations; reduce the exposure

to solar radiation, wind desiccation, and soil erosion; or

prevent soil crust formation [Eldridge and Greene, 1994;

Smit and Rethman, 2000; Greene et al., 2001]. Moreover,

plant individuals located in the middle of vegetated patches

are protected against fires and grazing.

[96] As noted, it has been also found that water and/or

nutrient availability are higher in the areas located under the

canopy of existing plant individuals than in the surrounding

bare soil [Charley and West, 1975]. These nutrient-rich

areas are known as ‘‘fertility islands’’ or ‘‘resource islands’’

[Schlesinger et al., 1990]. Mechanisms commonly invoked

to explain the formation of these heterogeneous distribu-

tions of resources include the ability of the canopies to trap

nutrient-rich airborne soil particles, the accumulation of

sediments transported by wind and water, the sheltering

effect of vegetation against the erosive action of wind

and water, and the presence of nitrogen-fixing species

[Garcia-Moya and McKell, 1970; Charley, 1972; Archer,

1989; Schlesinger et al., 1990; Breman and Kessler, 1995;

Okin et al., 2001; Li et al., 2007]. Sometimes, fertility

islands lead to the formation of aperiodic vegetation patterns

in the form of stable spatial configuration corresponding to

isolated vegetation patches (spots), usually named ‘‘local-

ized structures’’ or ‘‘localized patches’’ [Lejeune et al.,

2002; Meron et al., 2007]. Other examples of ecosystem

engineers relevant to pattern formation include the ability of

deep-rooted plants to facilitate shallow-rooted species by

increasing surface soil moisture through ‘‘hydraulic lift’’

mechanisms [Richards and Caldwell, 1987], the reduction

in fire pressure resulting from the encroachment of woody

vegetation at the expenses of grass fuel [e.g., Anderies et al.,

2002; van Langevelde et al., 2003; D’Odorico et al.,

2006a], and the ability of alpine/subalpine vegetation and

desert shrubs to maintain warmer microclimate conditions

and reduce frost-induced mortality.

[97] Similar facilitative mechanisms exist also in wetland

environments, including salt marshes, where vegetation

may prevent salt accumulation by limiting soil evaporation

(shading effect), riparian corridors, and wetland forests,

where vegetation can favor the aeration of anoxic soils

through soil drainage by plant uptake and transpiration

[Wilde et al., 1953; Chang, 2002; Ridolfi et al., 2006]. It

has also been found that on cobble beaches, dense stands of

spartina alterniflora occupying the lower intertidal zone

can protect other plant communities from intense wave

action [Bruno, 2000; van de Koppel et al., 2006].

[98] On the other hand, competitive or inhibitory effects

typically occur within a longer range. Competition for water

and nutrients is generally exerted via the root system

[Aguilera and Lauenroth, 1993; Belsky, 1994; Breman

and Kessler, 1995; Breshears et al., 1997; Martens et al.,

1997; Couteron and Lejeune, 2001]. In fact, the lateral roots

extend beyond the edges of the crown [Casper et al., 2003;

Barbier et al., 2008] and extract water and nutrients from

the intercanopy areas [Martens et al., 1997; Lejeune et al.,


19 of 36

RG1005

2004]. Hence, the typical range of competitive interactions

is larger than that of facilitation. For example, for trees and

shrubs, the ratio between the radii of the footprint of canopy

and root systems may be as small as 1/10 [Lejeune et al.,

2004]. Thus, the root system is able to deplete soil resources

(i.e., water and nutrients) from the intercanopy areas and to

compete for resources with other plant individuals [Lefever

and Lejeune, 1997; Yokozawa et al., 1998; Lejeune et al.,

1999; Lejeune and Tlidi, 1999; Couteron and Lejeune,

2001; Lejeune et al., 2002; Rietkerk et al., 2004; Yizhaq et

al., 2005; Barbier et al., 2006]. This competition for

resources usually reduces the growth rate of competing

individuals thereby leading to a net effect of inhibition

through competition [Shnerb et al., 2003].

[99] Thus, facilitative interactions occur within the range

of the crown area and are mostly associated with positive

feedbacks induced by the canopy (e.g., mulching, shading,

protection against fires, grazing, and wind action), while

negative interactions are exerted mainly as resource com-

petition by roots and typically occur at larger distances

(Figure 8). The sign and range of these interactions justifies

the choice of kernel functions shaped as in Figure 7: the

kernel shows positive interactions within the range of the

canopy scale (short-range) and negative interactions within

the range of the typical lateral root length (long range), while

the magnitude of these interactions vanishes (i.e., w ! 0)

as the distance between the interacting plants exceeds the

typical extent of lateral roots. Nevertheless, as already

noticed at the end of section 2.1.2, spatial patterns can also

emerge when the kernel is ‘‘upside/down’’ with respect to the

case of Figure 7. In fact, in some cases, facilitation can occur

at a long range (e.g., buffering from intense wave action in

intertidal communities [van de Koppel et al., 2006]), and

competition can occur at a short range (e.g., competition

for light typically due to interactions among canopies [Caylor

et al., 2005; van de Koppel et al., 2006]).

[100] Although all these mechanisms of cooperation and

competition are generally isotropic (i.e., they operate in the

same way in all directions), in some environments the

presence of a slope or of a dominant wind direction may

lead to anisotropy in the spatial dynamics. In fact, if the

wind regime exhibits a prevailing direction, asymmetry may

emerge in the cooperative and competitive mechanisms. For

example, the persistent existence of a cone-shaped wind

shadow downwind of tree/shrub clumps would provide a

favorable protected environment for the establishment and

growth of other plant individuals (short-range positive

feedback), while plant individuals located at larger distances

would remain with no protection and would consequently

be prone to higher mortality rates [Puigdefabregas et al.,

1999; Yokozawa et al., 1999; Okin and Gillette, 2001].

Similarly, in section 2.1.4 we discussed the role of an

advective flow on the dynamics of two diffusive species

and its role in the phenomenon of differential flow insta-

bility. Advective flows can originate as an effect of runoff in

sloping terrains. During intense rainfall events, water and

sediments run off bare areas and are intercepted and trapped

by vegetated patches. This supplementary input of limiting

resources favors plant growth on the uphill side of vegetated

patches, thereby securing more efficient trapping during

subsequent rainstorm events. Runoff and erosion are there-

fore viewed as a fundamental mechanism to maintain

striped configuration over hillsides [Thiery et al., 1995;

Dunkerley, 1997a, 1997b; Okayasu and Aizawa, 2001;

Sherratt, 2005; Saco et al., 2006; Barbier et al., 2008].

Thus, rainfall onto an unvegetated area generates overland

flow, which transports water in the downhill direction until

it reaches a vegetated area, where it infiltrates into the

ground and is taken up by vegetation. The relatively moist

soil on the uphill side of a stripe creates opportunities for

uphill expansion of the vegetation band at the expenses of

the downhill side, which remains deprived of the resources

necessary for vegetation survival. The overall dynamics

lead to the uphill migration of vegetated bands [Sherratt,

2005]. A similar mechanism can explain the banded patterns

of trees in the Tierra del Fuego (Argentina, see Figure 1h)

where a sawtooth pattern of tree heights is observed in the

wind direction [Puigdefabregas et al., 1999]. Taller trees

provide more protected favorable conditions for seedling

establishment and tree growth in the leeward direction. At

the same time, the strong winds uproot and kill the taller

upwind trees leading to an overall downwindmigration of the

sawtooth pattern.

[101] Unlike the case of neural and differential flow

models, the application of Turing instability to ecohydro-

logical systems is not straightforward. These models are

generally used when the state of the system is described by

more than one state variable and when spatial dynamics can

be modeled as diffusion processes. In most of the existing

models (see section 4), two or three state variables are used,

including vegetation biomass, subsurface, and surface water.

The diffusive character of plant encroachment has been

invoked by a number of studies, which argued that the

propagation of vegetation fronts due to seed dispersal or to

clonal reproduction can be expressed as a diffusion process

[e.g., von Hardenberg et al., 2001;Murray, 2002]. The use of

diffusion in the modeling of surface overland flow has been

justified using the shallow water theory, i.e., assuming that

overland flow occurs with only a thin layer of water [Gilad

et al., 2004]. The diffusive character of subsurface flow is

often justified by recalling the diffusive nature of Darcy’s

law (i.e., the proportionality between water flux and water

potential gradients). However, the diffusivity of unsaturated

subsurface flows is expected to be small and to be unable to

lead to significant soil moisture patterns at the patch scale.

Moreover, it has been noted [Barbier et al., 2008] that the

existence of wetter soils in vegetated areas should result from

mechanisms leading to the concentration of resources be-

neath plant canopies, while soil moisture diffusion would

operate in the opposite direction. Nevertheless, even the use

of diffusion in the description of biomass spreading is

problematic. Diffusive models treat biomass as a ‘‘green

slime,’’ i.e., as if it was not rooted into a place as a series of

individuals. Even vegetative/clonal reproducers could not

rigorously be described as diffusers because the only place

where a real gradient exists is at the edge of a patch. As an


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RG1005

TABLE2.EcohydrologicalTuringModelsProposedto

Explain

PatternForm

ationa

Authors

Model

andVariables

EmergingPatterns

ComparisonWithData

NotesandComments

HilleRisLamberset

al.[2001]

Threevariables(surfacewater,soilwater,andplantdensity).

Adetailedstabilityanalysisinvestigates

theconditionsfor

theform

ationofspatiallynonuniform

structures.

Notspecified.

None.

Themodel

confirm

sthat

itisthe

combinationofpositiveand

negativefeedbacksthat

generates

vegetationpatterns.Other

processes

(herbivores,plantdispersal,and

rainfall)arenottheprimaryfactors

that

generatepatterns.

Rietkerket

al.[2002]

Evolutionofthemodel

byHilleRisLamberset

al.[2001].

Thediffusionterm

ofsurfacewater

canbereplacedwith

adriftterm

switchingthemodel

into

adifferential

flow

model.

Spots,labyrinths,gaps,

andstripes.

Niger.

Transitionfrom

spotsto

labyrinthsand

togapswithincreasingrainfall

amount.Theparam

eter

rangeused

insimulationsindicates

that

patterns

tendto

developmore

frequentlyon

fine-texturedsoils.

vandeKoppel

etal.[2002]

Evolutionofthemodel

byHilleRisLamberset

al.[2001].

Themodel

also

accountsforaterm

describingthevariation

inherbivore

populationdependingonthevariationsofplant

biomass.Graphical

andnumerical

resolutionofequations.

Notspecified.

None.

Graphical

resolutionofequations:two

param

etersarecomputed,P(local

vegetation)andPavg(spatialaverage

ofvegetation).Patternsem

ergewhen

P>Pavg.

Meronet

al.[2004]

Twovariables(plantbiomassandwater).Thesystem

has

two

uniform

stationarysolutionsforlow

andhighvalues

ofrainfall.

Numerical

simulationsshow

nonuniform

solutionsfor

interm

ediate

values

ofrainfall.

Spots,labyrinths,andgaps.

NorthernNegev,

goodagreem

ent

betweenreal

dataand

numerical

simulations.

Themodel

revealsdifferentranges

of

precipitationwherevegetation

patternsmay

coexistwithuniform

states

(baresoilorcompletely

vegetated

soil)andalso

withother

pattern

states.

vandeKoppel

etal.[2006]

Twovariables(plantbiomassandwrack

biomass).Numerical

simulationsanalyze

theem

ergence

ofvegetationpatternsusing

differentfunctiondescribingcooperativeandcompetitive

interactions.

Spots.

Spatialstructuresof

CarexStricta

tussocks

inMaine,

USA.

Model

forwetlands.

GuttalandJayaprakash

[2007]

Evolutionofthemodel

byRietkerket

al.[2002],introducinga

param

eter

that

capturestheeffectsofseasonal

adaptationsofplants

tothedry

andwet

season.Resultsobtained

withnumerical

simulations.

Spots,gaps,andlabyrinths.

None.

Thepattern

dependsonthedurationof

thewet

seasoneven

withfixed

total

annual

precipitation.Thedistribution

within

theyearplaysafundam

ental

role

indeterminingpattern

form

ation

andshape.

ZengandZeng[2007]

Threevariables(m

assdensity

oflivingleaves,available

soilwetness,

andmassdensity

ofwiltedanddeadleaves).A

linearstabilityanalysis

isdeveloped

toobtain

thesetofparam

etersleadingto

spatialinstability.

Theresultsarechecked

throughnumerical

simulations.

Spots,gaps,labyrinths,

andstripes.

None.

Theauthors

stress

therelevance

of

vegetationpatternsin

theprocess

ofdesertification:they

arean

indicatorthat

theecosystem

isat

thevergeofacatastrophic

shift.

aUnless

differentlyspecified,thecomparisonwithdataisonly

qualitative.


21 of 36

RG1005

example, Thompson and Katul [2008] proved that the use of

suitable seed dispersal kernels to represent plant biomass

spread leads a better representation of biomass spreading than

those provided by diffusion models. In sections 4.1.1–4.1.3

and 4.2 we provide an overview of a number of ecological

models of pattern formation, and we will use the concepts

presented in this section and section 2 to discuss their main

properties.

4. MODELS FOR VEGETATION PATTERNFORMATION

[102] In this section we review the ecohydrology litera-

ture on mechanisms and models of vegetation pattern

formation. We first analyze deterministic models of sym-

metry-breaking instability and discuss the underlying biotic

and abiotic mechanisms. Then we discuss some stochastic

models of noise-induced pattern formation in ecohydrology

based on nonequilibrium phase transitions or on random

switching between deterministic dynamics.

4.1. Deterministic Models

4.1.1. Turing-Like Instability Models[103] Only relatively few authors have used Turing mod-

els to explain vegetation patterns (Table 2). The models by

Meron et al. [2004] and van de Koppel et al. [2006] involve

two variables, namely, plant biomass and water [Meron et

al., 2004] or plant biomass and wrack biomass (i.e., algae)

[van de Koppel et al., 2006]. All the other models consid-

ered in this review involve three variables, namely, plant

biomass, surface water, and soil water [HilleRisLambers et

al., 2001; Rietkerk et al., 2002; van de Koppel et al., 2002;

Guttal and Jayaprakash, 2007], except for the case of Zeng

and Zeng [2007], which involves living leaves, dead leaves,

and soil moisture as state variables.

[104] As noted in section 3, the diffusive character of the

spatial dynamics of vegetation depends on seed dispersal

and clonal reproduction [HilleRisLambers et al., 2001;

Rietkerk et al., 2002; van de Koppel et al., 2002; Meron

et al., 2004; Guttal and Jayaprakash, 2007; Zeng and Zeng,

2007], while for dead biomass the diffusive dynamics are

ascribed to horizontal transport by wind, herbivores, or

anthropogenic disturbances [van de Koppel et al., 2006;

Zeng and Zeng, 2007]. Meron et al. [2004] considered (soil)

water as the limiting resource and justified the assumption

of its diffusive behavior using Darcy’s law. Other Turing

models considering soil water as a state variable (coupled

with vegetation dynamics) do not explicitly invoke Darcy’s

law [HilleRisLambers et al., 2001; Rietkerk et al., 2002; van

de Koppel et al., 2002; Guttal and Jayaprakash, 2007; Zeng

and Zeng, 2007]. Turing models accounting for the spatial

and temporal dynamics of surface water [HilleRisLambers

et al., 2001; Rietkerk et al., 2002; van de Koppel et al.,

2002; Guttal and Jayaprakash, 2007] approximate overland

flow as a diffusive process [Bear and Verruyt, 1990]; in

these models, surface water flows from bare soil to vege-

tated areas driven by pressure gradients associated with the

higher soil infiltration capacities in the vegetated soil patches.

Thus, this framework assumes that a positive feedback exists

between vegetation and soil moisture (vegetation ! higher

infiltration ! enhancement of water transport from bare

soil ! more vegetation), which favors transport of water to

vegetated patches. The magnitude of this transport depends

on the soil texture, the presence of physical and/or biological

crusts, and the possible development of some microreliefs as

a result of the accumulation of sediment mounds beneath the

vegetation canopies. Moreover, recent microscale measure-

ments of soil infiltration within vegetation patches [Ravi

et al., 2007, 2008] have shown that relatively small soil

infiltration may occur in the middle of vegetation patches

where finer soil particles are generally found. In this case,

overland flow would likely occur toward the edges of the

vegetated patches both from the surrounding bare soil and

from the middle of vegetation mounds. Thus, the patterns of

overland flow may be more complex than what is usually

believed.

[105] Even though the models by Rietkerk et al. [2002],

van de Koppel et al. [2002], and Guttal and Jayaprakash

[2007] are based on the same Turingmodel [HilleRisLambers

et al., 2001], they stress different aspects of processes

associated with the formation of vegetation patterns. For

example, Rietkerk et al. [2002] extended the model of

HilleRisLambers et al. [2001] replacing the diffusion term

with an advective term, generating a differential flow

model to study patterns on slopes. Numerical simulations

point out that vegetation stripes only emerge if a minimum

terrain slope is present, while the model by HilleRisLambers

et al. [2001] can only generate spots, labyrinths, and gaps.

The model by van de Koppel et al. [2002] investigates how

the pattern geometry changes as the system approaches a

bifurcation point, where it is more susceptible to a discon-

tinuous response (often called ‘‘catastrophic shift’’) to a

continuous change in environmental parameters. The model

by HilleRisLambers et al. [2001] allows one to understand

that the spatial redistribution of surface water prevents

irreversible vegetation collapse, but van de Koppel et al.

[2002] found that if the presence of herbivores is taken into

account, the reduction of vegetation cover beyond a threshold

level can lead the system to a desert state. The model by

Guttal and Jayaprakash [2007] investigates the seasonal

variation of plant growth due to plant response to the dry

and wet season. In nature, as the season changes, various

physical factors affecting the vegetation growth can also

change. This means that the dynamics of vegetation are

different during the wet and the dry season, and the alterna-

tion of different dynamics can affect the spatial arrangements

of plants. This kind of behavior is usually investigated with

stochastic models (see section 2.2.2), while Guttal and

Jayaprakash [2007] use a deterministic approach based on

a Turing model.

[106] As noted in section 2.1.3, approximations of Turing

and neural models lead to the same analytical expressions of

the conditions determining the emergence of symmetry-

breaking instability. The relation existing between these two

types of models suggests that activation/inhibition (or

facilitation/competition) processes explicitly represented in


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RG1005

TABLE

3.EcohydrologicalContinuousKernel-BasedModelsProposedto

Explain

PatternForm

ationa

Authors

Model

andVariables

EmergingPatterns

ComparisonWithData

NotesandComments

Lefever

andLejeune[1997]

Continuousmodel

foronly

onevariable

(vegetation).Kernel

representedbythe

combinationofthreeGaussiankernels

(cooperation,inhibition,andmortality).

System

withtwohomogeneoussteadystates.

Alinearstabilityanalysisisperform

edto

search

forconditionsunder

whichthe

establishmentofuniform

vegetationisim

possible.

Labyrinthsandspotsin

isotropic

system

sandstripes

inanisotropic

system

s.

Tiger

bush

inSomalia

andstripes

inJordan.

Themodel

predictstwokindsofstriped

structures:parallelandperpendicularto

groundslope,

inagreem

entwithobservations

insitu.

Lejeuneet

al.[1999]

Sam

emodel

asLefever

andLejeune[1997].

Anadditional

analysisiscarriedoutto

investigatetherole

offeedbackson

vegetationpattern

form

ation.Itisdem

onstrated

that

intheabsence

ofcooperativefeedbacks,

vegetationpatternscannotform

.

Spotsandstripes

inisotropic

environmentsandstripes

inanisotropic

environments.

Tiger

bush

inBurkinaFaso.

Theauthors

developadetailedanalysisof

aerial

photographsoftiger

bush

innorthwest

BurkinaFasoto

calibrate

theparam

etersof

themodel

and,in

particular,to

estimatethe

ranges

ofthecooperativeandinhibitory

interactionsbetweenplants.

LejeuneandTlidi[1999]

Sam

emodel

asLefever

andLejeune[1997]but

thebiarm

onic

approxim

ationofneuralmodels

(equation(18))isusedto

runnumerical

simulationsin

isotropic

environments.

Spotsandlabyrinths.

Tiger

bush

insouthwestNiger.

Thebiarm

onic

approxim

ationusedhereleads

tothesamemodel

asCouteronandLejeune

[2001].

Lefever

etal.[2000]

Based

onLefever

andLejeune[1997],

Lejeuneet

al.[1999],andLejeuneandTlidi[1999].

Spotsandstripes.

Tiger

bush

inBurkinaFaso.

Numerical

simulationssuggestthat

asource

ofanisotropy(e.g.,slope)

predominantly

affectsinhibitory

rather

than

cooperative

interactionssince

hexagonal

spotsobtained

inisotropic

conditionsarereplacedbystripes

when

inhibitionissufficientlyanisotropic.

CouteronandLejeune[2001]

Continuousmodel

forasingle

variable

representing

averagephytomass.Themodel

originates

from

thebiarm

onic

approxim

ationofthemodel

byLejeuneandTlidi[1999].Numerical

simulations

arerunto

investigatetheshapeofvegetationpatterns

withvaryingecological

param

eters.

Spotsandlabyrinths.

BurkinaFasoandNiger.

Comparisonbetweenspectral

analysesof

digitized

photographsandpatternsresulting

from

simulations.Thisanalysisallowsthe

authors

tocalibrate

themodel.

Lejeuneet

al.[2002]

Sam

emodel

asCouteronandLejeune[2001]withmore

mathem

atical

details(linearstabilityanalysis);

one-dim

ensional

andtwo-dim

ensional

analyses.

General

clustering.

South

Americaand

westAmerica.

None.

Lejeuneet

al.[2004]

Sam

emodel

asCouteronandLejeune[2001].Theauthors

pointoutthedestabilizingeffect

oftheLaplacian

term

andthestabilizingeffect

ofthedouble

Laplacian.

Adetailedstabilityanalysisisrunto

studythepattern

selection,that

is,theem

ergence

ofspotted

orstriped

configurations.

Spots,zigzagstripes,andgaps.

Manyarid

regionsofAfrica,

Australia,

NorthAmerica,

andtheMiddle

East.

Alongarainfalltransect

ofdecreasingrainfall

themodel

showsthetransitionfrom

completely

vegetated

groundto

spots

(vegetationgaps),then

stripes,then

spots

again(vegetationclumps),andfinally

baresoil.

aUnless


qualitative.


23 of 36

RG1005

TABLE

4.EcohydrologicalDiscreteKernel-BasedModelsProposedto

Explain

PatternForm

ationa

Authors

Model

andVariables

EmergingPatterns

ComparisonWithData

NotesandComments

Sato

andIwasa

[1993]

Discretemetapopulationmodel.Single

variable

representingthemedium

heightofthecohort

oftreesstandingin

each

cell.

Ingeneral,when

atree

ismuch

higher

than

thewindwardtrees,itisproneto

windfallanddies.

Stripes

(saw

-toothed

travelingwaves).

Subalpineforests

ofcentral

Japan,

northeastUSA.

Theauthors

notedthat

theinclusionof

disturbancesin

thedeterministicmodel

leadsto

more

realisticsolutions.This

isdonebySatake

etal.[1998].

Thiery

etal.[1995]

Discretecellularautomatamodel.Eachcell

representsastateofatree

corresponding

todifferentmetaboliccharacteristics.

Aconvolutionmatrixrunsover

thedomain

andrepresentscompetitionorsynergyin

ananisotropic

environment.

Irregularstripes.

SouthwestNiger.

Inorder

tovalidatethemodel,aerial

photographsarescanned

andcompared

withnumerical

results.Thesimulated

stripes

migrate

upward,andthissuggests

pattern

changes

atthetopofreal

landscapes

wherebandsconverge.

CominsandHassell[1996]

Discretemetapopulationmodel.Threevariables

representingtwoparasitoid

speciesandthe

density

ofhostspecies(trees).

Somerulesdefinetheinteractionsbetween

speciesandtheinteractionsbetweenneighboringcells.

Spotsandlabyrinths.

None.

Thesimulationsgenerateself-organization

only

when

thespeciesshow

very

differentdispersalrates,as

inaTuringmodel.

Dunkerley

[1997a]

Discretecellularautomatamodel

for

anisotropic

environments.

Therulesdescribethepartitioningofwater

flow

toneighboringcellsdependingonlandcover

andthewater-controlled

dynam

icsofvegetation.

Stripes.

Australia,

Niger,

andMexico.

Stripes

occuronly

withgentleslopes.

Stripes

canbedestroyed

bya

long-lastingim

pactofgrazers,even

ifnodataexistto

evaluatethiseffect.

Dunkerley

[1997b]

Sam

emodel

asDunkerley

[1997a]

withtheadditionofrules

representingtheeffect

ofdroughtonthesoilwater

content

andofgrazingonthevegetationbiomass.

Stripes.

Australia.

Grazingim

pacthas

nosustained

effect.

Themodel

showsthat

droughtis

potentially

more

destructivethan

grazingpressure.

Keymer

etal.[1998]

Discretemetapopulationmodel.Single

binaryvariable

representing

thestateofthepatch

(occupiedorem

pty).A

single

rule

accountingforextinctionandcolonization.Therule

accountsfor

differentranges

ofinteractionsbetweenneighboringcells.

General

clustering.

None.

Numerical

simulationsshow

that

spatial

patternscanem

ergeonly

withlocal

interactions,that

is,when

therange

ofinteractionsbetweencellsisnot

toolargebecause

theirnet

effect

isto

includecooperativeeffects.

Yokozawaet

al.[1998]

Discretecellularautomatamodel.Single

variable

representing

thesize

ofeach

individual.A

single

rule

defines

theevolution

ofthevariable

dependingonagrowth

andacompetitionterm

.Competitioncanbesymmetricorasymmetric.

General

clustering.

ForestsofnorthernJapan.

Numerical

simulationsshow

that

coarser

andmore

uniform

patches

ariseunder

symmetricalcompetition.Thisisin

agreem

entwithearlierstudies[Kubota

andHara,1995]that

foundsymmetric

competitionbetweentreesin

coniferousforests.

Puigdefabregaset

al.[1999]

Discretemetapopulationmodel.Thepopulationin

each

cellgrows

proportionally

toagrowth

rate

ordeclines

byself-thinningand

winddieback.Winddiebackdependsonaprotectionfactor

from

windwardneighbors.Numerical

simulationsstartfrom

arandom

distributionofindividuals.

Stripes.

Tierradel

Fuego.

Numerical

simulationslead

tostriped

configurations(perpendicularto

wind

direction).Larger

tree

growth

rateslead

tolonger

wavelengths,andan

increase

ofwindstrength

leadsto

shorter

wavelengthsin

agreem

entwithfieldobservations.

Yokozawaet

al.[1999]

Sam

emodel

asYokozawaet

al.[1998].A

sourceofexternal

asymmetry,i.e.,unidirectional

wind,isadded.

Stripes.

SubalpineregionsofJapan

andUnited

States.

Comparisonbetweensymmetricand

asymmetriccompetition.The

wave-shaped

spatialpattern

ismuch

clearerunder

symmetricalcompetition.


24 of 36

RG1005

neural models of pattern formation exist also in the dynam-

ics underlying the spatial interactions in Turing models.

This aspect of Turing models is addressed by van de Koppel

and Rietkerk [2004], who built a model able to reproduce

different interactions between plants. This study showed

that vegetation patterns emerge only with short-range co-

operation and long-range inhibition, in agreement with the

results obtained with neural models (section 2.1.2). Simi-

larly, Zeng and Zeng [2007] explicitly described the spatial

interactions in their Turing model as the result of facilitation

and competition processes.

4.1.2. Kernel-Based Models[107] Two main types of kernel-based models can be

found in the ecohydrology literature, namely continuous

and discrete spatially extended models. Continuous models

(Table 3) use a continuous representation of the kernel

describing the spatial interactions among individuals, while

discrete models (Table 4) use a discrete kernel defined over

a discrete spatially extended domain represented as a lattice

of square cells. Because continuous kernels can be defined

with continuous mathematical functions, they lend them-

selves to an analytical description (section 2.1.2), which is

often not possible with discrete kernels.

[108] The first continuous kernel-based model of vegeta-

tion patterns [Lefever and Lejeune, 1997] describes the

interactions among plants through three different functions,

representing short-range cooperative effects (higher infiltra-

tion rate, shading, mulching, and less soil crusting under the

canopy), long-range competition for resources, and a tox-

icity effect accounting for the mortality rates existing with

higher densities of vegetation. The combination of these

three components gives the kernel of equation (12). These

authors studied the stability of the system in isotropic and

anisotropic conditions. Numerical simulations lead to

striped, spotted, and labyrinthine configurations in isotropic

landscapes and to banded patterns in anisotropic conditions.

This model was used to explain tiger bush dynamics and

favorably compared [Lejeune and Tlidi, 1999; Lejeune et

al., 1999] with real data from tiger bush vegetation in Niger

and Burkina Faso. A kernel-based model was developed by

D’Odorico et al. [2006c] to show how vegetation patterns

can be due to the ability of spatial dynamics to enhance

productivity and vegetation cover in arid landscapes, with

respect to the case of ecosystems with no spatial interac-

tions. As noted, the main difference between the model by

D’Odorico et al. [2006c] and that by Lefever and Lejeune

[1997] is that in the former model, nonlinearities are local,

while in the latter they modulate the spatial interactions.

[109] Lefever et al. [2000] provided a biarmonic approx-

imation of this kernel-based model [Lefever and Lejeune,

1997] with an approach similar to the one presented in

section 2.1.2. This biarmonic model is used by Couteron

and Lejeune [2001] to account for short-range cooperative

effects (shading, mulching, and protection against herbi-

vores [Lejeune et al., 2002]) expressed by the diffusion term

(propagation) and for long-range competitive interactions

(exerted via lateral roots [Couteron and Lejeune, 2001;

Lejeune et al., 2002]) represented by the biarmonic term

Authors

Model

andVariables

EmergingPatterns

ComparisonWithData

NotesandComments

Adleret

al.[2001]

Discretecellularautomatonmodel.Twoplantspeciesandagrazing

component.Differenttypes

ofinteractionsbetweenspecies

(competitionandcooperation).Numerical

simulationsallow

one

toobtain

differentspatialconfigurationsdependingonthe

interactionrules.

General

clustering.

USA,South

Africa,

andTierradel

Fuego.

Thesimulationssuggestthat

neighborhood

interactionsat

theindividual

plantscale

rather

than

grazingeffectsorunderlying

environmentalheterogeneities

lead

topatchystructures.

LanzerandPillar[2002]

Discretecellularautomatonmodel.Eachcellcanbeoccupiedwith

oneoutofninepossible

states

(aplantspeciesorbareground).

Arule

defines

thepersistence

ofthestatein

thecelloritschange

into

another

possible

statedependingonneighborhoodfrequencies.

General

clustering.

Atlanticheathland

intheNetherlands.

Alargesetofreal

dataallowsoneto

model

theinteractionsbetweenspeciesfitting

theneighborhoodfrequency

ofagiven

statearoundeverysampledplant.

Rietkerket

al.[2004]

Sam

emodel

asThiery

etal.[1995]butdeveloped

hereonly

for

isotropic

environments.Anisotropic

convolutionmatrixmodeling

short-rangecooperationandlong-rangeinhibitionrunsover

the

whole

domain.

Spotsandlabyrinths.

Niger,Israel,Ivory

Coast,

French

Guiana,

andSiberia.

Thesystem

isbistable;that

is,forsomeranges

of

resourceinputitshowstwodifferentstable

configurations.When

thesystem

approaches

acritical

valueofresourceinput,itcanundergo

catastrophic

shifts.

Feagin

etal.[2005]

Discretecellularautomatonmodel.A

convolutionmatrixmodels

facilitationeffects.A

gradientmatrixsimulatestheeffect

ofa

sloped

terrainandthecompetitiveeffectsdueto

the

interceptionofrunoff.

Stripes.

Texas.

Only

thesimulationsaccountingforasloped

terraingenerateplantpatterns.Thesimulated

patternsarecomparable

withthefielddata.

EstebanandFairen

[2006]

Discretecellularautomatonmodel.Thedomainrepresentsa

gentlysloped

surface.

Arule

defines

soilmoisture

storage

andtheam

ountofbiomassdependingonthesoilmoisture.

Stripes.

Tiger

bush

intheSahel.

Iftheannual

rainfallisbelow

acrossover

value,

asudden

catastrophic

shiftfrom

patchystructure

andbaresoilisobserved.Upslopemigratingbands.

aUnless


qualitative.

TABLE4.(continued)


25 of 36

RG1005

(inhibition). This ‘‘propagation-inhibition’’ model [Couteron

and Lejeune, 2001], leads to the spotted/labyrinthine

patterns in isotropic conditions and to banded patterns in

anisotropic landscapes. These patterns were compared with

those from aerial photographs over West Africa: a good

agreement was found overall between modeled and observed

patterns [Barbier et al., 2008]. The same model was also

used by Lejeune et al. [2002, 2004].

[110] In the discrete kernel-based models the spatial

domain is subdivided into a discrete number of cells, with

each cell representing either a single plant or a group of

plants. Two major types of discrete neural models can be

found in the literature, namely, cellular automaton (CA)

models and metapopulation models. In CA models (Table 4)

the values of the state variables in each cell change accord-

ing to a set of rules, which use a spatial kernel to modulate

the interactions among neighboring cells [Dunkerley,

1997b]. These models lend themselves to the representation

with simple rules of the physical processes underlying the

dynamics of the system. For example, the model developed

by Thiery et al. [1995] to explain the formation of tiger bush

in Niger uses an asymmetric kernel to account for the

anisotropy associated with the presence of a slope and the

consequent occurrence of water runoff and sediment flow.

This kernel expresses the interplay between cooperation and

inhibition. Cooperation is induced by the same synergistic

mechanisms of facilitation described in section 3 (i.e.,

shading, mulching, and enhancement of soil infiltration),

while competition is caused by the interception of water at

the uphill side of a vegetation stripe, which results in water

deficit at the downhill side. A similar model was developed

by Rietkerk et al. [2004], using a symmetric discrete kernel

to describe the spatial dynamics of vegetation in isotropic

environments. This CA model showed how vegetation

patterns (spots and labyrinths) may emerge as a result of

the spatial interactions on flat terrains, in the absence of any

prescribed landscape heterogeneity and anisotropy. This

model [Rietkerk et al., 2004] showed how the pattern

geometry may depend on the strength of the mutual

interactions among plants: the model can generate spots

with relatively weak positive interactions and labyrinths

when these interactions become relatively strong. This

aspect is of great importance to relate the different pattern

shapes to the different levels of cooperation. Thus, pattern

geometry can be also explained as a result of the strength of

the cooperative interaction that the existing species can

develop. Similar models were proposed to explain vegeta-

tion pattern formation in Texas [Feagin et al., 2005] and

tiger bush dynamics in the Sahel [Esteban and Fairen,

2006]. Both models are anisotropic in that they account

for the existence of a slope and for its role in the process of

pattern formation. Other authors used similar anisotropicmod-

els to investigate vegetation patterns in Australia [Dunkerley,

1997a, 1997b]. The models by Dunkerley [1997a, 1997b]

were also used to investigate the role of grazing and drought

in the formation of vegetation patterns and showed that

spatially organized configurations of vegetation emerge only

with gentle slopes. Moreover, these models were able to

demonstrate how this spatial organization enhances the

ability of the system to recover after a disturbance. Thus,

aridland vegetation would increase its resilience by develop-

ing spatial interactions, which result in the commonly ob-

served patchy distribution of vegetation.

[111] The model by Yokozawa et al. [1998] considers a

different type of anisotropy, which is not associated with a

topographic slope but with a possible asymmetric competi-

tion between individuals. In this model the discrete kernel

function can account for both a symmetric and an asym-

metric competition. This study shows how patterns can

emerge with both symmetric and asymmetric competition.

However, symmetric competition leads to more homoge-

neous patterns (i.e., patterns with features of about the same

size), which are more similar to those observed in conifer-

ous forests. These results confirmed experimental results

[Kubota and Hara, 1995] showing that competition be-

tween trees is generally symmetric. The situation is different

in the presence of anisotropy induced by an external driver

(e.g., a prevailing wind). In this case, a modified version of

the model [Yokozawa et al., 1999] shows that the same

patterns can be generated either with symmetric or asym-

metric competition.

[112] Metapopulation models (Table 4) are based on a

very simplified framework and are generally used to sim-

ulate the dynamics of very complex systems. These models

can give a preliminary idea of the basic aspects of the

system’s behavior, which can be further investigated with

more complex models. Hanski [2004] used metapopulation

models to study the dynamics of populations living in

highly fragmented landscapes, where only a small fraction

of the total area offers suitable habitat for a given species.

This is often the case in arid and semiarid environments,

where the landscape typically exhibits a highly heteroge-

neous distribution of resources. Thus, metapopulation mod-

els seem to provide a suitable framework to examine the

spatial dynamics of dryland plant ecosystems.

[113] In a metapopulation model the domain is usually

subdivided into cells similarly to the case of CA kernel-

based models; however, a single cell represents a relatively

large portion of the landscape, where a group of individuals

exist, the so-called ‘‘metapopulation.’’ Each metapopulation

has its own dynamics, which can be independent of those in

other cells. Spatial patterns emerge when interactions based

on short-range cooperation and long-range competition

exist among cells [Keymer et al., 1998]. For example, in

the works by Puigdefabregas et al. [1999], Iwasa et al.

[1991], and Sato and Iwasa [1993], patterns are induced by

metapopulations interacting through a positive feedback in a

windy environment, where leeward trees benefit from the

sheltering provided by upwind trees, while in the work by

Comins and Hassell [1996] the authors investigate the

patterns emerging from the interactions between vegetation

and different parasitoid species.

4.1.3. Differential Flow Instability Models[114] The idea that vegetation patterns could be induced

by a drift (section 2.1.4) was first formulated by Klausmeier

[1999] in a model based on differential flow instability. This


26 of 36

RG1005

model accounts for two variables, namely, vegetation bio-

mass and (surface) water. Plant dispersal is represented as a

diffusion process, while surface water undergoes advective

flow in the downhill direction. This model does not invoke

diffusive processes to mimic overland flow. This framework

leads to the formation of vegetation stripes migrating uphill

on sloped terrains, while on flat terrains (i.e., with no drift),

patterns do not emerge because the system dynamics reduce

to those of a bivariate Turing model with only one diffusing

variable. Sherratt [2005] and Ursino [2005] used the same

framework to investigate changes in the dynamics of

vegetation stripes with varying environmental parameters

(e.g., rainfall, plant mortality, and slope). They found that

the wavelength of stripe sequences is a decreasing function

of rainfall and an increasing function of slope [Sherratt, 2005].

The slope plays a crucial role in determining the conditions for

pattern formation: Ursino [2005] found that a minimum

terrain slope is needed for stripes to emerge. Some of these

properties are in agreement with experimental observations.

[115] More recently, other modeling frameworks (Table 5)

have been proposed to explain vegetation pattern formation

as a drift-induced process. For example, Gilad et al. [2004]

accounted for the dynamics of three state variables, namely,

plant biomass, soil water, and surface water depth. The

model accounts for two feedbacks between biomass and

water and shows how patterns result from mechanisms of

ecosystem engineering. The two feedbacks are associated

with the short-range positive interaction resulting from the

increase in infiltration induced by vegetation and long-

range negative interaction due to plant competition for soil

water uptake. In this model, all variables exhibit a diffusive

behavior ascribed to plant seed dispersal, the diffusive

nature of soil-water transport in a nonsaturated soil [Hillel,

1998], and the shallow-water theory for surface water.

Shallow-water theory also accounts for an advective term

associated with the slope-dependent water flow. Thus,

because this model is a combination of a Turing and a

differential flow model, patterns may arise either from

Turing instability or differential flow instability. When

applied to flat terrains, this framework becomes a Turing

model, in that the drift term becomes zero and the spatial

dynamics are contributed only by diffusion processes. In

this case, Turing instability may lead to the formation of

spots, gaps, or labyrinths. In the case of sloped terrains the

model can generate patterns through the mechanism of

differential flow instability, leading to the formation of

stripes or spots (with low rainfall values). The model was

recently used byGilad et al. [2007] to investigate the role of

the two feedbacks in the process of pattern formation.

These authors found that the same type of patterns may

emerge when either one of these feedbacks is predominant.

Yizhaq et al. [2005] used the same model to relate the

resilience of striped vegetation to the wavelength and found

that patterns with high wave numbers correspond to sys-

tems with higher productivity that are more biologically

productive but less resilient to environmental changes.

Moreover, starting from the model by Gilad et al. [2004],

Meron et al. [2007] addressed problems like biomass-water

relationships in spot-like vegetation patches, formation

mechanisms of ring-like vegetation patches, and resilience

of vegetation patches.

[116] Other authors [Okayasu and Aizawa, 2001; von

Hardenberg et al., 2001] previously developed similar

models obtained by adding a drift term to a Turing model

to account for the flow of water and the transport of

sediments in the downhill direction. Both models show

the transition between spots and labyrinths in the absence of

drift, and the formation of bands when the drift is included.

A comparison between pattern formation with or without

drift is also given by Shnerb et al. [2003], who used a model

obtained as a combination of a differential flow model

(accounting for the drift) with a neural model. These

authors showed how both drift and cooperative interactions

may induce pattern formation. Further studies on the role of

sediment transport are given by Saco et al. [2006], where

the models by HilleRisLambers et al. [2001] and Rietkerk

et al. [2002] were extended explicitly accounting for the

dynamic effect of erosion-deposition processes caused by

an overland water flow on a hillslope. The simulated

configuration evolves into a profile with stepped micro-

topography, in agreement with field data in shrubland

communities in Australia [Dunkerley and Brown, 1999].

4.2. Stochastic Models

[117] Despite the pervasive presence of random fluctua-

tions in environmental processes, applications of the theo-

ries of noise-induced pattern formation in ecohydrology are

still rare. Early studies on the ability of noise to enhance

order in the spatial distribution of vegetation are given by

Satake et al. [1998] and Yokozawa et al. [1998, 1999]. In

particular, Satake et al. [1998] modified a previous discrete

metapopulation formulation of the neural model [Iwasa et

al., 1991; Sato and Iwasa, 1993] to show how, when one or

more suitable parameters are interpreted as random varia-

bles, the spatial patterns generated by the model become

more distinct and regular. This constructive effect of noise is

observed when the noise intensity (e.g., variance) exceeds a

certain minimum level; however, as the noise intensity

exceeds another critical threshold, the random drivers de-

stroy the spatial organization of the system (destructive

effect of noise). As noted in section 2.2, this noise-induced

behavior corresponds to a reentrant phase transition in that

the order-forming effect of noise is observed only within a

limited range of noise intensities. These results were found

by Satake et al. [1998] through numerical simulations of dis-

crete (meta)population dynamics, without using an analytical/

theoretical framework.

[118] D’Odorico et al. [2007b] capitalized on the theory of

noise-induced nonequilibrium phase transitions (section 2.2)

to analyze possible mechanisms of fire-induced pattern for-

mation in savannas. These authors modeled the temporal and

spatial changes in shrub (or tree) vegetation, u, using only one

stochastic differential equation with the same form as (34),

@u

@t¼ f uð Þ þ g uð Þxcp þ dr2u; ð41Þ


27 of 36

RG1005

TABLE

5.EcohydrologicalModelsofDifferentialFlow

InstabilityProposedto

Explain

PatternForm

ationa

Authors

Model

andVariables

EmergingPatterns

ComparisonWithData

NotesandComments

Klausm

eier

[1999]

Twovariables(surfacewater

andplantbiomass).Themodel

exhibits

multiple

stable

states,andtwocasesareanalyzed

(hillsides

andflat

ground).A

linearstabilityanalysisisused

todeterminewhether

regularpatternscanform

.

Stripes.

Niger.

Linearstabilityanalysisshowsthat

regularpattern

form

ationis

impossible

onflat

groundand

that

thewavelength

ofthe

patternsincreaseswith

decreasingwater

input.

vonHardenberget

al.[2001]Twovariables(plantbiomassandwater

available

toplants).

Themodel

exhibitstwodifferentstable

states

becomingunstable

forinterm

ediate

values

ofprecipitation.Themodel

isstudied

throughthestabilityanalysisofuniform

solutionsandintegrating

equationsnumerically.

Spots,labyrinths,andgaps

forflat

landscapes

and

stripes

onaslope.

Niger

andnorthernNegev.Differentpatterns(spots,labyrinths,

andgaps)

emergealongarainfall

gradient,andthisallowsthe

introductionofaridity

classesdependingonthe

vegetationstate:

uniform

vegetation,vegetationpatterns,

orbaresoil.

Okayasu

andAizawa[2001]

Threevariables(w

ater

onthegroundsurface,

water

inthe

soil,andplantbiomass).Thesystem

has

twodifferent

stable

states.A

linearstabilityanalysisinvestigates

the

instabilityofparticularwavenumbers.Numerical

simulationsanalyze

spatialpatternsem

ergingwithdifferentsetsofparam

eters.


(withoutsurfaceflow)and

stripes

withdownslopeflow.

Somalia

andnorthwest

BurkinaFaso.

Ingoodclim

ate,

spotsem

erge

over

hillslopes,whilethey

becomestripes

forsevere

clim

ates.Thestripes

migrate

upwardandthewavenumber

ofperiodic

patters

decreases

astheexternal

environmentbecomes

worse.

Shnerbet

al.[2003]

Twovariables(available

water

density

anddensity

ofshrubbiomass).

Alinearanalysisshowsthestabilityofthehomogeneousstatewithout

water

flow,andnumerical

simulationsshow

theem

ergence

ofspatial

patternswhen

adownhillflow

ispresent.

General

clustering

tendingto

astriped

configuration.

Israel.

Patternsem

ergeonly

ifapositive

feedbackbetweenplantsis

present.Thepatternsare‘‘frozen’’

(i.e.,nouphillmigration).

Giladet

al.[2004]

Threevariables(density

ofbiomass,soilmoisture,anddepth

of

surfacewater).Thesystem

has

twodifferentuniform

solutions.

Theirlinearstabilityisstudiedanalytically,

whilenonuniform

solutionsareobtained

bysolvingequationsnumerically

forflat

topography

andonaslope.


(noslope)

andstripes

and

spots(slope).

None.

Themodel

isusedto

study

conditionsthat

maketheecosystem

resilientto

environmentalchanges.

Everyenvironmentalparam

eter

presentsan

optimal

rangefor

whichplantsarecapable

of

restoringthehabitatsaftera

disturbance.

Sherratt[2005]

EvolutionbyKlausm

eier

[1999].Mathem

atical

analysis.A

linearstability

analysisisusedto

determineunstable

modes.Analysisofthedependence

ofstriped

patternsonecological

param

eters(rainfall,plantloss,andslope).

Stripes.

Aerialphotosfrom

Africa,

Australia,

andMexico.

Themodel

generates

anuphill

migrationofstripes

provided

that

theeigenvalues

corresponding

tounstable

modes

arecomplex

numbers.

Ursino[2005]

Themodel

byKlausm

eier

[1999]isreinterpretedto

accountforrelevantsoil

properties.A

detailedstabilityanalysisisrunin

order

toobtain

thecritical

stabilityconditionsdependingonecological

param

eterslikerainfall,terrain

slope,

andsoilparam

eterslikeconductivity.

Banded

configurations.

Differentsitesin

America

andAustralia.

Themodel

pointsoutthecrucial

role

ofsoilphysics

inplant

development.

Yizhaqet

al.[2005]

Evolutionofthemodel

byGiladet

al.[2004],thoughtheauthors

analyze

only

thesituationonaslope.

Numerical

simulationsarerunto

investigatethe

resilience,water

consumption,andbiomassproductivityofbandswithdifferent

wavenumbersunder

thesamerainfallconditions.

Stripes.

None.

Water

consumptionper

unit

biomassdecreases

asthepattern’s

wavenumber

increases,so

banded

vegetationismore

productivefor

higher

wavenumbers,butitisless

resilientto

environmentalchanges.


28 of 36

RG1005

where the local dynamics are expressed by a logistic growth

term f(u) = a(u + �)(umax � u) (the parameter a measures

the reproduction rate of the logistic growth), with umax

being the shrub carrying capacity and � being a parameter

preventing u from remaining locked at u = 0 after a severe

fire kills all the shrubs. The second term on the right-hand

side accounts for loss in shrub biomass associated with

random fire occurrences and is modeled as discrete

sequence of events occurring at random times, ti, with each

event having a random magnitude, w. A compound Poisson

noise, xcp = wd(t � ti), is used, where d() is the Dirac delta

function, and the times, t, between two consecutive

occurrences, t = ti+1 � ti (i = 1, 2, . . .), are modeled as an

exponentially distributed random variable with mean hti = 1/l.To account for the dependence of fire occurrences on the fuel

load (i.e., locally available grass biomass), which is inversely

related to the shrub biomass, u, the fire frequency, l, isexpressed as a function of u, l(u) = l0 + bu (with b < 0). The

variable w is an exponentially distributed random variable

(independent of t) with mean w0. The multiplicative function

g(u) =min(u,w) accounts for the fact that the amount of shrub

biomass removed by any fire is either u or w, whichever isless. The diffusion term in (41) expresses the encroachment

of shrub vegetation as a diffusion process. Both g(u) and l(u)determine the multiplicative character of the stochastic

forcing in (41).

[119] Using the mean field approximation (see section 2.2)

the steady state probability distribution of u is calculated

[D’Odorico et al., 2007b] as a function of the spatial mean,hui, of u

pst u; huið Þ ¼ C

r uð Þ expu

w0

�Z

l uð Þr uð Þ du

� �; ð42Þ

with r(u) = a(u + e)(1 � u) + d(hui � u). The solution of

the self-consistency equation (36) with steady state prob-

ability distribution given by (42) gives the mean value, hui,of u. Figure 14a shows a plot of hui as a function of the

diffusivity, d. It is observed that when the spatial coupling is

relatively strong (i.e., high values of d), the self-consistency

equation exhibits three solutions. Numerical simulations

show that two of them are stable and the other is unstable.

Thus, in these conditions the system may have two different

possible mean values and hence two possible steady state

probability distributions of u [D’Odorico et al., 2007b]. It

can be shown [see also Porporato and D’Odorico, 2004]

that the emergence of these multiple statistically steady

states is induced by the combined effect of spatial coupling

and the nonlinearity associated with the positive feedback

between stochastic forcing (i.e., fires) and the state of the

system (i.e., shrub biomass). As noted in section 2.2 this

effect of ergodicity breaking is induced by the multiplicative

character of the noise in (41). Patterns may emerge as

ordered states, resulting from nonequilibrium phase transi-

tions that break the ergodicity of the system. An example is

provided in Figure 14b: it can be observed that these

multiple (statistically) steady states emerge when the

variance of the random forcing exceeds a critical value;

Authors

Model

andVariables

EmergingPatterns

ComparisonWithData

NotesandComments

Saco

etal.[2006]

Partially

based

onthemodelsbyHilleRisLamberset

al.[2001]and

Rietkerket

al.[2002].Threevariables(plantbiomass,soilmoisture,

andoverlandflow).Themodel

iscoupledwithaphysicallybased

model

oftheevolutionoflandform

s.Numerical

simulationsanalyze

thespatial

evolutionofvegetationandmicrotopography.

Stationarybandsand

migratingbands.

Australia.

Theinitiallyplanar

hillslopeevolves

into

aprofile

withstepped

microtopography.

Giladet

al.[2007]

Evolutionofthemodel

byGiladet

al.[2004],buthereadetailedlinearstability

analysisisdeveloped

toanalyze

thestabilityofthetwouniform

stationary

solutions.Numerical

simulationsarerunto

investigatetheeffectsofdisturbances

andchanges

inphysicalparam

eters.

Spotsandstripes.

None.

Numerical

simulationsmainly

analyze

thesoilwater

distributiondepending

onphysicalparam

eters(infiltration,

precipitation,evaporation,and

grazingstress)to

investigatethe

role

ofplantsas

ecosystem

engineers.

Meronet

al.[2007]

Sam

emodel

byGiladet

al.[2004]andGiladet

al.[2007].

Spots(localizedstructures).

NorthernNegev.

Thepreviousmodelsarefurther

investigated

inorder

toaddress

avariety

ofproblemsrelatedto

patchiness,resilience,anddiversity

ofdrylandvegetation.

aUnless


qualitative.

TABLE

5.(continued)


29 of 36

RG1005

with increasing values of the variance a second phase

transition occurs, which leads to the disappearance of the

stable states shown in Figure 14a and hence of the spatial

patterns.

[120] An ecohydrological model of patterns induced by

the random switching between deterministic dynamics was

suggested by D’Odorico et al. [2006b], who showed how

patterns may emerge as an effect of random interannual

fluctuations of precipitation. This model describes the

dynamics of dryland vegetation, u, as a repeated switch

between two deterministic processes corresponding to

stressed conditions (years of drought or ‘‘state 1’’) and

unstressed conditions (wet years or ‘‘state 2’’). In each of

these states the system’s dynamics are expressed by a neural

model (equation (11)), with the same kernel function (12),

w, for the two states. This function accounts for short-range

facilitation and long-range competition resulting from the

spatial interactions among plant individuals. The local

dynamics are expressed as a linear decay (i.e., f1(u) =

�a1u) in state 1 and as a logistic growth (i.e., f2(u) =

a2u(1 � u)) in state 2. Neither one of these states is capable

of leading to pattern formation. However, patterns may

emerge from the random switching between the two states

caused by interannual fluctuations of precipitation [D’Odorico

et al., 2006b] as shown in Figure 15 for the case in which the

system is in state 1 with probability P1 and in state 2 with

probability P2 = 1 � P1. The process can be studied in terms

of mean dynamics because the response of woody vegetation

to water stressed or unstressed condition is slow if compared

to the year-to-year rainfall variability [e.g., Barbier et al.,

2006].

5. CONCLUSIONS

[121] In a number of landscapes around the world the

vegetation cover frequently exhibits a remarkable spatial

organization with either periodic or random mosaics of

gaps, spots, bands, circles, or geometrically irregular vege-

tation patches. In several instances these organized config-

urations of vegetation exhibit absolutely spectacular

patterns, which have drawn the attention of geographers,

ecologists, physicists, and mathematicians. In fact, in the

recent past a number of scientists have developed some

metrics for the quantitative characterization of vegetation

patterns and have related them to environmental factors,

including climate, topography, soil properties, dominant

plant species, and disturbance regime. Moreover, the wide-

spread occurrence of some conspicuous patterns of vegeta-

tion has also stimulated the interest in the mechanisms

underlying vegetation pattern formation.

[122] This paper has provided a review of the major

theories explaining the formation of self-organized patterns

in vegetation. The focus has been only on patterns that are

not induced by preexisting heterogeneity in the soil sub-

strate (nor in other abiotic drivers); rather, this review

concentrated on self-emerging patterns, which result from

properties inherent to the dynamics of vegetation and to

their coupling with the environmental conditions (e.g.,

limiting resources and disturbance). The analytical and

Figure 14. (a) Steady state average woody biomass, hui,as a function of the diffusion coefficient, d, for a = 0.45,l0 = 0.65, b/l0 = �0.9, w0 = 0.4, and e = 0.0001. The linesrepresent the steady states of mean vegetation, hui,depending on spatial coupling, d. The continuous linesrepresent the stable states, while the dashed line representsthe unstable state. From D’Odorico et al. [2007b].(b) Example of model-generated pattern. Same parametersas in Figure 14a, d = 0.3. From D’Odorico et al. [2007b].

Figure 15. Dependence of mean and standard deviation ofdryland vegetation, u, on the probability, P2, of being in un-stressed conditions. The parameters are q2/q1 = 2.0, b2/b1 =0.25, a1/a2 = 1.454, and b1q1

2/a1 = 0.4. For 0.2 < P2 < 0.64the spatial standard deviation is larger than zero, suggestingthe emergence of spatially heterogeneous vegetation. Thepatterns generated by the model are shown in the insets. FromD’Odorico et al. [2006b].


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RG1005

semianalytical theories commonly invoked to explain the

self-organized patterning of vegetation are based on the

ability of the spatial dynamics of vegetation to destabilize

the homogeneous state of the system, thereby leading to

pattern formation. Known as ‘‘symmetry-breaking instabil-

ity’’ this mechanism can be obtained with a few different

models. This review has classified the models of symmetry-

breaking instability used in ecohydrology as either deter-

ministic or stochastic. Three major classes of analytical de-

terministic models have been discussed in sections 2.1.1,

2.1.2, and 2.1.4, namely, Turing’s models, kernel-based

models, and differential flow models. In all these models,

patterns emerge as a result of symmetry-breaking instability

in the presence of activator-inhibitor dynamics.

[123] This review considered stochastic models in which

noise plays a fundamental role in determining the emer-

gence of vegetation patterns. Two different mechanisms of

noise-induced pattern formation were reviewed: (1) a model

based on an effect of ergodicity-breaking phase transition

associated with a noise-induced bifurcation in the steady

state probability distribution of the state variable and (2) a

model in which patterns emerge from the random switching

between deterministic dynamics. In this second case the

mechanism of pattern formation remains deterministic, but

the emergence of patterns is triggered by the random

switching.

[124] A few major open issues emerge from this review:

[125] 1. Most of the theories presented in this paper have

not been quantitatively validated in the field. We are not

aware of any conclusive experimental evidence that the

vegetation patterns observed in many regions of the world

are actually induced by mechanisms of symmetry-breaking

instability.

[126] 2. The fact that the same types of patterns can be

generated by different models suggests that it is really

difficult to test the models just by comparing simulated

and observed patterns. This is due to the fact that the

amplitude equations, which determine important properties

of the pattern geometry, belong to only a few major classes.

Thus, the claim of relating patterns to processes by devel-

oping process-based models capable of reproducing the

observed patterns is probably too ambitious. Though veg-

etation patterns are interesting, and even striking at times,

there is very little we can learn from them in terms of the

processes that shape the landscape. There does not appear to

be any pattern that uniquely belongs to a certain type of

model that codifies a specific set of processes. Because most

models differ in the choice of the limiting factors (nutrients,

soil moisture, and surface water) and in the representation of

the spatial dynamics (diffusion, overland flow, and root

uptake), more confidence can be placed on those models

that have been actually validated in the field with direct

measurements assessing the significance of the invoked

processes [Barbier et al., 2008].

[127] 3. Most of the existing literature on self-organized

pattern formation is based on deterministic mechanisms. All

these contributions invoke one of the three deterministic

mechanisms discussed in sections 2.1.1, 2.1.2, and 2.1.4,

while only a handful of studies have investigated the

possible emergence of vegetation patterns as a noise-in-

duced effect. New interesting questions exist on the role of

random drivers in the process of vegetation self-organiza-

tion. Thus, the field of stochastic ecohydrology seems to

offer more ‘‘room’’ for the development of new significant

research on vegetation pattern formation.

[128] ACKNOWLEDGMENTS. We thank two anonymous

reviewers and Gregory S. Okin for useful comments and sugges-

tions. Support from Cassa di Risparmio di Torino Foundation and

NSF awards DEB 0717360 and EAR 0746228 is gratefully

acknowledged.

[129] The Editor responsible for this paper was Daniel Tartakovsky.

He thanks Greg Okin and two anonymous technical reviewers.

REFERENCES

Acosta, A., S. Diaz, M. Menghi, and M. Cabido (1992), Commu-nity patterns at different spatial scales in the grasslands of Sierrasde Cordoba, Argentina, Rev. Chilena Hist. Nat., 65(2), 195–207.

Adejuwon, J. O., and F. A. Adesina (1988), Vegetation patternsalong the forest-savanna boundary in Nigeria, Singapore J. Trop.Geol., 9(1), 18–32.

Adler, P. B., D. A. Raff, and W. K. Lauenroth (2001), The effect ofgrazing on the spatial heterogeneity of vegetation, Oecologia,128(4), 465–479.

Aguiar, M. R., and O. E. Sala (1999), Patch structure, dynamicsand implications for the functioning of arid ecosystems, TrendsEcol. Evol., 14(7), 273–277.

Aguilera, M. O., and W. K. Lauenroth (1993), Seedling establish-ment in adult neighborhoods intraspecific constraints in the re-generation of the bunchgrass Bouteloua-Gracilis, J. Ecol., 81(2),253–261.

Anderies, J. M., M. A. Janssen, and B. H. Walker (2002), Grazingmanagement, resilience, and the dynamics of a fire-driven range-land system, Ecosystems, 5(1), 23–44.

Archer, S. (1989), Have southern Texas savannas been convertedto woodlands in recent history?, Am. Nat., 134(4), 545–561.

Augustine, D. J. (2003), Spatial heterogeneity in the herbaceouslayer of a semi-arid savanna ecosystem, Plant Ecol., 167(2),319–332.

Bak, P. (1996), How Nature Works: The Science of Self-OrganizedCriticality, Copernicus, New York.

Bak, P., and C. Tang (1989), Earthquakes as a self-organized cri-tical phenomenon, J. Geophys. Res., 94(B11), 15,635–15,637.

Bak, P., C. Tang, and K. Wiesenfeld (1988), Self-organized criti-cality, Phys. Rev. A, 38(1), 364–374.

Bak, P., K. Chen, and M. Creutz (1989), Self-organized criticalityin the game of life, Nature, 342(6251), 780–782.

Bak, P., K. Chen, and C. Tang (1990), A forest-fire modeland some thoughts on turbulence, Phys. Lett. A, 147(5–6),297–300.

Barbier, N., P. Couteron, J. Lejoly, V. Deblauwe, and O. Lejeune(2006), Self-organized vegetation patterning as a fingerprint ofclimate and human impact on semi-arid ecosystems, J. Ecol.,94(3), 537–547.

Barbier, N., P. Couteron, R. Lefever, V. Deblauwe, and O. Lejeune(2008), Spatial decoupling of facilitation and competition at theorigin of gapped vegetation patterns, Ecology, 89(6), 1521–1531.

Bear, J., and A. Verruyt (1990), Modeling Groundwater Flow andPollution: Theory and Applications of Transports in PorousMedia, D. Reidel, Dordrecht, Netherlands.

Belnap, J., J. R. Welter, N. B. Grimm, N. Barger, and J. A. Ludwig(2005), Linkages between microbial and hydrologic processes inarid and semiarid watersheds, Ecology, 86(2), 298–307.


31 of 36

RG1005

Belsky, A. J. (1994), Influences of trees on savanna productivity—Tests of shade, nutrients, and tree-grass competition, Ecology,75(4), 922–932.

Bena, I. (2006), Dichotomous Markov noise: Exact results for out-of-equilibrium systems, Int. J. Mod. Phys. B, 20(20), 2825–2888.

Benzi, R., A. Sutera, and A. Vulpiani (1985), Stochastic resonancein the Landau-Ginzburg equation, J. Phys. A Math. Gen., 18(12),2239–2245.

Bergkamp, G., A. Cerda, and A. C. Imeson (1999), Magnitude-frequency analysis of water redistribution along a climate gradi-ent in Spain, Catena, 37(1–2), 129–146.

Bernd, J. (1978), The problem of vegetation stripes in semi-aridAfrica, Plant Res. Dev., 8, 37–50.

Bhark, E. W., and E. E. Small (2003), Association between plantcanopies and the spatial patterns of infiltration in shrubland andgrassland of the Chihuahuan Desert, New Mexico, Ecosystems,6(2), 185–196.

Boaler, S. B., and C. A. H. Hodge (1962), Vegetation stripes inSomaliland, J. Ecol., 50, 465–474.

Boaler, S. B., and C. A. H. Hodge (1964), Observations on vegeta-tion arcs in the Northern Region, Somali Republic, J. Ecol., 52,511–544.

Boeken, B., and D. Orenstein (2001), The effect of plant litter onecosystem properties in a Mediterranean semi-arid shrubland,J. Veg. Sci., 12(6), 825–832.

Bolliger, J. (2005), Simulating complex landscapes with a genericmodel: Sensitivity to qualitative and quantitative classifications,Ecol. Complexity, 2(2), 131–149.

Bonanomi, G., M. Rietkerk, S. C. Dekker, and S. Mazzoleni(2008), Islands of fertility induce co-occurring negative and po-sitive plant-soil feedbacks promoting coexistence, Plant Ecol.,197(2), 207–218.

Borgogno, F., P. D’Odorico, F. Laio, and L. Ridolfi (2007), Effectof rainfall interannual variability on the stability and resilience ofdryland plant ecosystems, Water Resour. Res., 43, W06411,doi:10.1029/2006WR005314.

Breman, H., and J.-J. Kessler (1995), Woody Plants in Agro-Eco-systems of Semi-Arid Regions, Springer, Berlin.

Breshears, D. D., O. B. Myers, S. R. Johnson, C. W. Meyer, andS. N. Martens (1997), Differential use of spatially heterogeneoussoil moisture by two semiarid woody species: Pinus edulis andJuniperus monosperma, J. Ecol., 85(3), 289–299.

Bromley, J., J. Brouwer, A. P. Barker, S. R. Gaze, and C. Valentin(1997), The role of surface water redistribution in an area ofpatterned vegetation in a semi-arid environment, south-westNiger, J. Hydrol., 198(1–4), 1–29.

Bruno, J. F. (2000), Facilitation of cobble beach plant communitiesthrough habitat modification by Spartina alterniflora, Ecology,81(5), 1179–1192.

Brush, S. G. (1967), History of Lenz-Ising model, Rev. Mod. Phys.,39(4), 883–893.

Buceta, J., and K. Lindenberg (2002), Switching-induced Turinginstability, Phys. Rev. E, Part 2, 66(4), 046202.

Buceta, J., and K. Lindenberg (2003), Spatial patterns inducedpurely by dichotomous disorder, Phys. Rev. E, Part 1, 68(1),011103.

Buceta, J., K. Lindenberg, and J. M. R. Parrondo (2002a), Station-ary and oscillatory spatial patterns induced by global periodicswitching, Phys. Rev. Lett., 88(2), 024103.

Buceta, J., K. Lindenberg, and J. M. R. Parrondo (2002b), Patternformation induced by nonequilibrium global alternation of dy-namics, Phys. Rev. E, Part 2, 66(6), 069902.

Buceta, J., K. Lindenberg, and J. M. R. Parrondo (2002c), Spatialpatterns induced by random switching, Fluctuation Noise Lett.,2(1), L21–L29.

Burgman, M. A. (1988), Spatial-analysis of vegetation patterns insouthern western Australia—Implications for reserve design,Aust. J. Ecol., 13(4), 415–429.

Burke, I. C., et al. (1998), Plant-soil interactions in temperategrasslands, Biogeochemistry, 42(1–2), 121–143.

Carrillo, O., M. A. Santos, J. Garcia-Ojalvo, and J. M. Sancho(2004), Spatial coherence resonance near pattern-forming in-stabilities, Europhys. Lett., 65(4), 452–458.

Casper, B. B., H. J. Schenk, and R. B. Jackson (2003), Defining aplant’s belowground zone of influence, Ecology, 84(9), 2313–2321.

Castets, V., E. Dulos, J. Boissonade, and P. Dekepper (1990),Experimental-evidence of a sustained standing Turing-type none-quilibrium chemical-pattern, Phys. Rev. Lett., 64(24), 2953–2956.

Caylor, K. K., and H. H. Shugart (2006), Pattern and process insavanna ecosystems, in Dryland Ecohydrology, edited by P.D’Odorico and A. Porporato, pp. 259–282, Springer, Berlin.

Caylor, K. K., H. H. Shugart, and I. Rodriguez-Iturbe (2005), Treecanopy effects on simulated water stress in southern Africansavannas, Ecosystems, 8(1), 17–32.

Caylor, K. K., P. D’Odorico, and I. Rodriguez-Iturbe (2006), Onthe ecohydrology of structurally heterogeneous semiarid land-scapes, Water Resour. Res., 42, W07424, doi:10.1029/2005WR004683.

Chandrasekhar, S. (1961), Hydrodynamic and Hydromagnetic Sta-bility, Clarendon, Oxford, U. K.

Chang, M. (2002), Forest Hydrology: An Introduction to Waterand Forests, CRC Press, Boca Raton, Fla.

Charley, J. L. (1972), The role of shrubs in nutrient cycling, inWildland Shrubs: Their Biology and Utilization, Gen. Tech. Rep.INT-1, pp. 182–203, For. Serv., U.S. Dep. of Agric., Washington,D. C.

Charley, J. L., and N. E. West (1975), Plant-induced soil chemicalpatterns in some shrub-dominated semi-desert ecosystem ofUtah, J. Ecol., 63(3), 945–963.

Clos-Arceduc, A. (1956), Etude sur photographies aeriennes d’uneformation veg’tale sahelienne: La brousse tigree, Bull. Inst. Fon-dam. Afr. Noire, Ser. A, 18(3), 677–684.

Comins, H. N., and M. P. Hassell (1996), Persistence of multi-species host-parasitoid interactions in spatially distributed mod-els with local dispersal, J. Theor. Biol., 183(1), 19–28.

Cornet, A. F., J. P. Delhoume, and C. Montana (1988), Dynamicsof striped vegetation patterns and water balance in the Chihua-huan desert, in Diversity and Patterns in Plant Communities,edited by H. J. During, M. J. A. Werger, and J. H. Willems,pp. 221–231, SPB Academic, The Hague, Netherlands.

Couteron, P. (2002), Quantifying change in patterned semi-aridvegetation by Fourier analysis of digitized aerial photographs,Int. J. Remote Sens., 23(17), 3407–3425.

Couteron, P., and O. Lejeune (2001), Periodic spotted patterns insemi-arid vegetation explained by a propagation-inhibition model,J. Ecol., 89(4), 616–628.

Couteron, P., A. Mahamane, P. Ouedraogo, and J. Seghieri (2000),Differences between banded thickets (tiger bush) at two sites inWest Africa, J. Veg. Sci., 11(3), 321–328.

Cross, M. C., and P. C. Hohenberg (1993), Pattern-formation out-side of equilibrium, Rev. Mod. Phys., 65(3), 851–1112.

Dale, M. R. T. (1999), Spatial Pattern Analysis in Plant Ecology,Cambridge Univ. Press, Cambridge, U. K.

DiPrima, R. C., and H. L. Swinney (1981), Instabilities and trasi-tion in flow between concentric rotating cylinders, in Hydrody-namic Instabilities and the Transition to Turbulence, edited byH. L. Swinney and J. P. Gollub, pp. 139–180, Springer, Berlin.

D’Odorico, P., F. Laio, and L. Ridolfi (2005), Noise-induced sta-bility in dryland plant ecosystems, Proc. Natl. Acad. Sci. U. S. A.,102(31), 10,819–10,822.

D’Odorico, P., F. Laio, and L. Ridolfi (2006a), A probabilisticanalysis of fire-induced tree-grass coexistence in savannas, Am.Nat., 167(3), E79–E87.

D’Odorico, P., F. Laio, and L. Ridolfi (2006b), Vegetation patternsinduced by random climate fluctuations, Geophys. Res. Lett., 33,L19404, doi:10.1029/2006GL027499.

D’Odorico, P., F. Laio, and L. Ridolfi (2006c), Patterns as indica-tors of productivity enhancement by facilitation and competition


32 of 36

RG1005

in dryland vegetation, J. Geophys. Res., 111, G03010,doi:10.1029/2006JG000176.

D’Odorico, P., K. Caylor, G. S. Okin, and T. M. Scanlon (2007a),On soil moisture-vegetation feedbacks and their possible effectson the dynamics of dryland ecosystems, J. Geophys. Res., 112,G04010, doi:10.1029/2006JG000379.

D’Odorico, P., F. Laio, A. Porporato, L. Ridolfi, and N. Barbier(2007b), Noise-induced vegetation patterns in fire-prone savan-nas, J. Geophys. Res., 112, G02021, doi:10.1029/2006JG000261.

Dormann, S., A. Deutsch, and A. T. Lawniczak (2001), Fourieranalysis of Turing-like pattern formation in cellular automatonmodels, Future Gen. Comput. Syst., 17(7), 901–909.

Drossel, B., and F. Schwabl (1992), Self-organized criticality in aforest-fire model, Physica A, 191(1–4), 47–50.

Dubois-Violette, E., G. Durand, E. Guyon, P. Manneville, andP. Pieranski (1978), Instabilities in nematic liquid crystals, inSolid State Physics, suppl. 14, edited by L. Liebert, pp. 147–208, Academic, New York.

Dunkerley, D. L. (1997a), Banded vegetation: Development underuniform rainfall from a simple cellular automaton model, PlantEcol., 129(2), 103–111.

Dunkerley, D. L. (1997b), Banded vegetation: Survival underdrought and grazing pressure based on a simple cellular automa-ton model, J. Arid Environ., 35(3), 419–428.

Dunkerley, D. L., and K. J. Brown (1999), Banded vegetation nearBroken Hill, Australia: Significance of surface roughness andsoil physical properties, Catena, 37(1–2), 75–88.

Durrett, R. (1999), Stochastic spatial models, SIAM Rev, 41(4),677–718.

Eddy, J., G. S. Humphreys, D. M. Hart, P. B. Mitchell, and P. C.Fanning (1999), Vegetation arcs and litter dams: Similarities anddifferences, Catena, 37(1–2), 57–73.

Eldridge, D. J., and R. S. B. Greene (1994), Microbiotic soilcrusts—A review of their roles in soil ecological processes inthe rangelands of Australia, Aust. J. Soil Res., 32(3), 389–415.

Esteban, J., and V. Fairen (2006), Self-organized formation ofbanded vegetation patterns in semi-arid regions: A model, Ecol.Complexity, 3(2), 109–118.

Evans, D., and J. R. Ehlringer (1993), Broken nitrogen cycles inarid lands: Evidence from 15N of soils, Oecologia, 94, 314–317.

Feagin, R. A., X. B. Wu, F. E. Smeins, S. G. Whisenant, and W. E.Grant (2005), Individual versus community level processes andpattern formation in a model of sand dune plant succession, Ecol.Modell., 183(4), 435–449.

Fearnehough, W., M. A. Fullen, D. J. Mitchell, I. C. Trueman, andJ. Zhang (1998), Aeolian deposition and its effect on soil andvegetation changes on stabilised desert dunes in northern China,Geomorphology, 23(2–4), 171–182.

Fuentes, E. R., A. J. Hoffmann, A. Poiani, and M. C. Alliende(1986), Vegetation change in large clearings—Patterns in theChilean Matorral, Oecologia, 68(3), 358–366.

Garcia-Moya, E., and C. M. McKell (1970), Contribution of shrubsto the nitrogen economy of a desert-wash plant community, Ecol-ogy, 51(1), 81–88.

Garcia-Ojalvo, J., and J. M. Sancho (1999), Noise in SpatiallyExtended Systems, Springer, New York.

Gilad, E., J. von Hardenberg, A. Provenzale, M. Shachak, andE. Meron (2004), Ecosystems engineers: From pattern for-mation to habitat creation, Phys. Rev. Lett., 93, 098105.

Gilad, E., J. von Hardenberg, A. Provenzale, M. Shachak, andE. Meron (2007), A mathematical model of plants as ecosystemengineers, J. Theor. Biol., 244(4), 680–691.

Giles, R. H., and M. K. Trani (1999), Key elements of landscapepattern measures, Environ. Manage., 23, 477–481.

Greene, R. S. B. (1992), Soil physical-properties of three geo-morphic zones in a semiarid Mulga woodland, Aust. J. SoilRes., 30(1), 55–69.

Greene, R. S. B., P. I. A. Kinnell, and J. T. Wood (1994), Role ofplant cover and stock trampling on runoff and soil-erosion fromsemiarid wooded rangelands, Aust. J. Soil Res., 32(5), 953–973.

Greene, R. S. B., C. Valentin, and M. Esteves (2001), Runoffand erosion processes, in Banded Vegetation Patterning in Aridand Semiarid Environments: Ecological Processes and Con-sequences for Management, Ecol. Stud., vol. 149, edited byC. Valentin et al., pp. 52–76, Springer, New York.

Greig-Smith, P. (1979), Pattern in vegetation, J. Ecol., 67, 775–779.Greig-Smith, P., and M. J. Chadwick (1965), Data on pattern with-in plant communities. III. Acacia-Capparis semi-desert scrub inthe Sudan, J. Ecol., 53, 465–474.

Guichard, F., P. M. Halpin, G. W. Allison, J. Lubchenco, and B. A.Menge (2003), Mussel disturbance dynamics: Signatures ofoceanographic forcing from local interactions, Am. Nat.,161(6), 889–904.

Gunaratne, G. H., and R. E. Jones (1995), An invariant measure ofdisorder in patterns, Phys. Rev. Lett., 75(18), 3281–3284.

Guttal, V., and C. Jayaprakash (2007), Self-organization and pro-ductivity in semi-arid ecosystems: Implications of seasonality inrainfall, J. Theor. Biol., 248, 490–500.

Hanski, I. (2004), Metapopulation theory, its use and misuse, BasicAppl. Ecol., 5(5), 225–229.

Harper, K. T., and J. R. Marble (1988), A role for nonvascularplants in management of arid and semiarid rangeland, in Vegeta-tion Science Applications for Rangeland Analysis and Manage-ment, edited by P. T. Tueller, pp. 135–169, Kluwer Acad.,Dordrecht, Netherlands.

Hemming, C. F. (1965), Vegetation arcs in Somaliland, J. Ecol.,53, 57–67.

Henderson, T. C., R. Venkataraman, and G. Choikim (2004),Reaction-diffusion patterns in smart sensor networks, in Pro-ceedings of the 2004 IEEE International Conference on Robot-ica and Automation, pp. 654–658, Inst. of Electr. and Electron.Eng., New Orleans, La.

Herbut, I. (2007), A Modern Approach to Critical Phenomena,Cambridge Univ. Press, Cambridge, U. K.

Hiernaux, P., and B. Gerard (1999), The influence of vegetationpattern on the productivity, diversity and stability of vegetation:The case of ‘brousse tigree’ in the Sahel, Acta Oecol., 20(3),147–158.

Hillel, D. (1998), Environmental Soil Physics, Academic, SanDiego, Calif.

HilleRisLambers, R., M. Rietkerk, F. van den Bosch, H. H. T.Prins, and H. de Kroon (2001), Vegetation pattern formation insemi-arid grazing systems, Ecology, 82(1), 50–61.

Horsthemke, W., and R. Lefever (1984), Noise-Induced Transi-tions. Theory and Applications in Physics, Chemistry, and Biol-ogy, Springer, Berlin.

Issa, O. M., J. Trichet, C. Defarge, A. Coute, and C. Valentin(1999), Morphology and microstructure of microbiotic soil crustson a tiger bush sequence (Niger, Sahel), Catena, 37(1–2), 175–196.

Iwasa, Y., K. Sato, and S. Nakashima (1991), Dynamic modelingof wave regeneration (Shimagare) in subalpine Abies forests,J. Theor. Biol., 152(2), 143–158.

Jensen, H. J. (1998), Self-Organized Criticality, Cambridge Univ.Press, Cambridge, U. K.

Joffre, R., and S. Rambal (1993), How tree cover influences thewater-balance of Mediterranean rangelands, Ecology, 74(2),570–582.

Jones, C. G., J. H. Lawton, and M. Shachak (1994), Organisms asecosystem engineers, Oikos, 69(3), 373–386.

Kefi, S., M. Rietkerk, C. L. Alados, Y. Pueyo, V. P. Papanastasis,A. ElAich, and P. de Ruiter (2007a), Spatial vegetation patternsand imminent desertification in Mediterranean arid ecosystems,Nature, 449(7159), 213–217.

Kefi, S., M. Rietkerk, M. van Baalen, and M. Loreau (2007b),Local facilitation, bistability and transitions in arid ecosystems,Theor. Popul. Biol., 71(3), 367–379.

Keymer, J. E., P. A. Marquet, and A. R. Johnson (1998), Patternformation in a patch occupancy metapopulation model: A cellu-lar automata approach, J. Theor. Biol., 194, 79–90.


33 of 36

RG1005

Klausmeier, C. A. (1999), Regular and irregular patterns in semi-arid vegetation, Science, 284(5421), 1826–1828.

Krummel, J. R., R. H. Grardner, G. Sugihara, and R. V. O’Neill(1987), Landscape patterns in a disturbed environment, Oikos,48(3), 321–324.

Kubota, Y., and T. Hara (1995), Tree competition and speciescoexistence in a sub-boreal forest, northern Japan, Ann. Bot.,76(5), 503–512.

Lam, N. S.-N., and L. de Cola (1993), Fractals in Geography,Prentice Hall, Englewood Cliffs, N. J.

Lanzer, A. T. S., and V. D. Pillar (2002), Probabilistic cellularautomaton: Model and application to vegetation dynamics, Com-mun. Ecol., 3(2), 159–167.

Lefever, R., and O. Lejeune (1997), On the origin of tiger bush,Bull. Math. Biol., 59(2), 263–294.

Lefever, R., O. Lejeune, and P. Couteron (2000), Generic mod-elling of vegetation patterns. A case study of tiger bush inSub-Saharian Sahel, in Mathematical Models for BiologicalPattern Formation: Frontiers in Biological Mathematics, edi-ted by P. K. Maini and H. G. Othmer, pp. 83–112, Springer,New York.

Lejeune, O., and M. Tlidi (1999), A model for the explanation ofvegetation stripes (tiger bush), J. Veg. Sci., 10(2), 201–208.

Lejeune, O., P. Couteron, and R. Lefever (1999), Short range co-operativity competing with long range inhibition explains vege-tation patterns, Acta Oecol., 20(3), 171–183.

Lejeune, O., M. Tlidi, and P. Couteron (2002), Localized vegeta-tion patches: A self-organized response to resource scarcity,Phys. Rev. E, Part 1, 66(1), 010901.

Lejeune, O., M. Tlidi, and R. Lefever (2004), Vegetation spots andstripes: Dissipative structures in arid landscapes, Int. J. QuantumChem., 98(2), 261–271.

Leppanen, T. (2005), The theory of Turing pattern formation, inCurrent Topics in Physics in Honor of Sir Roger Elliot, edited byK. Kaski and R. A. Barrio, pp. 190–227, Imperial Coll. Press,London.

Li, J., G. S. Okin, L. Alvarez, and H. Epstein (2007), Quantitativeeffects of vegetation cover on wind erosion and soil nutrient lossin a desert grassland of southern New Mexico, USA, Biogeo-chemistry, 85(3), 317–332.

Loescher, H. W., S. F. Oberbauer, H. L. Gholz, and D. B. Clark(2003), Environmental controls on net ecosystem-level carbonexchange and productivity in a Central American tropical wetforest, Global Change Biol., 9(3), 396–412.

Ludwig, J. A., and D. J. Tongway (1995), Spatial-organization oflandscapes and its function in semiarid woodlands, Australia,Landscape Ecol., 10(1), 51–63.

Mabbutt, J. A., and P. C. Fanning (1987), Vegetation banding inarid Western Australia, J. Arid Environ., 12(1), 41–59.

Macfadyen, W. A. (1950a), Soil and vegetation in British Somali-land, Nature, 165, 121.

Macfadyen, W. A. (1950b), Vegetation patterns in the semi-desertplains of British Somaliland, Geogr. J., 116, 199–210.

Maestre, F. T., J. F. Reynolds, E. Huber-Sannwald, J. Herrick, andM. Stafford-Smith (2006), Understanding global desertification:Biophysical and socioeconomic dimensions of hydrology, inDryland Ecohydrology, edited by P. D’Odorico and A. Porporato,pp. 315–332, Springer, Berlin.

Malamud, B. D., G. Morein, and D. L. Turcotte (1998), Forestfires: An example of self-organized critical behavior, Science,281(5384), 1840–1842.

Mandelbrot, B. B. (1984), The Fractal Geometry of Nature, Free-man, New York.

Manor, A., and N. M. Shnerb (2008), Facilitation, competition, andvegetation patchiness: From scale free distribution to patterns,J. Theor. Biol., 253(4), 838–842.

Martens, S. N., D. D. Breshears, C. W. Meyer, and F. J. Barnes(1997), Scales of above-ground and below-ground competitionin a semi-arid woodland detected from spatial pattern, J. Veg.Sci., 8(5), 655–664.

Mauchamp, A., C. Montana, J. Lepart, and S. Rambal (1993),Ecotone dependent recruitment of a desert shrub, FluorensiaCernua, in vegetation stripes, Oikos, 68(1), 107–116.

Meinhardt, H. (1982), Models of Biological Pattern Formation,Academic, London.

Meron, E., E. Gilad, J. von Hardenberg, M. Shachak, and Y. Zarmi(2004), Vegetation patterns along a rainfall gradient, Chaos So-litons Fractals, 19(2), 367–376.

Meron, E., H. Yizhaq, and E. Gilad (2007), Localized structures indryland vegetation: Forms and functions, Chaos, 17(3), 037109,doi:10.1063/1.2767246.

Montana, C. (1992), The colonization of bare areas in 2-phasemosaics of an arid ecosystem, J. Ecol., 80(2), 315–327.

Montana, C., J. Lopez-Portillo, and A. Mauchamp (1990), Theresponse of 2 woody species to the conditions created by ashifting ecotone in an arid ecosystem, J. Ecol., 78(3), 789–798.

Morse, D. R., J. H. Lawton, M. M. Dodson, and M. H. Williamson(1985), Fractal dimension of vegetation and the distribution ofarthropod body lengths, Nature, 314(6013), 731–733.

Murray, J. D. (2002), Mathematical Biology, Springer, Berlin.Murray, J. D., and P. K. Maini (1989), Pattern formation mechan-isms—A comparison of reaction diffusion and mechanochemicalmodels, in Cell to Cell Signaling: From Experiments to Theore-tical Models, pp. 159–170, Academic, New York.

Neiring, W. A., R. H. Whittaker, and C. H. Lowe (1963), TheSaguaro: A population in relation to environment, Science,142, 15–23.

Okayasu, T., and Y. Aizawa (2001), Systematic analysis of periodicvegetation patterns, Prog. Theor. Phys., 106(4), 705–720.

Okin, G. S., and D. A. Gillette (2001), Distribution of vegetation inwind-dominated landscapes: Implications for wind erosion mod-eling and landscape processes, J. Geophys. Res., 106(D9),9673–9683.

Okin, G. S., B. Murray, and W. H. Schlesinger (2001), Degradationof sandy arid shrubland environments: Observations, processmodelling, and management implications, J. Arid Environ.,47(2), 123–144.

Okin, G. S., D. A. Gillette, and J. E. Herrick (2006), Multi-scalecontrols on and consequences of aeolian processes in landscapechange in arid and semi-arid environments, J. Arid Eviron.,65(2), 253–275.

Oster, G. F., and J. D. Murray (1989), Pattern-formation modelsand developmental constraints, J. Exp. Zool., 251(2), 186–202.

Parrondo, J. M. R., C. van den Broeck, J. Buceta, and F. J.DeLaRubia (1996), Noise-induced spatial patterns, PhysicaA, 224(1–2), 153–161.

Perry, J. N. (1998), Measures of spatial pattern for counts, Ecology,79(3), 1008–1017.

Platten, J. K., and J.-C. Legros (1984), Convection in Liquids,Springer, Berlin.

Porporato, A., and P. D’Odorico (2004), Phase transitions driven bystate-dependent Poisson noise, Phys. Rev. Lett., 92(11), 110601.

Puigdefabregas, J., F. Gallart, O. Biaciotto, M. Allogia, and G. delBarrio (1999), Banded vegetation patterning in a subantarcticforest of Tierra del Fuego, as an outcome of the interactionbetween wind and tree growth, Acta Oecol., 20(3), 135–146.

Ravi, S., P. D’Odorico, and G. S. Okin (2007), Hydrologic andaeolian controls on vegetation patterns in arid landscapes, Geo-phys. Res. Lett., 34, L24S23, doi:10.1029/2007GL031023.

Ravi, S., P. D’Odorico, L. Wang, and S. Collins (2008), Form andfunction of grass ring patterns in arid grasslands: The role ofabiotic controls, Oecologia, 158, 545 – 555, doi:10.1007/s00442-008-1164-1.

Richards, J. H., and M. M. Caldwell (1987), Hydraulic lift—Sub-stantial nocturnal water transport between soil layers by Artemi-sia-Tridentata roots, Oecologia, 73(4), 486–489.

Ridolfi, L., P. D’Odorico, and F. Laio (2006), Effect of vegetation-water table feedbacks on the stability and resilience of plantecosystems, Water Resour. Res., 42, W01201, doi:10.1029/2005WR004444.


34 of 36

RG1005

Ridolfi, L., F. Laio, and P. D’Odorico (2008), Fertility island for-mation and evolution in dryland ecosystems, Ecol. Soc., 13(1), 5.

Rietkerk, M., and J. van de Koppel (2008), Regular pattern forma-tion in real ecosystems, Trends Ecol. Evol., 23(3), 169–175.

Rietkerk, M., M. C. Boerlijst, F. van Langevelde, R. HilleRisLam-bers, J. van de Koppel, L. Kumar, H. H. T. Prins, and A. M. deRoos (2002), Self-organization of vegetation in arid ecosystems,Am. Nat., 160(4), 524–530.

Rietkerk, M., S. C. Dekker, P. C. de Ruiter, and J. van de Koppel(2004), Self-organized patchiness and catastrophic shifts in eco-systems, Science, 305(5692), 1926–1929.

Rodriguez-Iturbe, I. (2000), Ecohydrology: A hydrologic perspec-tive of climate-soil-vegetation dynamics, Water Resour. Res.,36(1), 3–9.

Rodriguez-Iturbe, I., and A. Rinaldo (2001), Fractal River Basins:Chance and Self-Organization, Cambridge Univ. Press, Cam-bridge, U. K.

Rohani, P., T. J. Lewis, D. Grunbaum, and G. D. Ruxton (1997),Spatial self-organization in ecology: Pretty patterns or robustreality?, Trends Ecol. Evol., 12(2), 70–74.

Rovinsky, A. B., and M. Menzinger (1992), Chemical-instabilityinduced by a differential flow, Phys. Rev. Lett., 69(8), 1193–1196.

Rovinsky, A. B., and A. M. Zhabotinsky (1984), Mechanism andmathematical-model of the oscillating bromate-ferroin-bromoma-lonic acid reaction, J. Phys. Chem., 88(25), 6081–6084.

Ruxton, G. D., and P. Rohani (1996), The consequences of sto-chasticity for self-organized spatial dynamics, persistence andcoexistence in spatially extended host-parasitoid communities,Proc. R. Soc. London, Ser. B, 263(1370), 625–631.

Saco, P. M., G. R. Willgoose, and G. R. Hancock (2006), Eco-geomorphology and vegetation patterns in arid and semi-aridregions, Hydrol. Earth Syst. Sci. Discuss., 3, 2559–2593.

Sagues, F., J. M. Sancho, and J. Garcia-Ojalvo (2007), Spatiotem-poral order out of noise, Rev. Mod. Phys., 79(3), 829–882.

Satake, A., T. Kubo, and Y. Iwasa (1998), Noise-induce regularityof spatial wave patterns in subalpine Abies forests, J. Theor.Biol., 195(4), 465–479.

Sato, K., and Y. Iwasa (1993), Modeling of wave regeneration insub-alpine Abies forests—Population-dynamics with spatialstructure, Ecology, 74(5), 1538–1550.

Scanlon, T. M., K. K. Caylor, S. A. Levin, and I. Rodriguez-Iturbe(2007), Positive feedbacks promote power-law clustering of Ka-lahari vegetation, Nature, 449(7159), 209–U4.

Schimel, D. S., E. F. Kelly, C. Yonker, R. Aguilar, and R. D. Heil(1985), Effects of erosional processes on nutrient cycling insemiarid landscapes, in Planetary Ecology, edited by D. E. Caldwell,J. A. Brierley, and C. L. Brierley, pp. 571–580, Van NostrandReinhold, New York.

Schlesinger, W. H., J. F. Reynolds, G. L. Cunningham, L. F. Huen-neke, W. M. Jarrell, R. A. Virginia, and W. G. Whitford (1990),Biological feedbacks in global desertification, Science,247(4946), 1043–1048.

Scholes, R. J., and S. R. Archer (1997), Tree-grass interactions insavannas, Annu. Rev. Ecol. Syst., 28, 517–544.

Sherratt, J. A. (2005), An analysis of vegetation stripe formation insemi-arid landscapes, J. Math. Biol., 51(2), 183–197.

Shnerb, N. M., P. Sarah, H. Lavee, and S. Solomon (2003), Re-active glass and vegetation patterns, Phys. Rev. Lett., 90(3),038101.

Slatyer, R. O. (1961), Methodology of a water balance study con-ducted on a desert woodland (Acacia aneura F. Muell.) commu-nity in central Australia, in Plant-Water Relationship in Arid andSemiarid Conditions, Arid Zone Res., vol. 16, pp. 15–26, U. N.Educ. Sci., and Cult. Organ., Paris.

Smit, G. N., and N. F. G. Rethman (2000), The influence of treethinning on the soil water in a semi-arid savanna of southernAfrica, J. Arid Environ., 44(1), 41–59.

Sole, R. V., and S. C. Manrubia (1995), Are rain-forests self-orga-nized in a critical-state?, J. Theor. Biol., 173(1), 31–40.

Soriano, A., O. E. Sala, and S. B. Perelman (1994), Patch structureand dynamics in Patagonian arid steppe, Vegetatio, 111(2), 127–135.

Spagnolo, B., D. Valenti, and A. Fiasconaro (2004), Noise in eco-systems: A short review, Math. Biosci., 1(1), 185–211.

Sprott, J. C., J. Bolliger, and D. J. Mladenoff (2002), Self-orga-nized criticality in forest-landscape evolution, Phys. Lett. A,297(3–4), 267–271.

Staliunas, K., and V. J. Sanchez-Morcillo (2000), Turing patterns innonlinear optics, Opt. Commun., 117(1–6), 389–395.

Thiery, J. M., J.-M. D’Herbes, and C. Valentin (1995), A modelsimulating the genesis of banded vegetation patterns in Niger,J. Ecol., 83(3), 497–507.

Thompson, S., and G. Katul (2008), Plant propagation fronts andwind dispersal: An analytical model to upscale from seconds todecades using superstatistics, Am. Nat., 171(4), 468–479.

Tongway, D. J., and J. A. Ludwig (1990), Vegetation and soilpatterning in semiarid mulga lands of Eastern Australia, Aust.J. Ecol., 15(1), 23–34.

Turing, A. M. (1952), The chemical basis of morphogenesis, Phi-los. Trans. R. Soc., Ser. B, 237, 37–72.

Ursino, N. (2005), The influence of soil properties on the formationof unstable vegetation patterns on hillsides of semiarid catch-ments, Adv. Water Res., 28(9), 956–963.

Valentin, C., J.-M. D’Herbes, and J. Poesen (1999), Soil and watercomponents of banded vegetation patterns, Catena, 37(1–2),1–24.

van de Koppel, J., and M. Rietkerk (2004), Spatial interactions andresilience in arid ecosystems, Am. Nat., 163(1), 113–121.

van de Koppel, J., et al. (2002), Spatial heterogeneity and irrever-sible vegetation change in semiarid grazing systems, Am. Nat.,159(2), 209–218.

van de Koppel, J., M. Rietkerk, N. Dankers, and P. M. J. Herman(2005), Scale-dependent feedback and regular spatial patterns inyoung mussel beds, Am. Nat., 165(3), E66–E77.

van de Koppel, J., A. H. Altieri, B. S. Silliman, J. F. Bruno, andM. D. Bertness (2006), Scale-dependent interactions and com-munity structure on cobble beaches, Ecol. Lett., 9(1), 45–50.

van den Broeck, C. (1983), On the relation between white shotnoise, Gaussian white noise, and the dichotomic Markov process,J. Stat. Phys., 31(3), 467–483.

van den Broeck, C., J. M. R. Parrondo, and R. Toral (1994), Noise-induced nonequilibrium phase transition, Phys. Rev. Lett., 73(25),3395–3398.

van den Broeck, C., J. M. R. Parrondo, R. Toral, and R. Kawai(1997), Nonequilibrium phase transitions induced by multiplica-tive noise, Phys. Rev. E, 55(4), 4084–4094.

van der Meulen, F., and J. W. Morris (1979), Striped vegetationpatterns in a Transvaal savanna, Geol. Ecol. Trop., 3, 253–266.

van Kampen, N. G. (1981), Ito versus Stratonovich, J. Stat. Phys.,24, 175–187.

van Langevelde, F., et al. (2003), Effects of fire and herbivory onthe stability of savanna ecosystems, Ecology, 84(2), 337–350.

Vetaas, O. R. (1992), Micro-site effects of trees and shrubs in drysavannas, J. Veg. Sci., 3(3), 337–344.

von Hardenberg, J., E. Meron, M. Shachak, and Y. Zarmi (2001),Diversity of vegetation patterns and desertification, Phys. Rev.Lett., 87(19), 198101.

Walker, B. H., D. Ludwig, C. S. Holling, and R. M. Peterman(1981), Stability of semi-arid savanna grazing systems, J. Ecol.,69(2), 473–498.

Watt, A. S. (1947), Pattern and process in plant community,J. Ecol., 35, 1–22.

Webster, R., and F. T. Maestre (2004), Spatial analysis of semi-aridpatchy vegetation by the cumulative distribution of patch bound-ary spacings and transition probabilities, Environ. Ecol. Stat.,11(3), 257–281.

West, N. E. (1990), Structure and function of microphytic soilcrusts in wildland ecosystems of arid to semi-arid regions, Adv.Ecol. Res., 20, 179–223.


35 of 36

RG1005

White, L. P. (1969), Vegetation arcs in Jordan, J. Ecol., 57, 461–464.White, L. P. (1970), Brousse-tigree patterns in southern Niger,J. Ecol., 58(2), 549–553.

White, L. P. (1971), Vegetation stripes on sheet wash surfaces,J. Ecol., 59(2), 615–622.

Wickens, G. E., and F. W. Collier (1971), Some vegetation patternsin Republic of Sudan, Geoderma, 6(1), 43–59.

Wilde, S. A., R. S. Steinbrenner, R. C. Dosen, and D. T. Pronin(1953), Influence of forest cover on the state of the ground watertable, Soil Sci. Soc. Am. Proc., 17, 65–67.

Wilson, J. B., and A. D. Q. Agnew (1992), Positive-feedbackswitches in plant communities, Adv. Ecol. Res., 23, 263–336.

Worrall, G. A. (1959), The Butana grass patterns, J. Soil Sci., 10,34–53.

Worrall, G. A. (1960), Patchiness in vegetation in the northernSudan, J. Ecol., 48, 107–115.

Yizhaq, H., E. Gilad, and E. Meron (2005), Banded vegetation: Bio-logical productivity and resilience, Physica A, 356(1), 139–144.

Yokozawa, M., Y. Kubota, and T. Hara (1998), Effects of competi-tion mode on spatial pattern dynamics in plant communities,Ecol. Modell., 106(1), 1–16.

Yokozawa, M., Y. Kubota, and T. Hara (1999), Effects of competi-tion mode on the spatial pattern dynamics of wave regenerationin subalpine tree stands, Ecol. Modell., 118(1), 73–86.

Zeng, Q.-C., and X. D. Zeng (1996), An analytical dynamic modelof grass field ecosystem with two variables, Ecol. Modell., 85(2–3), 187–196.

Zeng, X., and X. Zeng (2007), Transition and pattern diversity inarid and semiarid grassland: A modeling study, J. Geophys. Res.,112, G04008, doi:10.1029/2007JG000411.

Zeng, X., S. S. P. Shen, X. Zeng, and R. E. Dickinson (2004),Multiple equilibrium states and the abrupt transitions in a dyna-mical system of soil water interacting with vegetation, Geophys.Res. Lett., 31, L05501, doi:10.1029/2003GL018910.

Zeng, Y., and G. P. Malanson (2006), Endogenous fractal dynamicsat alpine treeline ecotones, Geogr. Anal., 38(3), 271–287.

Zhonghuai, H., Y. Lingfa, X. Zuo, and X. Houwen (1998), Noiseinduced pattern transition and spatiotemporal stochastic reso-nance, Phys. Rev. Lett., 81(14), 2854–2857.

��F. Borgogno, F. Laio, and L. Ridolfi, Dipartimento di Idraulica,

Trasporti ed Infrastrutture Civili, Politecnico di Torino, I-10129 Torino,Italy. ([email protected]; [email protected]; [email protected])P. D’Odorico, Department of Environmental Sciences, University of

Virginia, Box 400123, Charlottesville, VA 22903, USA. ([email protected])


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