MATHEMATICAL MODELS OF VEGETATION PATTERN FORMATION IN ECOHYDROLOGY F. Borgogno, 1 P. D’Odorico, 2 F. Laio, 1 and L. Ridolfi 1 Received 13 November 2007; accepted 24 October 2008; published 18 March 2009. [1] Highly organized vegetation patterns can be found in a number of landscapes around the world. In recent years, several authors have investigated the processes underlying vegetation pattern formation. Patterns that are induced neither by heterogeneity in soil properties nor by the local topography are generally explained as the result of spatial self-organization resulting from ‘‘symmetry-breaking instability’’ in nonlinear systems. In this case, the spatial dynamics are able to destabilize the homogeneous state of the system, leading to the emergence of stable heterogeneous configurations. Both deterministic and stochastic mechanisms may explain the self-organized vegetation patterns observed in nature. After an extensive analysis of deterministic theories, we review noise-induced mechanisms of pattern formation and provide some examples of applications relevant to the environmental sciences. 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. Click Here for Full Articl e 1 Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Turin, Italy. 2 Department 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 1 of 36 Paper number 2007RG000256 RG1005
36
Embed
Here Full MATHEMATICAL MODELS OF …pd6v/Publications_files/2009...of bare soil (Figure 1). Depending on the topography and other external conditions (wind direction, light exposure,
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
[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
2 of 36
RG1005
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
3 of 36
RG1005
[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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
4 of 36
RG1005
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.,
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
7 of 36
RG1005
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
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
8 of 36
RG1005
[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-
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
10 of 36
RG1005
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
11 of 36
RG1005
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
12 of 36
RG1005
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
[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-
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
13 of 36
RG1005
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
14 of 36
RG1005
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
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
15 of 36
RG1005
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-
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
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].
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
30 of 36
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
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
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
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.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
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,