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MATHEMATICAL MODELS OF VEGETATION
PATTERN FORMATION IN ECOHYDROLOGY
F. Borgogno,1 P. D’Odorico,2 F. Laio,1 and L. Ridolfi1
Received 13 November 2007; accepted 24 October 2008; published 18 March 2009.
[1] Highly organized vegetation patterns can be found in anumber of landscapes around the world. In recent years,several authors have investigated the processes underlyingvegetation pattern formation. Patterns that are inducedneither by heterogeneity in soil properties nor by the localtopography are generally explained as the result of spatialself-organization resulting from ‘‘symmetry-breakinginstability’’ in nonlinear systems. In this case, the spatialdynamics are able to destabilize the homogeneous state of
the system, leading to the emergence of stableheterogeneous configurations. Both deterministic andstochastic mechanisms may explain the self-organizedvegetation patterns observed in nature. After an extensiveanalysis of deterministic theories, we review noise-inducedmechanisms of pattern formation and provide someexamples of applications relevant to the environmentalsciences.
Citation: Borgogno, F., P. D’Odorico, F. Laio, and L. Ridolfi (2009), Mathematical models of vegetation pattern formation in
ecohydrology, Rev. Geophys., 47, RG1005, doi:10.1029/2007RG000256.
Ecohydrology is the science, which seeks to describe the hydrologic
mechanisms that underlie ecologic patterns and processes.
[Rodriguez-Iturbe, 2000, p. 3]
1. INTRODUCTION
[2] In many landscapes around the world the vegetation
cover is sparse and exhibits spectacular organized spatial
features [e.g.,Macfadyen, 1950b] that can be either spatially
periodic or random. Commonly denoted as ‘‘vegetation
patterns’’ [e.g., Greig-Smith, 1979; Lejeune et al., 1999],
these features can be found in many regions around the
world, including Somalia [Macfadyen, 1950b; Boaler and
Hodge, 1964], Burkina Faso and Sudan [Worrall, 1959,
1960; Wickens and Collier, 1971], South Africa [van der
Meulen and Morris, 1979], Niger [White, 1970; Adejuwon
and Adesina, 1988], Australia [Slatyer, 1961; Mabbutt and
Fanning, 1987; Burgman, 1988; Tongway and Ludwig,
1990; Ludwig and Tongway, 1995], Mexico [Cornet et al.,
1988; Montana et al., 1990; Acosta et al., 1992; Mauchamp
et al., 1993], United States [Fuentes et al., 1986], Argentina
[Soriano et al., 1994], Chile [Fuentes et al., 1986], Japan
[Sato and Iwasa, 1993], and Jordan [White, 1969]. Vegeta-
tion patterns are often undetectable on the ground but
became visible with the advent of aerial photography
[e.g., Macfadyen, 1950b]. Figures 1–4 show some exam-
ples of spectacular spatially periodic vegetation patterns that
can be found especially in arid and semiarid landscapes
around the world. These patterns exhibit amazing regular
configurations of vegetation stripes or spots separated by
bare ground areas. In some cases, patterns may spread over
relatively large areas (up to several square kilometers)
[White, 1971; Eddy et al., 1999; Valentin et al., 1999;
Esteban and Fairen, 2006] and can be found on different
soils and with a broad variety of vegetation species and life
forms (i.e., grasses, shrubs, or trees) [Worrall, 1959, 1960;
White, 1969, 1971; Bernd, 1978; Mabbutt and Fanning,
1987;Montana, 1992; Lefever and Lejeune, 1997; Bergkamp
et al., 1999; Dunkerley and Brown, 1999; Eddy et al., 1999;
Valentin et al., 1999].
[3] The study of vegetation patterns is motivated by their
widespread occurrence in dryland landscapes and by the
possibility to infer from their presence and features useful
information on the underlying processes, including the
susceptibility of the system to abrupt shifts to a desert
(i.e., unvegetated) state as a result of climate change or
anthropogenic disturbances [e.g., van de Koppel et al.,
2002; D’Odorico et al., 2006c]. However, there is no doubt
that the beauty of some natural patterns of vegetation
contributed to draw the attention of a number of scientists,
who remained fascinated by their breathtaking natural
features and, thus, engaged themselves in the observation,
understanding, and modeling of these spatially organized
distributions of vegetation.
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1Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnicodi Torino, Turin, Italy.
2Department of Environmental Sciences, University of Virginia,Charlottesville, Virginia, USA.
Copyright 2009 by the American Geophysical Union.
8755-1209/06/2007RG000256$15.00
Reviews of Geophysics, 47, RG1005 / 2009
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Paper number 2007RG000256
RG1005
[4] Early studies on vegetation patterns began to appear
in the 1950s and 1960s [e.g., Macfadyen, 1950a, 1950b;
Worrall, 1959, 1960; Boaler and Hodge, 1962, 1964;
Greig-Smith and Chadwick, 1965] and became increasingly
popular in recent years [e.g., Lefever and Lejeune, 1997;
Klausmeier, 1999; Lejeune and Tlidi, 1999; Couteron and
Lejeune, 2001; Buceta and Lindenberg, 2002; D’Odorico et
al., 2007b] (see also section 4 for more references). Two
major approaches have been followed in the study of
vegetation patterns, depending on whether the focus was
on their qualitative empirical description and characteriza-
tion or on the mechanistic understanding of the key pro-
cesses determining pattern formation.
[5] The first group of studies concentrated on the
qualitative analysis of vegetation patterns [e.g., Worrall,
1959, 1960; Boaler and Hodge, 1962, 1964; Greig-Smith
and Chadwick, 1965; Greig-Smith, 1979; Adejuwon and
Adesina, 1988; Burgman, 1988; Acosta et al., 1992; Aguiar
and Sala, 1999] and recognized the recurrence of some
main types of spatial configurations, exhibiting organized
distributions of either stripes, spots, or gaps. Stripes consist
of an alternation of fairly regular vegetated bands with stripes
Figure 1. Example of aerial photographs showing vegetation patterns (tiger bush). (a) Somalia (9�200N,48�460E), (b) Niger (13�210N, 2�50E), (c) Somalia (9�320N, 49�190E), (d) Somalia (9�430N, 49�170E),(e) Niger (13�240N, 1�570E), (f) Somalia (7�410N, 48�00E), (g) Senegal (15�60N, 15�160W), and(h) Argentina (54�510S, 65�170W). Google Earth imagery # Google Inc. Used with permission.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
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of bare soil (Figure 1). Depending on the topography and
other external conditions (wind direction, light exposure, or
land degradation) the stripes can be perpendicular or parallel
to the slope [Maestre et al., 2006]. When stripes emerge on
flat terrains, they become less regular and exhibit Y-shaped,
arc-shaped, or labyrinthine patterns (Figure 2). In particular,
arc-shaped stripes have been named ‘‘brousse tigree’’ [Clos-
Arceduc, 1956] or ‘‘tiger bush’’ [Bromley et al., 1997;
Hiernaux and Gerard, 1999; Couteron et al., 2000] because
of their resemblance to the tiger’s coat [Macfadyen, 1950a,
1950b] (see Figures 1a, 1b, 1e, and 1f). It has been argued that
stripes emerging on hillslopes tend tomigrate uphill [Worrall,
1959;Hemming, 1965;Montana, 1992; Valentin et al., 1999;
Sherratt, 2005], though their slow migration rate may limit
our ability to provide conclusive experimental evidence in
support of this mechanism. Spots and gaps can be viewed as
two complementary configurations. In fact, spots are little
round-shaped aggregations of vegetation interspersed within
a bare soil background (Figure 3), while gaps are round-
shaped bare soil islands surrounded by relatively homoge-
neous vegetation (Figure 4). Both spots and gaps can be
arranged in randomly distributed or more regular configu-
rations.
[6] In addition to a qualitative description, a number of
authors have provided a quantitative characterization of
vegetation patterns, often based on a variety of indices
and parameters as descriptors of the geometry of vegetated
soil patches and of their spatial arrangement [e.g., Dale,
1999]. These indicators are often used to classify the
different types of configurations and relate them to land-
scape or climate variables [e.g., Couteron, 2002; Barbier et
al., 2006; Caylor and Shugart, 2006; Okin et al., 2006]. For
example, several authors have used some of these geomet-
rical parameters (i.e., wavelength of stripes, bandwidths,
and periodicity of spots) to investigate the association
between pattern shape and mean annual rainfall, tempera-
ture, ground slope, wind, or other topographic variables
[Gunaratne and Jones, 1995; Perry, 1998; Giles and Trani,
1999; Okin and Gillette, 2001; Augustine, 2003; Webster
and Maestre, 2004]. This empirical approach is useful to
shed light on the relation between pattern geometry and the
‘‘external’’ environmental conditions. For example, it
allows one to predict the type of pattern that is more likely
to emerge under given climate, soil, and topographic con-
ditions or to understand how vegetation patterns are
expected to change in response to changes in the external
drivers. Moreover, these empirical studies provide some
criteria to test mathematical models of pattern formation
through the comparison of their results with ‘‘real-world’’
observations.Figure 2. Example of aerial photographs showing vegeta-tion patterns (labyrinths). (a) Senegal (15�220N, 15�210W)and (b) Senegal (15�140N, 15�70W). Google Earth imagery# Google Inc. Used with permission. (c) Aerial obliquephotograph of vegetation patterns from SW Niger (courtesyof Nicolas Barbier, Oxford University).
Figure 3. Example of aerial photographs showing vegeta-tion patterns (spots). (a) Zambia (15�380S, 22�460E),(b) Australia (15�430S, 133�100E), and (c) Australia(16�140S, 133�100E). Google Earth imagery # GoogleInc. Used with permission.
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[7] The second group of studies investigated the physical
mechanisms of pattern formation in vegetation and their
response to changes in environmental conditions and dis-
turbance regime. These studies related vegetation patterns to
underlying ecohydrological processes, mechanisms of spa-
tial redistribution of resources [e.g., Klausmeier, 1999;
Barbier et al., 2008; Ridolfi et al., 2008], the nature of
the spatial interactions existing among plant individuals
[e.g., Lefever and Lejeune, 1997; Zeng and Zeng, 2007;
Barbier et al., 2008], the stability and resilience of dryland
ecosystems [Rietkerk et al., 2002; van de Koppel and
Rietkerk, 2004], and the landscape’s susceptibility to de-
sertification under different climate drivers and management
conditions [e.g., von Hardenberg et al., 2001; D’Odorico et
al., 2006c]. Because vegetation patterns are observed even
when topography and soils do not exhibit any heterogeneity,
their formation represents an intriguing case of self-orga-
nized biological systems, which results from completely
intrinsic vegetation dynamics [Lejeune et al., 1999]. This
fact is particularly manifest in the case of periodic patterns
emerging in systems that do not display periodicity in
topography, landforms, or the spatial distribution of other
environmental drivers.
[8] To analyze these intrinsic processes and their role in
the emergence of vegetation patterns, it is important to
capitalize on the understanding of pattern-forming mech-
anisms gained in other fields, such as biology and physics.
In fact, the understanding of mechanisms frequently in-
voked to explain the formation of self-organized patterns
in vegetation originated from studies in other fields,
including fluid dynamics (e.g., the Rayleigh-Bernard con-
vection [Chandrasekhar, 1961; Cross and Hohenberg,
1993], convection in fluid mixtures [Platten and Legros,
1984], or the Taylor-Couette flow [DiPrima and Swinney,
1981]), electrodynamics (e.g., instabilities in nematic
liquid crystals [Dubois-Violette et al., 1978]), chemistry
(morphogenesis in chemical reactions [Turing, 1952]),
and biology (morphogenesis, patterns on animals’ coats
or skin, pigment patterns on shells, and hallucination
patterns [Murray, 2002]). This broad body of literature
inspired a number of studies proposing a variety of
ecological models to explain the fundamental mecha-
nisms conducive to vegetation pattern formation. One of
the early examples of process-based analyses is given by
Watt [1947], who invoked, among others, mechanisms of
reallocation of nutrients and water to explain the emer-
gence of patchy vegetation covers: this model suggested
that as nutrient and water availability decrease, plant
individuals tend to grow in clumps. The emergence of
these aggregated structures is motivated by the need to
concentrate the scarce resources (e.g., soil moisture and
soil nutrients) in smaller areas, thereby increasing the
likelihood of vegetation survival within vegetated
patches that are richer in resources. In the subsequent
years the idea that the mechanisms underlying vegetation
pattern formation are intrinsically dynamic and originate
from interactions among plant individuals was better ar-
ticulated and formalized [e.g., Greig-Smith, 1979; Wilson
and Agnew, 1992; Thiery et al., 1995]. These studies
paved the way to a new generation of models explaining
vegetation patterns as the result of self-organization
emerging from symmetry-breaking instability, i.e., as a
process in which the existence of both cooperative and
inhibitory interactions at two slightly different spatial
ranges may induce the appearance of heterogeneous dis-
tributions of vegetation with wavelengths determined by
the interactions between the two spatial scales [Lefever
and Lejeune, 1997; Lejeune and Tlidi, 1999; Lejeune et
al., 1999; Barbier et al., 2006; Rietkerk and van de
Koppel, 2008]. Three major classes of deterministic
models explain pattern formation as the result of self-
organized dynamics conducive to symmetry-breaking
instability: models based on (1) Turing-like instability
(hereafter named ‘‘Turingmodels’’ [Turing, 1952]), (2) short-
range cooperative and long-range inhibitory interactions
among individuals (hereafter named ‘‘kernel-based mod-
els’’ because short- and long-range spatial interactions are
expressed through a kernel function (see section 2)), and
(3) instability induced by differential flow rates between
Figure 4. Example of aerial photographs showing vegeta-tion patterns (gaps). (a) Senegal (15�90N, 14�360W) and(b) Senegal (15�120N, 14�540W). Google Earth imagery #
Google Inc. Used with permission.
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two interacting species in an activator-inhibitor system
[Klausmeier, 1999]. We stress that these models differ only
in the mathematical description of the dynamics, while they
exhibit essentially the same mechanism of pattern forma-
tion. In fact, in all of them, patterns are induced by
symmetry-breaking instability in activation-inhibition sys-
tems. To further stress this point, in this paper we will
demonstrate that the first two types of models lead to
patterns that are qualitatively the same; that is, patterns
emerging from Turing models can be expressed as a
particular class of those generated by neural models.
[9] More recently, some stochastic models have been
developed, which explain vegetation patterns as noise-
induced effects. In this case patterns emerge as a result
of the randomness inherent to environmental fluctuations
and disturbance regime. We will review two major mech-
anisms of noise-induced pattern formation based either on
random switching between alternative dynamics or on
phase transitions with breaking of ergodicity in systems
driven by different types of noise. We will then discuss the
feedback mechanisms and spatial interactions commonly
invoked to explain the emergence of vegetation patterns.
The rest of the review provides a synopsis and a critical
discussion of the broad literature on the theory of process-
based pattern formation in landscape ecohydrology.
2. MECHANISMS OF PATTERN FORMATION
2.1. Deterministic Mechanisms of Pattern Formation
[10] In this section we provide a mathematical descrip-
tion of the three major deterministic models of self-
organized pattern formation that are commonly invoked
to explain the spatial organization of vegetation. These
three models invoke the same physical mechanism, i.e.,
symmetry-breaking instability. Spatial interactions induce
this instability, while the resulting patterns are stabilized
by suitable nonlinear terms. In Turing and kernel-based
models, symmetry-breaking is the result of the interactions
between short-range activation and long-range inhibition,
i.e., of positive and negative feedbacks acting at different
spatial scales [e.g., Rietkerk and van de Koppel, 2008]. In
the third class of models (i.e., differential flow models),
symmetry-breaking emerges as a result of the differential
flow rate between two (or more) species.
[11] In Turing and differential flow models the nonlinear-
ities are local (i.e., they do not appear in the terms express-
ing spatial interactions), while in kernel-based models the
nonlinearities can be, in general, nonlocal; that is, they can
appear as multiplicative functions of the term accounting for
spatial interactions [e.g., Lefever and Lejeune, 1997]. In a
particular class of kernel-based models, known as ‘‘neural
models’’ [e.g., Murray and Maini, 1989], the nonlinearities
are only local and do not affect the spatial interactions. In
these models the nonlinear terms appear as additive func-
tions of the spatial interaction term. Here we will describe
the Turing and kernel-based models separately because they
use a different mathematical representation of the spatial
dynamics. However, to stress that Turing and neural models
invoke similar mechanisms of morphogenesis (namely,
symmetry-breaking instability induced by spatial interac-
tions in activation-inhibition systems and stabilization by
local nonlinearities) in section 2.1.3 we will show the
relation existing between the analytical frameworks used
by these two models.
2.1.1. Turing-Like Instability[12] In the study of nonlinear chemical systems, Turing
[1952] found that the diffusion of two species (reagents)
may lead to pattern formation when they have different
diffusivities. In the absence of diffusion both species reach
a stable and spatially uniform steady state, while diffusion
may be able to destabilize this state (‘‘diffusion-driven
instability’’) leading to the formation of spatial patterns.
Known as ‘‘Turing’s instability,’’ this mechanism seems to
be counterintuitive. In fact, diffusion is usually believed to
act as a homogenizing process, leading to the dissipation
of concentration gradients of the diffusing species. Con-
versely, Turing’s [1952] model shows that diffusion may
lead to the emergence of spatial heterogeneity in the
coupled nonlinear dynamics of two diffusing species. In
the literature on symmetry-breaking instability in chemis-
try and biology the two diffusive species are often named
‘‘activator’’ and ‘‘inhibitor,’’ and pattern emergence
requires (1) nonlinear local dynamics and (2) the inhibitor
to diffuse faster than the activator [e.g., Meinhardt, 1982;
Murray and Maini, 1989].
[13] In this section we will present Turing’s [1952] model
and determine the conditions leading to Turing’s instability.
We will also stress that patterns emerging from this insta-
bility are self-organized, in that they originate from the
internal dynamics of the system and are not imposed by
heterogeneities in the external drivers. Thus, this mecha-
nism has been often invoked to explain the emergence of
self-organized patterns also in fields other than chemistry,
such as physics and biology, in systems with two or more
diffusing species. Notable examples include convection in
fluid mixtures [Platten and Legros, 1984], the formation of
shell patterns from pigment diffusion [Murray, 2002], and
vegetation pattern formation from diffusion-induced insta-
bility in arid landscapes [e.g., HilleRisLambers et al., 2001].
The emergence of natural patterns from Turing’s instability
has been experimentally demonstrated in a chemical system
[Castets et al., 1990] and in nonlinear optics [e.g., Staliunas
and Sanchez-Morcillo, 2000]. We are not aware of any
similar experiment for the case of vegetation patterns. Thus,
although models based on Turing’s instability are capable of
generating vegetation patterns which resemble those ob-
served in nature, there is no conclusive experimental evi-
dence suggesting that the organized spatial configurations
of vegetation observed in nature do emerge from Turing’s
dynamics (see also the discussion in section 5). One of the
major challenges in the application of Turing’s activator-
inhibition model to the field of landscape ecohydrology
arises from the need to recognize two or more leading state
variables and to assess whether they do diffuse in space.
The diffusive character of the spatial dynamics of both
activator and inhibitor is fundamental to the development of
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a sound Turing-like model of pattern formation, in that
diffusion is crucially important to the emergence of sym-
metry-breaking instability in a Turing’s system.
[14] The mathematical description of Turing’s instability
will be presented here for the case of two species, u and v,
diffusing across a two-dimensional infinite domain {x, y}.
The dynamics of u and v are modeled by two differential
equations involving both diffusive terms and functions of
the local values of the state variables [e.g., Murray and
Maini, 1989; Murray, 2002; Henderson et al., 2004]:
@u
@t¼ f u; vð Þ þ r2u; ð1aÞ
@v
@t¼ g u; vð Þ þ dr2v; ð1bÞ
where t is time and f and g are the local reaction kinetics,
while d is the ratio, d = d2/d1, between the two diffusivities,
d1 and d2, of u and v, respectively, and r2() is the Laplace
operator (@2/@x2 + @2/@y2). All variables are in dimension-
less units.
[15] Turing [1952] demonstrated that this diffusive sys-
tem exhibits diffusion-driven instability if (1) in the absence
of diffusion the homogeneous steady state is linearly
stable (i.e., stable with respect to small perturbations)
and (2) when diffusion is present the homogeneous steady
state is linearly unstable. Thus, we first need to determine
the homogeneous steady state (u0, v0) as the solution of
equations (1) with r2u = r2v = 0 (homogeneous state)
and @u/@t = @v/@t = 0 (steady state); therefore f(u0, v0) = 0
and g(u0, v0) = 0. Then, we need to impose the condition
that this solution is stable in the absence of diffusion. To
this end, we can study the stability of (u0, v0) with respect
to small perturbations,
w ¼ u� u0v� v0
� �; ð2Þ
around the steady state. For small perturbations of the
steady homogeneous state (i.e., for jwj ! 0) the system (1)
can be linearized around (u0, v0). Using a linear Taylor’s
expansion we have
@w
@t¼ Jw; J ¼
@f@u
@f@v
@g@u
@g@v
0@
1A
u0 ;v0
; ð3Þ
where J is the Jacobian of the dynamical system (1).
[16] The solutions of this system are in the form w(x, y, t)/ est and express the temporal evolution of the perturbation
of the homogeneous steady state, where s is an eigenvalue
of the system (1), i.e., a solution of the secular polynomial
jJ � sI j ¼ 0; ð4Þ
with I being the identity matrix.
[17] When the real part of s, Re[s], is negative,
jwj tends to zero for t ! 1, and the steady homogeneous
state, (u0, v0), is linearly stable with respect to small
perturbations. From the analysis of equation (4) we obtain
that this condition is met when
@f
@uþ @g
@v< 0 and
@f
@u
@g
@v� @f
@v
@g
@u> 0; ð5Þ
with all derivatives being calculated in (u0,v0) [Murray,
2002].
[18] To study the effect of diffusion on the stability of (u0,
v0), we consider the full system (1) and use a Taylor’s
expansion to linearize this set of equations around the
homogeneous steady state, (u0, v0):
@w
@t¼ Jwþ Dr2w; D ¼ 1 0
0 d
� �: ð6Þ
[19] The solution of the system (6) can be written in the
form of a sum of Fourier modes,
w r; tð Þ ¼ Wkestþik�r; ð7Þ
with k = (kx, ky) being the wave number vector, r = (x, y)
being the coordinate vector, and Wk being the Fourier
coefficients [Murray, 2002].
[20] The relation between eigenvalues and wave numbers
(known as ‘‘dispersion relation,’’ see Figure 5) can be
obtained inserting equation (7) into (6) and searching for
nontrivial solutions. Setting k =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k2y
q, one obtains
jsI � J þ Dk2j ¼ 0: ð8Þ
[21] For the state (u0, v0) to be unstable with respect to
small perturbations, the solution of the dispersion relation
(8) should exhibit positive values of Re[s(k)], for some
wave number k 6¼ 0. Using equation (8), it is possible to
demonstrate that this condition is met when
d@f
@uþ @g
@v> 0 and
d@f
@uþ @g
@v
� �2
�4d@f
@u
@g
@v� @f
@v
@g
@u
� �> 0: ð9Þ
[22] The first of equations (5) combined with the first of
equations (9) implies that d 6¼ 1, indicating that the system
cannot be unstable with respect to small perturbations if
both species have the same diffusivity. If all four conditions
(5) and (9) hold, at least one eigenfunction is unstable with
respect to small perturbations and grows exponentially with
time as a consequence of the destabilizing effect of diffu-
sion. The dispersion relation (8) imposes a specific link
between eigenvalues, s, and wave numbers, k. The wave
number, kmax, corresponding to a maximum positive value
of Re[s] represents the most unstable mode of the system.
This implies that if Re[s(kmax)] > 0, this mode grows faster
than the others, and the state of the system for t ! 1 is
dominated by kmax, in the sense that as t ! 1, only kmax
dictates the length scale of the spatial pattern.
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[23] Because this linear stability analysis is developed in
the limit jwj ! 0 (i.e., under the assumption of small
perturbations of the homogeneous steady state), it cannot
provide any information on the state of the system when the
perturbation grows in amplitude. In the absence of non-
linearities in f(u, v) and g(u, v) the solutions (7) of the
linearized model would coincide with the exact solutions of
system (1) even away from the state (u0, v0). In this case,
equation (7) clearly shows that if the state (u0, v0) is
unstable the perturbations, w, grow indefinitely. Thus,
suitable nonlinear terms are needed to stabilize the pattern
through higher-order terms in the Taylor’s expansion, which
become important when the amplitude of the perturbation is
finite. In other words, the system reaches a steady config-
uration when the exponential growth of the eigenfunction is
limited by second-order terms (or higher) that come into
play once the perturbation has finite amplitude. In these
conditions the (nonlinear) stability of the system can be
partly studied through a more complex mathematical frame-
work based on the so-called amplitude equations, which
investigate the dynamics of the system in the neighborhood
of the most unstable mode. This review will not present
these nonlinear methods, and we refer the interested reader
to specific literature on this topic for further details [Cross
and Hohenberg, 1993; Leppanen, 2005].
[24] The above theory refers to the case of unconfined
spatial domains. In the case of confined (i.e., spatially
limited) systems the set of unstable eigenvalues is no longer
infinite, and only a discrete set of nonnull wave numbers
k 6¼ 0 may correspond to solutions of equation (6) with
Re[s (k)] > 0 (i.e., unstable perturbations of the homogeneous
steady state). Similarly to the case of unconfined domains,
the dynamics lead to pattern emergence when there are
unstable modes, and, again, the pattern geometry is deter-
mined by the wave number of the most unstable mode.
However, the effect of the finite size of the domain is
seldom accounted for in deterministic models of vegetation
self-organization, as these patterns typically stretch across
relatively large areas (several square kilometers) [White,
1971; Eddy et al., 1999; Valentin et al., 1999; Esteban and
Fairen, 2006] compared to the size of vegetation patches.
[25] We present here a simple example of an ecological
Turing model able to generate spatial patterns in a system
with two species, u (activator) and v (inhibitor). To this end,
we use equations (1) with local kinetic functions
f u; vð Þ ¼ u avu� eð Þ; ð10aÞ
g u; vð Þ ¼ v b� cu2v
; ð10bÞ
where a, b, c, and e are dimensionless positive constants.
[26] Equation (10a) describes the growth or the decay of
the activator and accounts for a positive interaction between
u and v. In fact, as v increases, the growth rate of species u
increases. Moreover, the growth rate of u increases with
increasing values of u. Equation (10b) is a generalized
logistic growth [e.g., Murray, 2002] with carrying capacity
b and a strong negative influence (inhibition) of species u
on the growth rate of v: in fact, as u increases, the second
term of the function g decreases as a second-order power
law.
[27] The homogeneous steady state of this system is u0 =
ab/ce and v0 = ce2/ba2. The derivatives of the two functions
(10) calculated in (u0, v0) are @f/@u = e, @f/@v = a3b2/c2e2,
@g/@u = �2g2e3/ba3, and @g/@v = �b, while the four
conditions (5) and (9) leading to diffusion driven instability
become e � b < 0, eb > 0, de � b > 0, and d2e2 � 6bde +
b2 > 0, respectively.
[28] Figure 6 shows an example in which these condi-
tions are met and patterns emerge from diffusion-driven
instability as a hexagonal arrangement of spots with wave-
length l ’ 2p/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1= 1� dð Þ½ � eþ b� 1þ dð Þ=d½ �
ffiffiffiffiffiffiffiffiffiffi2bde
p� �qin agreement with the wavelength of the most unstable
mode obtained through the dispersion relation (8).
2.1.2. Kernel-Based Models of Short-RangeCooperative and Long-Range Inhibitory Interactions[29] We classify as kernel-based models those modeling
frameworks in which spatial interactions are expressed
through a kernel function accounting both for short-range
and long-range coupling. In most models of self-organized
vegetation, patterns arise as a result of short-range cooper-
ation (or ‘‘activation’’) and long-range inhibition. In these
models, stable patterns emerge when spatial interactions cause
symmetry-breaking instability and the system converges to
an asymmetric state, which exhibits patterns. The conver-
gence to this state is due to suitable nonlinear terms, which
prevent the initial (linear) instability to grow indefinitely.
[30] A kernel-based model with multiplicative nonlinear-
ities (i.e., with nonlinearities embedded also in the spatial
interaction term) was developed by Lefever and Lejeune
[1997] to explain the formation of patterns in dryland
vegetation. This model will be discussed at the end of this
section. We first consider a particular type of kernel-based
models, whereby the nonlinearity is not in the spatial
coupling but in an additive term. These models are often
Figure 5. Generic dispersion relation for a two diffusivespecies monodimensional system. The extremes of therange of unstable Fourier modes are represented by k1 andk2, while kmax represents the most unstable Fourier mode.
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known as ‘‘neural models’’ because of their applications to
neural systems.
[31] Some of the most fascinating and complex pattern-
forming processes existing in nature are associated with
neural systems. Typical examples include the process of
pattern recognition, the transmission of visual information
to the brain, and stripe formation in the visual cortex
[Murray, 2002]. The framework of a neural model is
often used to represent other systems, including the case
of vegetation dynamics in spatially extended systems
[D’Odorico et al., 2006b].
[32] Neural models can, in general, be developed for
systems with more than one state variable. However, unlike
Turing models, pattern-forming symmetry-breaking insta-
bility can emerge even when the dynamics have only one
state variable. Thus we concentrate on the case of neural
models that are mathematically described usually by only
one state variable, say u, representing, for example, the
population density in a two-dimensional domain (x, y). At
any point, r = (x, y), of the domain the population density,
u(r), undergoes local dynamics (i.e., independent of spatial
interactions) expressed by a function, h(u), with a steady
state at u = u0 (i.e., h(u0) = 0). For the sake of simplicity we
will assume that the local dynamics exhibit only one steady
state. To express the effect of spatial interactions on the
dynamics of u, we account for the impact that individuals at
other points, r0 = (x0, y0), of the domain have on the
population density, u(r, t), at the location r. It is sensible
to assume that this impact depends on the relative position
of the two points r and r0. Because the strength of the
interactions with other individuals is likely to decrease with
the distance, a weighting function w(r, r0) is introduced to
describe how the effect of spatial interactions depends on r0
and r. We integrate r0 over the whole domain, W, to account
for the interactions of u(r, t) with individuals at any point r0
in W
@u
@t¼ h uð Þ þ
ZWw r; r0ð Þ u r0; tð Þ � u0½ �dr0: ð11Þ
[33] The right-hand side of (11) consists of two terms: the
first term, h(u), describes the local dynamics, i.e., the
dynamics of u that would take place in the absence of
spatial interactions with other points of the domain. The
second term expresses the spatial interactions and depends
both on the shape of the weighting function (or ‘‘kernel’’) and
on the values of u in the rest of the domain W. If w(r, r0) > 0,
the spatial interactions affect the dynamics of u(r) positively
or negatively depending on whether u(r0) is smaller or greater
than u0, respectively. The opposite happens when w(r, r0) < 0.
Notice how the dynamics expressed by (11) are not neces-
sarily bounded at u = 0, and a bound may need to be imposed
to ensure that u � 0, if in the model u represents population
density or vegetation biomass.
[34] When the processes underlying the spatial interac-
tions are homogeneous (i.e., they do not change from point
to point) and isotropic (i.e., they are independent of the
direction), the kernel function is independent of r and
exhibits axial symmetry. In this case, w is a function only
of the distance, z = jr0 � rj, between the two interacting
points (w(z) = w(jr0 � rj)). It will be shown that even thoughthe underlying mechanisms are homogeneous, they can lead
to pattern formation, i.e., to nonhomogeneous distributions
of the state variable.
[35] In neural models of pattern formation the interac-
tions between cells are typically represented by short-range
activation and long-range inhibition [Oster and Murray,
1989]. In this case the kernel is positive at small distances,
z, and becomes negative at greater distances (Figure 7). This
type of framework has been proposed as a model for spatial
interactions within plant communities [e.g., Lefever and
Lejeune, 1997; Yokozawa et al., 1999; Couteron and
Lejeune, 2001] in other kernel-based models. A kernel with
the shape illustrated in Figure 7 can be obtained, for
example, as the difference between two exponential func-
tions of the form
w zð Þ ¼ b1 exp � z
q1
� �2" #
� b2 exp � z
q2
� �2" #
; ð12Þ
with 0 < q1 < q2, while b1 and b2 are two coefficients
expressing the relative importance of the facilitation and
competition components of the kernel.
Figure 6. Spatial pattern emerging for the variable u in theTuring system in equations (10a) and (10b). The parametersare a = 22, b = 84, c = 113.33, e = 18, and d = 27.2. Theparameters a and c do not influence the emergence of spatialpatterns (see the end of section 2.1.1), while they influenceonly the shape of spatial patterns. The simulation is carriedout over a domain of 256 � 256 cells, each cell representinga spatial step Dx = Dy = 0.2. Equations are solvednumerically by means of finite difference method.
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[36] To intuitively understand how equation (11) with a
kernel w(z) shaped as in Figure 7 can lead to the emergence
of spatial patterns, we show how the spatial dynamics can
render unstable the spatially uniform steady state, u0,
similarly to the case of Turing’s instability discussed in
section 2.1.1. In fact, starting from a small heterogeneous
perturbation of the state, u0, each point, r, positively
interacts with the nearby points, r0, that are located at a
distance, z, such that w(z) > 0. Thus, small perturbations
with u > u0 tend to further increase u, while those with u <
u0 tend to decrease the value of u in the surrounding points,
thereby enhancing the heterogeneity. The integrated impact
of the interaction with all individuals in the neighborhood of
r may be able to induce pattern formation. While short-
range positive interactions activate the formation of patterns
through the instability of the uniform steady state, u0,
mechanisms of long-range inhibition represented by the
negative part of the kernel (Figure 7) prevent the perturba-
tion of the uniform state from growing indefinitely in space.
Thus, inhibition (along with suitable nonlinearities) is
needed to stabilize the pattern in a way that the perturbed
state can reach a steady configuration [Murray, 2002].
[37] To apply this framework to the case of vegetation
patterns we need to justify the use of a kernel with the shape
shown in Figure 7 and determine its parameters on the basis
of what is known about mutual interactions between plant
individuals. There is no doubt that these are challenging
tasks [Barbier et al., 2008]. The existing models of vege-
tation pattern formation invoking kernel-based activation-
inhibition frameworks [Lefever and Lejeune, 1997] recognize
that spatial interactions typical of dryland plant communities
exhibit short-range cooperative effects (facilitation) which
concentrate the resources in a relatively small area, thereby
providing more favorable conditions for plant establishment
and growth in the surroundings of existing plant individuals
[Rietkerk and van de Koppel, 2008] (Figure 8). At the same
time, as noted, long-range negative interactions are needed to
stabilize the pattern [Rietkerk and van de Koppel, 2008].
Thus, these models invoke root competition for water and
nutrients as the main mechanism of long-range inhibition.
Section 3 will provide more details on the ecosystem pro-
cesses determining short-range cooperation and long-range
competition.
[38] The dynamics expressed by equation (11) may lead
to pattern formation through mechanisms that resemble
those of Turing’s instability. In fact, patterns emerge as a
result of the spatial interactions, which destabilize the
uniform stable state, u0, of the local dynamics. To study
the stability of the state u = u0 with respect to infinitesimal
perturbations, we linearize equation (11) around the steady
state u = u0. Indicating with u = u � u0 the amplitude of the
(‘‘small’’) perturbation, we obtain
@u
@t¼ uh0 u0ð Þ þ
ZWw jr0 � rjð Þu r0; tð Þdr0; ð13Þ
where h0(u0) is the derivative of the function h(u), calculated
for u = u0.
[39] Solutions of equation (13) can be expressed in the
form of integral sums of the harmonics u(r, t) / exp[st +ik � r], where each harmonic is a solution of (13), k = (kx,
ky) is the wave number vector, and the growth factor, s, isan eigenvalue of equation (13). Substituting this solution
in equation (13), setting z = jr0 � rj, and canceling out the
exponential function, we obtain the dispersion relation,
that is, the relation between k and s in solutions of
equation (11) obtained as small perturbations of the state
u = u0,
s kð Þ ¼ h0 u0ð Þ þZWw zð Þ exp ik � z½ �dz ¼ h0 u0ð Þ þW kð Þ; ð14Þ
with k = jkj. If W is infinitely extended both in the x and y
directions, W(k) is the Fourier transform of the kernel
function. The dispersion relation obtained with the kernel
(12) is shown in Figure 9. Notice how the shape of the
dispersion relation is entirely determined by W(k), i.e., by
the effect of the kernel function on the spatial dynamics,
while the local dynamics affect equation (14) only through
the constant h0(u0). In fact, changes in this constant
Figure 7. Typical kernel which exhibits local activationand long-range inhibition.
Figure 8. Visualization of the positive and negativeinteractions typical of a tree.
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determine a vertical shift of the curves in Figure 9 without
modifying their shape. This vertical shift affects the sign
of s(k), thereby determining the stability/instability of
the system and the range of unstable modes. All modes
with s < 0 are linearly stable because they vanish as time,
t, passes. Conversely, all modes with s > 0 are linearly
unstable and tend to grow with time. However, even in
this case, when the amplitude of the unstable modes
becomes finite, the assumptions underlying this linear
stability analysis (i.e., that perturbations are ‘‘small’’/
infinitesimal) are no longer valid. Thus, the linear stability
analysis does not shed light on the state approached by the
system as an effect of the unstable modes. However, as
noted for Turing’s instability, the dominant wavelength of
patterns emerging from this instability is dictated by the
most unstable mode, kmax (which grows faster than the
other unstable modes, thereby determining some key
aspects of pattern geometry). This wavelength depends
only on the shape of the kernel function and is not affected
by the term h0(n0) (see equation (14)), even though h0(n0)
determines the stability of the system and the emergence
of spatial patterns: for relatively low values of h0(u0), s < 0
for all wave numbers k (see Figure 9), while as h0(u0)
increases above a critical value, s(kmax) becomes positive,
and the mode, kmax, is unstable. Larger values of h0(u0)
correspond to broader ranges of unstable wave numbers.
[40] Spatial interactions lead to pattern formation in
equation (11) when the following conditions are met:
[41] 1. In the absence of spatial interactions the uniform
steady state, u = u0, of the local dynamics is stable. The
linear stability analysis demonstrates that the stability of u =
u0 requires h0(u0) to be negative, as shown by equation (13)
when the integral term is set equal to zero.
[42] 2. In the presence of spatial interactions there should
be at least one wave number (kmax) associated (through the
dispersion relation) with a positive value of s.[43] 3. Because the mode k = 0 corresponds to a spatially
uniform perturbation of u = u0, instability does not lead to
the emergence of any spatial pattern if the most unstable
mode, kmax, is zero.
[44] Thus, in a neural model, patterns emerge from spatial
interactions when
h0 u0ð Þ < 0; W kmaxð Þ þ h0 u0ð Þ > 0; kmax > 0; ð15Þ
where kmax is the solution of W 0(kmax) = 0 with W 00(kmax) <
0. We also notice that if h(u) is a linear function of u,
equation (11) is also linear. Thus, solutions of equation (13)
are exact expressions (rather than approximations) of the
perturbed state of the system (u). In this case, because of the
linearity of (13), the perturbed state remains an exponential
function of s even when the amplitude of the perturbation is
no longer infinitesimal. In other words, if h(u) is linear and
the conditions (15) are met, the steady homogeneous state is
unstable, and any perturbation of u = u0 grows indefinitely
without ever reaching a steady configuration. Thus, a
suitable nonlinear function, h(u), is needed for the neural
model to have a steady state in which patterns emerge from
symmetry-breaking instability. In this case, as soon as the
initial perturbation of the steady homogeneous state grows
in amplitude, suitable nonlinear terms can come into play
and prevent the indefinite growth of the perturbation.
[45] In the particular case of the kernel function
expressed by (12) the dispersion relation becomes
s ¼ h0 u0ð Þ þW kð Þ
¼ h0 u0ð Þ þ pb1q21 exp � q21k2
2
� �� pb2q22 exp � q22k
2
2
� �; ð16Þ
while the most unstable mode is
kmax ¼ q1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln �c4ð Þc2 � 1
s; ð17Þ
with � = b2/b1 and c = q2/q1. The last two conditions (15)
can be rewritten as
h0 u0ð Þb1q
21
>p�c2 c2 � 1ð Þ
�c2ð Þc2
c2�1
and �c4 > 1: ð18Þ
[46] As noted, one of the first models of vegetation self-
organization [Lefever and Lejeune, 1997] used a kernel-
based framework that resembles that of equation (11), with
spatial interactions involving both short-range activation
and long-range inhibition. The model by Lefever and
Lejeune [1997] differs from a neural model in that the
nonlinearities are not strictly local but modulate the spatial
interactions.
[47] In some cases, the spatial interactions modulated by
the kernel function have only a limited effect (i.e., w(z)! 0)
at relatively large distances, z. Thus, depending on the
shape of w(z), conditions leading to pattern formation in
neural models can be formalized through a Taylor’s expansion
(for small values of z) of the integral term of equation (11)
to the fourth order. This approach leads to the so-called
Figure 9. Dispersion relation s(k) as a function of thewave number k for various values of the bifurcationparameter h0(u0). The critical value for h0(u0) that dis-criminates the situations of stability and instability isrepresented by ac.
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long-range diffusion (or biarmonic) approximation of the
neural model [Murray, 2002],
@u
@t� h uð Þ þ w0 u� u0ð Þ þ w2r2uþ w4r4u; ð19Þ
where r4 is the biarmonic operator (@4/@x4 + 2@4/@x2@y2 +@4/@y4), while wm are the mth-order moments of the kernel
function
wm ¼ 1
m!
ZWzmw zð Þdz; m ¼ 0; 2; 4; . . . ð20Þ
[48] In equation (19) we have assumed that the dynamics
are isotropic, i.e., that the kernel function has axial sym-
metry. Thus, because in this case the odd-order moments of
w(z) are zero, we have not included the odd-order terms in
the Taylor’s expansion. Table 1 reports the moments of w(z)
for the case of the kernel function (12) in one- and two-
dimensional domains.
[49] Because the moment w2 multiplies the Laplacian of
u, it modulates the effect of ‘‘short-range diffusion,’’ while
the moment w4 multiplies the biarmonic term, which
accounts for long-range interactions (‘‘long-range diffu-
sion’’). It can be shown that the diffusion term alone is
unable to lead to persistent patterns [e.g.,Murray, 2002] and
that the biarmonic term is needed in the series expansion to
obtain (with equation (19)) patterns that do not vanish with
time. In fact, the linear stability analysis of the state u = u0with respect to a perturbation J(r, t) / est+ik�r leads to the
dispersion relation
s ¼ h0 u0ð Þ þ w0 � 2w2k2 þ 4w4k
4: ð21Þ
[50] In the absence of the long-range diffusion (biar-
monic) term (i.e., when w4 = 0), the most unstable mode,
kmax, is zero, and no patterns emerge. In the case of the
biarmonic equation (19) (i.e., when w4 6¼ 0), the most
unstable mode can be easily obtained from equation (21)
as kmax =12
ffiffiffiffiffiffiffiffiffiffiffiffiffiw2=w4
p. Patterns emerge when kmax is real and
different from zero (i.e., w2 and w4 need to have the same
sign), and s(kmax) > 0,
s kmaxð Þ ¼ h0 u0ð Þ þ w0 �w22
4w4
> 0: ð22Þ
[51] In addition, the stability of u = u0 in the absence of
spatial dynamics requires h0(u0) to be negative as in the first
of equations (15). Moreover, in most ecohydrological appli-
cations u is always nonnegative. This condition is met when
w0 < 0. Because, in this case, w0 and h0(u0) are both
negative, equation (22) combined with the requirement that
w2 and w4 have the same sign imply that pattern formation
occurs only if w2 and w4 are also negative. However, the
condition that w0, w2, and w4 are negative is only necessary
and not sufficient for pattern formation as the condition (22)
would still need to be met for the instability to emerge.
[52] An ecohydrologic neural model of vegetation pat-
tern formation is given by D’Odorico et al. [2006b], where
a typical kernel accounting for short-range cooperation and
long-range inhibition (Figure 7) is used to describe the
spatial interactions. Here we want to show how spatial
patterns may also emerge when the kernel is ‘‘upside
down’’ with respect to the case of Figure 7, i.e., in the
presence of short-range inhibition and a long-range coop-
eration. We develop a numerical simulation of a simple
neural model, using equation (11) with local dynamics ex-
pressed by a generalized logistic function, h(u) = a(u0� u)u2,
where a is a positive constant. Because h0(u0) = �au02 < 0 for
any u0, in the absence of spatial interactions the homoge-
neous state u = u0 is linearly stable. The results of the
numerical simulation of equations (11) and (12) are shown
in Figure 10. In this case the nonlinearity of h(u) is capable of
limiting the growth of the perturbations of the homogeneous
state. However, we recall that only some suitable nonlinear
functions, h(u), can prevent the indefinite growth of these
perturbations. For example, when h(u) = a(u0 � u)u, the
TABLE 1. Moments of the Kernel Function in Equation (12)
in One- and Two-Dimensional Systems
Moment 1-D 2-D
w0
ffiffiffip
p(b1q1 � b2q2) p(b1q1
2 � b2q22)
w2
ffiffiffip
p=4(b1q13 � b2q2
3) p/2(b1q14 � b2q2
4)
w4
ffiffiffip
p=32(b1q15 � b2q2
5) p/12(b1q16 � b2q2
6)
Figure 10. Spatial pattern emerging for variable u in theneural system in equation (11). The parameters of the kernel(see equation (12)) are q1 = 1, q2 = 0.8317, b1 = 242.45, b2 =1046.2, and a = 0.01. The simulation is carried out over adomain of 256 � 256 cells, each cell representing a spatialstep Dx = Dy = 0.2. Equations are solved numerically bymeans of finite difference method.
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nonlinear terms are not able to constrain the growth of u,
which tends to ±1.
2.1.3. Relation Between Patterns Generated by Turingand Neural Models[53] The relation between neural models and Turing’s
systems is mentioned in few studies commenting on sim-
ilarities existing between the variety of patterns generated
by these two classes of models. For example, Dormann
et al. [2001] demonstrated that simple cellular automata
models of activator-inhibitor systems resembling simplified
neural models can lead to the emergence of patterns that
are very similar to those obtained with a reaction-diffusion
model (i.e., Turing models). Moreover, von Hardenberg
et al. [2001] pointed out that the same patterns emerging
from Turing-like instability can be obtained with neural
models, which account for only one state variable and one
dynamic equation. Thus, the relation between these two
classes of models has been described mostly qualitatively.
In this section we develop a mathematical framework to
show the link between Turing’s model and the biarmonic
approximation (19) of the neural model. To this end, we
first notice that in both models the spatial means, �u or �v, inthe asymptotic states reached by the system for t ! 1 are
the same, in the linear approximation, as the homogeneous
steady states, (u0, v0) (Turing) or u0 (neural model). In
fact, at t ! 1 the terms @/@t are zero. Expanding the
local functions (i.e., f, g, and h) on the right-hand sides of
equations (1) and (19) in Taylor’s series around (�u, �v) and(�u), respectively, and taking only the linear terms, we find
that these equations reduce to f(�u, �v) = g(�u, �v) = 0 and h(�u)= 0.
[54] Combining the same linear approximations of equa-
tions (1) at steady state, we obtain
w04r4uþ w0
2r2uþ w00 u� uoð Þ ¼ 0; ð23Þ
w04r4vþ w0
2r2vþ w00 v� voð Þ ¼ 0; ð24Þ
where
w00 ¼ fvgu � gvfuð Þ; w0
2 ¼ � gv þ dfuð Þ; w04 ¼ �d; ð25Þ
with fu = @f/@u, gu = @g/@u, etc. (calculated for (u0, v0)).
Equations (23) and (24) are the same as equation (19) at
t ! 1, with h(u) linearized around u = u0 and with w0 =
w00 � h0(u0), w2 = w2
0, and w4 = w40. Thus, at t ! 1 the
two equations of Turing’s model (i.e., (1a) and (1b))
reduce to the same equation as (23) or (24) with the same
coefficients. This equation is also the same as (19) at
steady state. In the case of equation (19), spatial dynamics
associated with short- and long-range diffusion induce the
formation of patterns when the condition (22) is met and
w4 and w2 have the same sign (see section 2.1.2). Because
w4 < 0 (see equation (25)), w2 needs to be negative. Using
equations (25) it is easy to show that these conditions lead
to the same relations (9) determined for the emergence of
diffusion-induced instability in Turing’s model. Thus, in
both classes of models the conditions determining the
formation of patterns as a result of spatial interactions are
the same. Moreover, using equations (25), it can be shown
that the most unstable mode, kmax = 12
ffiffiffiffiffiffiffiffiffiffiffiffiffiw2=w4
p, of the
biarmonic model is the same as the one obtained from the
dispersion relation for Turing’s model in conditions of
marginal stability (i.e., when s = 0), indicating important
commonalities in the steady state geometry of the patterns
generated by these two models. Thus, Turing’s model can
be viewed as a particular case of the neural model. In fact,
in a neural model the dynamics of only one species are
explicitly described, while Turing’s model describes the
dynamics of at least two species. This means that pattern
formation in a neural model imposes constraints only for
one species, while in a Turing model the constraints are
required for at least two species.
[55] A biarmonic approximation of the example of neu-
ral model presented in Figure 10 can be obtained from
equation (19) with moments calculated (equation (20) and
Table 1) using the same parameters b1,2 and q1,2 as in
Figure 10. Patterns generated by this biarmonic model are
shown in Figure 11. Using equations (25) it can be shown
that the linearization of the Turing model in Figure 6 leads
to the biarmonic model (19) with the same coefficients as
the example in Figure 11.
[56] It is possible to observe that the patterns generated
by these three models (Figures 6, 10, and 11) exhibit the
same wavelengths. As noted, the same wavelength in
Figures 6 and 11 is found because in this case the dispersion
Figure 11. Spatial pattern emerging for the variable u inthe long-range approximation of the kernel-based model inequation (11). The parameters are w0 = �1512, w2 =�405.6, w4 = �27.2, a = 1, and u0 = �0.906. Thesimulation is carried out over a domain of 256 � 256 cells,each cell representing a spatial step Dx = Dy = 0.2.Equations are solved numerically by means of finitedifference method.
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relation (equation (8)) of the Turing model is tangent to the
x axis.
2.1.4. Patterns Emerging From Differential FlowInstability[57] The third major deterministic mechanism of self-
organized pattern formation associated with symmetry-
breaking instability is due to differential flow. This mech-
anism resembles Turing’s dynamics, in that it involves two
diffusing species, u and v (‘‘activator’’ and ‘‘inhibitor,’’
respectively). However, unlike Turing’s model, diffusion is
not important to the destabilization of the homogeneous
state. In this case, one or both species are subjected to
advective flow (or ‘‘drift’’), and instability emerges as a
result of the differential flow rate of the two species
[Rovinsky and Menzinger, 1992]. While diffusion is not
fundamental to the emergence of differential flow instabil-
ity, it plays a crucial role in imposing an upper bound to
the range of unstable modes, k, and determines the
wavelength of the most unstable mode [Rovinsky and
Menzinger, 1992]. As a result of the drift, patterns gener-
ated by this process are not time-independent as those
associated with Turing’s instability. Rather, they exhibit
traveling waves in the flow direction. Self-organized
patterns of this type have been observed in nature mainly
in chemical systems (the ‘‘Belousov-Zhabotinsky reaction’’
[Rovinsky and Zhabotinsky, 1984]). The same mechanism
has been also invoked to explain ecological patterns
subject to drift, including banded vegetation [Klausmeier,
1999; Okayasu and Aizawa, 2001; von Hardenberg et al.,
2001; Shnerb et al., 2003; Sherratt, 2005]. We note that
this mechanism of pattern formation induced by differen-
tial flow is often classified as a Turing model in that in
both models the dynamics can be expressed by the same
set of reaction-advection-diffusion equations. In the case of
Turing models, instability is induced by the Laplacian
term, while in the case of differential flow instability it
is the gradient term that causes instability. For sake of
clarity, here we discuss the case of differential flow
instability separately.
[58] We introduce the mathematical model of differential
flow instability [e.g., Rovinsky and Menzinger, 1992] as-
suming that only one of the two species undergoes a drift,
and we orient the x axis in the direction of the advective
flow. The activator-inhibitor dynamics can be expressed as
@u
@t¼ f u; vð Þ þ p
@u
@xþ d1r2u; ð26aÞ
@v
@t¼ g u; vð Þ þ d2r2v; ð26bÞ
where p is the drift velocity and with d1 and d2 being
the diffusivities of u and v, respectively. Notice that when
p = 0, equations (26) can be written in the same form as
equations (1).
[59] When p 6¼ 0, the conditions on d1 and d2 for the
emergence of patterns from equation (26) are less restrictive
than those for Turing’s instability. To stress the fact that
patterns emerge from the differential flow rates of u and v
we first consider the conditions leading to instability in the
absence of diffusion and set d1 = d2 = 0. The homogeneous
steady state, (u0, v0), obtained as solution of the equation set
f(u0, v0) = g(u0, v0) = 0 is stable when the conditions (5) are
met. To determine the conditions in which the differential
flow destabilizes the state (u0, v0), we linearize f(u, v) and
g(u, v) around (u0, v0) and seek for solutions of the
linearized equations in the form of
u ¼ uþ u0; ð27aÞ
v ¼ vþ v0: ð27bÞ
We obtain
@u
@t¼ fuuþ fvvþ p
@u
@x; ð28aÞ
@v
@t¼ guuþ gvv: ð28bÞ
[60] The solution of system (28) can be expressed as a sum
(or integral sum in spatially infinite domains) of Fourier
modes, uk =Ukexp(st + ik � r) and vk =Vkexp(st + ik � r), withUk and Vk being the Fourier coefficients of the kth mode.
Because equations (28) need to be satisfied for each mode, k,
we have
sUk ¼ fuUk þ fvVk þ ipUkkx; ð29aÞ
sVk ¼ guUk þ gvVk : ð29bÞ
[61] Nontrivial solutions of system (29) exist when its
determinant is zero:
s2 � fu þ gv þ ipkxð Þs þ fugv � fvgu þ ipkxgv ¼ 0: ð30Þ
[62] Notice how in this case, s is a complex number. The
emergence of instability requires the real part of s to be
positive. Traveling wave patterns require that the imaginary
part of s is different from zero. It has been noticed
[Rovinsky and Menzinger, 1992] that equation (30) does
not lead to the selection of any finite value for the most
unstable wave number in that s is a monotonically increas-
ing function of k, and the wave number interval of the
unstable modes has no upper bound. However, the addition
to equation (29) of a diffusion term to either the first or the
second equation (or to both, as in equation (26)) imposes an
upper bound to the range of unstable modes. In this case the
most unstable mode corresponds to a finite value of the
wave number.
[63] We present, as an example of differential flow
instability, a model developed to study the formation of
patterns in young mussel beds [van de Koppel et al., 2005].
The model can be adopted also to describe a system
involving trees or grasses. Two (dimensionless) state vari-
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ables, representing nutrient concentration, u, and vegetation
density, v, are used. The dynamics of the two variables are
expressed as
@u
@t¼ 8 1� uð Þ � uvþ pruþ d1r2u; ð31aÞ
@v
@t¼ huv� d
v
1þ vþ d2r2v: ð31bÞ
[64] The first term on the right-hand side of the first
equation represents the rate of increase in nutrient concen-
tration, the second term accounts for the consumption of
nutrient by biomass, while the third term is the loss of
nutrients by advection; the fourth term models the spreading
of u by diffusion. The first term on the right-hand side of the
second equation represents the nutrient-dependent rate of
biomass growth, the second term represents the state-
dependent mortality rate, and the third term accounts for
the diffusion-like spatial spreading of biomass. The steady
homogeneous state (u0 = (h8 � d)/[h(8 � 1)], v0 = [8(d �h)]/(h8 � d)) is stable in the absence of drift and diffusion
when the conditions (5) are met. Drift-induced instability
occurs if the drift term is able to destabilize the homogeneous
state (u0, v0) even when the Laplacian terms are set equal to
zero. In this case the dispersion relation (30) provides the
range of Fourier modes that are destabilized by drift (see
Figure 12). As noted by Rovinsky and Menzinger [1992], in
the absence of a diffusion term the interval of the unstable
modes has no upper bound (equation (30)). When a diffusion
term is added to the first equation (i.e., d1 6¼ 0), the dispersion
relation becomes
s2 þ d1k2 � fu � gv � ipkx
s
þ fugv � fvgu þ ipkxgv � d1k2gv
¼ 0: ð32Þ
[65] The plot of this relation (see Figure 12) shows that in
this case the interval of the unstable wave numbers has an
upper bound and the most unstable mode has finite wave
number. When a diffusive term is added also to the second
equation (i.e., d1 6¼ 0, d2 6¼ 0) as in equation (29), the
dispersion relation becomes
s2 þ d1k2 þ d2k
2 � fu � gv � ipkx
s
þ fugv � fvgu þ ipkxgv � id2pkxk2 � d1k
2gv � d2k2fu þ d1d2k
4
¼ 0; ð33Þ
with no substantial differences in the amplitude of the
interval of unstable modes (see Figure 12).
[66] An example of spatial patterns emerging with this
model is shown in Figure 13.
2.2. Stochastic Models
[67] Pattern formation in ecology has been often associ-
ated with the deterministic mechanisms of symmetry-break-
ing instability described in sections 2.1.1, 2.1.2, and 2.1.4,
while random environmental drivers have been usually
considered to be only able to introduce noise in the ordered
states of the system. Thus, random environmental fluctua-
tions are usually believed to disturb the states of the system
and to destroy the patterns formed by deterministic dynam-
ics [e.g., Rohani et al., 1997]. However, it has been shown
that random fluctuations are able to also play a ‘‘construc-
tive’’ role in the dynamics of nonlinear systems, in that they
can induce new dynamical behaviors that did not exist in the
deterministic counterpart of the system [e.g., Horsthemke
and Lefever, 1984]. In particular, stochastic fluctuations
have been associated with the emergence of new ordered
states in dynamical systems, in both time [e.g., Horsthemke
Figure 12. Different dispersion relations for the case ofdifferential flow instability. The parameters for the boldcontinuous line are 8 = 0.72, h = 6.10, d = 5.14, p = �1.315,d1 = 1, and d2 = 2.
Figure 13. Spatial pattern emerging for the variable uusing the model (26). The parameters are h = 6.10, d = 5.14,8 = 0.72, p = �1.315, d1 = 0, and d2 = 1. The simulation iscarried out over a domain of 256 � 256 cells, each cellrepresenting a spatial step Dx = Dy = 0.8. Equations aresolved numerically by means of finite difference method.
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and Lefever, 1984] and space [Garcia-Ojalvo and Sancho,
1999]. Known as ‘‘noise-induced phase transitions,’’ these
‘‘constructive’’ effects of noise may occur in systems forced
by multiplicative noise (i.e., when there is a state depen-
dency in the impact of random fluctuations on the system).
[68] Thus, random environmental drivers are not neces-
sarily in contraposition to pattern formation. Indeed, it has
been shown that noise may induce pattern formation [van
den Broeck et al., 1994; Garcia-Ojalvo and Sancho, 1999;
Loescher et al., 2003; Sagues et al., 2007]. Although these
noise-induced mechanisms of pattern formation have been
investigated by the physics community for over a decade,
they have found only limited applications in ecohydrology.
This fact is quite surprising, in that environmental dynamics
are undoubtedly affected by random fluctuations, which
might have the potential of playing a fundamental role on
the composition and structure of plant ecosystems.
[69] We will present two major mechanisms of noise-
induced pattern formation, based either on nonequilibrium
phase transitions or on the random switching between
dynamics. We will also discuss the few existing examples
of ecohydrological models of noise-induced pattern forma-
tion (section 4).
2.2.1. Nonequilibrium Phase Transition Models[70] Recently, it has been found that patterns may also
emerge as ordered symmetry-breaking states induced by
noise in nonlinear, spatially extended systems [van den
Broeck et al., 1994, 1997; Parrondo et al., 1996]. These
ordered states result from phase transitions, which break the
ergodicity of the system. In the thermodynamics literature
these transitions are often referred to as ‘‘nonequilibrium
phase transitions’’ to stress the fundamental difference in the
role of noise (i.e., its ability to generate order) with respect
to the case of equilibrium phase transitions [van den Broeck
et al., 1994]. In these (nonlinear) systems, multiplicative
noise destabilizes a homogeneous steady state of the under-
lying deterministic dynamics thereby leading to an ordered
state that is stabilized by the spatial dynamics [Sagues et al.,
2007]. For noise to be able to induce phase transition with
breaking of ergodicity, it has to be ‘‘multiplicative’’; that is,
its effect on the dynamics needs to be modulated by a
(multiplicative) term, which depends on the state of the
system. However, it has been recently found that order can
also be induced by additive noise acting in concert with
multiplicative noise in spatially extended systems [Sagues
et al., 2007]. These symmetry-breaking states are purely
noise induced; that is, they are induced by local fluctuations
and do not occur in the deterministic counterpart of the
system. In fact, they vanish as the noise intensity (i.e., the
variance) drops below a critical value, suggesting that a
threshold needs to be exceeded by the noise intensity for
noise-induced patterns to emerge. At the same time, these
nonequilibrium phase transitions have been found to be
reentrant, in that the ordered phase is destroyed when the
noise intensity exceeds another threshold value. In other
words, the multiplicative noise has a ‘‘constructive’’ effect
only when the variance is within a certain interval of values.
Smaller or larger values of the variance correspond to
conditions in which noise is either too weak or too strong
to induce ordered states. van den Broeck et al. [1994, 1997]
used an approximated analytical framework to investigate
conditions leading to nonequilibrium phase transitions with
breaking of ergodicity. This framework, which is based on
mean field analysis, was first developed for the case of
Gaussian noise [van den Broeck et al., 1994, 1997; Parrondo
et al., 1996] and then applied to systems forced by Poisson
[Porporato and D’Odorico, 2004] and dichotomous [Bena,
2006] noise. The only applications of this mechanism to
landscape ecology we are aware of are based on a model of
fire-vegetation interaction in which random fire occurrences
are represented as Poisson noise [D’Odorico et al., 2007b].
[71] When the state of the system is determined by only
one state variable, u, its temporal dynamics can be, in
general, modeled by a differential equation expressing the
temporal variability of u at any point, (x, y), as the sum of
three terms: a function of local conditions (i.e., of the value
of u at (x, y)), a term representing a state-dependent noise,
and a term accounting for the spatial interactions with the
other points of the domain. These spatial interactions are
modeled as a diffusion process
@u
@t¼ f uð Þ þ g uð Þx tð Þ þ dr2u; ð34Þ
where d is a diffusivity coefficient, f(u) and g(u) are two
functions of u(x, y), and x(t) is the noise term. Equation (34)
is a stochastic partial differential equation (a stochastic
reaction-diffusion equation) where the multiplicative term is
interpreted in the Stratonovich sense [van Kampen, 1981].
The solution of this equation would provide the probability
distribution of u as a function of (x, y) and t. However, there
are no known exact methods for the integration of (34).
Thus, the analytical evidence for the existence of a noise-
induced transition comes from an approximated approach
based on the mean field (Weiss) approximation [van den
Broeck et al., 1994, 1997; Buceta and Lindenberg, 2003].
These approximated analytical results have been supported
by numerical simulations [van den Broeck et al., 1994,
1997; Buceta and Lindenberg, 2003; Porporato and
D’Odorico, 2004].
[72] First, a finite difference representation of the diffu-
sion term is used and equation (34) is rewritten for the
generic site, i, in a square lattice domain
@ui@t
¼ f uið Þ þ xig uið Þ þ d
4
Xj�n ið Þ
ui � uj
; ð35Þ
where ui and xi are the values of u and x at site i,
respectively; n(i) is the set of the 4 nearest neighbors, j, of
site i. The solution of equation (35) is impeded by the fact
that the dynamics of ui are coupled to those of the
neighboring points. In fact, the spatial interaction term in
(35) depends on the mean value, E[u]i, of u in the
neighborhood of i,P
j�n(i)(ui � uj)/4 = ui � E[u]i. Van
den Broeck et al. [1994] solved equation (35) using the
mean field approximation. To this end, they assumed that
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the local mean can be approximated by the spatial mean,
E[u], of u across the whole domain, i.e., E[u]i � E[u]. The
effectiveness of this approximation can be improved by
taking E[u]i ’ 12[ui + E[u]] to account for the dependence
of E[u]i on the local conditions, as explained by Sagues et
al. [2007]. It can be shown that this last approximation
corresponds to weakening the spatial coupling by using an
effective diffusivity deff = d/2 [Sagues et al., 2007].
Because under this assumption the dynamics of ui do not
depend on those of the neighboring points, it is possible to
determine exact expressions for the steady state probability
distributions, pst(u; E[u]), of u obtained from equation (35)
for the cases of Gaussian, dichotomous, or Poisson noise.
Exact expressions for these distributions are given by van
den Broeck [1983]. The distribution pst(s; E[u]) will
necessarily depend on a number of parameters of the
dynamics and on E[u], which remains unknown. To
determine E[u], a self-consistency condition is used:
E u½ � ¼ hui ¼Z þ1
�1u pst u;E u½ �ð Þdu ¼ F E u½ �ð Þ: ð36Þ
[73] Multiple solutions of this equation correspond to
the existence of multiple possible average values of u in
statistically steady state conditions, i.e., of multiple possi-
ble steady state probability distributions of u. Thus, the
emergence of multiple solutions of the self-consistency
equation indicates the occurrence of ergodicity- and sym-
metry-breaking nonequilibrium phase transitions. Numeri-
cal simulations are then used to confirm the validity of the
approximated results obtained from equation (36) and to
show the actual emergence of nonequilibrium pattern
formation. Alternatively, the occurrence of nonequilibrium
phase transitions can be studied through an approximated
analytical solution of equation (36) obtained using the
Taylor’s expansion of its right-hand side term [Sagues et
al., 2007].
[74] Zhonghuai et al. [1998] used the framework by van
den Broeck et al. [1994] to study noise-induced phase
transitions in generic two-variable systems exhibiting
Turing instability. When a control parameter of the kinetics
is perturbed by noise, new kinds of patterns arise (transi-
tion from single spiral to double spiral waves). A similar
study was developed by Carrillo et al. [2004] for the
analysis of pattern formation in chemical reactions and
fluid convection.
2.2.2. Patterns Induced by Random Switchingof Dynamics[75] In the mechanism of noise-induced pattern formation
described in section 2.2.1, ordered states of the system
emerge as a result of local, random fluctuations of the state
variable, which can be either spatially correlated or uncor-
related [Sagues et al., 2007]. In the other major class of
stochastic models, noise-induced patterns result from the
random switching between dynamics that simultaneously
occurs at all points of the spatial domain [Buceta and
Lindenberg, 2002; Buceta et al., 2002a, 2002b, 2002c].
This mechanism is based on the idea that if the random
switching between dynamics is global (i.e., it simulta-
neously occurs across the domain) and ‘‘rapid,’’ the
system behaves as if it was undergoing the average
dynamics obtained as a weighted mean of the two states.
In this context with ‘‘rapid’’ switching we mean that the
average residence time of the dynamics in either one of
the two dynamical states is much shorter than the time
needed by the system to reach the equilibrium state in
each dynamics.
[76] Thus, if we take as an example two Turing models
and we randomly and rapidly switch between them the
system experiences only the average Turing dynamics. We
can envision cases in which, separately, neither of the two
dynamics are able to lead to pattern formation, while their
average exhibits diffusion-induced symmetry-breaking in-
stability. In these conditions, patterns emerge from the
nonequilibrium random (global) alternation between dy-
namics. Similar models can be constructed using two
suitable biarmonic (or neural) models with the same spatial
dynamics term but different local dynamics [Buceta et al.,
2002a]. Separately, these models are unable to exhibit
symmetry-breaking instability. The random switching be-
tween them may lead to mean dynamics that are capable of
generating patterns.
[77] We consider as an example a system in which
Turing’s instability is induced by noise. To this end, we
consider two reaction-diffusion systems (1):
@u
@t¼ f1;2 u; vð Þ þ r2u; ð37aÞ
@v
@t¼ g1;2 u; vð Þ þ d1;2r2v; ð37bÞ
with f1,2(u, v) and g1,2(u, v) being two pairs functions
describing the dynamics of states 1 and 2. The system
switches between state 1, where the local kinetics are
expressed by f1 and g1 and the diffusion ratio is d1 (these three
quantities are hereinafter simply indicated as A1), and state 2,
where the kinetics are modeled by the functions f2 and g2 and
the diffusion ratio is d2 (indicated as A2). Neither one of the
equations (37) for state 1 or 2 meets all the conditions (5) and
(9). Thus, neither one of the two dynamics can separately lead
to pattern formation. Each control parameter (f, g, or d)
alternates dichotomously in a way that the temporal
evolution, A(t), of each parameter can be expressed as
A tð Þ ¼ A1L tð Þ þ A2 1� L tð Þ½ �; ð38Þ
with L(t) being a dichotomous variable assuming values 0
and 1. When the switching is fast in the sense discussed
before, L(t) can be replaced by its average value, L(t) ’hL(t)i, and in this case
A tð Þ ’ �A ¼ A1P1 þ A2P2; ð39Þ
with P1 = hLi and P2 = 1 � hLi.
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[78] The dynamics resulting from the fast switching
between the two states are then
@u
@t¼ �f u; vð Þ þ r2u; ð40aÞ
@v
@t¼ �g u; vð Þ þ �dr2v; ð40bÞ
where �f = f1P1 + f2P2, �g = g1P1 + g2P2, and �d = d1P1 +
d2P2. Patterns emerge if the average dynamics meet the
conditions (5) and (9).
[79] The emergence of switching-induced patterns
depends on the velocity of the alternation between the
two dynamics. Over a relatively long time, both dynamics
would lead to spatially homogeneous configurations; how-
ever, if the switching is sufficiently fast the system can
experience the average dynamics (40). In fact, in these
conditions, the homogeneous steady state can never be
reached, and the system always remains in a nonequilibrium
configuration, which can be described by the mean of the
two states. The separation between slow and fast switching
can be defined using a control parameter, r, representing the
ratio between an ‘‘external’’ time scale, text, associated with
the random switching (i.e., average time the system spends
in each configuration), and an ‘‘internal’’ time scale, tint,
associated with the time needed by the system to reach
equilibrium in each of the two states. If r = text/tint ! 1, no
switching-induced instability emerges. On the contrary,
when r < 1 the dynamics can be described in average
terms as in equations (40), and patterns may emerge if
equations (5) and (9) are met. The following patterns have
been noted:
[80] 1. Patterns may emerge from random alternation of
dynamics even when both of them have the same homoge-
neous steady state. In this case, it has been found that the
random switching leads to spotted structures, while when
the two dynamics have different steady states the random
alternation may lead to labyrinthine-striped configurations
[Buceta et al., 2002a].
[81] 2. A similar model of pattern formation can be
developed using a random switching between two biar-
monic [Buceta and Lindenberg, 2002] or two neural models
[D’Odorico et al., 2006a], as discussed in section 4.2.
[82] 3. Although the random switching can play a crucial
role in this process of pattern formation, similar patterns
emerge when the switching mechanism is deterministic. In
fact, Buceta and Lindenberg [2002] showed that periodic
alternation of dynamics can also lead to pattern formation.
Thus, unlike the mechanism described in section 2.2.1,
patterns emerging from nonequilibrium dynamics associ-
ated with global alternation of dynamics are not noise
induced sensu strictu. We refer the interested reader to Bena
[2006] for a more detailed discussion of ordered states
induced by periodic and random drivers.
[83] In the specific case of ecohydrology the random
switching can be caused, for example, by interannual
rainfall variability and the consequent alternation of stressed
and unstressed conditions in vegetation [D’Odorico et al.,
2005, 2006b]. An example is discussed in section 4.
2.2.3. Case Study: Turing Instability Inducedby Random Switching[84] As an example of patterns induced by random
switching between two Turing models we consider the
case of a system that switches between the two states:
(1) state 1 expressed by equations (37) with f1(u, v) = avu2
and g1(u, v) = bv (with a = 27.5, b = 105, and d1 = 41.25)
and (2) state 2 with f2(u, v) = �eu and g2(u, v) = �cu2v2
(with e = 90, c = 566.67, and d2 = 20). The system is in
state 1 with probability P1 = 0.8 and in state 2 with
probability P2 = 0.2; using equations (40) we then have�f = u(P1avu � P2e), �g = v(P1b � P2cu
2v), and �d = P1d1 +
P2d2 (as in equations (10)). It is possible to show that unlike
the dynamics in states 1 and 2, the average dynamics
associated with the process of fast switching can induce
pattern formation through Turing instability. In fact, in this
case the average dynamics are the same as those of the
deterministic model presented at the end of section 2.1.1,
and the spatial configuration arising with this last model has
the same features as those shown in Figure 6.
2.3. Other Mechanisms of Noise-Induced PatternFormation
[85] Other authors have investigated the role of noise in the
process of pattern formation in ecohydrology [e.g.,Ruxton and
Rohani, 1996; Satake et al., 1998; Durrett, 1999; Spagnolo et
al., 2004]. For example, Spagnolo et al. [2004] proposed
multivariate models of noise-induced pattern formation in
generalized Lotka-Volterra systems with multiplicative
(Gaussian) noise and diffusion-like spatial interactions. Con-
sistently with van den Broeck et al. [1994], they found that
multiplicative noise can play a constructive role in the non-
linear dynamics in that it can lead to a variety of dynamical
behaviors including pattern formation. However, apart from
these few examples, there have been only limited ecohydro-
logical applications of some recent theories of noise-induced
order.
[86] Analyzing the geometrical features of patchy land-
scapes, it has been observed that in a number of natural
systems patterns may exhibit order at all scales; that is, the
spatial organization has no characteristic scale, and features
of all sizes exist [Morse et al., 1985; Krummel et al., 1987;
Sole and Manrubia, 1995; Malamud et al., 1998; Guichard
et al., 2003; Caylor and Shugart, 2006; Kefi et al., 2007b,
2007a]. Known as ‘‘fractals,’’ objects exhibiting (statistically)
the same geometrical properties at all scales are ubiquitous
in nature, as evidenced by the recurrence of power laws in
the probability distributions of geometrical features of
natural systems [Mandelbrot, 1984; Lam and de Cola,
1993; Rodriguez-Iturbe and Rinaldo, 2001]. The presence
of fractality is often associated with criticality [e.g., Bak,
1996], i.e., the state of systems close to a phase transition
[e.g., Herbut, 2007]. A few theoretical frameworks have
been developed to relate these patterns to the underlying
stochastic processes. In particular, in this review we briefly
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mention the theory of the self-organized criticality [Bak et
al., 1988].
[87] The concept of self-organized criticality provides a
rather general framework explaining some general features
of complex systems in which scale invariance has been
observed to emerge both in time and in space [Bak, 1996].
These systems are generally open, nonlinear, dissipative,
and with many degrees of freedom. They consist of many
components, with each component interacting with its
neighbors (internal dynamics). The overall dynamics result
from the combined effect of local interaction and external
stochastic drivers: at any given time the external driver may
activate some components of the system if some threshold
conditions are exceeded. Through a network of local inter-
actions the activation can spread to other components,
thereby propagating the signal until the system relaxes in
a marginally stable state in which all components are again
below threshold conditions. The system remains in this state
until the external driver activates some other components
thereby spreading through other pathways. Thus, an impor-
tant feature of these systems is that external drivers acting in
conjunction with local interactions lead to nonlocal changes
that propagate throughout the system (‘‘avalanche’’ effect).
Through a number of simple cellular automata models [Bak
et al., 1988; Bak and Tang, 1989; Bak et al., 1989, 1990;
Bak, 1996], P. Bak and coworkers demonstrated that these
dynamics are typical of systems that self-organize them-
selves in critical states (i.e., in states close to a phase
transition) which do not exhibit any dominant scale either
in space or time. Self-organized criticality (SOC) occurs
when the critical state is attained with no parameter tuning.
[88] Examples relevant to the study of vegetation pattern
formation are based on discrete (cellular automata) models.
In these systems the emergence of self-organized criticality
has been investigated mostly through numerical simula-
tions, though theoretical frameworks based on mean field
theory (section 2.2.1), group theory, and Langevin equa-
tions have been also developed and applied [e.g., Jensen,
1998]. The forest fire model [Bak et al., 1990; Drossel and
Schwabl, 1992] is a classical example of dynamics sugges-
tive of self-organized criticality, which could be relevant to
the study of vegetation patterns observed in nature [e.g.,
Malamud et al., 1998]. Developed as a ‘‘toy model’’ to
study SOC mechanisms, this model did not claim to
reproduce forest dynamics that are ecologically realistic.
Moreover, it requires parameter tuning [e.g., Jensen, 1998].
[89] Sole and Manrubia [1995] found scale invariance
(i.e., power laws) in the distribution of gap sizes in the
rainforests of Barro Colorado island. These authors devel-
oped a simple spatially extended cellular automata model to
show how scale-invariant order may spontaneously emerge
in these systems from SOC dynamics, i.e., as a result of the
tendency of the system to relax in a critical state. A few
other models have been developed to explain scale invari-
ance in vegetation patterns as a sign of self-organized
criticality [Sprott et al., 2002; Bolliger, 2005; Zeng and
Malanson, 2006; Kefi et al., 2007a; Manor and Shnerb,
2008].
[90] More recently, Scanlon et al. [2007] explained the
emergence of scale-invariant order in savanna vegetation
[Caylor and Shugart, 2006] without invoking the occur-
rence of criticality. These authors showed how power law
scaling found in the cluster size distribution of vegetation
[Caylor and Shugart, 2006] can result from the interplay of
positive feedbacks with local interactions and global con-
straints in an Ising-like model [e.g., Brush, 1967].
[91] In addition to the mechanisms reported in the
previous paragraphs, there is a relatively rich body of lit-
erature on pattern formation based on spatiotemporal
stochastic resonance [Benzi et al., 1985; Loescher et al.,
2003; Spagnolo et al., 2004], coherence resonance [e.g.,
Sagues et al., 2007], noise-induced phenomena in excit-
able systems [Sagues et al., 2007], and front dynamics
in the presence of external noise [Garcia-Ojalvo and
Sancho, 1999]. However, these more recent theories of
noise-induced order in spatially extended systems have had
only limited applications to the environmental sciences.
3. BIOECOLOGICAL MECHANISMS LEADING TOVEGETATION PATTERNS
[92] A common feature of the three deterministic models
of symmetry-breaking instability presented in section 2.1 is
that the emergence of periodic vegetation patterns arises
from the balance between positive (activation) and negative
(inhibition) interactions [e.g., Shnerb et al., 2003; Rietkerk
and van de Koppel, 2008]. For example, in the neural model
both pattern emergence and pattern geometry are deter-
mined by the interplay between short-range facilitation (or
cooperation) and long-range competition (or inhibition), as
reflected by the kernel shape shown in Figure 7 [Lejeune et
al., 2004; Rietkerk and van de Koppel, 2008]. The dominant
wavelength of the resulting pattern is typically larger than
the range of either facilitative or competitive interactions
between individuals because of the ability of spatial insta-
bility to amplify local processes and to induce pattern
formation at the patch scale [Couteron and Lejeune, 2001].
[93] Cooperative and synergistic short-range effects are
usually associated with positive feedbacks resulting from
the ability of some species or functional types to create
environmental conditions that favor plant establishment,
growth, and survival. These feedbacks typically operate
within a short range. For example, cooperation among
neighboring individuals may lead to the concentration of
resources in vegetated areas where plant individuals find
more favorable conditions for establishment and survival
[Charley and West, 1975; Schlesinger et al., 1990; Greene,
1992; Wilson and Agnew, 1992; Bhark and Small, 2003;
D’Odorico et al., 2007a]. The aerial parts of plant individ-
uals that have already established in a certain patch may
favor the growth of other plants in the same area by limiting
soil moisture losses associated with evapotranspiration
[Vetaas, 1992; Thiery et al., 1995; Lejeune et al., 2004;
Zeng et al., 2004; D’Odorico et al., 2007b] either through a
mulching effect (i.e., soil evaporation limited by wilted
leaves and litter) or shading (i.e., when the foliage shades
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the ground surface thereby limiting evaporation) [Zeng and
Zeng, 1996; Scholes and Archer, 1997; Zeng et al., 2004;
Caylor et al., 2006; Borgogno et al., 2007]. Moreover, the
formation of physical and biological crusts on bare soil
may further reduce the infiltration of surface water [e.g.,
Fearnehough et al., 1998]. Physical crusts, typically 1–
3 mm thick, are generated by rain splashing [Issa et al.,
1999; Esteban and Fairen, 2006], while biological crusts
are formed by micro-organisms such as cyanobacteria,
which exude mucilaginous secretions that bind together soil
grains and organic fractions [Issa et al., 1999; Meron et al.,
2004; Belnap et al., 2005]. These crusts greatly reduce the
soil infiltration capacity thereby decreasing the soil moisture
available in the underlying soil layers, with consequent
limitations on the establishment and growth of perennial
vegetation [Fearnehough et al., 1998]. Because soil crusts,
on the other hand, seldom develop beneath vegetation
canopies owing to the reduced raindrop impact [Boeken
and Orenstein, 2001; Meron et al., 2004; Borgogno et al.,
2007] and the limited light available to the photosynthetic
activity of biological crusts [Walker et al., 1981; Greene,
1992; Joffre and Rambal, 1993; Greene et al., 1994, 2001],
a positive feedback exists between presence of vegetation
and absence of crusts. Conversely, in some arid and semi-
arid areas the absence of biological crusts leads to increased
erosion and loss of organic matter, fine soil particles, and
nutrients [Schimel et al., 1985; Harper and Marble, 1988].
In fact, cyanobacterial-lichen soil crusts have been shown to
be the dominant source of nitrogen in a cold desert and a
grassland ecosystem in southern Utah [Evans and Ehlringer,
1993]. Moreover, there is some discussion as to whether
biological crusts cause reduced infiltration [West, 1990] or
whether, as a result of the formation of microtopography,
they limit runoff and surface erosion [Belnap et al., 2005].
[94] In vegetated areas the protection against evapotrans-
piration and soil crust formation enhances surface water
infiltration which, in turn, favors vegetation growth. The
associated increase in root density, in turn, enhances the soil
infiltration capacity [Walker et al., 1981; Greene, 1992;
Joffre and Rambal, 1993; Greene et al., 1994, 2001;
HilleRisLambers et al., 2001; Okayasu and Aizawa,
2001; Gilad et al., 2004; Yizhaq et al., 2005; Borgogno
et al., 2007]. Moreover, a dense canopy of established
plants provides protection against herbivores (e.g., birds),
thereby favoring plant reproduction and growth [Lejeune
et al., 2002] in densely vegetated areas (propagation by
reproduction effect) where higher rates of seed production
and germination occur [e.g., Lefever and Lejeune, 1997;
Lejeune and Tlidi, 1999; Lejeune et al., 1999; Lefever et al.,
2000; Couteron and Lejeune, 2001].
[95] Species able to modify the abiotic environment, re-
distribute resources, and facilitate the growth of other species
as well as their own are known as ‘‘ecosystem engineers’’
[Jones et al., 1994; Gilad et al., 2007; Bonanomi et al.,
2008]. For example, the improvement of conditions exist-
ing in the microenvironment underneath the canopy of so-
called ‘‘nurse plants’’ [Neiring et al., 1963; Kefi et al.,
2007b] favors the establishment and growth of other plants
[e.g., Garcia-Moya and McKell, 1970; Burke et al., 1998;
Aguiar and Sala, 1999]. Vegetation cover may decrease the
amplitude of temperature fluctuations; reduce the exposure
to solar radiation, wind desiccation, and soil erosion; or
prevent soil crust formation [Eldridge and Greene, 1994;
Smit and Rethman, 2000; Greene et al., 2001]. Moreover,
plant individuals located in the middle of vegetated patches
are protected against fires and grazing.
[96] As noted, it has been also found that water and/or
nutrient availability are higher in the areas located under the
canopy of existing plant individuals than in the surrounding
bare soil [Charley and West, 1975]. These nutrient-rich
areas are known as ‘‘fertility islands’’ or ‘‘resource islands’’
[Schlesinger et al., 1990]. Mechanisms commonly invoked
to explain the formation of these heterogeneous distribu-
tions of resources include the ability of the canopies to trap
nutrient-rich airborne soil particles, the accumulation of
sediments transported by wind and water, the sheltering
effect of vegetation against the erosive action of wind
and water, and the presence of nitrogen-fixing species
[Garcia-Moya and McKell, 1970; Charley, 1972; Archer,
1989; Schlesinger et al., 1990; Breman and Kessler, 1995;
Okin et al., 2001; Li et al., 2007]. Sometimes, fertility
islands lead to the formation of aperiodic vegetation patterns
in the form of stable spatial configuration corresponding to
isolated vegetation patches (spots), usually named ‘‘local-
ized structures’’ or ‘‘localized patches’’ [Lejeune et al.,
2002; Meron et al., 2007]. Other examples of ecosystem
engineers relevant to pattern formation include the ability of
deep-rooted plants to facilitate shallow-rooted species by
increasing surface soil moisture through ‘‘hydraulic lift’’
mechanisms [Richards and Caldwell, 1987], the reduction
in fire pressure resulting from the encroachment of woody
vegetation at the expenses of grass fuel [e.g., Anderies et al.,
2002; van Langevelde et al., 2003; D’Odorico et al.,
2006a], and the ability of alpine/subalpine vegetation and
desert shrubs to maintain warmer microclimate conditions
and reduce frost-induced mortality.
[97] Similar facilitative mechanisms exist also in wetland
environments, including salt marshes, where vegetation
may prevent salt accumulation by limiting soil evaporation
(shading effect), riparian corridors, and wetland forests,
where vegetation can favor the aeration of anoxic soils
through soil drainage by plant uptake and transpiration
[Wilde et al., 1953; Chang, 2002; Ridolfi et al., 2006]. It
has also been found that on cobble beaches, dense stands of
spartina alterniflora occupying the lower intertidal zone
can protect other plant communities from intense wave
action [Bruno, 2000; van de Koppel et al., 2006].
[98] On the other hand, competitive or inhibitory effects
typically occur within a longer range. Competition for water
and nutrients is generally exerted via the root system
[Aguilera and Lauenroth, 1993; Belsky, 1994; Breman
and Kessler, 1995; Breshears et al., 1997; Martens et al.,
1997; Couteron and Lejeune, 2001]. In fact, the lateral roots
extend beyond the edges of the crown [Casper et al., 2003;
Barbier et al., 2008] and extract water and nutrients from
the intercanopy areas [Martens et al., 1997; Lejeune et al.,
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2004]. Hence, the typical range of competitive interactions
is larger than that of facilitation. For example, for trees and
shrubs, the ratio between the radii of the footprint of canopy
and root systems may be as small as 1/10 [Lejeune et al.,
2004]. Thus, the root system is able to deplete soil resources
(i.e., water and nutrients) from the intercanopy areas and to
compete for resources with other plant individuals [Lefever
and Lejeune, 1997; Yokozawa et al., 1998; Lejeune et al.,
1999; Lejeune and Tlidi, 1999; Couteron and Lejeune,
2001; Lejeune et al., 2002; Rietkerk et al., 2004; Yizhaq et
al., 2005; Barbier et al., 2006]. This competition for
resources usually reduces the growth rate of competing
individuals thereby leading to a net effect of inhibition
through competition [Shnerb et al., 2003].
[99] Thus, facilitative interactions occur within the range
of the crown area and are mostly associated with positive
feedbacks induced by the canopy (e.g., mulching, shading,
protection against fires, grazing, and wind action), while
negative interactions are exerted mainly as resource com-
petition by roots and typically occur at larger distances
(Figure 8). The sign and range of these interactions justifies
the choice of kernel functions shaped as in Figure 7: the
kernel shows positive interactions within the range of the
canopy scale (short-range) and negative interactions within
the range of the typical lateral root length (long range), while
the magnitude of these interactions vanishes (i.e., w ! 0)
as the distance between the interacting plants exceeds the
typical extent of lateral roots. Nevertheless, as already
noticed at the end of section 2.1.2, spatial patterns can also
emerge when the kernel is ‘‘upside/down’’ with respect to the
case of Figure 7. In fact, in some cases, facilitation can occur
at a long range (e.g., buffering from intense wave action in
intertidal communities [van de Koppel et al., 2006]), and
competition can occur at a short range (e.g., competition
for light typically due to interactions among canopies [Caylor
et al., 2005; van de Koppel et al., 2006]).
[100] Although all these mechanisms of cooperation and
competition are generally isotropic (i.e., they operate in the
same way in all directions), in some environments the
presence of a slope or of a dominant wind direction may
lead to anisotropy in the spatial dynamics. In fact, if the
wind regime exhibits a prevailing direction, asymmetry may
emerge in the cooperative and competitive mechanisms. For
example, the persistent existence of a cone-shaped wind
shadow downwind of tree/shrub clumps would provide a
favorable protected environment for the establishment and
growth of other plant individuals (short-range positive
feedback), while plant individuals located at larger distances
would remain with no protection and would consequently
be prone to higher mortality rates [Puigdefabregas et al.,
1999; Yokozawa et al., 1999; Okin and Gillette, 2001].
Similarly, in section 2.1.4 we discussed the role of an
advective flow on the dynamics of two diffusive species
and its role in the phenomenon of differential flow insta-
bility. Advective flows can originate as an effect of runoff in
sloping terrains. During intense rainfall events, water and
sediments run off bare areas and are intercepted and trapped
by vegetated patches. This supplementary input of limiting
resources favors plant growth on the uphill side of vegetated
patches, thereby securing more efficient trapping during
subsequent rainstorm events. Runoff and erosion are there-
fore viewed as a fundamental mechanism to maintain
striped configuration over hillsides [Thiery et al., 1995;
Dunkerley, 1997a, 1997b; Okayasu and Aizawa, 2001;
Sherratt, 2005; Saco et al., 2006; Barbier et al., 2008].
Thus, rainfall onto an unvegetated area generates overland
flow, which transports water in the downhill direction until
it reaches a vegetated area, where it infiltrates into the
ground and is taken up by vegetation. The relatively moist
soil on the uphill side of a stripe creates opportunities for
uphill expansion of the vegetation band at the expenses of
the downhill side, which remains deprived of the resources
necessary for vegetation survival. The overall dynamics
lead to the uphill migration of vegetated bands [Sherratt,
2005]. A similar mechanism can explain the banded patterns
of trees in the Tierra del Fuego (Argentina, see Figure 1h)
where a sawtooth pattern of tree heights is observed in the
wind direction [Puigdefabregas et al., 1999]. Taller trees
provide more protected favorable conditions for seedling
establishment and tree growth in the leeward direction. At
the same time, the strong winds uproot and kill the taller
upwind trees leading to an overall downwindmigration of the
sawtooth pattern.
[101] Unlike the case of neural and differential flow
models, the application of Turing instability to ecohydro-
logical systems is not straightforward. These models are
generally used when the state of the system is described by
more than one state variable and when spatial dynamics can
be modeled as diffusion processes. In most of the existing
models (see section 4), two or three state variables are used,
including vegetation biomass, subsurface, and surface water.
The diffusive character of plant encroachment has been
invoked by a number of studies, which argued that the
propagation of vegetation fronts due to seed dispersal or to
clonal reproduction can be expressed as a diffusion process
[e.g., von Hardenberg et al., 2001;Murray, 2002]. The use of
diffusion in the modeling of surface overland flow has been
justified using the shallow water theory, i.e., assuming that
overland flow occurs with only a thin layer of water [Gilad
et al., 2004]. The diffusive character of subsurface flow is
often justified by recalling the diffusive nature of Darcy’s
law (i.e., the proportionality between water flux and water
potential gradients). However, the diffusivity of unsaturated
subsurface flows is expected to be small and to be unable to
lead to significant soil moisture patterns at the patch scale.
Moreover, it has been noted [Barbier et al., 2008] that the
existence of wetter soils in vegetated areas should result from
mechanisms leading to the concentration of resources be-
neath plant canopies, while soil moisture diffusion would
operate in the opposite direction. Nevertheless, even the use
of diffusion in the description of biomass spreading is
problematic. Diffusive models treat biomass as a ‘‘green
slime,’’ i.e., as if it was not rooted into a place as a series of
individuals. Even vegetative/clonal reproducers could not
rigorously be described as diffusers because the only place
where a real gradient exists is at the edge of a patch. As an
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TABLE2.EcohydrologicalTuringModelsProposedto
Explain
PatternForm
ationa
Authors
Model
andVariables
EmergingPatterns
ComparisonWithData
NotesandComments
HilleRisLamberset
al.[2001]
Threevariables(surfacewater,soilwater,andplantdensity).
Adetailedstabilityanalysisinvestigates
theconditionsfor
theform
ationofspatiallynonuniform
structures.
Notspecified.
None.
Themodel
confirm
sthat
itisthe
combinationofpositiveand
negativefeedbacksthat
generates
vegetationpatterns.Other
processes
(herbivores,plantdispersal,and
rainfall)arenottheprimaryfactors
that
generatepatterns.
Rietkerket
al.[2002]
Evolutionofthemodel
byHilleRisLamberset
al.[2001].
Thediffusionterm
ofsurfacewater
canbereplacedwith
adriftterm
switchingthemodel
into
adifferential
flow
model.
Spots,labyrinths,gaps,
andstripes.
Niger.
Transitionfrom
spotsto
labyrinthsand
togapswithincreasingrainfall
amount.Theparam
eter
rangeused
insimulationsindicates
that
patterns
tendto
developmore
frequentlyon
fine-texturedsoils.
vandeKoppel
etal.[2002]
Evolutionofthemodel
byHilleRisLamberset
al.[2001].
Themodel
also
accountsforaterm
describingthevariation
inherbivore
populationdependingonthevariationsofplant
biomass.Graphical
andnumerical
resolutionofequations.
Notspecified.
None.
Graphical
resolutionofequations:two
param
etersarecomputed,P(local
vegetation)andPavg(spatialaverage
ofvegetation).Patternsem
ergewhen
P>Pavg.
Meronet
al.[2004]
Twovariables(plantbiomassandwater).Thesystem
has
two
uniform
stationarysolutionsforlow
andhighvalues
ofrainfall.
Numerical
simulationsshow
nonuniform
solutionsfor
interm
ediate
values
ofrainfall.
Spots,labyrinths,andgaps.
NorthernNegev,
goodagreem
ent
betweenreal
dataand
numerical
simulations.
Themodel
revealsdifferentranges
of
precipitationwherevegetation
patternsmay
coexistwithuniform
states
(baresoilorcompletely
vegetated
soil)andalso
withother
pattern
states.
vandeKoppel
etal.[2006]
Twovariables(plantbiomassandwrack
biomass).Numerical
simulationsanalyze
theem
ergence
ofvegetationpatternsusing
differentfunctiondescribingcooperativeandcompetitive
interactions.
Spots.
Spatialstructuresof
CarexStricta
tussocks
inMaine,
USA.
Model
forwetlands.
GuttalandJayaprakash
[2007]
Evolutionofthemodel
byRietkerket
al.[2002],introducinga
param
eter
that
capturestheeffectsofseasonal
adaptationsofplants
tothedry
andwet
season.Resultsobtained
withnumerical
simulations.
Spots,gaps,andlabyrinths.
None.
Thepattern
dependsonthedurationof
thewet
seasoneven
withfixed
total
annual
precipitation.Thedistribution
within
theyearplaysafundam
ental
role
indeterminingpattern
form
ation
andshape.
ZengandZeng[2007]
Threevariables(m
assdensity
oflivingleaves,available
soilwetness,
andmassdensity
ofwiltedanddeadleaves).A
linearstabilityanalysis
isdeveloped
toobtain
thesetofparam
etersleadingto
spatialinstability.
Theresultsarechecked
throughnumerical
simulations.
Spots,gaps,labyrinths,
andstripes.
None.
Theauthors
stress
therelevance
of
vegetationpatternsin
theprocess
ofdesertification:they
arean
indicatorthat
theecosystem
isat
thevergeofacatastrophic
shift.
aUnless
differentlyspecified,thecomparisonwithdataisonly
qualitative.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
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example, Thompson and Katul [2008] proved that the use of
suitable seed dispersal kernels to represent plant biomass
spread leads a better representation of biomass spreading than
those provided by diffusion models. In sections 4.1.1–4.1.3
and 4.2 we provide an overview of a number of ecological
models of pattern formation, and we will use the concepts
presented in this section and section 2 to discuss their main
properties.
4. MODELS FOR VEGETATION PATTERNFORMATION
[102] In this section we review the ecohydrology litera-
ture on mechanisms and models of vegetation pattern
formation. We first analyze deterministic models of sym-
metry-breaking instability and discuss the underlying biotic
and abiotic mechanisms. Then we discuss some stochastic
models of noise-induced pattern formation in ecohydrology
based on nonequilibrium phase transitions or on random
switching between deterministic dynamics.
4.1. Deterministic Models
4.1.1. Turing-Like Instability Models[103] Only relatively few authors have used Turing mod-
els to explain vegetation patterns (Table 2). The models by
Meron et al. [2004] and van de Koppel et al. [2006] involve
two variables, namely, plant biomass and water [Meron et
al., 2004] or plant biomass and wrack biomass (i.e., algae)
[van de Koppel et al., 2006]. All the other models consid-
ered in this review involve three variables, namely, plant
biomass, surface water, and soil water [HilleRisLambers et
al., 2001; Rietkerk et al., 2002; van de Koppel et al., 2002;
Guttal and Jayaprakash, 2007], except for the case of Zeng
and Zeng [2007], which involves living leaves, dead leaves,
and soil moisture as state variables.
[104] As noted in section 3, the diffusive character of the
spatial dynamics of vegetation depends on seed dispersal
and clonal reproduction [HilleRisLambers et al., 2001;
Rietkerk et al., 2002; van de Koppel et al., 2002; Meron
et al., 2004; Guttal and Jayaprakash, 2007; Zeng and Zeng,
2007], while for dead biomass the diffusive dynamics are
ascribed to horizontal transport by wind, herbivores, or
anthropogenic disturbances [van de Koppel et al., 2006;
Zeng and Zeng, 2007]. Meron et al. [2004] considered (soil)
water as the limiting resource and justified the assumption
of its diffusive behavior using Darcy’s law. Other Turing
models considering soil water as a state variable (coupled
with vegetation dynamics) do not explicitly invoke Darcy’s
law [HilleRisLambers et al., 2001; Rietkerk et al., 2002; van
de Koppel et al., 2002; Guttal and Jayaprakash, 2007; Zeng
and Zeng, 2007]. Turing models accounting for the spatial
and temporal dynamics of surface water [HilleRisLambers
et al., 2001; Rietkerk et al., 2002; van de Koppel et al.,
2002; Guttal and Jayaprakash, 2007] approximate overland
flow as a diffusive process [Bear and Verruyt, 1990]; in
these models, surface water flows from bare soil to vege-
tated areas driven by pressure gradients associated with the
higher soil infiltration capacities in the vegetated soil patches.
Thus, this framework assumes that a positive feedback exists
between vegetation and soil moisture (vegetation ! higher
infiltration ! enhancement of water transport from bare
soil ! more vegetation), which favors transport of water to
vegetated patches. The magnitude of this transport depends
on the soil texture, the presence of physical and/or biological
crusts, and the possible development of some microreliefs as
a result of the accumulation of sediment mounds beneath the
vegetation canopies. Moreover, recent microscale measure-
ments of soil infiltration within vegetation patches [Ravi
et al., 2007, 2008] have shown that relatively small soil
infiltration may occur in the middle of vegetation patches
where finer soil particles are generally found. In this case,
overland flow would likely occur toward the edges of the
vegetated patches both from the surrounding bare soil and
from the middle of vegetation mounds. Thus, the patterns of
overland flow may be more complex than what is usually
believed.
[105] Even though the models by Rietkerk et al. [2002],
van de Koppel et al. [2002], and Guttal and Jayaprakash
[2007] are based on the same Turingmodel [HilleRisLambers
et al., 2001], they stress different aspects of processes
associated with the formation of vegetation patterns. For
example, Rietkerk et al. [2002] extended the model of
HilleRisLambers et al. [2001] replacing the diffusion term
with an advective term, generating a differential flow
model to study patterns on slopes. Numerical simulations
point out that vegetation stripes only emerge if a minimum
terrain slope is present, while the model by HilleRisLambers
et al. [2001] can only generate spots, labyrinths, and gaps.
The model by van de Koppel et al. [2002] investigates how
the pattern geometry changes as the system approaches a
bifurcation point, where it is more susceptible to a discon-
tinuous response (often called ‘‘catastrophic shift’’) to a
continuous change in environmental parameters. The model
by HilleRisLambers et al. [2001] allows one to understand
that the spatial redistribution of surface water prevents
irreversible vegetation collapse, but van de Koppel et al.
[2002] found that if the presence of herbivores is taken into
account, the reduction of vegetation cover beyond a threshold
level can lead the system to a desert state. The model by
Guttal and Jayaprakash [2007] investigates the seasonal
variation of plant growth due to plant response to the dry
and wet season. In nature, as the season changes, various
physical factors affecting the vegetation growth can also
change. This means that the dynamics of vegetation are
different during the wet and the dry season, and the alterna-
tion of different dynamics can affect the spatial arrangements
of plants. This kind of behavior is usually investigated with
stochastic models (see section 2.2.2), while Guttal and
Jayaprakash [2007] use a deterministic approach based on
a Turing model.
[106] As noted in section 2.1.3, approximations of Turing
and neural models lead to the same analytical expressions of
the conditions determining the emergence of symmetry-
breaking instability. The relation existing between these two
types of models suggests that activation/inhibition (or
facilitation/competition) processes explicitly represented in
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TABLE
3.EcohydrologicalContinuousKernel-BasedModelsProposedto
Explain
PatternForm
ationa
Authors
Model
andVariables
EmergingPatterns
ComparisonWithData
NotesandComments
Lefever
andLejeune[1997]
Continuousmodel
foronly
onevariable
(vegetation).Kernel
representedbythe
combinationofthreeGaussiankernels
(cooperation,inhibition,andmortality).
System
withtwohomogeneoussteadystates.
Alinearstabilityanalysisisperform
edto
search
forconditionsunder
whichthe
establishmentofuniform
vegetationisim
possible.
Labyrinthsandspotsin
isotropic
system
sandstripes
inanisotropic
system
s.
Tiger
bush
inSomalia
andstripes
inJordan.
Themodel
predictstwokindsofstriped
structures:parallelandperpendicularto
groundslope,
inagreem
entwithobservations
insitu.
Lejeuneet
al.[1999]
Sam
emodel
asLefever
andLejeune[1997].
Anadditional
analysisiscarriedoutto
investigatetherole
offeedbackson
vegetationpattern
form
ation.Itisdem
onstrated
that
intheabsence
ofcooperativefeedbacks,
vegetationpatternscannotform
.
Spotsandstripes
inisotropic
environmentsandstripes
inanisotropic
environments.
Tiger
bush
inBurkinaFaso.
Theauthors
developadetailedanalysisof
aerial
photographsoftiger
bush
innorthwest
BurkinaFasoto
calibrate
theparam
etersof
themodel
and,in
particular,to
estimatethe
ranges
ofthecooperativeandinhibitory
interactionsbetweenplants.
LejeuneandTlidi[1999]
Sam
emodel
asLefever
andLejeune[1997]but
thebiarm
onic
approxim
ationofneuralmodels
(equation(18))isusedto
runnumerical
simulationsin
isotropic
environments.
Spotsandlabyrinths.
Tiger
bush
insouthwestNiger.
Thebiarm
onic
approxim
ationusedhereleads
tothesamemodel
asCouteronandLejeune
[2001].
Lefever
etal.[2000]
Based
onLefever
andLejeune[1997],
Lejeuneet
al.[1999],andLejeuneandTlidi[1999].
Spotsandstripes.
Tiger
bush
inBurkinaFaso.
Numerical
simulationssuggestthat
asource
ofanisotropy(e.g.,slope)
predominantly
affectsinhibitory
rather
than
cooperative
interactionssince
hexagonal
spotsobtained
inisotropic
conditionsarereplacedbystripes
when
inhibitionissufficientlyanisotropic.
CouteronandLejeune[2001]
Continuousmodel
forasingle
variable
representing
averagephytomass.Themodel
originates
from
thebiarm
onic
approxim
ationofthemodel
byLejeuneandTlidi[1999].Numerical
simulations
arerunto
investigatetheshapeofvegetationpatterns
withvaryingecological
param
eters.
Spotsandlabyrinths.
BurkinaFasoandNiger.
Comparisonbetweenspectral
analysesof
digitized
photographsandpatternsresulting
from
simulations.Thisanalysisallowsthe
authors
tocalibrate
themodel.
Lejeuneet
al.[2002]
Sam
emodel
asCouteronandLejeune[2001]withmore
mathem
atical
details(linearstabilityanalysis);
one-dim
ensional
andtwo-dim
ensional
analyses.
General
clustering.
South
Americaand
westAmerica.
None.
Lejeuneet
al.[2004]
Sam
emodel
asCouteronandLejeune[2001].Theauthors
pointoutthedestabilizingeffect
oftheLaplacian
term
andthestabilizingeffect
ofthedouble
Laplacian.
Adetailedstabilityanalysisisrunto
studythepattern
selection,that
is,theem
ergence
ofspotted
orstriped
configurations.
Spots,zigzagstripes,andgaps.
Manyarid
regionsofAfrica,
Australia,
NorthAmerica,
andtheMiddle
East.
Alongarainfalltransect
ofdecreasingrainfall
themodel
showsthetransitionfrom
completely
vegetated
groundto
spots
(vegetationgaps),then
stripes,then
spots
again(vegetationclumps),andfinally
baresoil.
aUnless
differentlyspecified,thecomparisonwithdataisonly
qualitative.
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TABLE
4.EcohydrologicalDiscreteKernel-BasedModelsProposedto
Explain
PatternForm
ationa
Authors
Model
andVariables
EmergingPatterns
ComparisonWithData
NotesandComments
Sato
andIwasa
[1993]
Discretemetapopulationmodel.Single
variable
representingthemedium
heightofthecohort
oftreesstandingin
each
cell.
Ingeneral,when
atree
ismuch
higher
than
thewindwardtrees,itisproneto
windfallanddies.
Stripes
(saw
-toothed
travelingwaves).
Subalpineforests
ofcentral
Japan,
northeastUSA.
Theauthors
notedthat
theinclusionof
disturbancesin
thedeterministicmodel
leadsto
more
realisticsolutions.This
isdonebySatake
etal.[1998].
Thiery
etal.[1995]
Discretecellularautomatamodel.Eachcell
representsastateofatree
corresponding
todifferentmetaboliccharacteristics.
Aconvolutionmatrixrunsover
thedomain
andrepresentscompetitionorsynergyin
ananisotropic
environment.
Irregularstripes.
SouthwestNiger.
Inorder
tovalidatethemodel,aerial
photographsarescanned
andcompared
withnumerical
results.Thesimulated
stripes
migrate
upward,andthissuggests
pattern
changes
atthetopofreal
landscapes
wherebandsconverge.
CominsandHassell[1996]
Discretemetapopulationmodel.Threevariables
representingtwoparasitoid
speciesandthe
density
ofhostspecies(trees).
Somerulesdefinetheinteractionsbetween
speciesandtheinteractionsbetweenneighboringcells.
Spotsandlabyrinths.
None.
Thesimulationsgenerateself-organization
only
when
thespeciesshow
very
differentdispersalrates,as
inaTuringmodel.
Dunkerley
[1997a]
Discretecellularautomatamodel
for
anisotropic
environments.
Therulesdescribethepartitioningofwater
flow
toneighboringcellsdependingonlandcover
andthewater-controlled
dynam
icsofvegetation.
Stripes.
Australia,
Niger,
andMexico.
Stripes
occuronly
withgentleslopes.
Stripes
canbedestroyed
bya
long-lastingim
pactofgrazers,even
ifnodataexistto
evaluatethiseffect.
Dunkerley
[1997b]
Sam
emodel
asDunkerley
[1997a]
withtheadditionofrules
representingtheeffect
ofdroughtonthesoilwater
content
andofgrazingonthevegetationbiomass.
Stripes.
Australia.
Grazingim
pacthas
nosustained
effect.
Themodel
showsthat
droughtis
potentially
more
destructivethan
grazingpressure.
Keymer
etal.[1998]
Discretemetapopulationmodel.Single
binaryvariable
representing
thestateofthepatch
(occupiedorem
pty).A
single
rule
accountingforextinctionandcolonization.Therule
accountsfor
differentranges
ofinteractionsbetweenneighboringcells.
General
clustering.
None.
Numerical
simulationsshow
that
spatial
patternscanem
ergeonly
withlocal
interactions,that
is,when
therange
ofinteractionsbetweencellsisnot
toolargebecause
theirnet
effect
isto
includecooperativeeffects.
Yokozawaet
al.[1998]
Discretecellularautomatamodel.Single
variable
representing
thesize
ofeach
individual.A
single
rule
defines
theevolution
ofthevariable
dependingonagrowth
andacompetitionterm
.Competitioncanbesymmetricorasymmetric.
General
clustering.
ForestsofnorthernJapan.
Numerical
simulationsshow
that
coarser
andmore
uniform
patches
ariseunder
symmetricalcompetition.Thisisin
agreem
entwithearlierstudies[Kubota
andHara,1995]that
foundsymmetric
competitionbetweentreesin
coniferousforests.
Puigdefabregaset
al.[1999]
Discretemetapopulationmodel.Thepopulationin
each
cellgrows
proportionally
toagrowth
rate
ordeclines
byself-thinningand
winddieback.Winddiebackdependsonaprotectionfactor
from
windwardneighbors.Numerical
simulationsstartfrom
arandom
distributionofindividuals.
Stripes.
Tierradel
Fuego.
Numerical
simulationslead
tostriped
configurations(perpendicularto
wind
direction).Larger
tree
growth
rateslead
tolonger
wavelengths,andan
increase
ofwindstrength
leadsto
shorter
wavelengthsin
agreem
entwithfieldobservations.
Yokozawaet
al.[1999]
Sam
emodel
asYokozawaet
al.[1998].A
sourceofexternal
asymmetry,i.e.,unidirectional
wind,isadded.
Stripes.
SubalpineregionsofJapan
andUnited
States.
Comparisonbetweensymmetricand
asymmetriccompetition.The
wave-shaped
spatialpattern
ismuch
clearerunder
symmetricalcompetition.
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neural models of pattern formation exist also in the dynam-
ics underlying the spatial interactions in Turing models.
This aspect of Turing models is addressed by van de Koppel
and Rietkerk [2004], who built a model able to reproduce
different interactions between plants. This study showed
that vegetation patterns emerge only with short-range co-
operation and long-range inhibition, in agreement with the
results obtained with neural models (section 2.1.2). Simi-
larly, Zeng and Zeng [2007] explicitly described the spatial
interactions in their Turing model as the result of facilitation
and competition processes.
4.1.2. Kernel-Based Models[107] Two main types of kernel-based models can be
found in the ecohydrology literature, namely continuous
and discrete spatially extended models. Continuous models
(Table 3) use a continuous representation of the kernel
describing the spatial interactions among individuals, while
discrete models (Table 4) use a discrete kernel defined over
a discrete spatially extended domain represented as a lattice
of square cells. Because continuous kernels can be defined
with continuous mathematical functions, they lend them-
selves to an analytical description (section 2.1.2), which is
often not possible with discrete kernels.
[108] The first continuous kernel-based model of vegeta-
tion patterns [Lefever and Lejeune, 1997] describes the
interactions among plants through three different functions,
representing short-range cooperative effects (higher infiltra-
tion rate, shading, mulching, and less soil crusting under the
canopy), long-range competition for resources, and a tox-
icity effect accounting for the mortality rates existing with
higher densities of vegetation. The combination of these
three components gives the kernel of equation (12). These
authors studied the stability of the system in isotropic and
anisotropic conditions. Numerical simulations lead to
striped, spotted, and labyrinthine configurations in isotropic
landscapes and to banded patterns in anisotropic conditions.
This model was used to explain tiger bush dynamics and
favorably compared [Lejeune and Tlidi, 1999; Lejeune et
al., 1999] with real data from tiger bush vegetation in Niger
and Burkina Faso. A kernel-based model was developed by
D’Odorico et al. [2006c] to show how vegetation patterns
can be due to the ability of spatial dynamics to enhance
productivity and vegetation cover in arid landscapes, with
respect to the case of ecosystems with no spatial interac-
tions. As noted, the main difference between the model by
D’Odorico et al. [2006c] and that by Lefever and Lejeune
[1997] is that in the former model, nonlinearities are local,
while in the latter they modulate the spatial interactions.
[109] Lefever et al. [2000] provided a biarmonic approx-
imation of this kernel-based model [Lefever and Lejeune,
1997] with an approach similar to the one presented in
section 2.1.2. This biarmonic model is used by Couteron
and Lejeune [2001] to account for short-range cooperative
effects (shading, mulching, and protection against herbi-
vores [Lejeune et al., 2002]) expressed by the diffusion term
(propagation) and for long-range competitive interactions
(exerted via lateral roots [Couteron and Lejeune, 2001;
Lejeune et al., 2002]) represented by the biarmonic term
Authors
Model
andVariables
EmergingPatterns
ComparisonWithData
NotesandComments
Adleret
al.[2001]
Discretecellularautomatonmodel.Twoplantspeciesandagrazing
component.Differenttypes
ofinteractionsbetweenspecies
(competitionandcooperation).Numerical
simulationsallow
one
toobtain
differentspatialconfigurationsdependingonthe
interactionrules.
General
clustering.
USA,South
Africa,
andTierradel
Fuego.
Thesimulationssuggestthat
neighborhood
interactionsat
theindividual
plantscale
rather
than
grazingeffectsorunderlying
environmentalheterogeneities
lead
topatchystructures.
LanzerandPillar[2002]
Discretecellularautomatonmodel.Eachcellcanbeoccupiedwith
oneoutofninepossible
states
(aplantspeciesorbareground).
Arule
defines
thepersistence
ofthestatein
thecelloritschange
into
another
possible
statedependingonneighborhoodfrequencies.
General
clustering.
Atlanticheathland
intheNetherlands.
Alargesetofreal
dataallowsoneto
model
theinteractionsbetweenspeciesfitting
theneighborhoodfrequency
ofagiven
statearoundeverysampledplant.
Rietkerket
al.[2004]
Sam
emodel
asThiery
etal.[1995]butdeveloped
hereonly
for
isotropic
environments.Anisotropic
convolutionmatrixmodeling
short-rangecooperationandlong-rangeinhibitionrunsover
the
whole
domain.
Spotsandlabyrinths.
Niger,Israel,Ivory
Coast,
French
Guiana,
andSiberia.
Thesystem
isbistable;that
is,forsomeranges
of
resourceinputitshowstwodifferentstable
configurations.When
thesystem
approaches
acritical
valueofresourceinput,itcanundergo
catastrophic
shifts.
Feagin
etal.[2005]
Discretecellularautomatonmodel.A
convolutionmatrixmodels
facilitationeffects.A
gradientmatrixsimulatestheeffect
ofa
sloped
terrainandthecompetitiveeffectsdueto
the
interceptionofrunoff.
Stripes.
Texas.
Only
thesimulationsaccountingforasloped
terraingenerateplantpatterns.Thesimulated
patternsarecomparable
withthefielddata.
EstebanandFairen
[2006]
Discretecellularautomatonmodel.Thedomainrepresentsa
gentlysloped
surface.
Arule
defines
soilmoisture
storage
andtheam
ountofbiomassdependingonthesoilmoisture.
Stripes.
Tiger
bush
intheSahel.
Iftheannual
rainfallisbelow
acrossover
value,
asudden
catastrophic
shiftfrom
patchystructure
andbaresoilisobserved.Upslopemigratingbands.
aUnless
differentlyspecified,thecomparisonwithdataisonly
qualitative.
TABLE4.(continued)
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(inhibition). This ‘‘propagation-inhibition’’ model [Couteron
and Lejeune, 2001], leads to the spotted/labyrinthine
patterns in isotropic conditions and to banded patterns in
anisotropic landscapes. These patterns were compared with
those from aerial photographs over West Africa: a good
agreement was found overall between modeled and observed
patterns [Barbier et al., 2008]. The same model was also
used by Lejeune et al. [2002, 2004].
[110] In the discrete kernel-based models the spatial
domain is subdivided into a discrete number of cells, with
each cell representing either a single plant or a group of
plants. Two major types of discrete neural models can be
found in the literature, namely, cellular automaton (CA)
models and metapopulation models. In CA models (Table 4)
the values of the state variables in each cell change accord-
ing to a set of rules, which use a spatial kernel to modulate
the interactions among neighboring cells [Dunkerley,
1997b]. These models lend themselves to the representation
with simple rules of the physical processes underlying the
dynamics of the system. For example, the model developed
by Thiery et al. [1995] to explain the formation of tiger bush
in Niger uses an asymmetric kernel to account for the
anisotropy associated with the presence of a slope and the
consequent occurrence of water runoff and sediment flow.
This kernel expresses the interplay between cooperation and
inhibition. Cooperation is induced by the same synergistic
mechanisms of facilitation described in section 3 (i.e.,
shading, mulching, and enhancement of soil infiltration),
while competition is caused by the interception of water at
the uphill side of a vegetation stripe, which results in water
deficit at the downhill side. A similar model was developed
by Rietkerk et al. [2004], using a symmetric discrete kernel
to describe the spatial dynamics of vegetation in isotropic
environments. This CA model showed how vegetation
patterns (spots and labyrinths) may emerge as a result of
the spatial interactions on flat terrains, in the absence of any
prescribed landscape heterogeneity and anisotropy. This
model [Rietkerk et al., 2004] showed how the pattern
geometry may depend on the strength of the mutual
interactions among plants: the model can generate spots
with relatively weak positive interactions and labyrinths
when these interactions become relatively strong. This
aspect is of great importance to relate the different pattern
shapes to the different levels of cooperation. Thus, pattern
geometry can be also explained as a result of the strength of
the cooperative interaction that the existing species can
develop. Similar models were proposed to explain vegeta-
tion pattern formation in Texas [Feagin et al., 2005] and
tiger bush dynamics in the Sahel [Esteban and Fairen,
2006]. Both models are anisotropic in that they account
for the existence of a slope and for its role in the process of
pattern formation. Other authors used similar anisotropicmod-
els to investigate vegetation patterns in Australia [Dunkerley,
1997a, 1997b]. The models by Dunkerley [1997a, 1997b]
were also used to investigate the role of grazing and drought
in the formation of vegetation patterns and showed that
spatially organized configurations of vegetation emerge only
with gentle slopes. Moreover, these models were able to
demonstrate how this spatial organization enhances the
ability of the system to recover after a disturbance. Thus,
aridland vegetation would increase its resilience by develop-
ing spatial interactions, which result in the commonly ob-
served patchy distribution of vegetation.
[111] The model by Yokozawa et al. [1998] considers a
different type of anisotropy, which is not associated with a
topographic slope but with a possible asymmetric competi-
tion between individuals. In this model the discrete kernel
function can account for both a symmetric and an asym-
metric competition. This study shows how patterns can
emerge with both symmetric and asymmetric competition.
However, symmetric competition leads to more homoge-
neous patterns (i.e., patterns with features of about the same
size), which are more similar to those observed in conifer-
ous forests. These results confirmed experimental results
[Kubota and Hara, 1995] showing that competition be-
tween trees is generally symmetric. The situation is different
in the presence of anisotropy induced by an external driver
(e.g., a prevailing wind). In this case, a modified version of
the model [Yokozawa et al., 1999] shows that the same
patterns can be generated either with symmetric or asym-
metric competition.
[112] Metapopulation models (Table 4) are based on a
very simplified framework and are generally used to sim-
ulate the dynamics of very complex systems. These models
can give a preliminary idea of the basic aspects of the
system’s behavior, which can be further investigated with
more complex models. Hanski [2004] used metapopulation
models to study the dynamics of populations living in
highly fragmented landscapes, where only a small fraction
of the total area offers suitable habitat for a given species.
This is often the case in arid and semiarid environments,
where the landscape typically exhibits a highly heteroge-
neous distribution of resources. Thus, metapopulation mod-
els seem to provide a suitable framework to examine the
spatial dynamics of dryland plant ecosystems.
[113] In a metapopulation model the domain is usually
subdivided into cells similarly to the case of CA kernel-
based models; however, a single cell represents a relatively
large portion of the landscape, where a group of individuals
exist, the so-called ‘‘metapopulation.’’ Each metapopulation
has its own dynamics, which can be independent of those in
other cells. Spatial patterns emerge when interactions based
on short-range cooperation and long-range competition
exist among cells [Keymer et al., 1998]. For example, in
the works by Puigdefabregas et al. [1999], Iwasa et al.
[1991], and Sato and Iwasa [1993], patterns are induced by
metapopulations interacting through a positive feedback in a
windy environment, where leeward trees benefit from the
sheltering provided by upwind trees, while in the work by
Comins and Hassell [1996] the authors investigate the
patterns emerging from the interactions between vegetation
and different parasitoid species.
4.1.3. Differential Flow Instability Models[114] The idea that vegetation patterns could be induced
by a drift (section 2.1.4) was first formulated by Klausmeier
[1999] in a model based on differential flow instability. This
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model accounts for two variables, namely, vegetation bio-
mass and (surface) water. Plant dispersal is represented as a
diffusion process, while surface water undergoes advective
flow in the downhill direction. This model does not invoke
diffusive processes to mimic overland flow. This framework
leads to the formation of vegetation stripes migrating uphill
on sloped terrains, while on flat terrains (i.e., with no drift),
patterns do not emerge because the system dynamics reduce
to those of a bivariate Turing model with only one diffusing
variable. Sherratt [2005] and Ursino [2005] used the same
framework to investigate changes in the dynamics of
vegetation stripes with varying environmental parameters
(e.g., rainfall, plant mortality, and slope). They found that
the wavelength of stripe sequences is a decreasing function
of rainfall and an increasing function of slope [Sherratt, 2005].
The slope plays a crucial role in determining the conditions for
pattern formation: Ursino [2005] found that a minimum
terrain slope is needed for stripes to emerge. Some of these
properties are in agreement with experimental observations.
[115] More recently, other modeling frameworks (Table 5)
have been proposed to explain vegetation pattern formation
as a drift-induced process. For example, Gilad et al. [2004]
accounted for the dynamics of three state variables, namely,
plant biomass, soil water, and surface water depth. The
model accounts for two feedbacks between biomass and
water and shows how patterns result from mechanisms of
ecosystem engineering. The two feedbacks are associated
with the short-range positive interaction resulting from the
increase in infiltration induced by vegetation and long-
range negative interaction due to plant competition for soil
water uptake. In this model, all variables exhibit a diffusive
behavior ascribed to plant seed dispersal, the diffusive
nature of soil-water transport in a nonsaturated soil [Hillel,
1998], and the shallow-water theory for surface water.
Shallow-water theory also accounts for an advective term
associated with the slope-dependent water flow. Thus,
because this model is a combination of a Turing and a
differential flow model, patterns may arise either from
Turing instability or differential flow instability. When
applied to flat terrains, this framework becomes a Turing
model, in that the drift term becomes zero and the spatial
dynamics are contributed only by diffusion processes. In
this case, Turing instability may lead to the formation of
spots, gaps, or labyrinths. In the case of sloped terrains the
model can generate patterns through the mechanism of
differential flow instability, leading to the formation of
stripes or spots (with low rainfall values). The model was
recently used byGilad et al. [2007] to investigate the role of
the two feedbacks in the process of pattern formation.
These authors found that the same type of patterns may
emerge when either one of these feedbacks is predominant.
Yizhaq et al. [2005] used the same model to relate the
resilience of striped vegetation to the wavelength and found
that patterns with high wave numbers correspond to sys-
tems with higher productivity that are more biologically
productive but less resilient to environmental changes.
Moreover, starting from the model by Gilad et al. [2004],
Meron et al. [2007] addressed problems like biomass-water
relationships in spot-like vegetation patches, formation
mechanisms of ring-like vegetation patches, and resilience
of vegetation patches.
[116] Other authors [Okayasu and Aizawa, 2001; von
Hardenberg et al., 2001] previously developed similar
models obtained by adding a drift term to a Turing model
to account for the flow of water and the transport of
sediments in the downhill direction. Both models show
the transition between spots and labyrinths in the absence of
drift, and the formation of bands when the drift is included.
A comparison between pattern formation with or without
drift is also given by Shnerb et al. [2003], who used a model
obtained as a combination of a differential flow model
(accounting for the drift) with a neural model. These
authors showed how both drift and cooperative interactions
may induce pattern formation. Further studies on the role of
sediment transport are given by Saco et al. [2006], where
the models by HilleRisLambers et al. [2001] and Rietkerk
et al. [2002] were extended explicitly accounting for the
dynamic effect of erosion-deposition processes caused by
an overland water flow on a hillslope. The simulated
configuration evolves into a profile with stepped micro-
topography, in agreement with field data in shrubland
communities in Australia [Dunkerley and Brown, 1999].
4.2. Stochastic Models
[117] Despite the pervasive presence of random fluctua-
tions in environmental processes, applications of the theo-
ries of noise-induced pattern formation in ecohydrology are
still rare. Early studies on the ability of noise to enhance
order in the spatial distribution of vegetation are given by
Satake et al. [1998] and Yokozawa et al. [1998, 1999]. In
particular, Satake et al. [1998] modified a previous discrete
metapopulation formulation of the neural model [Iwasa et
al., 1991; Sato and Iwasa, 1993] to show how, when one or
more suitable parameters are interpreted as random varia-
bles, the spatial patterns generated by the model become
more distinct and regular. This constructive effect of noise is
observed when the noise intensity (e.g., variance) exceeds a
certain minimum level; however, as the noise intensity
exceeds another critical threshold, the random drivers de-
stroy the spatial organization of the system (destructive
effect of noise). As noted in section 2.2, this noise-induced
behavior corresponds to a reentrant phase transition in that
the order-forming effect of noise is observed only within a
limited range of noise intensities. These results were found
by Satake et al. [1998] through numerical simulations of dis-
crete (meta)population dynamics, without using an analytical/
theoretical framework.
[118] D’Odorico et al. [2007b] capitalized on the theory of
noise-induced nonequilibrium phase transitions (section 2.2)
to analyze possible mechanisms of fire-induced pattern for-
mation in savannas. These authors modeled the temporal and
spatial changes in shrub (or tree) vegetation, u, using only one
stochastic differential equation with the same form as (34),
@u
@t¼ f uð Þ þ g uð Þxcp þ dr2u; ð41Þ
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TABLE
5.EcohydrologicalModelsofDifferentialFlow
InstabilityProposedto
Explain
PatternForm
ationa
Authors
Model
andVariables
EmergingPatterns
ComparisonWithData
NotesandComments
Klausm
eier
[1999]
Twovariables(surfacewater
andplantbiomass).Themodel
exhibits
multiple
stable
states,andtwocasesareanalyzed
(hillsides
andflat
ground).A
linearstabilityanalysisisused
todeterminewhether
regularpatternscanform
.
Stripes.
Niger.
Linearstabilityanalysisshowsthat
regularpattern
form
ationis
impossible
onflat
groundand
that
thewavelength
ofthe
patternsincreaseswith
decreasingwater
input.
vonHardenberget
al.[2001]Twovariables(plantbiomassandwater
available
toplants).
Themodel
exhibitstwodifferentstable
states
becomingunstable
forinterm
ediate
values
ofprecipitation.Themodel
isstudied
throughthestabilityanalysisofuniform
solutionsandintegrating
equationsnumerically.
Spots,labyrinths,andgaps
forflat
landscapes
and
stripes
onaslope.
Niger
andnorthernNegev.Differentpatterns(spots,labyrinths,
andgaps)
emergealongarainfall
gradient,andthisallowsthe
introductionofaridity
classesdependingonthe
vegetationstate:
uniform
vegetation,vegetationpatterns,
orbaresoil.
Okayasu
andAizawa[2001]
Threevariables(w
ater
onthegroundsurface,
water
inthe
soil,andplantbiomass).Thesystem
has
twodifferent
stable
states.A
linearstabilityanalysisinvestigates
the
instabilityofparticularwavenumbers.Numerical
simulationsanalyze
spatialpatternsem
ergingwithdifferentsetsofparam
eters.
Spots,labyrinths,andgaps
(withoutsurfaceflow)and
stripes
withdownslopeflow.
Somalia
andnorthwest
BurkinaFaso.
Ingoodclim
ate,
spotsem
erge
over
hillslopes,whilethey
becomestripes
forsevere
clim
ates.Thestripes
migrate
upwardandthewavenumber
ofperiodic
patters
decreases
astheexternal
environmentbecomes
worse.
Shnerbet
al.[2003]
Twovariables(available
water
density
anddensity
ofshrubbiomass).
Alinearanalysisshowsthestabilityofthehomogeneousstatewithout
water
flow,andnumerical
simulationsshow
theem
ergence
ofspatial
patternswhen
adownhillflow
ispresent.
General
clustering
tendingto
astriped
configuration.
Israel.
Patternsem
ergeonly
ifapositive
feedbackbetweenplantsis
present.Thepatternsare‘‘frozen’’
(i.e.,nouphillmigration).
Giladet
al.[2004]
Threevariables(density
ofbiomass,soilmoisture,anddepth
of
surfacewater).Thesystem
has
twodifferentuniform
solutions.
Theirlinearstabilityisstudiedanalytically,
whilenonuniform
solutionsareobtained
bysolvingequationsnumerically
forflat
topography
andonaslope.
Spots,labyrinths,andgaps
(noslope)
andstripes
and
spots(slope).
None.
Themodel
isusedto
study
conditionsthat
maketheecosystem
resilientto
environmentalchanges.
Everyenvironmentalparam
eter
presentsan
optimal
rangefor
whichplantsarecapable
of
restoringthehabitatsaftera
disturbance.
Sherratt[2005]
EvolutionbyKlausm
eier
[1999].Mathem
atical
analysis.A
linearstability
analysisisusedto
determineunstable
modes.Analysisofthedependence
ofstriped
patternsonecological
param
eters(rainfall,plantloss,andslope).
Stripes.
Aerialphotosfrom
Africa,
Australia,
andMexico.
Themodel
generates
anuphill
migrationofstripes
provided
that
theeigenvalues
corresponding
tounstable
modes
arecomplex
numbers.
Ursino[2005]
Themodel
byKlausm
eier
[1999]isreinterpretedto
accountforrelevantsoil
properties.A
detailedstabilityanalysisisrunin
order
toobtain
thecritical
stabilityconditionsdependingonecological
param
eterslikerainfall,terrain
slope,
andsoilparam
eterslikeconductivity.
Banded
configurations.
Differentsitesin
America
andAustralia.
Themodel
pointsoutthecrucial
role
ofsoilphysics
inplant
development.
Yizhaqet
al.[2005]
Evolutionofthemodel
byGiladet
al.[2004],thoughtheauthors
analyze
only
thesituationonaslope.
Numerical
simulationsarerunto
investigatethe
resilience,water
consumption,andbiomassproductivityofbandswithdifferent
wavenumbersunder
thesamerainfallconditions.
Stripes.
None.
Water
consumptionper
unit
biomassdecreases
asthepattern’s
wavenumber
increases,so
banded
vegetationismore
productivefor
higher
wavenumbers,butitisless
resilientto
environmentalchanges.
RG1005 Borgogno et al.: VEGETATION PATTERN FORMATION
28 of 36
RG1005
where the local dynamics are expressed by a logistic growth
term f(u) = a(u + �)(umax � u) (the parameter a measures
the reproduction rate of the logistic growth), with umax
being the shrub carrying capacity and � being a parameter
preventing u from remaining locked at u = 0 after a severe
fire kills all the shrubs. The second term on the right-hand
side accounts for loss in shrub biomass associated with
random fire occurrences and is modeled as discrete
sequence of events occurring at random times, ti, with each
event having a random magnitude, w. A compound Poisson
noise, xcp = wd(t � ti), is used, where d() is the Dirac delta
function, and the times, t, between two consecutive
occurrences, t = ti+1 � ti (i = 1, 2, . . .), are modeled as an
exponentially distributed random variable with mean hti = 1/l.To account for the dependence of fire occurrences on the fuel
load (i.e., locally available grass biomass), which is inversely
related to the shrub biomass, u, the fire frequency, l, isexpressed as a function of u, l(u) = l0 + bu (with b < 0). The
variable w is an exponentially distributed random variable
(independent of t) with mean w0. The multiplicative function
g(u) =min(u,w) accounts for the fact that the amount of shrub
biomass removed by any fire is either u or w, whichever isless. The diffusion term in (41) expresses the encroachment
of shrub vegetation as a diffusion process. Both g(u) and l(u)determine the multiplicative character of the stochastic
forcing in (41).
[119] Using the mean field approximation (see section 2.2)
the steady state probability distribution of u is calculated
[D’Odorico et al., 2007b] as a function of the spatial mean,hui, of u
pst u; huið Þ ¼ C
r uð Þ expu
w0
�Z
l uð Þr uð Þ du
� �; ð42Þ
with r(u) = a(u + e)(1 � u) + d(hui � u). The solution of
the self-consistency equation (36) with steady state prob-
ability distribution given by (42) gives the mean value, hui,of u. Figure 14a shows a plot of hui as a function of the
diffusivity, d. It is observed that when the spatial coupling is
relatively strong (i.e., high values of d), the self-consistency
equation exhibits three solutions. Numerical simulations
show that two of them are stable and the other is unstable.
Thus, in these conditions the system may have two different
possible mean values and hence two possible steady state
probability distributions of u [D’Odorico et al., 2007b]. It
can be shown [see also Porporato and D’Odorico, 2004]
that the emergence of these multiple statistically steady
states is induced by the combined effect of spatial coupling
and the nonlinearity associated with the positive feedback
between stochastic forcing (i.e., fires) and the state of the
system (i.e., shrub biomass). As noted in section 2.2 this
effect of ergodicity breaking is induced by the multiplicative
character of the noise in (41). Patterns may emerge as
ordered states, resulting from nonequilibrium phase transi-
tions that break the ergodicity of the system. An example is
provided in Figure 14b: it can be observed that these
multiple (statistically) steady states emerge when the
variance of the random forcing exceeds a critical value;
Authors
Model
andVariables
EmergingPatterns
ComparisonWithData
NotesandComments
Saco
etal.[2006]
Partially
based
onthemodelsbyHilleRisLamberset
al.[2001]and
Rietkerket
al.[2002].Threevariables(plantbiomass,soilmoisture,
andoverlandflow).Themodel
iscoupledwithaphysicallybased
model
oftheevolutionoflandform
s.Numerical
simulationsanalyze
thespatial
evolutionofvegetationandmicrotopography.
Stationarybandsand
migratingbands.
Australia.
Theinitiallyplanar
hillslopeevolves
into
aprofile
withstepped
microtopography.
Giladet
al.[2007]
Evolutionofthemodel
byGiladet
al.[2004],buthereadetailedlinearstability
analysisisdeveloped
toanalyze
thestabilityofthetwouniform
stationary
solutions.Numerical
simulationsarerunto
investigatetheeffectsofdisturbances
andchanges
inphysicalparam
eters.
Spotsandstripes.
None.
Numerical
simulationsmainly
analyze
thesoilwater
distributiondepending
onphysicalparam
eters(infiltration,
precipitation,evaporation,and
grazingstress)to
investigatethe
role
ofplantsas
ecosystem
engineers.
Meronet
al.[2007]
Sam
emodel
byGiladet
al.[2004]andGiladet
al.[2007].
Spots(localizedstructures).
NorthernNegev.
Thepreviousmodelsarefurther
investigated
inorder
toaddress
avariety
ofproblemsrelatedto
patchiness,resilience,anddiversity
ofdrylandvegetation.
aUnless
differentlyspecified,thecomparisonwithdataisonly
qualitative.
TABLE
5.(continued)
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
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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.
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�������������������������F. Borgogno, F. Laio, and L. Ridolfi, Dipartimento di Idraulica,
Trasporti ed Infrastrutture Civili, Politecnico di Torino, I-10129 Torino,Italy. (fabio.borgogno@polito.it; francesco.laio@polito.it; luca.ridolfi@polito.it)P. D’Odorico, Department of Environmental Sciences, University of
Virginia, Box 400123, Charlottesville, VA 22903, USA. (paolo@virginia.edu)
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