ARTICLE IN PRESS
Theoretical Population Biology 71 (2007) 367–379
www.elsevier.com/locate/tpb
Local facilitation, bistability and transitions in arid ecosystems
Sonia Kéfia,, Max Rietkerka, Minus van Baalenb, Michel Loreauc
a
Department of Environmental Sciences, Utrecht University, Copernicus Institute, P.O. Box 80115, 3508 TC, Utrecht, The Netherlands
Laboratoire d’Ecologie—UMR 7625, Université Pierre et Marie Curie, Bât. A, 7ème Etage, Case 237, 7 Quai St.-Bernard, 75252 Paris Cedex 05, France.
c
Department of Biology, McGill University, 1205 ave Docteur Penfield, Montreal, Qué., Canada
b
Received 1 June 2006
Available online 29 September 2006
Abstract
Arid ecosystems are liable to undergo sudden discontinuous transitions from a vegetated to a desert state as a result of human pressure
and climate change. A predictive framework about the conditions under which such transitions occur is lacking. Here, we derive and
analyze a general model describing the spatial dynamics of vegetation in arid ecosystems considering local facilitation as an essential
process. We investigate the conditions under which continuous or discontinuous transitions from a vegetated to a desert state are likely to
occur. We focus on arid ecosystems but our approach is sufficiently general to be applied to other ecosystems with severe environmental
conditions. The model exhibits bistability and vegetation patchiness. High local facilitation decreases the risk of discontinuous
transitions. Moreover, for arid ecosystems where local facilitation is a driving process, vegetation patchiness indicates proximity to a
transition point, but does not allow distinguishing between continuous and discontinuous transitions.
r 2006 Elsevier Inc. All rights reserved.
Keywords: Bistability; Cellular automaton; Continuous transition; Desertification; Discontinuous transition; Pair approximation; Spatial selforganization; Vegetation patchiness
1. Introduction
In the extreme conditions of arid areas, amelioration of
the micro-environment underneath the canopy of a ‘nurse
plant’ (Niering et al., 1963) significantly improves the
conditions for establishment and growth of other plants
(e.g., Mc Auliffe, 1988; Pugnaire et al., 1996; Aguiar and
Sala, 1999; Holzapfel and Mahall, 1999; Armas and
Pugnaire, 2005). Vegetation cover plays the role of physical
protection that decreases thermal amplitudes, radiation
exposures, wind dessication and soil erosion and prevents
soil crust formation (Noy-Meir, 1973; Callaway and
Walker, 1997; Greene et al., 2001). Moreover, water and/
or nutrient availability are higher under the canopy of
certain woody species and perennial grasses as compared to
areas deprived of vegetation, leading to the development of
‘islands of fertility’ (Jackson and Caldwell, 1993; Smith
Corresponding author. Fax: +31302532746.
E-mail addresses: kefi@geo.uu.nl (S. Kéfi), rietkerk@geo.uu.nl
(M. Rietkerk), minus.van.baalen@ens.fr (M. van Baalen),
michel.loreau@mcgill.ca (M. Loreau).
0040-5809/$ - see front matter r 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.tpb.2006.09.003
et al., 1994; Schlesinger et al., 1996). Here, we use the term
local facilitation to describe this assortment of physical and
biological mechanisms that impact the space below and
close to the canopy of a plant. In this term, local stresses
the spatial component of these mechanisms, and facilitation
refers to their positive effects on other plants (Stachowicz,
2001; Bruno et al., 2003). Water is the main limited
resource in arid ecosystems. Combined with the harsh
conditions for establishment and survival of individuals,
this prevents the full extension of the vegetation cover (e.g.,
Aguiar and Sala, 1999).
In line with these mechanisms, vegetation cover in arid
ecosystems is characterized by vegetation patchiness,
meaning that sparsely populated, or bare, patches coexist
with dense vegetation patches (Aguiar and Sala, 1999).
Vegetation patches of different sizes, shapes and spatial
distributions have been observed in arid ecosystems
throughout the world (e.g., Aguiar and Sala, 1999;
Rietkerk et al., 2000; Alados et al., 2004; Webster and
Maestre, 2004). These observations have raised questions
about the origin of this patchiness and the mechanisms that
maintain it.
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Besides its influence on the spatial organization of
vegetation, local facilitation constitutes a positive feedback
between vegetation abundance and local environmental
conditions that can theoretically lead to bistability
(Rietkerk and van de Koppel, 1997; Scheffer et al., 2001;
van de Koppel et al., 2001, 2002). Bistable systems can be
in one of two alternative stable states depending on the
initial conditions (Connel and Sousa, 1983). Bistable
systems potentially undergo sudden, discontinuous transitions, corresponding to switches from one stable state to
the other because of small, continuous changes in parameter values (Scheffer et al., 2001). For example, with
decreasing rainfall, such a system may shift from a patched
vegetation state to a bare state (von Hardenberg et al.,
2001; Rietkerk et al., 2002). An increase in rainfall to
values for which this shift occurs does not necessarily lead
to a recovery of the vegetation state. This hysteresis effect
is specific to discontinuous transitions and does not occur
along continuous transitions, where no bistability takes
place. Hence, it has been hypothesized that discontinuous
transitions are associated with vegetation patchiness, and
that certain shapes of patches could indicate imminent
discontinuous transition towards the bare state (Rietkerk
et al., 2004).
Local facilitation by sessile organisms is also ubiquitous
in physically harsh habitats other than arid ecosystems,
such as salt marshes, alpine areas and intertidal zones
(Bertness and Callaway, 1994; Bertness and Leonard, 1997;
Stachowicz, 2001; Callaway et al., 2002; van de Koppel et
al., 2005). The objective of this work is to gain insight into
the link between bistability, vegetation patchiness and local
facilitation in harsh environments. More specifically, we
investigate the conditions under which local facilitation
leads to continuous or discontinuous transitions under
environmental and human pressure changes, and how these
transitions are associated with vegetation patchiness. This
is important for understanding and predicting the effect of
global change on physically harsh habitats, including
desertification in the case of arid ecosystems. In this paper,
we focus on arid ecosystems, although our approach
remains sufficiently general to be applied to other
ecosystems with severe environmental conditions.
Therefore, we constructed a lattice-structured model for
arid ecosystems including local facilitation as an essential
process. Each cell directly interacts with its nearest
neighbors, which corresponds to a plant’s sphere of
influence. Besides the full spatial model, two analytic
approximations, a mean field (ignoring spatial interactions;
Durret and Levin, 1994) and a pair approximation (Sato
and Iwasa, 2000; van Baalen, 2000), leading to two systems
of ordinary differential equations, were derived and
analyzed for conditions leading to bistability. Comparing
the results of the mean field and of the pair approximation
models allows understanding the role of local interactions
on the bistability of the system. Comparing the results of
the pair approximation model and of the full spatial model
made it possible to assess whether continuous and
discontinuous transitions are associated with different
vegetation patchiness.
2. The model
Consider an arid ecosystem as a lattice-structured
habitat, consisting of an infinite number of sites, each
roughly the size of a square meter. Each site can be in one
of three possible states: vegetated (denoted +), unoccupied
(denoted 0) and degraded (denoted ); there is no other
sub-structure within a site. The difference between unoccupied and degraded sites is that the latter cannot be
colonized whereas in the former seeds can germinate. The
basic task is then to formulate a model that allows us to
keep track of the areas covered by vegetation and degraded
soil, on the basis of the fundamental processes that occur at
the site level.
The fundamental processes that occur in our model are
colonization (events that change {0}-sites into {+}-sites),
mortality (events that change {+}-sites into {0}-sites), soil
degradation (events that change {0}-sites into {}-sites)
and recovery (events that change {}-sites into {0}-sites).
At the site level, our model is fully stochastic. This means
that each of the possible events that applies to a site in a
given state does not occur with certainty but only with a
given probability per unit time (i.e., a rate). These rates can
be dependent not only on global vegetation cover but also
on local configurations (state of the z nearest neighbors).
Here we define neighborhood of a site as its four nearest
neighbors (von Neumann neighborhood, z ¼ 4).
Below, we give mathematical specifications for these
rates. These formulae allow us to derive differential
equations that describe the changes in expected cover (or
mean densities) if we disregard spatial structures. These
differential equations, or mean field model, can be easily
analyzed to study the conditions for bistability. Not
surprisingly, ignoring the spatial structure can lead to
incorrect conclusions about the outcome of the population
dynamics, as we show using stochastic spatial simulations
of a cellular automaton. To better understand the role of
spatial structure we formulate and analyze a so-called pair
approximation of the cellular automaton that tracks local
correlations as well as global averages.
2.1. Basic events and their associated rates
Global density of sites in a given state (which is
equivalent to a fraction of the total available area) is
denoted by ri where i is either +, 0 or . To model
colonization and facilitation we also need a way to
represent local densities: following Matsuda et al. (1992)
we represent these by the probabilities of finding a
neighbor of a given site i in state j, in mathematical form
qj|i (where i and j are either +, 0 or ). In what follows, all
symbols stand for parameters and variables that have nonnegative values (Appendix A).
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S. Kéfi et al. / Theoretical Population Biology 71 (2007) 367–379
2.1.1. Colonization
Plants reproduce by spreading seeds across the lattice,
but the recruitment of a new individual has a probability of
being successful only if the seeds have arrived at an
unoccupied site. Research on seed dispersal indicates that
the seed bank is mainly concentrated in the vegetation
patches (Mauchamp et al., 2001; Montaña et al., 2001; van
Rheede van Oudtshoorn and van Rooyen, 1999). A nonnegligible part of the seed rain, however, is dispersed
further away by animals and surface runoff (van Rheede
van Oudtshoorn and van Rooyen, 1999). Thus, we suppose
that a fraction of the seeds produced by a vegetation site is
locally dispersed, while the rest is globally dispersed. We
represent the rate of colonization of an unoccupied site by
w{0,+}, and we assume that
(1)
wf0;þg ¼ b drþ þ ð1 dÞqþj0 GðÞ,
where b is a positive constant, which represents the
intrinsic seed production rate per vegetation site multiplied
by the survival and the germination probabilities; d is the
fraction of seeds that disperses globally and G is a function
that describes how germination depends on competition for
resources. The spatial scale at which competition operates
in arid ecosystems remains an area of current research.
Existing studies have already shown that the spatial scale
of competition is larger than the one of facilitation (e.g.,
Lejeune et al., 1999; Rietkerk et al., 2002). For simplicity,
we made competition occur at a global scale by assuming
that G() is a linear function of global vegetation cover
Gðrþ Þ ¼ grþ ,
(2)
where e is the maximum germination rate and g represents
the strength of density dependence. We chose goe, so that
G() is always positive. For brevity, we write be as b and bg
as c. Then
(3)
wf0;þg ¼ drþ þ ð1 dÞqþj0 ðb crþ Þ.
2.1.2. Mortality
Just as germination turns an unoccupied site into a
vegetation site, mortality turns it back into an unoccupied
site again. We assume that mortality is density-independent
(e.g., Klausmeier, 1999; von Hardenberg et al., 2001;
Rietkerk et al., 2002; Meron et al., 2004)
wfþ;0g ¼ m,
(4)
where m is a vegetation characteristic that may be influenced
by anthropogenic factors such as grazing by cattle.
2.1.3. Degradation
An unoccupied site can not only be recolonized by
vegetation, it can also be eroded by rain or wind. As for
mortality, we assume that degradation of the soil of an
unoccupied site occurs at a density-independent rate
wf0;g ¼ d;
(5)
369
where d is a positive constant depending on soil type and
climatic characteristics. Plant seeds cannot germinate in
these sites.
2.1.4. Local facilitation
Because of the positive effect of vegetation on its microenvironment, regeneration of a degraded site is assumed to
be faster when there are more vegetation sites in the
neighborhood. We assume regeneration of a degraded site
occurs at rate w{,0}
wf;0g ¼ r þ fqþj ,
(6)
where r is a positive constant corresponding to the
spontaneous regeneration rate of a site, supposed to be a
function of rainfall (the more rainfall, the easier the
regeneration) and soil type. Facilitation is represented by
f which is the maximum effect of the neighborhood on
regeneration realized when all nearest neighbors sites are
{+}. Note that another facilitative term could have been
included in the degradation rate, where vegetation could
prevent the degradation of recolonizable sites in its
neighborhood. However, in order to isolate the effect of
local positive interactions, and to keep the model as
simple as possible, we decided to retain only one type of
facilitative effect in the model.
Whenever possible we set parameters to realistic values,
based on the literature (see Appendix A). For parameters
for which no values could be found in the literature, we
were careful to assign values with a realistic order of
magnitude (Table 1, Appendix A).
2.2. Vegetation dynamics
Incorporating the transition rates as defined above, we can
derive the exact equations for global vegetation dynamics
(changes in densities represent expected changes in cover),
drþ
¼ rþ wfþ;0g þ r0 wf0;þg ,
dt
dr0
¼ rþ wfþ;0g r0 wf0;þg þ wf0;g þ r wf;0g ,
dt
(7)
(8)
dr
¼ r0 wf0;g r wf;0g .
(9)
dt
These equations are simple bookkeeping equations based on
the fact that transition rates transfer sites from one type to
another. It can be shown that this set of equations represent
the expected dynamics of mean densities provided that the
number of sites is sufficiently large (Matsuda et al., 1992;
van Baalen and Rand, 1998; van Baalen, 2000). At this
stage, however, we cannot calculate these differential
equations, as they depend on (at least some) local densities
qj|i which depend on the spatial structures that develop.
2.2.1. Mean field analysis
The so-called mean field approximation assumes that
there is no spatial structure. Hence, local densities are
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identical to global densities, qi|j ¼ ri. Vegetation dynamics
is then given by
drþ
¼ rþ m þ r0 drþ þ ð1 dÞrþ ðb crþ Þ,
dt
(10)
dr0
¼ rþ m r0 drþ þ ð1 dÞrþ ðb crþ Þ þ d þ r r þ f rþ ,
dt
(11)
dr
(12)
¼ r0 d r r þ f r þ .
dt
Using the fact that global densities sum to unity, we can
represent one density in terms of the others. Therefore,
only two non-linear differential equations are necessary to
describe the system. For instance r0 ¼ 1r+r yields
drþ
¼ rþ ðb crþ Þð1 rþ r Þ mrþ ,
dt
(13)
dr
¼ dð1 rþ r Þ ðr þ f rþ Þr .
dt
(14)
2.2.2. Spatial simulations
Simulations of the full spatial model (or cellular
automata) were carried out using a stochastic asynchronous update algorithm (discrete time). The initial conditions were generated randomly for given initial values of
r+, r0 and r. We assume that the lattice is isotropic,
meaning that there is no preferential direction (no slope for
example). As previously described, interactions between
the cells of the cellular automaton were either with the four
nearest neighbors or with the entire lattice. See figure
legends for further details about the size of the lattices and
the boundary conditions.
2.2.3. Pair approximation
The pair approximation method makes it possible to
analyze lattice models by constructing a dynamical system
of ordinary differential equations (Matsuda et al., 1992;
Sato and Iwasa, 2000; van Baalen, 2000) that tracks not
only global densities but also local correlations. More
specifically, differential equations are derived for the
densities of pairs of neighbors: rij represents the probability
of finding in a given pair of the lattice one neighbor in state
i and the other in state j. We do not assume any asymmetry
in the interaction between sites, meaning that rij ¼ rji (Sato
and Iwasa, 2000). Since formally qj|i is a conditional
probability (given that one neighbor is in state i, what is the
probability to find the other in state j?) we can express it in
terms of densities of pairs and ‘singletons,’ as
rij
qjji ¼ .
(15)
ri
Making use again of conservation laws to reduce the
number of equations (see Appendix B), and assuming
that the dynamics of pairs does not depend on the densities of triplets or higher-order configurations (van
Baalen, 2000) we can obtain our pair approximation
model from
drþþ
¼ 2r0þ w0;þ 2rþþ wþ;0 ,
dt
(16)
drþ
¼ r0 w0;þ þ r0þ w0; rþ ðwþ;0 þ w;0 Þ,
dt
(17)
dr
¼ 2r0 w0; 2r w;0 ,
dt
(18)
drþ
¼ r0 w0;þ rþ wþ;0 ,
dt
(19)
dr
(20)
¼ r0 w0; r w;0
dt
after substituting the transition rates as defined above
(Appendix B). This is a set of five closed equations that can
be studied using standard methods for ordinary differential
equations. Note that here the transition rates are not
affected by z, the number of nearest neighbors; but z
intervenes in the equation of pairs (Appendix B; Sato and
Iwasa, 2000).
3. Results
The mean field model has one trivial equilibrium,
corresponding to a system deprived of vegetation that we
call desert (r+ ¼ 0, r0 ¼ r/(d+r), r ¼ d/(d+r)) which is
stable if bom(d+r)/r. Non-trivial equilibria are solutions
of a cubic equation (Appendix B), so that there are in total
three possible outcomes for the mean field model: (1) the
desert equilibrium (r+ ¼ 0) is the only globally stable
equilibrium; (2) a vegetation equilibrium (r+40) is the
only globally stable equilibrium; (3) the vegetation and the
desert equilibria coexist and their domains of attraction are
separated by an unstable equilibrium (Fig. 1). In this latter
case, the system is bistable. Analysis of the mean field
model suggests that the system is bistable for a certain
range of parameter values (Fig. 1). However, this result
stems for the assumption that there is no spatial structure.
In spatially explicit simulations, vegetation patches are
present over a certain range of parameter space (Fig. 2).
We define a vegetation patch, or cluster of vegetation, as a
group of vegetation sites connected to each other by
neighborhood distances, where the neighborhood of a site
is identical to that used in the model, i.e. the four nearest
neighbors (Pascual et al., 2002). Starting from favorable
environmental conditions (high b) and high facilitation (f),
low mortality (m) or low degradation (d), the vegetation
cover is homogeneous (meaning that there is no conspicuous spatial structure). When environmental conditions
deteriorate, clusters of bare cells appear and remain in the
vegetation cover, which then becomes more fragmented
and is eventually composed of a few isolated patches of
vegetation in a bare soil matrix before the system becomes
a desert. High facilitation (f), low mortality rate (m) and
low degradation rate (d) favor the maintenance of the
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371
Fig. 1. Effects of the parameters on the bifurcation diagrams of the mean field model. On the ordinate, (r+)eq is the global density of vegetation sites at
equilibrium. For clarity, only non-trivial equilibria are traced. A lower b value reflects more severe environmental conditions. In black: stable equilibria, in
gray: unstable equilibria. The system is bistable for a certain range of environmental conditions. (a): Effect of facilitation (f). m ¼ 0.1, d ¼ 0.1, c ¼ 0.3,
d ¼ 0.2, r ¼ 0.05. (b): Effect of soil type (r and d). m ¼ 0.01, c ¼ 0.3, d ¼ 0.1, f ¼ 0.3. 1: unsusceptible soil r ¼ 0.09, d ¼ 0.1, 2: r ¼ 0.03, d ¼ 0.3, 3:
susceptible soil r ¼ 0.01, d ¼ 0.9. (c): Effect of mortality (m). d ¼ 0.1, r ¼ 0.01, c ¼ 0.3, d ¼ 0.2, f ¼ 0.3. (d): Effect of competition intensity (c). m ¼ 0.1,
d ¼ 0.1, d ¼ 0.1, r ¼ 0.01, f ¼ 0.9. See Table 1 for the interpretation of the parameters that are not defined here.
Fig. 2. Spatial organization of the vegetation predicted by the full spatial model depending on plant (f and m) and soil (d and r) characteristics. The lattices
are characterized by a gradient in two parameters: facilitation (f), degradation (d) or regeneration (r) on the vertical axis, and b on the horizontal axis. A
lower b value reflects more severe environmental conditions. Each cell of the lattice is submitted to a different combination of the two parameters. The
simulations were run on a lattice of 40,000 (200 200) cells, with absorbing boundaries (for this type of analysis, large lattices were required so that the
variation step of the parameters from one site to the next one was not too large). In black: sites occupied by vegetation, in gray: recolonizable cells, in
white: degraded cells. Vegetation patches are present over a certain range of parameter space. (a) Effect of facilitation (f), and environmental conditions
(b). m ¼ 0.1, r ¼ 0, d ¼ 0.2, c ¼ 0.3, d ¼ 0.1. (b) Effect of mortality (m), and environmental conditions (b). m varies from value 0.005 to 0.3, f ¼ 0.9, r ¼ 0,
d ¼ 0.2, c ¼ 0.3, d ¼ 0.1. (c) Effect of the degradation rate (d) of a recolonizable site, and environmental conditions (b). m ¼ 0.1, f ¼ 0.9, r ¼ 0, c ¼ 0.3,
d ¼ 0.1, d varies from 0.02 to 1. (d) Effect of the regeneration rate (r) of a degraded site, and environmental conditions (b). m ¼ 0.1, f ¼ 0.9, d ¼ 0.2,
c ¼ 0.3, d ¼ 0.1. See Table 1 for the interpretation of the parameters that are not defined here.
vegetation in the system. An increase in facilitation (f) leads
to an increase in recolonizable sites in the neighborhood of
vegetation sites. These {o}-sites can then be recolonized by
seeds dispersed by their occupied neighbors, thus forming
clusters of vegetation sites (Fig. 2(a)). When mortality (m)
decreases, vegetation sites are more likely to recolonize {o}sites before they die (Fig. 2(b)). In a similar way, when
degradation (d) decreases, {o}-sites have a higher chance of
being recolonized by vegetation before being degraded
(Fig. 2(c)). The regeneration rate of a degraded site that has
no vegetation in its neighborhood (r) seems to hardly affect
the spatial organization of the vegetation (Fig. 2(d)).
Analysis of the pair approximation model provides
analytical insight into the behavior of the full spatial
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model. Simulation results of the pair approximation model
and the full spatial model closely resemble each other,
while they both differ from the results of the mean field
model (Fig. 3(a) and (b)). The ratio c++ ¼ r++/r2+ was
calculated to measure the patchiness of the vegetation sites
(Iwasa, 2000; van Baalen, 2000; Hui et al., 2006). In case
there is no spatial correlation in the system r++ ¼ r2+ and
c++ ¼ 1. If vegetation sites aggregate, the probability that
two nearest neighbors are {+} is higher than in a random
lattice (i.e., a lattice in which the population is uniformly
distributed) and c++41. We then talk about patchiness, or
clustering, of the vegetation. Simulations of both the full
spatial model and the pair approximation model show that
when f6¼0 or d6¼1, vegetation clustering occurs (c++41)
(Fig. 3(c)). From random initial conditions, vegetation sites
clump together despite the stochasticity of the full spatial
model, and form vegetation patches.
For stability analyses of the pair approximation model,
we investigated the conditions for which an initially rare
vegetation state (r+ E0) can increase in the system. This is
determined by a so-called invasion condition (Appendix C;
Iwasa et al., 1998). If the condition is satisfied, vegetation
initially increases and is always maintained in the system at
equilibrium (Iwasa et al., 1998). On the contrary, if the
condition is not satisfied, the system can either be a desert
at equilibrium, or be bistable. In case the state degraded
(r E0) is initially rare in the system, the condition for
invasion of such sites can also be derived. This condition is
always satisfied (Appendix C), meaning that there are
always degraded sites in the system at equilibrium. Thus,
the same three outcomes as those of the mean field model
are possible for the pair approximation model, i.e., a stable
desert system, a stable vegetation system or a bistable
system (Fig. 4).
Before going on with the results, we now explain
intuitively how bistability arises in the mean field as well
as in the pair approximation models. We call the transition
rate from a vegetation to degraded site loss rate, and the
transition rate from a degraded to vegetation site recovery
rate. If the recovery rate in the absence of vegetation (i.e.,
such that r+ E0), the basic recovery rate, is greater than the
loss rate, then vegetation always invades the system and
maintains itself. However, in case the loss rate exceeds the
basic recovery rate, the balance between recovery and loss
rates depends on both global and local densities of
vegetation, and bistability may arise. Now, the system is
bistable if the recovery rate can outbalance the loss rate for
certain values of global and local densities of vegetation.
From this general explanation, the influence of changing
parameter values on the occurrence of bistability in both
models can be understood.
If a region of bistability is crossed when changing the
bifurcation parameters, the system undergoes a discontinuous transition. A higher mortality (m) favors discontinuous transitions (Fig. 4(c)). Systems with high degradation
rate (d) and low basic regeneration rate (r) are also more
likely to undergo discontinuous transitions (Fig. 4(b)). The
intensity of competition (c) modifies the slope of the upper
branch of the bifurcation diagram (i.e., the resistance of the
system, Walker et al., 2002) but does not influence the type
of transition (continuous or discontinuous) (Fig. 4(d)).
This is because competition intensity hardly influences the
system at low vegetation densities. In the pair approximation model the effect of facilitation is reversed compared to
the mean field model and facilitation favors continuous
transitions (Fig. 4(a)). So, if environmental conditions
worsen in a system with high local facilitation, the
vegetation density gradually decreases until it reaches zero.
After the loss of vegetation in the whole system, the
desert system is composed of {}- and {0}-sites. Now, if
environmental conditions improve again, the appearance
of only one vegetation site in the system is already sufficient
Fig. 3. Comparison of the full spatial model with the two approximations. Evolution of site densities and clustering intensity during time. Example of a
simulation for m ¼ 0.1, c ¼ 0.3, b ¼ 0.4, d ¼ 0.2, f ¼ 0.8, r ¼ 0, d ¼ 0. For the cellular automaton, the simulation was run on a lattice of 50 50 cells with
periodic boundary conditions. See Table 1 for the interpretation of the parameters that are not defined here. Simulations of the full spatial model and of
the pair approximation model closely resemble each other while they both differ from the results of the mean field model. (a) Global environment. Global
density of vegetated sites (r+) in gray, and global density of degraded sites (r) in black. Normal lines: pair approximation model, thick lines: cellular
automaton, dashed lines: mean field model. (b) Local environment of vegetation sites. q+|+ in gray and q|+ in black. Normal lines: pair approximation
model, thick lines: cellular automaton, dashed lines: mean field model. (c) Variation of the clustering intensity (c++). Grey: pair approximation model,
black: cellular automaton.
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373
Fig. 4. Effect of the parameters on the bifurcation diagram of the pair approximation model. On the ordinate, (r+)eq is the global density of vegetation
sites at equilibrium. For clarity, only non-trivial equilibria are traced. A lower b value reflects more severe environmental conditions. In black: stable
equilibria, in gray: unstable equilibria. Parameters have a similar effect on the bifurcation diagrams of the pair approximation model as the mean field
model, except for facilitation. Higher facilitation (f) decreases the risk of discontinuous transitions in the pair approximation model. (a): Effect of
facilitation (f). m ¼ 0.1, d ¼ 0.1, c ¼ 0.2, d ¼ 0.1, r ¼ 0.01. (b): Effect of soil type (r and d). m ¼ 0.2, c ¼ 0.02, d ¼ 0.1, f ¼ 1. 1: unsusceptible soil r ¼ 0.04,
d ¼ 0.1, 2: r ¼ 0.02, d ¼ 0.2, 3: susceptible soil r ¼ 0.01, d ¼ 0.4. (c): Effect of mortality (m). d ¼ 0.1, r ¼ 0.01, c ¼ 0.2, d ¼ 0.1, f ¼ 0.9. The spatial
organization observed at the points indicated by the arrows on (c) are displayed Fig. 6. (d): Effect of competition intensity (c): m ¼ 0.1, d ¼ 0.1, d ¼ 0.1,
r ¼ 0.01, f ¼ 0.9. See Table 1 for the interpretation of the parameters that are not defined here. Bifurcation diagrams for the pair approximation model
were numerically derived with the software package CONTENT (Kuznetsov and Levitin, 1997).
Fig. 5. State diagrams: comparison of the mean field (a, b, c) and the pair approximation models (d, e, f). Gray: vegetation system at equilibrium, black:
bistability, white: desert system at equilibrium. Decreasing facilitation (f) from left to right. A lower b value reflects more severe environmental conditions.
m is the mortality rate of a vegetation site. Spatial interactions lead to a strong reduction of the bistability area in the pair approximation model as
compared to the mean field model. r ¼ 0.01, c ¼ 0.2, d ¼ 0.1, d ¼ 0.1. (a) and (d): f ¼ 0.9, (b) and (e): f ¼ 0.3, (c) and (f): f ¼ 0.1. See Table 1 for the
interpretation of the parameters that are not defined here.
for vegetation recovery. Indeed, the probability is high that
empty neighbors of this vegetation site also become
occupied by vegetation. Thus, patches of vegetation can
easily form and maintain themselves even from very low
initial vegetation densities.
Comparing the state diagrams of the mean field with the
pair approximation model (Fig. 5), the desert and the
vegetation domains are larger with spatial structure than
without. This leads to a strong reduction of the bistability
domain in the pair approximation model, regardless of the
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Fig. 6. Spatial organization of the vegetation predicted by the full spatial model along discontinuous (top) and continuous transitions (bottom) In black:
cells occupied by vegetation, in gray: recolonizable cells, in white: degraded cells. Decreasing b from left to right (a lower b value reflects more severe
environmental conditions). The simulations were run on lattices of 50 50 cells with periodic boundary conditions, which was appropriate for the
observation of the shapes of the patches. When inspected by eye, the sequence of patch shape along discontinuous transition looks similar to the sequence
of patch shape along continuous transition. r ¼ 0, c ¼ 0.2, d ¼ 0.1, d ¼ 0.1, f ¼ 0.9. Top row: Discontinuous transition, m ¼ 0.2, (a): b ¼ 0.8, (b):
b ¼ 0.62, (c): b ¼ 0.57. Bottom row: Continuous transition, m ¼ 0.1, (d): b ¼ 0.5, (e): b ¼ 0.35, (f): b ¼ 0.3. See Table 1 for the interpretation of the
parameters that are not defined here.
intensity of local facilitation. As explained in the previous
paragraph, in the area where the pair approximation
predicts the maintenance of vegetation and where the mean
field approximation predicts bistability, low mortalities
leave enough time for patches to arise from vegetation sites
because of local facilitation. At high mortalities (m)
however, vegetation sites disappear before having had time
to form patches: local processes, namely local facilitation
(f) and local seed dispersal (1–d) become a constraint and
vegetation goes always extinct in the pair approximation
model whereas it can maintain itself for high initial
vegetation densities in the mean field model.
Vegetation patches were followed along both continuous
and discontinuous transitions using the full spatial model.
Vegetation patches arise along continuous and discontinuous transitions. The sequences of patch shape along
continuous as compared to discontinuous transition look
similar when inspected by eye (Fig. 6).
4. Discussion
Our model exhibits two main properties: bistability
under certain environmental conditions and vegetation
clustering. These two properties arise in an isotropic
environment as a result of soil characteristics (d and r),
plant characteristics (m, d and b) and plant interactions
(f and g).
Vegetation clustering occurs when local processes are
included in the model (f6¼0 or d6¼1). When environmental
conditions become more severe, the vegetation cover
decreases, and vegetation sites clump together, forming
patches. This clustering is the result of internal dynamics
and arises from random initial conditions, despite the
stochasticity of the full spatial model. It may therefore be
considered as a form of spatial self-organization (Perry,
1995; Rohani et al., 1997).
The pair approximation model is capable of predicting
the behavior of the full spatial model even when the mean
field model fails. The mean field model is in particular
unsuccessful in predicting the bistability area, which is
always smaller in the pair approximation than in the mean
field model. This illustrates the effect of local interactions
on the stability of the system. In the mean field model, a
higher positive feedback (f) allows vegetation to survive
under harsher environmental conditions and favors discontinuous transitions. This result is in line with the mean
field models results of Rietkerk and van de Koppel (1997),
Lefever and Lejeune (1997), Lejeune and Tlidi (2002) and
Meron et al. (2004). However, in our pair approximation
model, which takes local interactions into account, a
stronger facilitation (f) diminishes the risks of discontinuous transitions, because patches of vegetation can more
easily form, spread and maintain themselves. According to
both the mean field and the pair approximation models,
systems with more susceptible soils (high d, low r) and with
higher mortality (m) are more likely to be bistable.
Susceptible soils correspond to clay rather than sandy soils
(Walker et al., 2002). Low facilitation and high mortality
characterize herbaceous rather than woody vegetation and
annual rather than perennial vegetation (e.g., Pugnaire
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et al., 1996; Holzapfel and Mahall, 1999). Thus, our model
suggests that systems with dominant woody or perennial
vegetation and sandy soils are less likely to be bistable than
systems with dominant herbaceous or annual vegetation
and clayey soils.
In certain arid ecosystems, so-called regular vegetation
patterns are found. Bands (‘‘tiger’’ vegetation) are observed
on terrain with slopes above a critical threshold value.
Spots, labyrinths or gaps in uniform vegetation cover
(‘‘leopard’’ vegetation) are observed on less inclined terrain
(d’Herbès et al., 2001). Note that we reserve the term
pattern(s) for regular spatial organization of the vegetation
(following Levin and Segel, 1985). Various mathematical
models succeed in reproducing these regular patterns
(Lejeune et al., 1999; von Hardenberg et al., 2001; Lejeune
and Tlidi, 2002; Rietkerk et al., 2002; Meron et al., 2004).
These models assume that the driving mechanism of
regular vegetation pattern formation is the difference in
water infiltration and/or evaporation rates between vegetation and bare sites within the system (von Hardenberg et
al., 2001; Rietkerk et al., 2002; Meron et al., 2004).
Resource concentration underneath the vegetation, where
the infiltration rate is high and the evaporation rate is low,
causes resource depletion farther away, leading to scaledependent feedback. These models also exhibit bistability
of certain regular vegetation patterns and a homogeneous
bare state for a certain range of parameter values (Lejeune
et al., 1999; Rietkerk et al., 2002; Meron et al., 2004). These
models have provided insight into how such striking
regular patterns can arise in ecosystems because the
patterns’ occurrence can be linked with specific mechanisms leading to scale-dependent feedbacks. In our model,
facilitation is considered as a combination of physical and
biological processes occurring in the local environment of a
plant only, and does not trigger effects farther away.
Therefore, local facilitation and competition are two
independent, decoupled mechanisms, contrary to models
based on scale-dependent feedback, where facilitation’s
intensity and spatial scale are closely coupled with
competition’s intensity and spatial scale. Local facilitation
and scale-dependent feedback both include local positive
feedbacks, due to different causes which are well-established empirically. However, they differ in the causes and
spatial scale of competition, which has not been well
investigated yet in arid ecosystems.
Our general model results are in line with those of the
models based on scale-dependent feedback in that we
obtain bistability (under certain environmental conditions)
and spatial self-organization of the vegetation (Lefever and
Lejeune, 1997; Lejeune et al., 1999; von Hardenberg et al.,
2001; Lejeune and Tlidi, 2002; Rietkerk et al., 2002; Meron
et al., 2004). However, the spatial organization is different
in our system. The patches are not regularly distributed in
space, and do not have easily recognized (by eye) geometric
shapes (circles, labyrinths, etc.). So, the two possible
mechanisms, namely local facilitation and scale-dependant
feedback, lead to two different spatial organizations of the
375
vegetation in models, and these outcomes can be connected
to observations in the field. Therefore, we hypothesize that,
in the field, systems with vegetation patches that are not
regular and which do not have easily recognized geometric
shapes are systems were local facilitation rather than scaledependent feedback operates. Conversely, systems with
regular vegetation patterns could be systems that are driven
by scale-dependent feedback.
The link between spatial self-organization and discontinuous transitions, as hypothesized by Rietkerk et al.
(2004) for scale-dependent feedback systems, does not
occur in our model. In our model, both continuous and
discontinuous transitions are associated with vegetation
patchiness, and the two types of transitions cannot be
distinguished merely by observing changes in the shape of
vegetation patches. Vegetation patchiness and the type of
transition are disconnected from each other in our model.
Thus, in the case of systems driven by local facilitation,
clusters of vegetation in a bare matrix indicate proximity to
a transition to desert, but further knowledge on the
stability of the system is required to assess whether such
a transition will be discontinuous. On the other hand, the
link between self-organized pattern formation and discontinuous transitions in case of coupled scale-dependent
feedback remains an area of future research. Studies
remain to be done on which facilitative mechanism occurs
where and under which conditions in arid ecosystems. Our
study already suggests that observation of spatial organization of vegetation in the field could give information
about the type of facilitative mechanism involved. This, in
turn, could provide relevant information about a possible
link between such organization and the type of transition.
Finally, the mechanism of local facilitation takes diverse
forms in harsh environments depending on the type of
stress that predominates. In subalpine and alpine plant
communities, at high elevations surrounding vegetation
maintains warmer temperatures, prevents from frost, and
protects from strong winds and desiccation (Choler et al.,
2001; Callaway et al., 2002). In salt marshes, species
tolerant of extreme edaphic conditions prevent salt
accumulation in the soil because their cover limits surface
evaporation by shading (Bertness and Hacker, 1994).
Certain species also aerate the anoxic soils by rhizosphere
oxidation (Bertness and Leonard, 1997). In interdidal
areas, aquatic and marine vascular plants reduce physical
disturbance of the flow and enhance depositions and
stabilization of the substrate (Bruno, 2000). In rock shores,
shading by conspecifics renders temperature less extreme
and variable and dislodgments of sessile invertebrates such
as mussels by waves are reduced in dense clumps of
individuals (Bertness and Leonard,1997). These ecosystems
are also characterized by a certain degree of clustering. The
facilitative mechanisms operating in these ecosystems could
be modeled in the same way as we have done in our model.
So, our model could be further generalized to study the link
between local facilitation, spatial organization and type
of transition in a wider range of harsh environments.
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A remaining issue is the ability to identify intrinsically
bistable systems as well as systems that can become bistable
through external forces. For this purpose, finding indicators of discontinuous transitions would be of general
interest in ecology and relevant for ecosystem management
and restoration.
Acknowledgments
The manuscript greatly benefited from the comments of
Peter de Ruiter. We would like to thank Maarten Eppinga
and Yolanda Pueyo for interesting discussions, and Ryan
Chaves, Birka Wicke and Joseph Wilde-Ramsing for
helping improving the clarity of the manuscript. The
research of S.K. and M.R. is supported by a personal
VIDI grant of the Netherlands Organization of Scientific
Research/Earth and Life Sciences (NWO-ALW) to M. R.
Appendix A. Estimation of the parameter values
The parameters and their definitions are displayed in
Table 1.
The average lifespan of the vegetation of arid ecosystems
is very variable (Mauchamp et al., 2001), depending on the
species. Precise information on species is difficult to find in
the literature. The lifespan of perennial grasses may be a
few years while most of the shrubs and trees are long-lived
(50–120 years). For example, a mature adult of Acacia
Aneura, the tree component dominant in Australian
banded landscapes (d’Herbès et al., 2001), is usually
more than 100 years old and can live until 200 years. Thus
the mortality rate, m, ranges between 0.005 and 1 in our
model.
Accelerated erosion caused by decreased vegetation
cover or increased precipitation occurs at a time scale of
10–50 years. Other degradation processes such as saliniza-
tion, acidification, and decomposition of organic matter
occur at a shorter time scale (respectively: 0.1, 0.5–5 and
5–20 years) (Walker et al., 2002). Thus the degradation
rate, d, ranges between 0.02 and 1 in our simulations.
There appear to be no specific studies about seed
production in arid ecosystems (Montaña et al., 2001). We
have not found even an order of magnitude estimate in the
literature. It is however known that, in arid ecosystems,
recruitment of new individuals is a very rare event, even
more when the vegetation is dominated by long-lived
species. This is partly due to the great number of seeds lost
in the system, either because they do not arrive at a
recolonizable site, or because they leave the system with
water runoff or animals. To reflect these phenomena, b (the
intrinsic seed production rate per vegetation site survival
probability germination probability) is kept between 0
and 1.
The few studies which have been published on seeds in
arid ecosystems all concern seed dispersal. The seed bank is
patchily distributed and mainly concentrated in the
vegetation patches. It has been shown that 90% of the
seeds of F. Cernua, the dominant vegetation cover of bands
in Mexico, are beneath the adult crown, while 10% of the
seeds are dispersed by runoff water, wind, domestic and
wild animals (Montaña et al., 2001). This order of
magnitude is confirmed by other studies (Montaña et al.,
1990).
A study in northern Burkina Faso (Rasmussen et al.,
2001) showed that the vegetation took 20–30 years to
recover after the drought of the 1970s. In the studied area,
the soil is sandy and the mean annual rainfall is around
500 mm. Therefore, this recovery time is an optimal value,
rather than an average value. Another study (Walker et al.,
2002) suggests indeed the century rather than the decade as
a recovery time scale. Thus the recovery rate, r, ranges
between 0 and 0.05 in our simulations.
Table 1
Table of parameters
Class
Symbol
Interpretation
Parameter
m
f
b
e
b
d
g
c
r
d
z
Mortality rate of a vegetated cell
Local facilitation, i.e., maximum effect of neighbors on the regeneration of a degraded site realized when all the
nearest neighbors are {+}-sites
Intrinsic seed production rate per vegetated cell survival probability germination probability
Establishment probability of seeds on a {0}-site in a system without competition
be
Fraction of seeds globally dispersed
Competitive effect of the global density of {+}-sites on the establishment of new individuals
bg
Regeneration rate of a {}-site without vegetation in its neighborhood
Degradation rate of {0}-sites
Number of nearest neighbors of a cell (here, z ¼ 4)
r+
r
r0
rsl
c++
Density of vegetated sites
Density of degraded sites
Density of recolonizable sites
Density of pairs of neighboring sites sl (s and l can be +, 0 or )
Clustering intensity of the vegetated patches
Variable
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The parameters c (competition) and f (facilitation)
correspond to two different types of effects of the
vegetation on the system. Their values are kept between
0, for no effect, and 1, for a maximal effect. They could not
really be extracted from the literature. Intuitively, high
values of c and f may however correspond to vegetation
dominated by trees and shrubs while low values may
represent a system dominated by perennial grasses.
The parameter e is the probability of recruitment of a
seed on a {0}-site in a system without competition (i.e., a
system with no vegetation). This parameter reflects the
‘‘quality of the environment’’. We varied it between 0 and
1, the two possible extreme values, to see the effect of a
wide range of different environmental conditions on the
system.
Appendix B. Systems of equations
B.1. The pair approximation model
The four conservation equations are:
rþ þ r0 þ r ¼ 1,
(21)
rþþ þ rþ0 þ rþ ¼ rþ ,
(22)
r00 þ r0þ þ r0 ¼ r0 ,
(23)
r þ r0 þ rþ ¼ r .
(24)
In the system, there are nine variables in total. Three are
singletons variables (rs), and six are distinct doublet
variables (rss0 ), since rss0 ¼ rs0 s. Because there are also 4
conservation equations, we need five (9–4) equations to
solve the system. We chose dr++/dt, dr+/dt, dr/dt,
dr+/dt, and dr/dt.
The principle of deriving the pair approximation
equations from Eqs. (16–20) is explained in detail in Iwasa
(2000) and van Baalen (2000). Using the conservation
equations Eqs. (21–24) and the transition probabilities Eqs.
(3–6) leads to the following closed system of equations:
drþ
¼ dðrþ rþ rþþ Þ þ ðr r rþ Þ
dt
ðrþ rþ rþþ Þ
z1
ð1 dÞ
drþ þ
z
ð1 rþ r Þ
ðb crþ Þ
f z 1 rþ
f
þm ,
rþ r þ þ
z
z
r
dr
z 1 rþ
f
¼ 2dðr r rþ Þ 2r r þ
,
z
r
dt
(27)
drþ
r rþ rþþ
¼ drþ þ ð1 dÞ þ
dt
1 rþ r
ðb crþ Þð1 rþ r Þ mrþ ,
r
dr
¼ dð1 rþ r Þ r þ f þ r ,
r
dt
ð28Þ
(29)
B.2. The mean field model
The mean field model (Eqs. (13) and (14)) has one trivial
equilibrium (r+ ¼ 0, r0 ¼ r/(d+r), r ¼ d/(d+r)) which is
stable if bom(d+r)/r.
Non trivial equilibria are solutions of the system:
b r
m b r br
r3þ r2þ 1 þ
þ rþ
c f
c c f fc
br mr md
þ
¼ 0,
cf
m
ð30Þ
r ¼ 1 rþ
b crþ
with the constraints:rþ 2 ½0; 1 and r 2 ½0; 1.
Appendix C. Invasion analysis for the pair approximation
model
C.1. Invasion of the degraded state in a system dominated by
the vegetation state
Let us suppose that the lattice is in a state such that
degraded sites are rare in the system (rE0). We ask
whether the initially rare state ‘‘degraded’’ can increase
in the system. This is determined by the sign of dr/dt.
r increases if
dr
¼ dð1 rþ r Þ ðr þ fqþj Þr 40.
(31)
dt
This condition is always satisfied when r tends to 0,
meaning that there are always some degraded sites in the
lattice at equilibrium.
C.2. Invasion of the vegetation state in a system dominated
by the degraded state.
ð25Þ
drþþ
¼ 2ðrþ rþ rþþ Þ
dt
!
1d
z 1 ðrþ rþ rþþ Þ
þ ð1 dÞ
drþ þ
z
z
ð1 rþ rÞ
ðb crþ Þ 2rþþ m,
377
ð26Þ
In case the vegetated state is initially rare in the system,
its density increases if dr+/dt40
Using the approximation r+E0 and Eq. (19), we obtain
the following invasion condition:
qojþ 4
m
dr
.
ð1 dÞb ðd þ rÞð1 dÞ
(32)
This condition depends on the local density q0|+. To
estimate this quantity near the equilibrium, we followed the
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following procedure (Iwasa et al., 1998):
the equilibrium values of the global and local densities
of the ‘‘resident’’ state (r and q|) can be calculated
by using Eqs. (18), (20), the Bayes’ formula (i.e.,
d
Eq. (15)) and by neglecting r+. We obtain: r dþr
,
d
qj dþr;
these results and the approximation r+E0 provide
an autonomous system of differential equations of
the two variables q|+ and q+|+. With x ¼ q+|+, y ¼ q|+
and w ¼ 1q|+q+|+, this system can be written as
follows
b
r
¼ 0,
2wð1 dÞ mx bxwð1 dÞ bxd
z
dþr
drb
z 1 wðd þ rÞ
ðd þ ð1 dÞ
z
r
ðd þ rÞ2
f
r
¼ 0,
y rþ
bywð1 dÞ ydb
z
d þr
dw þ
x and y can then be expressed as
x¼
y¼
2bwð1 dÞ
r
zðm þ bwð1 dÞ þ db dþr
Þ
bdr
z1 wðdþrÞ
wd þ ðdþrÞ
2 ðd þ ð1 dÞ z
r Þ
dbr
r þ fz þ bwð1 dÞ þ dþr
.
So that using w ¼ 1xy, we can write
1¼wþ
þ
2bwð1 dÞ
r
zðm þ bwð1 dÞ þ db dþr
Þ
bdr
z1 wðdþrÞ
wd þ ðdþrÞ
2 ðd þ ð1 dÞ z
r
dbr
r þ fz þ bwð1 dÞ þ dþr
¼ funcðwÞ.
ð33Þ
func is continuous and differentiable on [0,1]. Moreover:
8wA[0,1] func’(w)40, func(0)o1 and func(1)41
Thus, there is one unique solution of Eq. (33) between 0
and 1 (Intermediate values theorem).
The invasion condition is thus:
m
dr
w4 ð1dÞb
ðdþrÞð1dÞ
(using Eq. (32) and the fact that
w ¼ 1q|+q+|+ ¼ qo|+)
m
dr
3f ðwÞ4f
ð1 dÞb ðd þ rÞð1 dÞ
m
dr
2b
b
dþr
m
3
þ
2zm
ð1 dÞb
d
m
dr
d
drb
1d b dþr þ zðdþrÞ ðz 1Þm þ dþr
þ
r
dr
f
þ þ mo1.
ð34Þ
ð1 dÞðd þ rÞ z
When the invasion condition (34) is satisfied, vegetation
is always maintained in the system at equilibrium. On the
contrary, when the invasion condition (34) is not satisfied,
the system is either a desert at equilibrium or bistable.
Numerical simulations can then be done to decide between
desert or bistable system.
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