Post on 31-Oct-2021
Habitat Dynamics in Drylands:
The Role of Collective Modes
E. Gilad
Department of Physics, Ben-Gurion University, Beer Sheva, 84105, Israel
Department of Solar Energy and Environmental Physics, BIDR, Ben Gurion University,
Sede Boker Campus 84990, Israel
gilade@bgu.ac.il
J. von Hardenberg
ISAC-CNR, Corso Fiume 4, I-10133 Torino, Italy
CIMA, Universita di Genova e della Basilicata, Via Cadorna 7, 17100 Savona, Italy
jost@cima.unige.it
A. Provenzale
ISAC-CNR, Corso Fiume 4, I-10133 Torino, Italy
CIMA, Universita di Genova e della Basilicata, Via Cadorna 7, 17100 Savona, Italy
A.Provenzale@isac.cnr.it
M. Shachak
Mitrani Department of Desert Ecology, BIDR, Ben Gurion University, Sede Boker Campus
84990, Israel
shachak@bgu.ac.il
E. Meron1
Department of Solar Energy and Environmental Physics, BIDR, Ben Gurion University,
Sede Boker Campus 84990, Israel
– 2 –
Department of Physics, Ben-Gurion University, Beer Sheva, 84105, Israel
ehud@bgu.ac.il
Received ; accepted
Elements to appear in the expanded online edition: Appendices A,B, Tables 1, 2, Figs. 1,3,
4,7,10, 12,13. This manuscript was prepared with the AASTEX LATEX macros v5.2.
1Corresponding author
– 3 –
ABSTRACT
Dryland landscapes are mosaics of patches that differ in resource concentra-
tion, biomass production and species richness. Understanding their structure and
dynamics often calls for the identification of key species that modulate the abi-
otic environment, redistribute resources and facilitate the growth of other species.
Shrubs in drylands are excellent examples; they self-organize to form patterns
of mesic patches which provide habitats for herbaceous species. In this paper
we present a mathematical model for studying the dynamics of habitats created
by plant ecosystem engineers in drylands. Using a pattern formation approach
we study conditions and mechanisms for habitat creation, habitat diversity, and
habitat response to environmental changes. We highlight the roles of two collec-
tive modes of habitat dynamics, associated with the engineering effects of plant
species and the self-organizing spatial patterns they form, and identify emergent
mechanisms of habitat dynamics.
Subject headings: collective dynamics, pattern formation, ecosystem engineer, habitats
dynamics, diversity, drylands
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1. Introduction
A fundamental problem in understanding global ecosystem aspects, such as biodiversity,
productivity and stability, is how local properties at the individual and single plant-patch
level, where laboratory and field experiments can be carried out, are up-scaled to the
ecosystem or landscape level (Loreau et al. 2001). This problem has two facets, conceptual
and technical. Conceptually, at any higher level of organization, emergent phenomena may
appear. Emergent properties involve collective dynamics of species individuals and therefore
cannot be understood by studying individuals’ properties alone. The concept of emergent
properties has a long history in ecology (Odum 1969, 1971) and it has been adopted
in many fields of physics ranging from superconductivity through pattern formation to
network dynamics (Anderson 1972, 1995). A striking example in water limited ecosystems
is banded vegetation on hill-slopes, where positive water-biomass feedbacks and intraspecific
competition at the level of the individual patch lead to self-organizing band patterns at the
landscape level (Valentin et al. 1999).
The self-organized dynamics at the ecosystem or landscape level usually involve
interactive activities of many species spanning a diversity of traits in respect to resource
acquisition and distribution. There are cases, however, where a few species have greater
impacts on the trophic web structure than others, through their roles in energy flow and
nutrient cycling. Such species have been named “keystone species” (Power et al. 1996), as
their introduction or removal can dramatically alter the ecosystem’s behavior. Recently, a
non-trophic organismal trait, termed ecosystem engineering, was suggested as an additional
organizational trait that links species and ecosystems (Jones et al. 1994, 1997; Olding-Smee
et al. 2003). While keystone species are assumed to directly affect the biotic component,
ecosystem engineers affect the physical environment. Plant ecosystem engineers modulate
the landscape by creating patches and patterns of biomass and resources, thereby affecting
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species diversity and ecosystem functioning (Perry 1998; Gilad et al. 2004; Shachak et al.
2005).
From a technical point of view, the attempt of addressing the up-scaling problem
from individual patches to the whole ecosystem is plagued with the difficulty of studying
large systems characterized by long length and time scales. In the context of water limited
ecosystems, significant progress in this direction has recently been made by the introduction
of spatially explicit mathematical models of vegetation growth (Lefever & Lejeune 1997;
Klausmeier 1999; von Hardenberg et al. 2001; Okayasu & Aizawa 2001; Rietkerk et al. 2002;
Shnerb et al. 2003; Gilad et al. 2004). The input information used in formulating these
models is at the level of a single plant-patch, while the predictive power extends to the
patchwork created at the landscape level. These models have reproduced various vegetation
patterns observed in the field (Rietkerk et al. 2004), predicted the possible coexistence of
different stable patterns under given environmental conditions, and identified a generic
sequence of pattern states along the rainfall gradient (von Hardenberg et al. 2001; Meron et
al. 2004).
In this paper we use a mathematical modelling approach to study emergent mechanisms
of habitat creation and change in drylands, where the limiting resource is water. We focus
on plant species that act as ecosystem engineers by trapping soil water under the patches
they form. The water enriched patches provide micro-habitats for other species. In modeling
plant ecosystem engineers we capture two collective modes representing interspecific and
intraspecific dynamics. The interspecific mode is associated with the micro-habitats that are
created by the ecosystem engineer, which other species can benefit from. The intraspecific
mode is associated with the spatial patterns formed by the ecosystem engineer, which
reflect cooperative dynamics of many individuals over long distances. Taking into account
these collective modes is important for understanding the impact of abiotic or human
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induced environmental disturbances on species diversity. Such disturbances may change
the spatio-temporal patterns of the ecosystem engineer and thus the habitats it forms. The
implications for species diversity inferred in this way may differ and even contradict those
inferred by merely considering the tolerance limits of individual species.
A brief account of the model to be presented here has been given in (Gilad et al. 2004).
In this paper we give a full account of the model, provide detailed model analysis, and
present new results concerning habitat types, habitat diversity and scenarios for habitat
change.
2. Background
Water limited landscapes are a mosaic of patches that differ in resource concentration,
biomass production and species richness. Redistribution of the water resource by biotic
and abiotic factors is a major factor in the formation of this patchwork. An abiotic
factor contributing to water redistribution is dust and sediment deposition in rocky
watersheds (Yair & Shachak 1987). The higher water infiltration rates in deposition patches
induce source-sink water flow relationships and enrich the water content in these patches.
Biotic factors include cyanobacteria that form partially impermeable biogenic crusts,
limiting infiltration and favoring surface runoff, and plants, particularly shrubs, that act
as sinks for the surface-water flow. The accumulation of litter and dust under the shrub
forms a mound with high infiltration capacity, which not only absorbs direct rainfall but
also intercepts the runoff water generated by the soil crust. Reduced evaporation under the
shrub canopy further contributes to local water accumulation. The water-enriched patch
and the accumulated dust and litter create an island of fertility under the shrub canopy,
characterized by a concentration of water, nutrient and organic matter that is higher than
in the surrounding crusted soil.
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Organisms capable of modulating the landscape and redistributing resources, such
as shrubs in drylands, have been termed ”ecosystem engineers”. There is an increasing
recognition that ecosystem engineers have important implications for biodiversity and for
ecosystem function (Olding-Smee 1988; Jones et al. 1994; Shachak & Jones 1995; Gurney
& Lawton 1996; Jones et al. 1997; Laland et al. 1999). Studies in the Negev desert, for
example, indicate that the concentration of resources, particularly soil water, under shrub
patches significantly increases species richness and density of annual plants (Boeken &
Shachak 1994).
The implications of the shrub patchwork for species richness at the landscape scale,
however, is largely unexplored. At this scale collective species dynamics, manifested
in the form of symmetry breaking spatial patterns, become important. Depending on
environmental conditions (rainfall, topography, grazing stress, etc.) and species traits,
different vegetation patterns have been observed, including banded or labyrinthine patterns,
spotted patterns, and gaps in uniform vegetation (Valentin et al. 1999; Rietkerk et al. 2002;
Meron et al. 2004; Rietkerk et al. 2004). The self organization of vegetation in water limited
systems to form spatial patterns has recently been attributed to an instability (Cross &
Hohenberg 1993; Murray 1993) of uniform vegetation which has been produced by several
independent models (Lefever & Lejeune 1997; von Hardenberg et al. 2001; Okayasu &
Aizawa 2001; Rietkerk et al. 2002; Gilad et al. 2004). The mechanism of the instability
is based on a positive feedback between biomass and resources. A common realization of
such a feedback is the increased infiltration of surface water at vegetation patches discussed
above. Another significant realization of a biomass-water feedback is water uptake by
plants’ roots; the larger the biomass the longer the roots and the larger amount of soil-water
the roots take up. This feedback is not captured by earlier mathematical models.
The biomass-water positive feedback is also responsible for the coexistence of different
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stable vegetation states, uniform or patterned, under the same environmental conditions
(von Hardenberg et al. 2001; Meron et al. 2004; Rietkerk et al. 2004). Coexistence of stable
states over limited parameter ranges (precipitation rate, grazing stress, etc.) may explain
desertification processes (von Hardenberg et al. 2001; Meron et al. 2004), and catastrophic
shifts in general (Scheffer et al. 2001; Scheffer & Carpenter 2003; Rietkerk et al. 2004), as
transitions between coexisting states. Coexistence of stable states also gives rise to a high
pattern diversity as mixtures of the different stable states in various spatial distributions
can be realized (Meron et al. 2004).
Spatial vegetation patterns are important agents in resource concentration processes
on the landscape level. Vegetation patterns such as spots, bands and gaps differ in the
proportion of bare soil to vegetative cover and thus in the resource patches they form.
Environmental changes or human disturbances may induce transitions from one pattern
state of the ecosystem engineer to another, thereby affecting the habitats it forms. This
landscape-scale mechanism of habitat change has not been explored yet.
3. A model for patch dynamics
We introduce a model for a single plant species where the limiting resource is water.
A ”patch” in the context of the model is defined to be an area covered by the plant, which
generally differs in its water content from the surrounding bare soil. Two main positive
feedback mechanisms between plant biomass and water are explicitly modelled: increased
water infiltration rates at vegetation patches (hereafter ”infiltration feedback”) and water
uptake by plants’ roots (hereafter ”uptake feedback”). Both feedbacks are responsible for
the appearance of symmetry breaking patterns of biomass and soil water at the landscape
scale. They control, however, different complementary properties of the ecosystem induced
by the plant: habitat creation and resilience to disturbances.
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The model contains three dynamical variables: (a) the biomass density variable,
B(r, t), representing the plant’s biomass above ground level and has units of [kg/m2], (b)
the soil-water density variable, W (r, t), describing the amount of soil water available to
the plants per unit area of ground surface in units of [kg/m2], and (c) the surface water
variable, H(r, t), describing the height of a thin water layer above ground level in units of
[mm]. Rainfall and topography are introduced parametrically; thus vegetation feedback on
climate and soil erosion are assumed to be negligible. The model equations are:
BT = GBB (1−B/K)−MB + DB∇2B
WT = IH −N (1−RB/K) W −GW W + DW∇2W
HT = P − IH + DH∇2(H2
)+ 2DH∇H · ∇Z + 2DHH∇2Z , (1)
where the subscript T denotes partial time derivative, r = (X,Y ) and ∇2 = ∂2X + ∂2
Y .
The quantity GB [yr−1] represents biomass growth rate, while K [kg/m2] is the maximum
standing biomass. The quantity GW [yr−1] represents the soil water consumption rate,
the quantity I [yr−1] represents the infiltration rate of surface water into the soil and the
parameter P [mm/yr] stands for the precipitation rate. The parameter N [yr−1] represents
soil water evaporation rate, while R describes the reduction in soil-water evaporation rate
due to shading 1. The parameter M [yr−1] describes the rate of biomass loss due to mortality
and different kinds of disturbances (e.g. grazing, fire). The term DB∇2B represents seed
dispersal while the term DW∇2W describes soil water transport in non-saturated soil (Hillel
1998). Finally, the non-flat ground surface height is described by the topography function
Z(r) while the parameter DH [m2/yr (kg/m2)−1] represents the phenomenological bottom
friction coefficient between the surface water and the ground surface.
1Reduced evaporation by shading is another positive feedback mechanism between
biomass and water that is captured by the model. Throughout this study, however, we
keep this feedback constant by fixing the value of R.
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While the equations for B and W are purely phenomenological (resulting from
modelling processes at the single patch scale), the equation for H was motivated by shallow
water theory. The shallow water approximation is based on the assumption of a thin
layer of water where pressure variations are very small and the motion becomes almost
two-dimensional (Weiyan 1992). The derivation of the equation for H is described in online
Appendix A.
The two positive feedback processes are modelled in the equations through the explicit
forms of the infiltration rate term I and the growth rate term GB. The infiltration feedback
is modelled by assuming a monotonously increasing dependence of I on biomass; the bigger
the biomass the higher the infiltration rate and the more soil-water available to the plants.
This form of the infiltration feedback mirrors the fact that in many arid regions infiltration
is low far from vegetation patches due to the presence of the biogenic crust. Conversely,
the presence of shrubs destroys the cyanobacterial crust and favor water infiltration. The
uptake feedback is modelled by assuming a monotonously increasing dependence of root
length on biomass; the bigger the biomass the longer the roots and the bigger amount of
soil-water the roots take up from the soil.
The explicit dependence of the infiltration rate of surface water into the soil on the
biomass density is chosen as (HilleRisLambers et al. 2001; Rietkerk et al. 2002):
I = AB(r, t) + Qf
B(r, t) + Q, (2)
where A [yr−1], Q [kg/m2] and f are constant parameters. Two distinct limits of this
term, illustrated in online Fig. 1, are noteworthy. When B → 0, this term represents the
infiltration rate in bare soil, I = Af . When B À Q it represents infiltration rate in fully
vegetated soil, I = A. The parameter Q represents a reference biomass beyond which the
plant approaches its full capacity to increase the infiltration rate. The difference between
the infiltration rates in bare and vegetated soil (hereafter the ”infiltration contrast”) is
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quantifed by the parameter f , defined to span the range 0 < f < 1. When f ¿ 1 the
infiltration rate in bare soil is much smaller than the rate in vegetated soil. Such values can
model bare soils covered by biological crusts (Campbell et al. 1989; West 1990). As f gets
closer to 1, the infiltration rate becomes independent of the biomass density B, representing
non-crusted soil where the infiltration is high everywhere. The parameter f measures the
strength of the positive feedback due to increased infiltration at vegetation patches. The
smaller f the stronger the feedback effect. As we shall see below, this feedback plays a
crucial role in determining the engineering behavior of shrubs.
The growth rate GB has the form:
GB(r, t) = Λ
∫
Ω
G(r, r′, t)W (r′, t)dr′ , (3)
G(r, r′, t) =1
2πS20
exp
[− |r− r′|2
2[S0(1 + EB(r, t))]2
],
where the integration is over the entire domain Ω and the kernel G (r, r′, t) is normalized
such that for B = 0 the integration over the entire domain equals unity. According to
this form, the biomass growth rate depends not only on the amount of soil water at the
plant location, but also on the amount of soil water in the neighborhood spanned by the
plant’s roots. The roots length [m] is given by S0(1 + EB(r, t)), where E [(kg/m2)−1] is the
roots’ extension per unit biomass, beyond some minimal root length S0. The parameter
Λ has units of [(kg/m2)−1 yr−1] and represents the plant’s growth rate per unit amount of
soil water. The parameter E measures the strength of the positive feedback due to water
uptake by the roots. The larger E the stronger the feedback effect.
The soil water consumption rate at a point r is similarly given by
GW (r, t) = Γ
∫
Ω
G(r′, r, t)B(r′, t)dr′ . (4)
Note that G(r′, r, t) 6= G(r, r′, t). The soil water consumption rate at a given point is due
to all plants whose roots extend to this point. The parameter Γ has units [(kg/m2)−1yr−1]
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and it measures the soil water consumption rate per unit biomass.
The parameter values used in this paper are summarized in online Table 1. They are
chosen to describe shrub species and are taken or deduced from (Hillel 1998; Sternberg
& Shoshany 2001; Rietkerk et al. 2002). The model solutions described here are robust
and do not depend on delicate tuning of any particular parameter. The precipitation
parameter represents mean annual rainfall in this paper and assumes constant values. This
approximation is justified for species, such as woody shrubs, whose growth time scales are
much larger than the typical rainfall time scale. The effects of temporal rainfall variability
will be explored in a future study.
It is convenient to study the model equations using non-dimensional variables and
parameters as defined in online Table 2. The model equations can now be written in the
following non-dimensional form:
bt = Gbb(1− b)− b + δb∇2b
wt = Ih− ν(1− ρb)w −Gww + δw∇2w
ht = p− Ih + δh∇2(h2) + 2δh∇h · ∇ζ + 2δhh∇2ζ . (5)
The infiltration term has the non dimensional form:
I = αb(r, t) + qf
b(r, t) + q. (6)
The growth rate term Gb is written as:
Gb(r, t) = ν
∫
Ω
g(r, r′, t)w(r′, t)dr′ , (7)
g(r, r′, t) =1
2πexp
[− |r− r′|2
2(1 + ηb(r, t))2
],
and the soil water consumption rate can be similarly written as:
Gw(r, t) = γ
∫
Ω
g(r′, r, t)b(r′, t)dr′ , (8)
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where r = (x, y) and r′ = (x′, y′).
The non-dimensional precipitation parameter p can be used to define an aridity
parameter, a, as
a = p−1 =MN
ΛP. (9)
This parameter captures the effects of four factors on aridity: precipitation, evaporation,
mortality or grazing, and biomass growth rate per unit amount of soil water. It extends
an earlier definition (Lefever & Lejeune 1997) in including the two pluvial parameters,
precipitation rate and evaporation rate.
4. Model solutions describing basic landscape states
The model has two homogeneous stationary solutions representing bare soil and
uniform coverage of the soil by vegetation. Their existence and linear stability ranges for
plane topography are shown in the bifurcation diagram displayed in Fig. 2. The bifurcation
parameter is the inverse of the aridity parameter, p. The linear stability analysis leading to
this diagram is described in online Appendix B.
The bare soil solution, denoted in Fig. 2 by B, is given by b = 0, w = p/ν and
h = p/αf . It is linearly stable for p < pc = 1 and it loses stability at p = 1 to uniform
perturbations 2. Note that the instability threshold at pc = 1 is independent of all other
non-dimensional parameters. We will further discuss the significance of this threshold at
the end of the section. The uniform vegetation solution, denoted by E , exists for p > 1
in the case of a supercritical bifurcation and for p > p1 (where p1 < 1) in the case of a
subcritical bifurcation. It is stable, however, only beyond another threshold, p = p2 > p1.
2The bifurcation is subcritical (supercritical) depending whether the quantity 2ην/[ν(1−ρ) + γ] is greater (lower) than unity.
– 14 –
As p is decreased below p2 the uniform vegetation solution loses stability to non-uniform
perturbations in a finite wavenumber (Turing like) instability (Cross & Hohenberg 1993).
These perturbations grow to form large amplitude patterns. The following sequence of basic
patterns has been found at decreasing precipitation values for plane topography (see panels
A-C in Fig. 2): gaps, stripes and spots. The sequence of basic landscape states, uniform
vegetation, gaps, stripes, spots and bare soil, as the precipitation parameter is decreased,
has been found in earlier vegetation models (Lefever et al. 2000; von Hardenberg et al.
2001; Okayasu & Aizawa 2001; Rietkerk et al. 2002; Gilad et al. 2004).
Any pair of consecutive landscape states along the rainfall (precipitation) gradient has
a range of bistability (coexistence of two stable states): bistability of bare soil with spots,
spots with stripes, stripes with gaps, and gaps with uniform vegetation (see online Fig. 10).
Bistability of different landscape states has at least three important implications: (i) it
implies hysteresis, which has been used to elucidate the irreversibility of desertification
phenomena (von Hardenberg et al. 2001; Scheffer et al. 2001), (ii) it implies vulnerability
to desertification and (iii) it increases landscape diversity as patterns involving spatial
mixtures of two distinct landscape states become feasible.
The basic landscape states persist on slopes with two major differences: stripes, which
form labyrinthine patterns on a plane, reorient perpendicular to the slope direction to form
parallel bands, and the patterns travel uphill (typical speeds for the parameters used in
this paper are of the order of centimeters per year). Online Fig. 3 shows the development
of bands travelling uphill from an unstable uniform vegetation state. Travelling bands on
a slope have been found in earlier models as well (Thiery et al. 1995; Lefever & Lejeune
1997; Dunkerley 1997; Klausmeier 1999; von Hardenberg et al. 2001; Okayasu & Aizawa
2001; Rietkerk et al. 2002; Gilad et al. 2004). Another difference is the coexistence of
multiple band solutions with different wavenumbers in wide precipitation ranges (Yizhaq
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et al. 2005). The landscape states on plane and slope topographies predicted by this and
earlier models are consistent with field observations (Valentin et al. 1999; Rietkerk et al.
2004; Lejeune et al. 2004).
The sequence of landscape states described above represents the states that appear
along the rainfall gradient as p is proportional to P . We could alternatively use Eq. (9) to
express the same sequence of landscape states, but in reverse order, as a function of the
grazing (mortality) rate M = (PΛ/N)p−1, as shown in Fig. 2E.
The bare state instability threshold, p = PΛ/MN = 1, implies that only four
parameters affect the bare-state stability: the precipitation rate P , the mortality or grazing
rate, M , the evaporation rate, N , and the biomass growth rate per unit amount of soil
water, Λ. The effects of the four parameters on the bare-state stability threshold are
shown in online Fig. 4. Noteworthy is the hyperbolic relationship between P and Λ (online
Fig. 4c). The bare-state stability threshold designates the point at which the system can
spontaneously recover from desertification. A region that has gone through desertification
due to a precipitation drop can spontaneously recover by seedling with species that have
Λ values sufficiently high to cross the stability threshold. This practice, however, becomes
increasingly hard to implement as the system becomes dryer for the required up shift in Λ
is getting larger as the rectangles in online Fig. 4c illustrate.
5. Habitat creation by ecosystem engineers
The model equations (5) are now used to study micro-habitat creation by plant
ecosystem engineers. We define a ”micro-habitat” of a given species as a small area in the
physical domain, at the scale of a single patch of the ecosystem engineer, that is mapped into
the fundamental niche of that species in niche space (Hutchinson 1957). The micro-habitats
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created by an ecosystem engineer is characterized by soil-water density higher than the
density that would exist in the absence of the ecosystem engineer. We reserve the term
”habitat” for a local area at the landscape (many patches) scale characterized by a given
pattern of the ecosystem engineer and its associated micro-habitats. The definition of
habitats generally include additional resources besides soil water - light, nutrients, soil
pH, etc., which are not explicitly modelled in equations (5). In drylands, however, the
water resource can often be considered the limiting factor, justifying the use of the model
equations for studying habitat creation.
The micro-habitats to be considered here are results of trapping soil water by plant
ecosystem engineers. In the following we use the model equations to study conditions for
engineering effects and for ecosystem resilience, focusing on the parameters that control the
two positive feedbacks, η and f .
5.1. Engineering effects
The engineering capacity of a plant species is measured by its ability to capture surface
water, either by increasing the water infiltration rate under its canopy, or by intercepting
runoff water by the soil mounds it forms. Both ways can be simulated using the model but
for simplicity we concentrate here on the first.
Figure 5 shows spatial profiles of b, w and h for a single patch of the ecosystem engineer
at decreasing values of η, representing species with different root extension properties,
and for two extreme values of f . The value f = 0.1 models high infiltration rates under
engineer’s patches and low infiltration rates in bare soil, which may result from a biological
crust covering the bare soil (or from a litter layer formed by the engineer). The value
f = 0.9 models high infiltration rates everywhere. This case may describe, for example,
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un-crusted sandy soil. Engineering effects resulting in soil water concentration appear only
in the case of (i) low infiltration in bare soil, (ii) engineer species with rather limited root
extension capabilities, η = 3.5, η = 2 (frames B and C in Fig. 5). The soil water density
under an engineer’s patch in this case exceeds the soil-water density level of bare soil (shown
by the dotted lines), thus creating opportunities for species that require this extra amount
of soil water to colonize the water-enriched patch.
We conclude that ecosystem engineering, resulting in water enriched patches or
micro-habitats, is obtained when the infiltration feedback is strong (small f) and the uptake
feedback is weak (small η).
5.2. Resilience
While a weak uptake feedback enhances the engineering ability, it reduces the resilience
of the ecosystem engineer (and all dependent species) to disturbances. Fig. 5F shows
the response of an engineer species with the highest engineering ability to concentrate
water (η = 2, Fig. 5C) to a disturbance that strongly reduces the infiltration contrast
(f = 0.9). In the following we refer to crust removal, but other disturbances that reduce
the infiltration contrast, such as erosion of bare soil, will have similar effects. The engineer,
and consequently the micro-habitat it forms, disappear altogether for two reasons: (i)
surface water infiltrate equally well everywhere and the plant patch is no longer effective in
trapping water, (ii) the engineer’s roots are too short to collect water from the surrounding
area.
Resilient ecosystem engineers are obtained with uptake feedbacks of intermediate
strength (η = 3.5) as Fig. 5E shows. Removal of the crust (by increasing f) destroys
the micro-habitats (soil-water density is smaller than the bare-soil’s value) but the
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engineer persists. Once the crust recovers the ecosystem engineer resumes its capability to
concentrate water and the micro-habitats recover as well. It is also of interest to comment
that when the uptake feedback is too strong, the engineer persists but it no longer functions
as an ecosystem engineer (Fig. 5A,D).
Fig. 6 provides another view of the resilience of the engineering effect. Shown in this
figure are two graphs of the maximum soil water density under an engineer patch as a
function of the coverage of the surrounding crust, which is parametrized by f . One graph
(solid line) describes a resilient engineering effect (η = 3.5) while the other graph (dashed
line) describes a non-resilient engineering effect (η = 2). Also shown is the soil water level
in bare soil (dotted line). The resilient engineer survives highly damaged crusts but loses
the engineering ability to concentrate water at some f = fe (the crossing point of the solid
line and the dotted line). The non-resilient engineer has better engineering abilities in the
case of weakly damaged crusts (dashed line above solid line) but does not survive crust
damages beyond f = fc.
Resilience to climatic fluctuations, such as droughts, is achieved by a wide coexistence
range of stable spots and stable bare-soil (see online Fig. 10A and the discussion in
section 7.1). The larger η (or uptake feedback) the wider this coexistence range becomes.
We can conclude that resilient ecosystem engineers call for a compromise in the
strength of the uptake feedback; a weak feedback (small η) favors engineering, as the
engineer consumes less water, while a strong feedback (large η) favors resilience both to
disturbances such as crust removal and to climatic fluctuations such as drought.
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6. Habitat diversity
At the scale of a single patch of the ecosystem engineer we found two types of soil-water
distributions, representing different micro-habitats. Online Fig. 7 shows possible infiltration
conditions giving rise to the two distributions. For moderately low infiltration rates in bare
soil (f = 0.1), the maximum of the soil water density coincides with the maximum of the
engineer’s biomass (online Fig. 7A) whereas for very low infiltration rates (f = 0.01) the
soil water density attains maximal values at the edge of the patch (online Fig. 7B). The
two types of micro-habitats provide different niches with respect to water availability and
shading. Species which are sensitive to the absence of sunlight are expected to prefer the
micro-habitat of online Fig. 7B whereas species which are not sensitive are expected to be
found in both micro-habitats.
Habitat diversity on a landscape scale is determined by the diversity of spatial patterns
the ecosystem engineers form. A high diversity of pattern solutions can be obtained in
parameter regimes giving rise to irregular solutions. One possible mechanism leading to
irregular solutions is spatial chaos. So far, however, we have not identified chaotic solutions
and therefore we will not address this possibility here. Irregular patterns can also result
from coexistence of stable states (e.g. bistability), where spatial mixtures of the coexisting
states form stationary or long lived patterns. We demonstrate two aspects of these patterns;
the first pertains to the dominant role that arbitrary initial conditions have in shaping the
asymptotic patterns, and the second to the irregular soil-water distributions that can result
from these patterns. Fig. 8 shows the time evolution of two identical initial conditions
(leftmost frames), one in a parameter range where spots are the only stable state (upper
frames) and the other in a coexistence range of spots and bare soil (lower frames). In the
former case the system evolves towards a spot pattern and the initial pattern has little
effect. In the latter case the initial pattern has a strong imprint on the asymptotic pattern;
– 20 –
the system becomes sensitive to random factors and the asymptotic patterns will generally
show high landscape diversity. Fig. 9A,B show biomass and soil-water distributions in a
coexistence range of stripes and gaps. As the one-dimensional cut in Fig. 9C shows the
soil-water distribution is rather irregular, creating a diversity of micro-habitats as compared
with uniform or regular periodic patterns.
7. Habitat response to environmental changes
Climatic or human induced environmental changes, such as alternation in rainfall
regime or biomass harvesting, can affect the patterns formed by the ecosystem engineer and
consequently the habitats they provide for other organisms. The most significant pattern
changes are those involving transitions between different biomass pattern states. We focus
here on two cases of pattern transitions induced by gradual precipitation decreases or
disturbances to biomass patches. The first case is a ”catastrophic shift” (Scheffer et al.
2001; Scheffer & Carpenter 2003; Rietkerk et al. 2004) that involves an abrupt drop in
the soil water concentration under the biomass patch. The second case, which we term a
”resource-gain shift”, involves soil-water gain as a result of a pattern transition. The two
cases are discussed and illustrated below.
7.1. Catastrophic shifts
Consider a coexistence range of stable spots and stable bare-soil (online Fig. 10A –
coexistence of the stable N and B branches over the precipitation range p0 < p < pc). The
soil-water densities associated with these two states are shown in online Fig. 10B. For a soil
initially covered with spots of biomass, a precipitation decrease below p0 will result in the
decay of the spotted vegetation and the convergence to the bare soil state (see the arrows
– 21 –
in online Figs. 10A,B). Along with this transition, the enriched soil moisture patches under
the spots disappear as the soil-water density drops down to the bare soil level.
A similar transition, from spot to bare-soil, can result from crust disturbances which
increase the water infiltration rate I away from biomass patches. As implied by Fig. 6, an
increase in the value of f beyond fc, for an engineer species characterized by short roots
(η = 2), results in the decay of the ecosystem engineer and the drop of the soil water density
to the bare soil level (see the arrow in Fig. 6).
7.2. Resource-gain shifts
We illustrate this case by considering a transition from a banded biomass and water
pattern induced by the ecosystem engineer to a spotted pattern on a uniform slope as
precipitation decreases. Snapshots of the pattern transition and the associated soil-water
distributions are shown in Fig. 11. Surprisingly, the transition involves the appearance of
spot patches with higher soil-water densities, despite the lower precipitation value. This
counter-intuitive result can be explained as follows. The spot pattern self-organizes to form
an hexagonal pattern. As a result each spot ”sees” a bare area uphill which is twice as
large as the bare area between successive bands, and therefore absorbs more runoff. The
higher biomass density of spots (due to water uptake from all directions) also increases the
infiltration rate at the patch.
A transition from bands to spots involving soil-water gain can also be induced by
a local disturbance (e.g. clear-cutting) at a given precipitation value corresponding to a
coexistence range of stable bands and stable spots. The transition occurs by expansion of
the spot pattern downhill into the area occupied by the banded pattern as illustrated in
online Fig. 12.
– 22 –
The resource gain shifts described above reflect emergent mechanisms of habitat
dynamics resulting from the two collective modes associated with ecosystem engineering
and spatial symmetry breaking. Overlooking the roles of these modes, when inferring the
impact of environmental changes on habitat dynamics, may result in misleading conclusions.
8. Discussion
We presented here a mathematical model for the dynamics of dryland vegetation that
allows up-scaling processes occuring at the single plant-patch level to system behaviors at the
landscape level, and thereby exploring emergent mechanisms of habitat creation, diversity
and change. The model captures the biomass-water feedback due to water uptake by
plants’ roots (unlike earlier models of this kind), in addition to the two other biomass-water
feedbacks, increased infiltration at vegetation patches and reduced evaporation by shading.
These feedbacks are tightly related to collective inter and intra-specific species dynamics.
In the following we discuss these relations and the implications they bear on ecosystem
processes, stability and diversity, as predicted and elucidated by the model. We conclude by
discussing the limitations of the model and by delineating further directions of development.
8.1. Biomass and resource patterns
The model predicts the possible development of vegetation patterns at the landscape
scale due to intra-specific competition over the limited water resource. The pattern
formation mechanism is the positive feedback between biomass and water that acts at
the single patch scale. Vegetation patterns, which appear at the landscape scale, reflect
collective individual dynamics involving long spatial correlations of biomass and soil-water.
Vegetation patterns have been produced by earlier models that captured the biomass-water
– 23 –
feedback. Most of these models, and the model introduced here, predict the same sequence
of patterns along aridity gradients; uniform vegetation, gaps, stripes, spots and bare soil, as
Fig. 2 shows. Moreover, most models predict coexistence ranges of different stable states;
uniform vegetation with gaps, gaps with stripes, stripes with spots and spots with bare
soil. This apparent agreement among most models is due to the universal nature of the
instabilities that are responsible for the pattern formation phenomena.
The difference between the present model and earlier ones lies in the details of the
biomass and the water-resource spatial distributions. By capturing the uptake feedback
in the model equations and controlling its strength relative to the infiltration feedback, a
whole range of interspecific plant interactions, from negative (or competitive) to positive
(or facilitative), can be studied. Competitive interactions occur when the uptake feedback
dominates and the high water uptake depletes the amount of soil water under the vegetation
patch and its surroundings (Fig. 5D). Facilitation or ecosystem engineering occurs when
the infiltration feedback dominates, leading to soil-water accumulation under vegetation
patches (Fig. 5C). Ecosystem engineering represents an interspecific mode of collective
dynamics whereby one species, the ecosystem engineer, strongly affects the dynamics of
other species by creating water-rich micro-habitats.
Both modes of collective species dynamics, the intraspecific pattern mode and the
interspecific engineering mode, are realized in vegetation patterns of ecosystem engineers.
The coupling between the two modes offers a mechanism for species-richness change that
has not been explored yet. Fig. 11 illustrates an example of this mechanism by showing
soil-water enrichment (resource-gain shift) resulting from banded patterns transforming
into spotted patterns on a slope in response to a precipitation down shift.
There is another aspect to the landscape modulation that ecosystem engineers induce
besides resource modulation and habitat creation. The basic patterns, spots, stripes and
– 24 –
gaps, represent different forms of landscape fragmentation, which bear on population,
community and ecosystem level processes (Fahrig 2003). Spot patterns (Fig. 2A) create
open landscapes through which organisms and materials can move or flow with limited
hindrance. Gap patterns (Fig. 2C), on the other hand, form closed landscapes through
which movements of organisms and materials are highly restricted. The effect of landscape
fragmentation on the dynamics of populations, communities and ecosystems is of great
concern to ecologists (Debinski & Holt 2000). Most studies, however, refer to fragmentation
induced by human factors (Harrison & Bruna 1999). Our model studies suggest that
natural processes such as biomass pattern formation are also important agents in landscape
fragmentation.
8.2. Engineering strength and resilience
Ecosystem engineering in the present context refers to the ability of plant species
to trap and concentrate the water resource under biomass patches. It can be quantified
by the difference between the maximal value of the soil water density under an engineer
patch and the soil water density at bare soil, far from the patch (see online Fig. 10B) 3.
Strong engineering is obtained when the infiltration feedback dominates. In this case the
infiltration contrast between bare soil and vegetated soil is high (f ¿ 1) and the capacity
of the plant to extend its roots as it grows is weak (η ∼ O(1)) (see Fig. 5C).
In our model, resilience refers to the ranges of environmental conditions where the
ecosystem engineer survives and reproduces, that is, to the size of its fundamental niche
(Hutchinson 1957). Fixing all environmental conditions (precipitation, slope, etc.) except
3When this difference is negative, i.e. in water deprived patches, the engineering can be
regarded as negative (Jones et al. 1997)
– 25 –
the infiltration contrast, the resilience of the system can be quantified by the range of f
values within the fundamental niche.
A significant prediction of the model is a tradeoff between engineering strength and
resilience, as Fig. 6 shows. An engineer with high strength (η = 2) has low resilience to
disturbances that reduce the infiltration contrast (increase f), e.g. crust removal. As f
increases past a critical value fc the engineer patch disappears altogether, leaving behind
a bare soil with no water enriched micro-habitats and no ability for the system to recover
(arrow pointing downwards). In contrast, a plant with lower engineering strength (η = 3.5)
survives severe reduction of the infiltration contrast (see Fig. 5E which corresponds to
f = 0.9). The engineering effect no longer exists for f > fe, but once the disturbances
disappear and the infiltration contrast builds up again, positive engineering resumes.
This tradeoff is significant for understanding the stability and functioning of water
limited ecosystems on micro (patch) and macro (watershed) spatial scales (Shachak et al.
1998). The soil-moisture accumulation at an engineer patch accelerates litter decomposition
and nutrient production, and culminates in the formation of ”fertility islands” (Charley
& West 1975). Watershed-scale disturbances that are incompatible with the resilience
of dominant engineer species can destroy the engineer patches and the fertility islands
associated with them, thereby damaging the stability and functioning of the ecosystem.
8.3. Self-organized heterogeneity and species diversity
Spatial heterogeneity, along with species differentiation in niche space, are considered to
be major factors in inducing species coexistence and diversity (Tilman 1990, 1994; Tilman
& Kareiva 1997; Chesson et al. 2004; Silvertown 2004). Different spatial locations are then
mapped into different fundamental niches (Hutchinson 1957) and provide micro-habitats
– 26 –
for different species. The two collective modes, whose overall effect is the formation of
self-organized patterns of water enriched biomass patches, induce spatial heterogeneity even
under conditions of homogeneous environments. This form of heterogeneity has not been
studied so far in the context of species diversity.
The heterogeneity induced by the two collective modes create micro-habitats
characterized by a few niche variables including soil moisture and grazing stress4. The
model, and further extensions thereof as discussed below, provide means for studying the
niche space probed by these micro-habitats and thus species diversity.
Periodic patterns, such as hexagonal spot patterns, probe small volumes in niche space,
as the same micro-habitats repeat themselves, and therefore are associated with low species
diversity. Strictly periodic patterns are never expected to be observed in nature because
of the long time scales needed for approaching these asymptotic states. According to the
model, however, under environmental conditions where spot patterns constitute the only
stable state of the system, any initial biomass distribution will evolve on a relatively short
time scale into a space filling spot pattern as Fig. 8 (top) demonstrates. These patterns,
even though not periodic, are still expected to give rise to rather low species diversity.
Higher species diversity can be realized in more arid conditions where coexistence of
stable bare-soil and spots states occurs. Under these conditions engineer patterns can be
highly irregular, and therefore can probe larger volumes in niche space. Fig. 8 (bottom)
shows the time evolution of the same initial state as in the top part. The initial state
remains ”frozen” apart of patch size adjustments because of the stability of the bare-soil
4Assuming engineer’s biomass patches can reduce grazing of understorey herbaceous
species, soil-water distributions as in online Figs. 7A and 7B imply micro-habitats char-
acterized by different grazing stresses.
– 27 –
state and the depletion of soil water in the patch vicinity. This implies that highly irregular
initial biomass distributions, dictated by random factors, will remain highly irregular also
asymptotically.
8.4. Further model development
We restricted our analysis to the rather artificial circumstances of homogeneous systems
with respect to soil properties, topography, rainfall, evaporation, mortality, etc. We did
so to highlight the roles of self-organizing patterns of biomass and soil water in breaking
the spatial symmetry of the system and in creating habitats. Heterogeneous factors can
easily be incorporated into the model by introducing spatially dependent parameters. The
coupling of this heterogeneity to coexistence of stable states is expected to intensify the
diversity of possible habitats discussed in Sections 6 and 8.3.
The model in its present form takes into account the major feedbacks between
vegetation and water but leaves out a few other feedbacks. The atmosphere affects
vegetation through the precipitation and evaporation rate parameters, but the vegetation is
assumed to have no feedback on the atmosphere. Organic nutrients effects are parametrized
by the biomass growth rate, but litter decomposition (Moro et al. 1997) that feedbacks on
nutrient concentrations is not considered. Finally, ground topography, parametrized by
the function ζ(r), affects runoff and water concentration, but topography changes due to
soil erosion by water flow are neglected. In drylands, where the vegetation is sparse and
the limiting resource is water, the vegetation feedbacks on the atmosphere and on organic
nutrients are often of secondary importance. Soil erosion processes, however, can play
important roles, e.g. in desertification, and restrict the circumstances the model applies to.
Another limitation of the model is the elimination of the soil depth dimension which
– 28 –
restricts the variety of roots effects the model can capture. Thus, engineering by hydraulic
lifts (Caldwell et al. 1998) is not captured by the model. Finally, we consider in this model
local seed dispersal, ruling out dispersion by wind or animals.
While extensions of the model to include the factors discussed above are interesting
and significant, they may require considerable modifications of the model (especially the
modeling of feedbacks with the atmosphere, and soil-erosion). A more direct extension
of the model, that is already underway, is the inclusion of additional biomass variables
representing herbaceous species which benefit from the habitats created by the ecosystem
engineer. Each additional species is modeled by a biomass equation similar to that of
the engineer but differing considerably in the values of various species parameters such as
growth rate, mortality, maximum standing biomass, etc. In this context, the introduction
of environmental heterogeneities and temporal fluctuations by means of space and time
dependent model parameters is particularly significant in studying tradeoffs between species
and species richness. Including additional biomass variables will also allow studying systems
with competing ecosystem engineers, and the effects of this competition on habitat creation
along environmental gradients.
Acknowledgements
This research was supported by the James S. McDonnell Foundation (grant No.
220020056) and by the Israel Science Foundation (grant No. 780/01).
– 29 –
Online Appendices
A. Derivation of the surface water equation
In order to derive a dynamical equation for the surface water depth above the ground,
d(x, y, t), we will use the continuity and momentum equations in the shallow water
approximation (Weiyan 1992). Neglecting vertical accelerations and internal friction and
in the case of a shallow fluid layer (as it is the case for surface runoff), the shallow-water
momentum equations are written on the horizontal plane (x, y) and as a function of time t
as
Du
Dt= −g∇ (ζ + d) +
1
ρF , (A1)
where u = (u, v) is the two-dimensional, depth-averaged fluid velocity [m/s], function of
x, y and t, d(x, y, t) is the depth of the surface water layer, ζ(x, y) is the height of the soil
surface, g is the acceleration of gravity [m s−2], ρ is the (constant) fluid density [kg/m3] and
F represents bottom friction [kg/m2 s−2].
The shallow-water continuity equation is written as
ρ
[∂d
∂t+
∂
∂x(ud) +
∂
∂y(vd)
]= P − Iρd , (A2)
where P is the average precipitation measured in units of [kg/m2 s−1] and I is infiltration
rate measured in units of [s−1]. Since ρ is constant, we scale it out using h, and define the
new quantity H = ρd, which has units of [kg/m2] (Note that for water [kg/m2]=[mm]). We
can thus write
∂H
∂t+
∂
∂x(Hu) +
∂
∂y(Hv) = P − IH . (A3)
We can simplify the above equations by noticing that on the time scales of interest here
the fluid velocity adjusts rapidly to changes in inputs and outputs, and fluid accelerations
– 30 –
(Du/Dt = 0) can be neglected. If we further assume a linear (Rayleigh) bottom friction
term of the form F = −ku, where k is a constant coefficient, from the momentum equation
we obtain
u = −κ
(∂Z
∂x+
∂H
∂x
)
v = −κ
(∂Z
∂y+
∂H
∂y
)(A4)
where we have defined κ = g/k and we have introduced the rescaled soil-height variable
Z = ρζ, in analogy to what has been done for the water depth.
The final equation for H is obtained by substituting Eqs. (A4) into Eq. (A3) and
collecting terms:
∂H
∂t= P − IH +
κ
2∇2H2 + κ∇H · ∇Z + κH∇2Z (A5)
Different forms for the bottom friction term F lead to different forms of the dynamical
equation for H. For example, bottom friction of the form F = −ku/dp, where p is an
integer, would lead to the following equation for H:
∂H
∂t= P − IH +
κ
(p + 2)∇2Hp+2 + κ∇ (
Hp+1) · ∇Z + κHp+1∇2Z . (A6)
Although quantitatively different, these alternate forms for the H equation do not bring in
any qualitatively new behavior.
B. Linear stability analysis
Linear stability analysis of the model homogeneous stationary solutions for plane
topography is performed by adding an infinitesimal spatial perturbation and linearizing
around the solution in order to calculate the linear growth rate. We denote the homogeneous
– 31 –
stationary solution by U0 = (b0, w0, h0)T and define the perturbed solution as
U(r, t) = U0 + δU(r, t) , (B1)
where U = (b, w, h)T and δU = (δb, δw, δh)T . The perturbation dependence on time and
space is:
δU(r, t) = a(t)eik·r + c.c , (B2)
where a(t) = [ab(t), aw(t), ah(t)]T . Substitution of the perturbed solution (B1) into the
model equations Eq. (5) gives (time and space dependence of the perturbation is omitted
for convenience):
(δb)t = Gb|U0+δU (b0 + δb) [1− (b0 + δb)]− (b0 + δb) + δb∇2(b0 + δb)
(δw)t = I|U0+δU (h0 + δh)− ν [1− ρ(b0 + δb)] (w0 + δw)− Gw|U0+δU (w0 + δw) + δw∇2(δw)
(δh)t = p− I|U0+δU (h0 + δh) + δh∇2[(h0 + δh)2
], (B3)
where the infiltration term Eq. (6) reads
I|U0+δU =b0 + qf + δb
b0 + q + δb=
b0 + qf
b0 + q+
q (1− f)
(b0 + q)2 δb +O(δb2) . (B4)
In order to evaluate the terms Gb|U0+δU and Gw|U0+δU we expand the kernels, as defined
in Eq. (7), up to first order in δb:
g (r, r′) = g0 (r, r′) + g1 (r, r′) δb (r) +O (δb2
)
g (r′, r) = g0 (r′, r) + g1 (r′, r) δb (r′) +O (δb2
), (B5)
and
g0 (r, r′) = g0 (r′, r) = g (r, r′)|b=b0=
1
2πe−|r−r′|2/2σ2
g1 (r, r′) = g1 (r′, r) =∂
∂b[g (r, r′)]
∣∣∣∣b=b0
=η
2πσ3|r− r′|2 e−|r−r′|2/2σ2
, (B6)
where σ ≡ 1 + ηb0 and |r− r′|2 = (x− x′)2 + (y − y′)2.
– 32 –
After expanding the kernels and substituting the perturbed solution, we get up to first
order
Gb|U0+δU = ν
∫g (r, r′) w (r′) dr′
≈ ν
∫ [g0 (r, r′) + g1 (r, r′) δb (r)
][w0 + δw (r′)] dr′ (B7)
= νw0
∫g0 (r, r′) dr′ + ν
∫g0 (r, r′) δw (r′) dr′ + νw0δb (r)
∫g1 (r, r′) dr′ ,
where the integration is over the entire domain. Solving the integrals in Eq. (B7) results in
the following expression for Gb|U0+δU
Gb|U0+δU = νw0σ2 + νσ2e−k2σ2/2δw (r) + 2νησw0δb (r) . (B8)
Applying the same procedure to Gw|U0+δU we get
Gw|U0+δU = γb0σ2 + γσe−k2σ2/2
[1 + ηb0
(3− σ2k2
)]δb(r) . (B9)
Substituting Eqs. (B4), (B8) and (B9) into Eq. (B3) and linearizing around U0 can now be
performed in a straightforward manner. The equations for the unperturbed state U0 are
0 = νb0 (1− b0) (1 + ηb0)2 w0 − b0 (B10)
0 = αh0b0 + qf
b0 + q− ν (1− ρb0) w0 − γw0b0 (1 + ηb0)
2 (B11)
0 = p− αh0b0 + qf
b0 + q, (B12)
which are the equations for the homogeneous stationary solutions (b0, w0, h0)T .
At first order in the perturbation δU we get the governing equations for the linear
spatio-temporal dynamics of the perturbation:
∂
∂t[δU(r, t)] = J (k) δU(r, t) , (B13)
where J (k) ∈ R3×3. After eliminating the spatial part of the perturbation (i.e. eik·r) and
assuming exponential time dependence for the perturbation (i.e. aj(t) ∼ eλt) we get the
– 33 –
following eigenvalue problem:
λ a(t) = J (k) a(t) , (B14)
where the entries of the Jacobian matrix J (k) are given by
J11 = σνw0 [1− 2b0 + ηb0 (3− 4b0)]− 1− δbk2
J12 = νb0(1− b0)σ2e−k2σ2/2
J13 = 0
J21 = αh0q(1− f)
(b0 + q)2+ ρνw0 − γw0σe−k2σ2/2
[1 + ηb0
(3− σ2k2
)]
J22 = −ν(1− ρb0)− γb0σ2 − δwk2
J23 = αb0 + qf
b0 + q
J31 = −αh0q(1− f)
(b0 + q)2
J32 = 0
J33 = −αb0 + qf
b0 + q− 2δhh0k
2 . (B15)
By solving this eigenvalue problem we obtain the dispersion relation λ(k) as shown in online
Fig. 13.
– 34 –
REFERENCES
Anderson P. W. 1972. More is Different. Science 177:393–396.
Anderson P. W. 1995. Physics: The opening to complexity. Proceedings of the National
Academy of Sciences 92:6653–6654.
Boeken, B. & M. Shachak. 1994. Desert plant communities in human made patches –
implications for management. Ecological Applications 4:702–716.
Caldwell M.M., Dawson T.E. & J.H. Richards. 1998. Hydraulic Lift: consequences of water
efflux for the roots of plants. Oecologia 113:151161.
Campbell S. E., Seeler J. S., & S. Glolubic. 1989. Desert crust formation and soil
stabilization. Arid Soil Research and Rehabilitation 3:217–228.
Cross M. C. & P. C. Hohenberg. 1993. Pattern formation outside of equilibrium. Reviews
of Modern Physics 65:851–1112.
Charley, J. & N. West. 1975. Plant-induced soil chemical patterns in some shrub-dominated
semi-desert ecosystems in Utah. Journal of Ecology 63:945–963.
Chesson P., Gebauer R., Schwinning S., Huntly N., Wiegand K., Ernest M., Morgan E., Sher
A., Novoplansky A. & J. F. Weltzin. 2004. Resource pulses, species interactions, and
diversity maintenance in arid and semi-arid environments. Oecologia 141:236–253.
Debinski, D. M. and R. D. Holt. 2000. A survey and overview of habitat fragmentation
experiments. Conservation Biology 14:342–355.
Dunkerley D. L. 1997. Banded vegetation: development under uniform rainfall from a
simple cellular automaton model. Plant Ecology 129:103–111.
– 35 –
Fahrig, L. 2003. Effects of habitat fragmentation on biodiversity. Annual Review of Ecology
and Systematics 34:487–515.
Gilad E., von Hardenberg J., Provenzale A., Shachak M. & E. Meron. 2004. Ecosystem
Engineers: From Pattern Formation to Habitat Creation. Physical Review Letters
93:0981051.
Gurney W. S. C. & J. H. Lawton. 1996. The population dynamics of ecosystem engineers.
Oikos 76:273–283.
Harrison S. & E. Bruna. 1999. Habitat fragmentation and large-scale conservation: What
do we know for sure? Ecography 22:225–32.
Hillel D. 1998. Environmental Soil Physics. Academic Press, San Diego.
HilleRisLambers R., Rietkerk M., Van den Bosch F., Prins H.H.T., & H. de Kroon. 2001.
Vegetation pattern formation in semi-arid grazing systems. Ecology 82:50–61.
Hutchinson G.E. 1957. Cold Spring Harbor Symposia on Quantitative Biology. 22: 415–427.
Reprinted in 1991. Classics in Theoretical Biology. Bull. of Math. Biol. 53: 193–213.
Jones C. G., Lawton J. H. & M. Shachak. 1994. Organisms as Ecosystem engineers. Oikos
69:373–386.
Jones C. G., Lawton J. H. & M. Shachak. 1997. Positive and negative effects of organisms
as ecosystem engineers. Ecology 78:1946–1957.
Klausmeier C. A. 1999. Regular and irregular patterns in semiarid vegetation. Science
284:1826–1828.
Laland K. N., Odling-Smee F. J. & M. W. Feldman. 1999. Evolutionary consequences of
niche construction and their implications for ecology. Proceedings of the National
Academy of Sciences 96:10242–10247.
– 36 –
Lefever R. & O. Lejeune. 1997. On the Origin of Tiger Bush. Bulletin of Mathematical
Biology 59:263–294.
Lefever R., Lejeune O. & P. Couteron. 2000. Generic modelling of vegetation patterns.
A case study of Tiger Bush in sub-Saharian Sahel. in Mathematical Models for
Biological Pattern Formation, edited by P.K. Maini & H.G. Othmer, IMA Volumes
in Mathematics and its Applications 121:83–112. Springer, New York.
Lejeune O., Tlidi M. & R. Lefever. 2004. Vegetation spots and stripes: dissipative structures
in arid landscapes. International Journal of Quantum Chemistry 98:261–271.
Loreau M., Naeem S., Inchausti P., Bengtsson J., Grime J. P., Hector A., Hooper D. U.,
Huston M. A., Raffaelli D., Schmid B., Tilman D. & D. A. Wardle. 2001. Biodiversity
and ecosystem functioning: current knowledge and future challenges. Science 294:
804–808.
Meron E., Gilad E., von Hardenberg J., Shachak M. & Y. Zarmi. 2004. Vegetation patterns
along a rainfall gradient. Chaos, Solitons and Fractals 19:367–376.
Moro M.J., Pugnaire F.I., Haase P., and J. Puigdefabregas. 1997. Mechanisms of interaction
between Retama sphaerocarpa and its understorey layer in a semiarid environment.
Ecography 20: 175-184.
Murray, J. D. 1993. Mathematical Biology, Second Corrected Edition (Biomathematics Vol.
19), Springer-Verlag, Berlin Heidelberg.
Odum E. P. 1969. The strategy of ecosystem development. Science 164:266–270.
Odum E. P. 1971. Fundamentals of Ecology. 3rd edn. WB Saunders, Philadelphia.
Okayasu T. & Y. Aizawa. 2001. Systematic analysis of periodic vegetation patterns.
Progress of Theoretical Physics 106:705–720.
– 37 –
Odling-Smee F. J. 1988. Niche-Constructing Phenotypes. In: The role of behavior in
evolution. Ed. H.C. Plotkin. Cambridge, MIT Press.
Odling-Smee F.J., Laland K.N. & M. W. Feldman. 2003. Niche Construction: The Neglected
Process in Evolution. Monographs in Population Biology. 37. Princeton University
Press.
Perry, D,A. 1998. The scientific basis of forestry. Annual Review of Ecology and Systematics
29:435–466.
Power, M. E., Tilman, D., Estes, J. A., Menge, B. A., Bond, W. J., Mills, L. S., Daily,
G., Castilla, J. C., Lubchenco, J., & R. T. Paine. 1996. Challenges in the quest for
keystones. Bioscience 46: 609–620.
Rietkerk M., Boerlijst M. C., Van Langevelde F., HilleRisLambers R., Van de Koppel J.,
Kumar L., Prins H.H.T. & A. M. De Roos. 2002. Self-organization of vegetation in
arid ecosystems. The American Naturalist 160:524–530.
Rietkerk M., Dekker S. C., de Ruiter P. C. & J. Van de Koppel. 2004. Self-organized
patchiness and catastrophic shifts in ecosystems. Science 305:1926–1029.
Scheffer M., Carpenter S., Foley J. A., Folke C. & B. Walkerk. 2001. Catastrophic shifts in
ecosystems. Nature 413:591–596.
Scheffer M. & S. Carpenter. 2003. Catastrophic regime shifts in ecosystems: linking theory
to observation. TRENDS in Ecology and Evolution 18:648–656.
Shachak M., Gosz J., Pickett S.T.A. & A. Perevolotsky. 2005. Toward a unified framework
for biodiversity studies. In: Shachak, M., J. Gosz, S.T.A. Pickett and A. Perevolotsky
(eds). Biodiversity in Drylands: Towards a Unified Framework. pp. 3–14. Oxford
University Press.
– 38 –
Shachak M. & C. G. Jones. 1995. Ecological Flow Chains and Ecological Systems: Concepts
for Linking Species and Ecosystem Perspectives. pp 280–294. In: Linking Species
and Ecosystems. Eds. Jones C. G. & J. H. Lawton. Chapman & Hall NY.
Shachak M., Sachs M. & I. Moshe. 1998. Ecosystem management of desertified shrublands
in Israel. Ecosystems 1:475–483.
Shnerb N. M, Sarah P., Lavee H. & S. Solomon. 2003. Reactive glass and vegetation
patterns. Physical Review Letters 90:0381011.
Silvertown J. 2004. Plant coexistence and the niche. TRENDS in Ecology and Evolution
19: 605–611.
Sternberg M. & M. Shoshany. 2001. Influence of slope aspect on Mediterranean woody
formation: Comparison of a semiarid and an arid site in Israel. Ecological Research
16:335–345.
Thiery J. M., d’Herbes J. M. & C. Valentin. 1995. A model simulating the genesis of banded
vegetation patterns in Niger. Journal of Ecology 83:497–507.
Tilman D. 1990. Constraints and tradeoffs: toward a predictive theory of competition and
succession. Oikos 58:3–15.
Tilman D. 1994. Competition and biodiversity in spatially structured habitats. Ecology
75:2–16.
Tilman D. & P. Kareiva eds. 1997. Spatial Ecology. Monographs in Population Biology 30
(Princeton University Press, Princeton).
Valentin, C., d’Herbes, J.M. & J. Poesen. 1999. Soil and water components of banded
vegetation patterns. Catena 37:1–24.
– 39 –
von Hardenberg J., Meron E., Shachak M. & Y. Zarmi. 2001. Diversity of vegetation
patterns and desertification. Physical Review Letters 87:198101.
Weiyan T. 1992. Shallow Water Hydrodynamics. Elsevier Science, New York.
West N. E. 1990. Structure and function in microphytic soil crusts in wildland ecosystems
of arid and semi-arid regions. Advances in Ecological Research 20:179–223.
Yair, A. & M. Shachak. 1987. Studies in watershed ecology of an arid area. In: Progress in
Desert Research. Wurtele, M. O. and L. Berkofsky (eds.) pp. 146–193. Rowman and
Littlefield Publishers.
Yizhaq H., Gilad E., & E. Meron. 2005. Banded vegetation: Biological Productivity and
Resilience. Physica A 356(1):139–144.
This manuscript was prepared with the AAS LATEX macros v5.2.
– 40 –
Table 1. Dimensional parameters and their values/range
Parameter Value/Range Units
K 1 kg/m2
Q 0.05 kg/m2
M 1.2 yr−1
A 40 yr−1
N 4 yr−1
E 3.5 (kg/m2)−1
Λ 0.032 (kg/m2)−1yr−1
Γ 20 (kg/m2)−1yr−1
DB 6.25× 10−4 m2/yr
DW 6.25× 10−2 m2/yr
DH 0.05 m2/yr (kg/m2)−1
S0 0.125 m
Z mm
P [0, 1000] kg/m2 yr−1
R [0, 1] -
f [0, 1] -
Note. — (Online) Table 1 contains a list of
parameters of the model, their numerical values
– 41 –
and their units. The parameters values were set
using (Sternberg & Shoshany 2001; Hillel 1998;
Rietkerk et al. 2002).
– 42 –
Table 2. Non-dimensional parameters and variables definitions
non-dimensional definition
quantity
b B/K
w ΛW/N
h ΛH/N
q Q/K
ν N/M
α A/M
η EK
γ ΓK/M
p ΛP/MN
δb DB/MS20
δw DW /MS20
δh DHN/MΛS20
ζ ΛZ/N
ρ R
t MT
x X/S0
– 43 –
Note. — (Online) Table 2 con-
tains the definitions of the non-
dimensional quantities (parame-
ters and variables) used for study-
ing the model equations (Eqs. 5–8).
– 44 –
Af
A
0 Q/K 0.2 0.4 0.6 0.8 1
infil
trat
ion
rate
[yr-1
]
B/K
Fig. 1.— (Online) The infiltration rate I = A(B + Qf)/(B + Q) as a function of biomass
density B. When the biomass is diminishingly small (B ¿ Q) the infiltration rate approaches
the value of Af . When the biomass is large (B À Q) the infiltration rate approaches A.
The infiltration contrast between bare and vegetated soil is quantified by the parameter f ,
where 0 < f < 1; when f = 1 the contrast is zero and when f = 0 the contrast is maximal.
Small f values can model biological crusts which significantly reduce the infiltration rates in
bare soils. Disturbances involving crust removal can be modeled by relatively high f values.
Parameter values used: Q = 0.1, K = 1 and A = 1.
– 45 –
A) B) C)
0
0.2
0.4
0.6
biom
ass
[kg/
m2 ]
precipitation pcp1 p2
D)
E
B
0
0.2
0.4
0.6
0.8
1
biom
ass
[kg/
m2 ]
mortality rate [yr-1] Mc M1M2
E)
E
B
Fig. 2.— Bifurcation diagrams for homogeneous stationary solutions of the model equations
(Eqs. 5–8) showing the biomass B vs. precipitation p (frame D) and mortality M (frame
E) for plane topography. The solution branches B and E denote, respectively, the bare-
soil and uniform-vegetation solutions. Solid lines represent linearly stable solutions, dashed
lines represent solutions which are unstable to homogeneous perturbations, and dotted line
represents solutions which are stable to homogeneous perturbations but unstable to non-
homogeneous ones. Also shown are the basic vegetation patterns along the precipitation
gradient, spots (frame A), stripes (frame B) and gaps (frame C), obtained by numerical
integration of the model equations. Dark shades of gray represent high biomass. The same
sequence of patterns but in a reverse order appears along the mortality axis. The parameters
used are given in online Table 1 and correspond to a woody shrub.
– 46 –
Fig. 3.— (Online) Development of vegetation bands traveling uphill from an unstable uni-
form vegetation state, obtained by numerical integration of the model equations (Eqs. 5–8)
at P = 600 mm/yr. The different frames A–D shows snapshots of this process at different
times (t in years). Dark shades of gray represent high biomass. The bands are oriented
perpendicular to the slope gradient and travel uphill with typical speed of a few centimeters
per year. The parameters used are given in online Table 1, the domain size is 5× 5 m2 and
the slope angle is 15o.
– 47 –
0
0.5
1
1.5
2
0 50 100 150 200 250
M
P
stable
unstable
A)
0
2.5
5
7.5
10
12.5
0 0.05 0.1 0.15N
Λ
unstable
stable
B)
0
50
100
150
200
250
0 0.05 0.1 0.15
P
Λ
unstable
stable
C)
∆Λ
∆P
∆Λ
∆P
0
5
10
15
20
0 0.4 0.8 1.2 1.6 2
N
M
stable
unstable
D)
Fig. 4.— (Online) Neutral stability curves for the bare soil solution obtained from the linear
stability result p = pc = ΛPMN
= 1, by plotting M = ΛN
P (frame A), N = PM
Λ at P = 100
mm/yr (frame B), P = (MN)Λ−1 (frame C), and N = (ΛP )M−1 at P = 150 mm/yr (frame
D). All other parameters are given in online Table 1. The bare-soil solution loses tability
as any of these curves is traversed. The rectangles in frame C show two upshifts in growth
rates that are required to cross the recovery threshold, pc, of a desertified system. The dryer
the system, the larger the required upshift.
– 48 –
η = 12 η = 3.5 η = 2
f=
0.1
A) bwh
B) bwh
C) bwh
f=
0.9
D) bwh
E) bwh
F) bwh
Fig. 5.— Spatial profiles of the variables b, w and h as affected by the parameters that
control the main positive biomass-water feedbacks, f (infiltration feedback) and η (uptake
feedback). The profiles are cross sections of two-dimensional simulations of the model equa-
tions (Eqs. 5–8). In all frames, the horizontal dotted lines denote the soil-water level at
bare soil. Strong infiltration feedback and week uptake feedback (frame C) leads to high
soil-water concentration reflecting strong engineering. Strong uptake feedback and weak in-
filtration feedback (frame D) results in soil-water depeletion and no engineering. While the
species characterized by η = 2 is the best engineer under conditions of strong infiltration
contrast (frame C), it leads to low system resilience; the engineer along with the micro-
habitat it forms completely disappear when the infiltration contrast is strongly reduced, e.g.
by crust removal (frame F). A species with somewhat stronger uptake feedback (η = 3.5)
still acts as an ecosystem engineer (frame B) and also survives disturbances that reduce the
infiltration contrast (frame E), thereby retaining the system’s resilience. Parameter values
are given in online Table 1 with P = 75 mm/yr. Frames A) and D) span a horizontal range
of 14 m while all other frames span 3.5 m.
– 49 –
15
30
45
0 0.1 fc 0.2 0.3 fe 0.4
Max
. soi
l wat
er [k
g/m
2 ]
f
η=2η=3.5
Fig. 6.— Maximal soil-water densities under ecosystem-engineer patches for two different
engineer species, η = 2 (dotted curve) and η = 3.5 (solid curve), as functions of the infiltra-
tion contrast, quantified by f . At low values of f (e.g. in the presence of a soil crust) both
species concentrate the soil-water resource under their patches beyond the soil-water level of
bare soil (horizontal dashed line), thereby creating water enriched micro-habitats for other
species (see Fig. 5). The engineer species with η = 2 outperforms the η = 3.5 species, but
does not survive low infiltration contrasts (resulting e.g. from crust removal); as f exceeds
a threshold value, fc = 0.15, the engineer’s patch dries out and the micro-habitat it forms
is irreversibly destroyed. The η = 3.5 species, on the other hand, survives low infiltration
contrasts and while its engineering capacity disappears above a second threshold fe = 0.38,
the engineering capacity is regained as the infiltration contrast builds up (e.g. by crust
recovery). Parameter values are given in online Table 1 with P = 75 mm/yr.
– 50 –
Fig. 7.— (Online) Two typical spatial distributions of soil water at the scale of a single
patch, obtained with different infiltration rates in bare soil. The upper frames show two-
dimensional soil water distributions (dark shades of gray represent high soil-water densities),
while the lower frames show the corresponding one-dimensional cross-sections along the patch
center. In the range of intermediate to low infiltration rates (frame A, f = 0.1) the soil-water
density is maximal at the center of the patch, whereas at very low infiltration rates (frame B,
f = 0.01) the maximum is shifted to the edge of the patch. Similar soil water distributions
can be obtained at different precipitation values. The soil-water distributions were obtained
by numerical integration of the model equations (Eqs. 5–8) at P = 75 mm/yr, with domain
size of 5 m2. All other parameters are given in online Table 1.
– 51 –
Fig. 8.— Bistability as a mechanism for pattern diversity. Snapshots of the time evolution
of the same initial state (leftmost frames) when a spot pattern is the only stable state
(upper frames, P = 300 mm/yr), and when spots stably coexist with bare soil (lower frames,
P = 75 mm/yr). In the former case the asymptotic state is independent of the initial state;
any initial patch configuration will converge to a spot pattern as this is the only stable
state of the system. In the latter case the initial patch configuration remains unchanged as
time evolves (appart of patch size adjustments), implying high sensitivity to random factors
that affect the initial state, and thus high pattern diversity. The images were obtained by
numerical integration of the model equations (Eqs. 5–8). The domain size is 7.5× 7.5 m2, t
in years and the parameters are given in online Table 1.
– 52 –
Fig. 9.— Biomass (A) and soil-water distributions (B,C) of an asymptotic pattern in a
coexistence range of stripes and gaps. Frame C shows the soil-water profile along the transect
denoted by the dashed line in B. The soil-water distribution is pretty irregular as is evident by
the variable grey shades in frame B and by the profile in frame C. Such irregular distributions
create a diversity of micro-habitats as compared with the cases of uniform or regular periodic
patterns. The images were obtained by numerical integration of the model equations (Eqs. 5–
8) at P = 950 mm/yr. The domain size is 7.5 × 7.5 m2 and the parameters used are given
in online Table 1.
– 53 –
0
0.2
0.4
0.6
0.8
p0
biom
ass
[kg/
m2 ]
precipitation
A)
pcp1
B
E
N
0
20
40
60
p0
soil
wat
er [k
g/m
2 ]
precipitation
B)
pcp1
B
E
N
Fig. 10.— (Online) Solution branches for the biomass density of an ecosystem engineer and
the soil-water density it induces along the low end of the precipitation axis. The branches
B and E denote, respectively, bare-soil and uniform-vegetation solutions. The branch Ndescribes a spot-pattern solution (see frame A in Fig. 2). In frame A it denotes the amplitude
of the biomass density, while in frame B it denotes the soil-water density at the center of a
biomass spot. The figure shows the coexistence range, p0 < p < pc, of stable spot-pattern
and bare-soil solutions, and illustrates the occurrence of a catastrophic shift (see arrow)
as the precipitation drops below p0. The figure also shows the range of engineering where
the value of the soil-water density under a vegetation spot (given by N ) exceeds the value
at bare soil (given by B). The N branch was calculated by numerical integration of the
model equations (Eqs. 5–8). The parameters used are given in online Table1. With these
parameters p0 corresponds to 50 mm/yr and pc to 150 mm/yr.
– 54 –
Fig. 11.— Resource-gain shift induced by a transition from vegetation bands (frame A) to
spots (frame D) on a slope, in response to a precipitation downshift from P = 225 mm/yr
(A) to P = 200 mm/yr (frames B–D). The transition is shown by displaying snapshots of
two-dimensional biomass patterns (frames A–D, with darker gray shades corresponding to
higher biomass density) at successive times (t in years), obtained by solving numerically
Eqs. 5–8. The lower frames show soil water profiles along the transects denoted by the
dashed lines in the corresponding upper frames. The soil-water density under spots is higher
than the density under bands despite the precipitation downshift. Domain size is 5× 5 m2,
slope angle is 15o and all other parameters are given in online Table 1.
– 55 –
Fig. 12.— (Online) Resource-gain shift induced by a transition from vegetation bands (frame
A) to spots (frame D) on a slope, in response to a clear-cutting disturbance. The transition
is shown in frames A–D by displaying snapshots of biomass patterns (darker gray shades
correspond to higher biomass density) at successive times (t in years). The patterns were
obtained by solving numerically Eqs. (5)–(8) at a precipitation value where both band and
spot patterns are stable. The initial cut of the uppermost band allows for more runoff to
accumulate at the band section just below it (frame A). As a result this section grows faster,
draws more water from its surrounding and induces vegetation decay at the nearby band
sections. The whole process continues repeatedly until the whole pattern transforms into a
spot pattern. Domain size is 5 × 5 m2, P = 225 mm/yr, slope angle is 15o and all other
parameters are given in online Table 1.
– 56 –
-0.005
-0.0025
0
0.0025
6.5 7 7.5 8 8.5 9
Re[
λ(k)
]
wavenumber k
p<p2
p=p2
p>p2
Fig. 13.— (Online) Linear growth-rate curves of non-homogeneous spatial perturbations
applied to the uniform vegetation solution of the model equations Eqs. (5)–(8) (denoted by
E in Fig. 2), around the point p = p2. When p > p2 all wavenumbers have negative growth
rates and any perturbation decays. At p = p2 a critical wavenumber becomes marginal;
perturbations having this wavenumber neither grow nor decay. When p < p2 there exists
a band of wavenumbers with positive growth rates; the uniform solution is unstable and
perturbations characterized by the critical wavenumber grow faster than all others.