# Integrating theory and experiments to link local mechanisms and ecosystem-
level consequences of vegetation patterns in drylands

Ricardo Martinez-Garcia1,∗, Ciro Cabal2, Justin M. Calabrese3,4,5,
Emilio Hernández-García6, Corina E. Tarnita2, Cristóbal López6,†, Juan A.
Bonachela7,†

( 1 ICTP South American Institute for Fundamental Research & Instituto de
Física Teórica - Universidade Estadual Paulista, São Paulo, SP, Brazil.
2 Department of Ecology and Evolutionary Biology, Princeton University,
Princeton, NJ, USA.
3 Center for Advanced Systems Understanding (CASUS), Görlitz, Germany.
4 Helmholtz-Zentrum Dresden Rossendorf (HZDR), Dresden, Germany.
5 Department of Ecological Modelling, Helmholtz Centre for Environmental
Research – UFZ, Leipzig, Germany.
6 IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB),
Palma de Mallorca, Spain.
7 Department of Ecology, Evolution, and Natural Resources, Rutgers University,
New Brunswick, NJ, USA
∗ Correspondence<EMAIL_ADDRESS>$\dagger$ equal contribution. )

###### Abstract

Self-organized spatial patterns of vegetation are frequent in water-limited
regions and have been suggested as important indicators of ecosystem health.
However, the mechanisms underlying their emergence remain unclear. Some
theories hypothesize that patterns could result from a water-mediated scale-
dependent feedback (SDF), whereby interactions favoring plant growth dominate
at short distances and growth-inhibitory interactions dominate in the long
range. However, we know little about how net plant-to-plant interactions may
shift from positive to negative as a function of inter-individual distance,
and in the absence of strong empirical support, the relevance of this SDF for
vegetation pattern formation remains disputed. These theories predict a
sequential change in pattern shape from gapped to labyrinthine to spotted
spatial patterns as precipitation declines. Nonetheless, alternative theories
show that the same sequence of patterns could emerge even if net interactions
between plants were always inhibitory (purely competitive feedbacks, PCF).
Importantly, although these alternative hypotheses lead to visually
indistinguishable patterns they predict very different desertification
dynamics following the spotted pattern, limiting their potential use as an
ecosystem-state indicator. Moreover, vegetation interaction with other
ecosystem components can introduce additional spatio-temporal scales that
reshape both the patterns and the desertification dynamics. Therefore, to make
reliable ecological predictions for a focal ecosystem, it is crucial that
models accurately capture the mechanisms at play in the system of interest.
Here, we review existing theories for vegetation self-organization and their
conflicting predictions about desertification dynamics. We further discuss
possible ways for reconciling these predictions and potential empirical tests
via manipulative experiments to improve our understanding of how vegetation
self-organizes and better predict the fate of the ecosystems where they form.

Keywords: ecological patterns, competition, scale-dependent feedback,
ecological transitions, spatial self-organization, mathematical models.

##  1 Introduction

From microbial colonies to ecosystems extending over continental scales,
complex biological systems often feature self-organized patterns, regular
structures that cover large portions of the system and emerge from nonlinear
interactions among its components (Meinhardt,, 1982; Camazine et al.,, 2003;
Sole and Bascompte,, 2006; Pringle and Tarnita,, 2017). Importantly, because
harsh environmental conditions provide a context in which self-organization
becomes important for survival, emergent patterns contain crucial information
about the physical and biological processes that occur in the systems in which
they form (Sole and Bascompte,, 2006; Meron,, 2018; Zhao et al.,, 2019).

A well-known example of ecological self-organization is vegetation pattern
formation in water-limited ecosystems (Deblauwe et al.,, 2008; Rietkerk and
van de Koppel,, 2008). Flat landscapes can show vegetation spots regularly
distributed on a matrix of bare soil, soil-vegetation labyrinths, and gaps of
bare soil regularly interspersed on a homogeneous layer of vegetation (Fig.
1). Importantly, water availability strongly influences the specific shape of
the pattern. In agreement with model predictions (von Hardenberg et al.,,
2001; Meron et al.,, 2004), a Fourier-based analysis of satellite imagery
covering extensive areas of Sudan revealed that more humid regions are
dominated by gapped patterns, whereas spotted patterns dominate in more arid
conditions (Deblauwe et al.,, 2011). However, imagery time series are not long
enough to observe whether vegetation cover in a specific region undergoes
these transitions between patterns when aridity increases over time.

Figure 1: Aerial images of self-organized vegetation patterns. In all panels,
vegetated regions are darker and bare-soil regions, lighter. a) Spot pattern
in Sudan (11∘34’55.2” N; 27∘54’47.52”E). b) Labyrinthine pattern in Mali
(12∘27’50”N; 3∘18’30”E). c) Gap pattern in Niger (11∘1’12”N; 28∘10’48”E). d)
Band pattern in Sudan (11∘3’0”N; 28∘17’24”E). Microsoft product screen shot(s)
reprinted with permission from Microsoft Corporation. Image Copyrights
$\copyright$2021 Maxar.

Following the spotted pattern, if precipitation continues to decrease, models
predict that patterned ecosystems undergo a transition to a desert state. This
observed correlation between pattern shape and water availability suggests
that the spotted pattern could serve as a reliable and easy-to-identify early-
warning indicator of this ecosystem shift (Scheffer and Carpenter,, 2003;
Rietkerk et al.,, 2004; Scheffer et al.,, 2009; Dakos et al.,, 2011, 2015).
This has reinforced the motivation to develop several models aiming to explain
both the formation of spatial patterns of vegetation and their dependence on
environmental variables (von Hardenberg et al.,, 2001; Rietkerk et al.,, 2002;
Meron et al.,, 2004; Borgogno et al.,, 2009; Martínez-García et al.,, 2014;
Gowda et al.,, 2014). Although some attempts to test model predictions with
satellite imagery exist (Weissmann et al.,, 2017; Bastiaansen et al.,, 2018),
theoretical studies using models remain the dominant approach to study this
hypothesized transition.

All these frameworks successfully reproduce the sequence of gapped,
labyrinthine, and spotted patterns found in satellite imagery (Fig. 2a).
However, they disagree in their predictions regarding the nature of the
desertification transition that follows the spotted pattern. For example,
Rietkerk et al., (2002) and von Hardenberg et al., (2001) predicted that
ecosystems may undergo abrupt desertification, including a hysteresis loop,
following the spotted pattern. Martínez-García et al., 2013a  and Yizhaq and
Bel, (2016) predicted that desertification could also occur gradually with
progressive loss of vegetation biomass. Using alternative modeling approaches,
other studies have supported the idea that whether an ecosystem will collapse
gradually or abruptly is system-dependent and determined by the intensity of
stochasticity (Weissmann et al.,, 2017), vegetation and soil type (Kéfi et
al., 2007b, ), colonization rates (Corrado et al.,, 2015), and intensity of
external stresses, such as grazing (Kéfi et al., 2007a, ). Because drylands
cover $\sim 40\%$ of Earth’s land surface and are home to $\sim 35\%$ of the
world population (Mortimore et al.,, 2009), determining whether they will
respond abruptly or gradually to aridification is critical both from an
ecosystem-management and socio-economic point of view.

Active lines of theoretical research aiming to address this question have
focused on understanding how different components of the ecosystem may
interact with each other to determine an ecosystem’s response to aridification
(Bonachela et al.,, 2015; Yizhaq and Bel,, 2016), as well as on designing
synthetic feedbacks (in the form of artificial microbiomes) that could prevent
ecosystems collapses or make such transitions smoother (Villa Martín et al.,,
2015; Conde-Pueyo et al.,, 2020; Vidiella et al.,, 2020). The question has
also attracted considerable attention from empirical researchers (Maestre et
al.,, 2016), and recent evidence suggests that certain structural and
functional ecosystem attributes respond abruptly to aridity (Berdugo et al.,,
2020). Despite current efforts, whether desertification is more likely to
occur gradually or abruptly remains largely unknown.

Here, we outline and rank strategies that will help answer this question. In
section 2, we discuss the ecological rationale behind existing models for
vegetation self-organization. We review such models in section 3 and summarize
their opposing predictions about the ecosystem collapse in section 4. In
section 5, we describe possible manipulative experiments and empirical
measures to select among the previously scrutinized models. Finally, in
section 6, we discuss different research lines that build on current knowledge
and discuss how to apply lessons learned from studying self-organized
vegetation patterns to other self-organizing systems.

##  2 Ecological rationale behind current models for vegetation spatial self-
organization

The net biotic interaction between plants, i.e., the effect that plants have
on each other’s survival, reproduction, and growth rate ultimately determines
vegetation spatial patterns. Depending on the number of individuals that form
each patch, we can classify non-random vegetation spatial patterns into two
families: segregated and aggregated patterns (Fig. 2b). Segregated patterns
emerge when competition is the dominant net biotic interaction and surviving
individuals avoid interacting with each other. In the long term, vegetation
arranges in a hexagonal lattice of patches with each individual representing
one patch (Pringle and Tarnita,, 2017). Aggregated patterns, on the other
hand, emerge when surviving plants form vegetation patches with several
individuals. Segregated patterns are very common in drylands and are expected
to emerge exclusively due to ecological interference or competition (Pringle
and Tarnita,, 2017). Aggregated patterns, which are the focus of this study,
are ecologically more intriguing because they might result from a richer set
of mechanisms (van de Koppel et al.,, 2008; Rietkerk and van de Koppel,, 2008;
Lee et al.,, 2021) and their ecological implications strongly depend on them
(Fig. 2c, d). Moreover, a direct evidence of which feedback type drives
aggregated patterns of vegetation in drylands remains elusive.

Existing theories for vegetation self-organization hypothesize two alternative
ways for the formation of aggregated patterns: scale-dependent feedbacks (SDF)
and purely competitive feedbacks (PCF). Both are based on the biophysical
effects of the plant canopy on the microclimate underneath and of the root
system on the soil conditions (Cabal et al., 2020b, ) (Fig. 2c), but they
differ in how the net interaction between plants changes with the inter-
individual distance. Next, we will briefly review the mechanisms that have
been suggested to underpin each of these two feedbacks and the type of
patterns they might create.

Scale-dependent feedbacks. Biotic facilitation is a very common interaction in
semiarid and arid ecosystems (Holmgren and Scheffer,, 2010). The SDFs invoked
to explain vegetation self-organization are caused by the facilitative effects
of plants nearby their stems, coupled with negative effects at longer
distances. Several ecological processes have been suggested to support these
SDFs. One is the positive effects of shade, which can overcome competition for
light and the effects of root competition for water, and lead to under-canopy
facilitation (Valladares et al.,, 2016). In this context, focal plants have an
overall facilitative effect in the area of most intense shade at the center of
the crown. This effect progressively loses intensity with distance to the
center of the crown and ultimately vanishes, leaving just simple below-ground
competition in areas farther from the plant. A complementary rationale is that
plants modify soil crust, structure, and porosity, and therefore enhance soil
water infiltration (Eldridge et al.,, 2000; Ludwig et al.,, 2004). Enhanced
water infiltration has a direct positive effect close to the plant because it
increases soil water content but, as a by-product, it has negative
consequences farther away from the plant’s insertion point because, by
increasing local infiltration, plants also reduce the amount of water that can
infiltrate further away in bare soil locations (Montaña,, 1992; Bromley et
al.,, 1997). The spatial heterogeneity in water availability due to plant-
enhanced infiltration is higher in sloped landscapes where down-slope water
runoff leads to the formation of banded vegetation patterns (Deblauwe et al.,,
2012; Valentin et al.,, 1999), but it is also significant in flat landscapes
and might cause the of emergence gaps, labyrinths, and spots of vegetation
(HilleRisLambers et al.,, 2001; Gilad et al.,, 2004; Okayasu and Aizawa,,
2001). (Fig. 2a).

Figure 2: Conceptual summary of existing theories for vegetation self-
organization, their emergent patterns and the type of desertification
processes they predict. a) Observed spatial patterns across a gradient of
rainfall, with more humid ecosystems showing gaps and more arid, spots. b)
Examples of spatial random, segregated, and aggregated patterns as defined in
the main text. Aggregated patterns may result both from PCF and SDF, whereas
segregated patterns emerge from PCF. c) Types of feedbacks invoked to explain
the emergence of self-organized vegetation patterns and d) the different
desertification transitions they predict.

Purely competitive feedbacks. On the other hand, competition for resources is
a ubiquitous interaction mechanism that affects the relation between any two
plants that are located in sufficient proximity. Above ground, plants compete
for light through their canopies; below ground, they compete for several soil
resources, including water and nitrogen, through their roots (Craine and
Dybzinski,, 2013). If only competitive mechanisms occur, we should expect
plants to have a negative effect on any other plant within their interaction
range (Fig. 2c) and the intensity of this effect to peak at intermediate
distances between vegetation patches(van de Koppel and Crain,, 2006; Rietkerk
and van de Koppel,, 2008). Because finite-range competition is the only
interaction required by PCF models to explain vegetation self-organization,
PCF is the most parsimonious feedback type that generates vegetation patterns.

##  3 Models for vegetation self-organization

Mathematical frameworks to vegetation self-organized are grouped into two main
categories: individual-based models (IBM) and partial differential equations
models (PDEM). IBMs describe each plant as a discrete entity whose attributes
change in time following a stochastic updating rule (DeAngelis and Yurek,,
2016; Railsback and Grimm,, 2019). PDEMs describe vegetation biomass and water
concentration as continuous fields that change in space and time following a
system of deterministic partial differential equations (Meron,, 2015). IBM are
the most convenient approach to study segregated patterns, where single
individuals are easy to identify and central to the formation of vegetation
patches (Bolker and Pacala,, 1999; Iwasa,, 2010; Wiegand and Moloney,, 2013;
Plank and Law,, 2015). Conversely, PDEMs are a better approximation to
aggregated patterns because they focus on a continuous measure of vegetation
abundance and describe the dynamics of patches that can spread or shrink
without any upper or lower limit on their size. Natural multi-individual
patches can change in size and shape depending on the interaction among the
individual plants within them, whereas the size of single-plant patches is
subject to stronger constraints (they usually grow, not shrink, and their
maximum size is bounded by plant physiology). Therefore, PDEMs represent a
simplification of the biological reality that is more accurate in aggregated
than in segregated patterns. Because we are only considering aggregated
patterns, we will focus our review of the mathematical literature on PDEMs.

###  3.1 Reaction-diffusion SDF models

In 1952, Turing showed that differences in the diffusion coefficients of two
reacting chemicals can lead to the formation of stable spatial heterogeneities
in their concentration (Turing,, 1952). In Turing’s original model, one of the
chemicals acts as an activator and produces both the second chemical and more
of itself via an autocatalytic reaction. The second substance inhibits the
production of the activator and therefore balances its concentration (Fig.
3a). Spatial heterogeneities in the concentrations can form if the inhibitor
diffuses much faster than the activator, so that it inhibits the production of
the activator at a long range and confines the concentration of the activator
locally (Fig. 3b). This activation-inhibition principle thus relies on a SDF
to produce patterns: positive feedbacks (autocatalysis) dominate on short
scales and negative, inhibitory feedbacks dominate on larger scales. In the
context of vegetation pattern formation, plant biomass acts as the self-
replicating activator. Water is a limiting resource and thus water scarcity is
an inhibitor of vegetation growth (Rietkerk and van de Koppel,, 2008; Meron,,
2012).

Figure 3: a) Schematic of the Turing activation-inhibition principle. The
activator, with diffusion coefficient $D_{a}$, produces the inhibitor at rate
$k_{ai}$ as well as more of itself at rate $k_{aa}$ through an autocatalytic
reaction. The inhibitor degrades the activator at rate $k_{ia}$ and diffuses
at rate $D_{i}>D_{a}$. b) Schematic of the pattern-forming process in a one-
dimensional system.

####  3.1.1 Two-equation water-vegetation dynamics: the generalized Klausmeier
model

Initially formulated to describe the formation of stripes of vegetation in
sloping landscapes (Fig. 1d), subsequent studies have generalized the
Klausmeier model to flat surfaces (Kealy and Wollkind,, 2011; Bastiaansen et
al.,, 2018; Eigentler and Sherratt,, 2020). Mathematically, the generalized
version of the Klausmeier model is given by the following equations:

$\displaystyle\frac{\partial w({\bm{r}},t)}{\partial t}$ $\displaystyle=$
$\displaystyle
R-l\,w({\bm{r}},t)-a\,g\left(w\right)f\left(v\right)v({\bm{r}},t)+D_{w}\mathrm{\nabla}^{2}w({\bm{r}},t),$
(1) $\displaystyle\frac{\partial v({\bm{r}},t)}{\partial t}$ $\displaystyle=$
$\displaystyle
a\,q\,g\left(w\right)f\left(v\right)v({\bm{r}},t)-m\,v({\bm{r}},t)+D_{v}\mathrm{\nabla}^{2}v({\bm{r}},t),$
(2)

where $w({\bm{r}},t)$ and $v({\bm{r}},t)$ represent water concentration and
density of vegetation biomass at location ${\bm{r}}$ and time $t$,
respectively. In Eq. (1), water is continuously supplied at a precipitation
rate $R$, and its concentration decreases due to physical losses such as
evaporation, occurring at rate $l$ (second term), and local uptake by plants
(third term). In the latter, $a$ is the plant absorption rate, $g(w)$
describes the dependence of vegetation growth on water availability, and
$f(v)$ is an increasing function of vegetation density that represents the
positive effect that the presence of plants has on water infiltration.
Finally, water diffuses with a diffusion coefficient $D_{w}$. Similarly, Eq.
(2) accounts for vegetation growth due to water uptake (first term), plant
mortality at rate $m$ (second term), and plant dispersal (last term). In the
plant growth term, the parameter $q$ represents the yield of plant biomass per
unit of consumed water; although the original model assumed for mathematical
simplicity linear absorption rate and plant response to water
($g(w)=w({\bm{r}},t)$ and $f(v)=v({\bm{r}},t)$), other biologically-plausible
choices can account for processes such as saturation in plant growth due to
intraspecific competition (Eigentler,, 2020).

The generalized Klausmeier model with linear absorption rate and plant
responses to water has three spatially-uniform equilibria obtained from the
fixed points of Eqs. (1)-(2): an unvegetated state $(v^{*}=0,w^{*}=R/l)$,
stable for any value of the rainfall parameter $R$; and two states in which
vegetation and water coexist at different non-zero values. Only the vegetated
state with higher vegetation biomass is stable against non-spatial
perturbations, and only for a certain range of values of $R$. The latter
suffices to guarantee bistability, that is, the presence of alternative stable
states (vegetated vs unvegetated), and hysteresis. For spatial perturbations,
however, the stable vegetated state becomes unstable within a range of $R$,
and the system develops spatial patterns.

####  3.1.2 Three-equation water-vegetation dynamics: the Rietkerk model

The Rietkerk model extends the generalized Klausmeier model by splitting Eq.
(1) into two equations (one for surface water, and another one for soil water)
and including a term that represents water infiltration. Moreover, the
functions that represent water uptake and infiltration are nonlinear, which
introduces additional feedbacks between vegetation, soil water, and surface
water. The model equations are as follows:

$\displaystyle\frac{\partial u(\mathbf{r},t)}{\partial t}$ $\displaystyle=$
$\displaystyle
R-\alpha\frac{v(\mathbf{r},t)+k_{2}\,w_{0}}{v(\mathbf{r},t)+k_{2}}u(\mathbf{r},t)+D_{u}\nabla^{2}u(\mathbf{r},t)$
(3) $\displaystyle\frac{\partial w(\mathbf{r},t)}{\partial t}$
$\displaystyle=$
$\displaystyle\alpha\frac{v(\mathbf{r},t)+k_{2}\,w_{0}}{v(\mathbf{r},t)+k_{2}}u(\mathbf{r},t)-g_{m}\,\frac{v(\mathbf{r},t)\,w(\mathbf{r},t)}{k_{1}+w(\mathbf{r},t)}-\delta_{w}\,w(\mathbf{r},t)+D_{w}\nabla^{2}w(\mathbf{r},t)$
(4) $\displaystyle\frac{\partial v(\mathbf{r},t)}{\partial t}$
$\displaystyle=$ $\displaystyle
c\,g_{m}\,\frac{v(\mathbf{r},t)\,w(\mathbf{r},t)}{k_{1}+w(\mathbf{r},t)}-\delta_{v}\,v(\mathbf{r},t)+D_{v}\nabla^{2}v(\mathbf{r},t)$
(5)

where $u(\mathbf{r},t)$, $w(\mathbf{r},t)$, and $v(\mathbf{r},t)$ are the
density of surface water, soil water, and vegetation, respectively. In Eq.
(3), $R$ is the mean annual rainfall, providing a constant supply of water to
the system; the second term accounts for infiltration; and the diffusion term
accounts for the lateral circulation of water on the surface. In Eq. (4), the
first term represents the infiltration of surface water into the soil, which
is enhanced by the presence of plants; the second term represents water
uptake; the third one accounts for physical losses of soil water, such as
evaporation; and the diffusion term describes the lateral circulation of
underground water. Finally, the first term in Eq. (5) represents vegetation
growth due to the uptake of soil water, which is a function that saturates for
high water concentrations; the second term accounts for biomass loss at
constant rate due to natural death or external hazards; and the diffusion term
accounts for plant dispersal. The meaning of each parameter in the equations,
together with the values used in Rietkerk et al., (2002) for their numerical
analysis, are provided in Table 1.

In the spatially uniform case, this model allows for two different steady
states: a vegetated state in which vegetation, soil water, and surface water
coexist at non-zero values; and an unvegetated (i.e., desert) state in which
only soil water and surface water are non-zero. Considering the
parameterization in Table 1, the stability of each of these states switches at
$R=1$. For $R<1$, only the unvegetated equilibrium is stable against non-
spatial perturbations, whereas for $R>1$ the vegetated equilibrium becomes
stable and the desert state, unstable. When allowing for spatial
perturbations, numerical simulations using the parameterization in Table 1
show the existence of spatial patterns within the interval $0.7\lesssim
R\lesssim 1.3$, which is in agreement with analytical approximations (Gowda et
al.,, 2016). Within this range of mean annual rainfall, the patterns
sequentially transition from gaps to labyrinths to spots with increasing
aridity. For $R\approx 0.7$, the system transitions abruptly from the spotted
pattern to the desert state.

Parameter | Symbol | Value
---|---|---
$c$ | Water-biomass conversion factor |  $10$ (g mm-1 m-2)
$\alpha$ | Maximum infiltration rate |  $0.2$ (day-1)
$g_{m}$ | Maximum uptake rate |  $0.05$ (mm g-1 m-2 day-1)
$w_{0}$ | Water infiltration in the absence of plants |  $0.2$ (-)
$k_{1}$ | Water uptake half-saturation constant |  $5$ (mm)
$k_{2}$ | Saturation constant of water infiltration |  $5$ (g m-2)
$\delta_{w}$ | Soil water loss rate |  $0.2$ (day-1)
$\delta_{v}$ | Plant mortality |  $0.25$ (day-1)
$D_{w}$ | Soil water lateral diffusion |  $0.1$ (m2 day-1)
$D_{v}$ | Vegetation dispersal |  $0.1$ (m2 day-1)
$D_{u}$ | Surface water lateral diffusion |  $100$ (m2 day-1)
Table 1: Typical parameterization of the Rietkerk model (Rietkerk et al.,,
2002).

The Rietkerk model assumes constant rainfall, homogeneous soil properties, and
only local and short-range processes. Therefore, all the parameters are
constant in space and time, and patterns emerge from SDFs between vegetation
biomass and water availability alone. This simplification is, however, not
valid for most ecosystems. Arid and semi-arid regions feature seasonal
variability in rainfall (Salem et al.,, 1989). Kletter et al., (2009) showed
that, depending on the functional dependence between water uptake and soil
moisture, stochastic rainfall might increase the amount of vegetation biomass
in the ecosystem compared to a constant rainfall scenario. Moreover, the
properties of the soil often change in space. A widespread cause of this
heterogeneity is soil-dwelling macrofauna, such as ants, earthworms, and
termites (Pringle and Tarnita,, 2017). Bonachela et al., (2015) found that
heterogeneity in substrate properties induced by soil-dwelling macrofauna, and
modeled by space-dependent parameters, might interact with SDFs between water
and vegetation and introduce new characteristic spatial scales in the pattern.
Finally, researchers have also extended the Rietkerk model to account for
long-range, nonlocal processes. For example, Gilad et al., (2004) introduced a
nonlocal mechanism in the interaction between vegetation biomass and soil
water of Eqs. (4)-(5) to model the water conduction of lateral roots towards
the plant canopy. As explained in the next section, although this model
accounts for nonlocal processes, it is conceptually very different from
kernel-based models.

###  3.2 Kernel-based SDF models

Kernel-based models are those in which all the feedbacks that control the
interaction between plants are encapsulated in a single nonlocal net
interaction between plants. The nonlocality in the net plant interaction
accounts for the fact that individual (or patches of) plants can interact with
each other within a finite neighborhood. Therefore, the vegetation dynamics at
any point of the space is coupled to the density of vegetation at locations
within the interaction range. Because all feedbacks are collapsed into a net
interaction between plants, kernel-based models do not describe the dynamics
of any type of water and use a single partial integro-differential equation
for the spatiotemporal dynamics of the vegetation. The kernel is often defined
as the addition of two Gaussian functions with different widths, with the
wider function taking negative values to account for the longer range of
competitive interactions (D’Odorico et al.,, 2006) (central panels of Fig. 2).

####  3.2.1 Models with additive nonlocal interactions

In the simplest kernel-based SDF models, the spatial coupling is introduced
linearly in the equation for the local dynamics (D’Odorico et al.,, 2006),

$\frac{\partial v({\bm{r}},t)}{\partial t}=h\left(v\right)+\int
d{\bm{r}^{\prime}}G\left({\bm{r}^{\prime}};{\bm{r}}\right)\left[v\left({\bm{r}^{\prime}},t\right)-v_{0}\right].$
(6)

The first term describes the local dynamics of the vegetation, i.e., temporal
changes in vegetation density at a location ${\bm{r}}$ due to processes in
which neighboring vegetation does not play any role. The integral term
describes any spatial coupling, i.e., changes in vegetation density at
${\bm{r}}$ due to vegetation density at neighbor locations ${\bm{r}}$’.
Assuming spatial isotropy, the kernel function $G({\bm{r}},{\bm{r}^{\prime}})$
decays radially with the distance from the focal location,
$\rvert{\bm{r}^{\prime}}-{\bm{r}}\rvert$, so
$G\left({\bm{r}^{\prime}},{\bm{r}}\right)=G(\rvert{\bm{r}^{\prime}}-{\bm{r}}\rvert)$.
Therefore, the dynamics of the vegetation density is governed by two main
contributions: first, if the spatial coupling is neglected, the vegetation
density increases or decreases locally depending on the sign of $h(v)$ until
reaching a uniform steady state $v_{0}$, solution of $h(v_{0})=0$; second, the
spatial coupling enhances or diminishes vegetation growth depending on the
sign of the kernel function (i.e., whether the spatial interactions affect
growth positively or negatively) and the difference between the local
vegetation density and the uniform steady state $v_{0}$.

Assuming kernels that are positive close to the focal location and negative
far from it (a la SDF), local perturbations in the vegetation density around
$v_{0}$ are locally enhanced if they are larger than $v_{0}$ and attenuated
otherwise. As a result, the integral term destabilizes the homogeneous state
when perturbed, and spatial patterns arise in the system. Long-range growth-
inhibition interactions, together with nonlinear terms in the local-growth
function $h(v)$, avoid the unbounded growth of perturbations and stabilize the
pattern. However, although this mechanism imposes an upper bound to vegetation
density, nothing prevents $v$ from taking unrealistic, negative values. To
avoid this issue, the model must include an artificial bound at $v=0$ such
that vegetation density is reset to zero whenever it becomes negative.

####  3.2.2 Models with multiplicative nonlocal interactions

A less artificial way to ensure that vegetation density remain always positive
is to modulate the spatial coupling with nonlinear terms. For example, the
pioneering model developed by Lefever and Lejeune, (1997) consists of a
modified logistic equation in which each of the terms includes an integral
term to encode long-range spatial interactions,

$\frac{\partial v({\bm{r}},t)}{\partial
t}=\beta\,\left(\omega_{1}*v\right)({\bm{r}},t)\left[1-\frac{\left(\omega_{2}*v\right)({\bm{r}},t)}{K}\right]-\eta\left(\omega_{3}*v\right)({\bm{r}},t)$
(7)

where $\beta$ is the rate at which seeds are produced (a proxy for the number
of seeds produced by each plant) and $\eta$ is the rate at which vegetation
biomass is lost due to spontaneous death and external hazards such as grazing,
fires, or anthropogenic factors (last term). The model assumes spatial
isotropy, and the symbol $*$ indicates a linear convolution operation:

$\left(\omega_{i}*v\right)({\bm{r}},t)=\int
d{\bm{r}}^{\prime}\omega_{i}({\bm{r}}-{\bm{r}}^{\prime};\ell_{i})v({\bm{r}}^{\prime},t)\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i=1,2,3$ (8)

in which each $\omega_{i}$ is a weighting function with a characteristic
spatial scale $\ell_{i}$ that defines the size of the neighborhood
contributing to the focal process. For instance,
$\omega_{1}({\bm{r}}-{\bm{r}}^{\prime};\ell_{1})$ defines the size of the
neighborhood that contributes to the growth of vegetation biomass at
${\bm{r}}$. Similarly, $\ell_{2}$ defines the scale over which plants inhibit
the growth of their neighbors, and $\ell_{3}$ the scale over which vegetation
density influences the spontaneous death rate of vegetation at the focal
location. Because the sign of the interaction is explicit in each term of Eq.
(7), the convolutions only represent weighted averages of vegetation biomass
and the weighting functions are always positive. In addition, because Lefever
and Lejeune, (1997) set the scale of the inhibitory interactions larger than
the scale of the positive interactions ($\ell_{2}>\ell_{1}$), the model
includes a SDF with short-range facilitation and long-range competition.

Expanding upon this work, several other models have introduced non-linear
spatial couplings via integral terms (Fernandez-Oto et al.,, 2014; Escaff et
al.,, 2015; Berríos-Caro et al.,, 2020), and others have expanded the integral
terms and studied the formation of localized structures of vegetation (Parra-
Rivas and Fernandez-Oto,, 2020).

###  3.3 Kernel-based PCF models.

In previous sections, we invoked the existence of SDFs in the interactions
among plants to explain the emergence of self-organized spatial patterns of
vegetation. Both theoretical and empirical studies, however, have highlighted
the importance of long-range negative feedbacks on pattern formation,
suggesting that short-range positive feedbacks might be secondary actors that
sharpen the boundaries of clusters rather than being key for the instabilities
that lead to the patterns (van de Koppel and Crain,, 2006; Rietkerk and van de
Koppel,, 2008; van de Koppel et al.,, 2008). Following these arguments,
Martínez-García et al., 2013a , (2014) proposed a family of purely competitive
models with the goal of identifying the smallest set of mechanisms needed for
self-organized vegetation patterns to form.

The simplest PCF models consider additive nonlocal interactions (Martínez-
García et al.,, 2014). Alternatively, nonlocal interactions can be represented
through nonlinear functions modulating either the growth or the death terms.
In both cases, the models develop the full sequence of patterns (gaps,
labyrinths, and spots). The model proposed in Martínez-García et al., 2013a
introduces competition through the growth term:

$\frac{\partial v({\bm{r}},t)}{\partial
t}=P_{\mbox{\tiny{E}}}\left(\widetilde{v},\delta\right)\beta\,v({\bm{r}},t)\left(1-\frac{v({\bm{r}},t)}{K}\right)-\eta\,v({\bm{r}},t),$
(9)

where $\beta$ and $K$ are the seed production rate and the local carrying
capacity, respectively. $\delta$ is the competition-strength parameter, and
$\widetilde{v}\left({\bm{r}},t\right)$ is the average density of vegetation
around the focal position ${\bm{r}}$:

$\widetilde{v}\left({\bm{r}},t\right)=\int
d{\bm{r}}^{\prime}\mathcal{G}\left(\rvert{\bm{r}^{\prime}}-{\bm{r}}\rvert\right)\,v\left({\bm{r}}^{\prime},t\right).$
(10)

where the kernel function $\mathcal{G}$, assumes spatial isotropy and weighs
how vegetation at a location ${\bm{r}^{\prime}}$ contributes to the average
vegetation density around ${\bm{r}}$, and is necessarily defined positive.
Because it is a weighting function, $\mathcal{G}$ plays the same role and has
the same properties described for the $\omega_{i}$ functions in section 3.2.
The model further assumes that vegetation losses occur at constant rate $\eta$
and vegetation grows through a three-step sequence of seed production, local
dispersal, and establishment (Calabrese et al.,, 2010), represented by the
three factors that contribute to the first term in Eq. (9). First, plants
produce seeds at a constant rate $\beta$, which leads to the a growth term
$\beta v({\bm{r}},t)$. Second, seeds disperse locally and compete for space
which defines a local carrying capacity $K$. Third, plants compete for
resources with other plants, which is modeled using a plant establishment
probability, $P_{\mbox{\tiny{E}}}$. Because the only long-range interaction in
the model is root-mediated interference, and competition for resources is more
intense in more crowded environments, $P_{E}$ is a monotonically decreasing
function of the nonlocal vegetation density $\tilde{v}({\bm{r}},t)$ defined in
Eq. (10). Moreover, $P_{\mbox{\tiny{E}}}$ also depends on the competition-
strength parameter, $\delta$, representing resource limitation. In the limit
$\delta=0$, resources are abundant, competition is weak, and
$P_{\mbox{\tiny{E}}}=1$. Conversely, in the limit $\delta\rightarrow\infty$,
resources are very scarce, competition is very strong, and
$P_{\mbox{\tiny{E}}}\rightarrow 0$.

In PCF models, spatial patterns form solely due to the long-range competition.
If patches are far enough from each other, individuals attempting to establish
in between patches compete with vegetation from more than one adjacent patch,
whereas individuals within a patch only interact with plants in that same
patch. As a result, competition is more intense in the regions between patches
than inside each patch, which stabilizes an aggregated pattern of vegetation
(Fig. 4) whose shape (gaps, labyrinths or spots) will depend on the model
parameterization. This same mechanism has been suggested to drive the
formation of clusters of competing species in the niche space (Scheffer and
van Nes,, 2006; Pigolotti et al.,, 2007; Hernández-García et al.,, 2009; Fort
et al.,, 2009; Leimar et al.,, 2013) and the aggregation of individuals in
models of interacting particles with density-dependent reproduction rates
(Hernández-García and López,, 2004).

Figure 4: In PCF models, patchy distributions of vegetation in which the
distance between patches is between one and two times the range of the
nonlocal interactions are stable. Individuals within each patch only compete
with the individuals in that patch (a,b), whereas individuals in between
patches compete with individuals from both patches (c). Color code: green
trees are focal individuals, and dashed circles limit the range of interaction
of the focal individual. Dark grey is used for individuals that interact with
the focal one, whereas light gray indicates individuals that are out of the
range of interaction of the focal individual.

##  4 Self-organized patterns as indicators of ecological transitions

Models using different forms for the net biotic interaction between neighbor
patches (SDF vs PCF) have successfully reproduced qualitatively the spatial
patterns of vegetation observed in water-limited ecosystems (Deblauwe et al.,,
2011). All these different models also predict that the spotted pattern
precedes a transition to an unvegetated state, which positions it as an early-
warning indicator of desertification transitions (Scheffer et al.,, 2009;
Dakos et al.,, 2011). Despite the similar prediction, models with different
underlying mechanisms for the formation of these patterns result in different
desertification processes.

The Rietkerk model from section 3.1.2, for example, predicts that self-
organized ecosystems eventually collapse following an abrupt transition that
includes a hysteresis loop (Fig. 5a). Abrupt transitions such as this one are
typical of bistable systems in which the stationary state depends on the
environmental and the initial conditions. Bistability is a persistent feature
of models for vegetation pattern formation, sometimes occurring also in
transitions between patterned states (von Hardenberg et al.,, 2001). It also
denotes the existence of thresholds in the system that trigger sudden, abrupt
responses in its dynamics. These thresholds are often created by positive
feedbacks or quorum-regulated behaviors, as is the case in populations subject
to strong Allee effects (Courchamp et al.,, 1999). In the Rietkerk model, as
rainfall decreases, the spatial distribution of vegetation moves through the
gapped-labyrinthine-spotted sequence of patterns (Fig. 5a). However, when the
rainfall crosses a threshold value the system responds abruptly, and all
vegetation dies. Setting up the simulations as indicated in Rietkerk et al.,
(2002) and using the parameterization of Table 1, this threshold is located at
$R\approx 0.55$ mm day-1. Once the system reaches this unvegetated state,
increasing water availability does not allow vegetation recovery until
$R\approx 0.70$ mm day-1, which results in a hysteresis loop and a region of
bistability ($R\in[0.55,0.70]$ in Fig. 5a). Bistability and hysteresis loops
make abrupt, sudden transitions like this one extremely hard to revert. Hence,
anticipating such abrupt transitions is crucial from a conservation and
ecosystem-management point of view (Scheffer et al.,, 2009; Dakos et al.,,
2011).

Extended versions of the Rietkerk model have suggested that the interaction
between vegetation and other biotic components of the ecosystem may change the
transition to the unvegetated state. Specifically, Bonachela et al., (2015)
suggested that soil-dwelling termites, in establishing their nests (mounds),
engineer the chemical and physical properties of the soil in a way that turns
the abrupt desertification into a two-step process (Fig. 5b). At a certain
precipitation level ($R\approx 0.75$ mm day-1 using the parameterization in
Table 1 and the same initial condition used for the original Rietkerk model),
vegetation dies in most of the landscape (T1 in Fig. 5b) but persists on the
mounds due to improved properties for plant growth created by the termites.
On-mound vegetation survives even if precipitation continues to decline, and
is finally lost at a rainfall threshold $R\approx 0.35$ mm day-1 (T2 in Fig.
5b). As a consequence of the two-step transition, the ecosystem collapse is
easier to prevent because a bare soil matrix with vegetation only on mounds
serves as an early-warning signal of desertification, and it is easier to
revert since termite-induced heterogeneity breaks the large hysteresis loop of
the original model into two smaller ones (compare the hysteresis loops in
Figs. 5a and 5b).

On the other hand, the PCF model in Martínez-García et al., 2013a , ()
discussed in section 3.3 predicts a smooth desertification in which vegetation
biomass decreases continuously in response to decreasing seed production rate
(a proxy for worsening environmental conditions). It is important to say that
considering local vegetation dispersal, and thus not introducing a diffusive
term in model Eq. (9), is also one of the assumptions leading to this smooth
transition. According to this model, the spotted pattern would persist as
precipitation declines, with vegetation biomass decreasing until it eventually
disappears (Fig. 5c). As opposed to the catastrophic shift described for the
Rietkerk model, smooth transitions such as the one depicted by this model do
not show bistability and do not feature hysteresis loops. This difference has
important socio-ecological implications, because it enables easier and more
affordable management strategies to restore the ecosystem after the collapse
(Villa Martín et al.,, 2015). Moreover, continuous transitions are also more
predictable because the density of vegetation is univocally determined by the
control parameter (seed production rate $\beta$ in Fig. 5c).

Figure 5: Although different models for vegetation pattern formation may
recover the same sequence of gapped-labyrinthine-spotted patterns from
different mechanism, the type of desertification transition that follows the
spotted pattern strongly depends on the model ingredients. a) Abrupt
desertification as predicted by the Rietkerk model (Rietkerk et al.,, 2002).
b) Two-step desertification process as predicted in Bonachela et al., (2015)
when soil-dwelling insects are added to the Rietkerk model. c) Progressive
desertification as predicted by the PCF model introduced in Martínez-García et
al., 2013a . For each panel, numerical simulations were conducted using the
same setup described in the original publications.

In the three scenarios discussed above, patterns have tremendous potential for
ecosystem management as an inexpensive and reliable early indicator of
ecological transitions (Scheffer et al.,, 2009; Dakos et al.,, 2011). However,
the different predictions that models make about the transition highlights the
need for tailored models that not only reproduce the observed patterns but do
so through the mechanisms relevant to the focal system. We have shown that
widespread spotted patterns can form in models accounting for very different
mechanisms (Fig. 5). Because ecosystems are highly complex, it is very likely
that spotted patterns observed in different regions emerge from very different
mechanisms (or combinations of them) and thus anticipate very different
transitions (Kéfi et al., 2007a, ; Kéfi et al., 2007b, ; Corrado et al.,,
2015; Weissmann et al.,, 2017). Therefore, a reliable use of spotted patterns
as early warning indicators of ecosystem collapse requires (i) mechanistic
models that are parameterized and validated by empirical observations of both
mechanisms and patterns; (ii) quantitative analyses of field observations; and
(iii) manipulative experiments.

##  5 Testing models for vegetation self-organization in the field

As depicted through this review, the study of self-organized vegetation
patterns has been mostly theoretical and empirical evidences of the self-
organization hypotheses explaining the formation of vegetation spatial
patterns are much less widespread. In this section, we explore the possible
empirical approaches to understanding the mechanisms responsible for the
formation and persistence of self-organized aggregated patterns and propose a
two-step protocol to this end.

At least three possible approaches exist to study existing self-organization
hypotheses for pattern formation in natural systems. First, the observation
and measurement of the spatial structure of patterns from aerial and satellite
photography (the observational approach). Second, the assessment of the net
interactions among plants or plant patches as a function of the distance
between them (the net-interaction approach). Third, the investigation of the
specific mechanisms underpinning that interaction (the selective mechanistic
approach). The observational approach has been relatively common, and is the
one that has motivated the development of most existing models (Borgogno et
al.,, 2009; Wu et al.,, 2000). Nevertheless, without a more detailed
understanding of the focal systems, one cannot discard whether agreement
between model predictions and natural observations is coincidental. The
mechanistic approach has been relatively common. Some studies have
investigated selectively the mechanisms potentially leading to pattern
formation in tiger bush banded vegetation in Niger (Valentin and Poesen,,
1999), vegetation rings in Israel (Sheffer et al.,, 2011; Yizhaq et al.,,
2019), or fairy circles in Namibia (Ravi et al.,, 2017) and Australia (Getzin
et al.,, 2021). To our knowledge, the net-interaction approach is conspicuous
by being absent, and researchers have scarcely measured directly the net
interaction among plants or vegetation patches and its variation with the
inter-plant distance in patterned ecosystems.

Because we are discussing here aggregated patterns, a first step in such
approach is to confirm that vegetation patches are formed by several
aggregated individuals. In the case of segregated patterns, we recommend the
use of point pattern analysis under the hypothesis of the dominance of
competition forces (Franklin,, 2010). Following this preliminary test, we
propose a two-step protocol to conduct future field reseearch on the emergence
and maintenance of vegetation spatial patterns.

First step: phenomenological investigation of the net interaction among plants
within the vegetated patch and in bare soil. This first step is needed because
a myriad of alternative mechanisms can explain the formation of spatial
patterns, some of them well aligned with the idea of vegetation self-
organization and others dependent on external biological or geological
factors. For example, in the case of fairy circles researchers have explored
the role of higher evaporation (Vlieghe and Picker,, 2019) and increased
termite activity (Juergens,, 2013) within the circles; spatial heterogeneity
in hydrological processes, such as increased infiltration in the circles of
bare soil (Ravi et al.,, 2017) or increased water runoff in the circles and in
the matrix (Getzin et al.,, 2021); and the geological emanation of toxic gases
(Naudé et al.,, 2011) and the presence of allelochemical substances (Meyer et
al.,, 2020) in the circles. By using a phenomenological approach as a first
step, researchers can discard many of these potential mechanisms, directing
their efforts towards more specific hypotheses in a second step. For this
first step, a simple experimental setup, based on mainstream methods to
measure plant biotic interaction would reveal whether PCF, SDF, or none of
them are responsible for the emergent pattern (Armas et al.,, 2004). Our
proposed experiment would compare a fitness proxy (e.g., growth, survival) of
experimental plants planted in the system under study, where we observe a
regular vegetation pattern and we assume that the vegetation vegetation
dynamics has reached a stationary state for the location-specific
environmental conditions. Each experimental block would consist of a plant
growing under-canopy (Fig. 6a), a plant growing in bare soil (i.e., between
two vegetation patches) (Fig. 6b), and a control plant growing in the same
ecosystem but artificially isolated from the interaction with pattern-forming
individuals (Fig. 6c). To isolate control plants from canopy interaction they
need to be planted in bare soil areas. To isolate them from below-ground
competition, one can excavate narrow, deep trenches in which a root barrier
can be inserted (Morgenroth,, 2008). To isolate them from the competition for
runoff water with the vegetation patches, root barriers should protrude from
the soil a few centimeters, preventing precipitation water to leave the area
and runoff water to enter. Comparing the fitness proxy of the control plant
with that of plants growing in vegetation patches or bare soil reveals the
sign and strength of the net biotic interaction. By replicating this
experimental block, we can statistically determine whether the pattern
formation results from a SDF, a PCF, or whether it involves a different
process. The SDF hypothesis would be validated if a predominantly positive net
interaction is observed under the canopy, and a negative interaction is
observed in bare soils. Conversely, the PCF would be validated if we observe a
negative net interaction in bare soils and under canopy (see Table 2). Any
other outcome in the spatial distribution of the sign of the net interaction
between plants would suggest that other mechanisms are at play, which could
include the action of different ecosystem components, such as soil-dwelling
macrofauna (Tarnita et al.,, 2017), or abiotic factors, such as micro-
topology.

Figure 6: Schematic representation of a simple experimental setup to test in the field whether the mechanism of spatial patterning is a purely competitive feedback (PCF) or a classic scale-dependent feedback (SDF). Plant (a) is an experimental plant growing under-canopy, (b) is growing in bare soil, and (c) is a control plant growing in artificial conditions, free from the biotic interaction using root barriers in bare soil areas of the same environment.  Under canopy vs control | Bare soil vs control | Outcome
---|---|---
$0/-$ | $-\ \ -$ | Purely competitive feedback
$+$ | $-$ | Scale-dependent feedback
Table 2: Testing the PCF versus SDF hypotheses in the experimental setup
introduced in Fig. 6. Double signs indicate stronger intensity. Indexes to
calculate the sign of the net interaction can be taken from Armas et al.,
(2004).

Second step: direct measurement of the biophysical processes responsible for
the pattern. After confirming the PCF, the SDF, finding an alternative spatial
distribution of plant interactions or rejecting the self-organizing
hypothesis, a second experimental step would test specific biophysical
mechanisms responsible for the measured interaction and driving the spatial
pattern. For example, PCF models hypothesize that spatial patterns are driven
by long-range below-ground competition for a limiting resource through the
formation of exclusion regions. As discussed in section 3.3, these exclusion
regions are territories between patches of vegetation in which the intensity
of competition is higher than within the patch (van de Koppel and Crain,,
2006), possibly because they present a higher density of roots (Fig. 4)
(Martínez-García et al., 2013a, ; Martínez-García et al.,, 2014). To test for
the existence of exclusion regions and confirm whether below-ground
competition is driving the spatial pattern, researchers can measure the
changes in root density using coring devices (Cabal et al.,, 2021) across soil
transects going from the center of a vegetated patch to the center of a bare
soil patch. Field tests and manipulative experiments to confirm that SDFs are
responsible for vegetation patterns are not easy to perform but researchers
can do a handful of analyses. For example, ecohydrological SDF models assume
that water infiltration is significantly faster in vegetation patches than in
bare soil areas (Rietkerk et al.,, 2002). Many empirical researchers have
tested this assumption in patterned vegetation using mini disk infiltrometers
to quantify unsaturated hydraulic conductivity, dual head infiltrometers to
measure saturated hydraulic conductivity, or buried moisture sensors connected
to data loggers to record volumetric soil moisture content (Yizhaq et al.,,
2019; Ravi et al.,, 2017; Getzin et al.,, 2021; Cramer and Barger,, 2013). The
measures should show higher infiltration rates and moisture under vegetated
patches than in bare soil. Note, however, that infiltration rates might be
very hard to measure due to small-scale soil heterogeneities. Ecohydrological
models make other assumptions that are less often considered in the field. For
instance, they assume that the lateral transport of water is several orders of
magnitude larger than vegetation diffusion (i.e., patch growth speed), which
might be true or not depending on soil properties. To test these assumptions,
field researchers need to measure the intensity of the water runoff and
compare it to a measure of the lateral growth rate of vegetation patches.
Water runoff is very challenging to measure directly, but estimates can be
calculated using the infiltration rates obtained with infiltrometers (Cook,,
1946). The lateral growth rate of vegetation patches can be estimated based on
drone or satellite images repeated over time (Trichon et al.,, 2018).
Combining measures of both water runoff and expansion rates of vegetation
patches, one can estimate approximated values for the relative ratio of the
two metrics.

##  6 Conclusions and future lines of research

As our ability to obtain and analyze large, high-resolution images of the
Earth’s surface increases, more examples of self-organized vegetation patterns
are found in water-limited ecosystems. Here, we have reviewed different
modeling approaches employed to understand the mechanistic causes and the
predicted consequences of those patterns. We have discussed how different
models, relying on different mechanisms, can successfully reproduce the
patterns observed in natural systems and that, however, each of these models
predicts very different ecosystem-level consequences of the emergent pattern.
This discrepancy limits the utility of the patterns as applied ecological
tools in the absence of explicit knowledge of underlying mechanisms. To solve
this issue, we claim that models need to move from their current
phenomenological formulation towards a more system-specific and mechanistic
one. This new approach must necessarily focus on isolating the system-
specific, key feedbacks for vegetation self-organization. We identify two main
directions for future research to develop this new approach to vegetation
pattern formation.

First, biologically-grounded studies should aim to combine system-specific
models with empirical measures of vegetation-mediated feedbacks. Existing
models for vegetation self-organization are mostly phenomenological and are
only validated qualitatively via the visual comparison of simulated and
observed (macroscopic) patterns. Experimental measures of the (microscopic)
processes and feedbacks central to most models of vegetation pattern formation
are hard to obtain, leading to arbitrary (free) parameter values and response
functions. For example, very few models incorporate empirically-validated
values of water diffusion and plant dispersal rates, despite the crucial role
of these parameters in the emergence of patterns. Instead, these models fine-
tune such values to obtain patterns similar in, for example, their wavelength,
to the natural pattern. Similarly, we are only beginning to understand how
plants rearrange their root system in the presence of competing individuals
(Cabal et al., 2020a, ), and hence kernel-based models still do not
incorporate realistic functional forms for the kernels. Instead, these models
use phenomenological functions to test potential mechanisms for pattern
formation by qualitatively comparing model output and target pattern, thus
limiting the potential of the models to make quantitative predictions. To
establish a dialogue between experiments and theory, models should develop
from a microscopic description of the system (DeAngelis and Yurek,, 2016;
Railsback and Grimm,, 2019). This approach allows for a more realistic and
accurate description of the plant-to-plant and plant-water interactions as
well as for a better reconciliation between model parameters and system-
specific empirical measures. Hopefully it will also allow researchers to unify
existing theoretical approaches to vegetation pattern formation (Martínez-
García et al.,, 2014). Subsequently, existing tools from mathematics,
statistical physics, and/or computer science can be used to reach a
macroscopic PDEM that captures the key ingredients of the microscopic
dynamics. Statistical physics, which was conceived to describe how observed
macroscopic properties of physical systems emerge from the underlying
microscopic processes, provides a compelling and well-developed framework to
make such a micro-macro connection.

Second, recent developments in remotely sensed imagery have enabled the
measurement of an ecosystem’s state indicators, which will allow researchers
to compare observed and simulated patterns quantitatively (Bastiaansen et
al.,, 2018). On the one hand, using existing databases of ecosystem responses
to aridity (Berdugo et al.,, 2020) and satellite imagery of vegetation
coverage (Deblauwe et al.,, 2011), researchers could conduct a model selection
analysis and classify existing models from more to less realistic depending on
whether (and how many) features of the focal ecosystem the model manages to
reproduce in the correct environmental conditions. For example, models could
be classified depending on whether, after proper parameterization, they can
predict ecosystem responses such as transitions between pattern types at the
correct aridity thresholds. To elaborate this model classification, the use of
Fourier analysis for identifying regularity in natural patterns, geostatistics
for quantifying spatial correlations, and time series analysis for tracking
changes in the ecosystem properties through time will be essential.

Beyond water-limited ecosystems, SDFs have been invoked to explain spatial
pattern formation in mussel beds (Rietkerk and van de Koppel,, 2008),
freshwater and salt marshes (van de Koppel et al.,, 2005; Van Wesenbeeck et
al.,, 2008; Zhao et al.,, 2021), and seagrasses (van der Heide et al.,, 2011;
Ruiz-Reynés et al.,, 2017). Conversely, nonlocal competition drives the
emergence of aggregated patterns in freshwater marshes (van de Koppel et al.,,
2008) and in theoretical models of population dynamics (Fuentes et al.,, 2003;
Dornelas et al.,, 2019; Maruvka and Shnerb,, 2006; Da Cunha et al.,, 2011;
Clerc et al.,, 2005; Maciel and Martinez-Garcia,, 2020). Understanding the
conditions in which negative feedbacks dominate over positive feedbacks,
finding the key features that distinguish the patterns caused by these
different feedbacks, and contrasting their divergent ecological consequences
constitutes an exciting venue for future research that has just started to
develop (Lee et al.,, 2021).

### Acknowledgments

We thank Robert M. Pringle, Rubén Juanes, and Ignacio Rodríguez-Iturbe for
various discussions at different stages of the development of this work. This
work is supported by FAPESP through grants ICTP-SAIFR 2016/01343-7, and
Programa Jovens Pesquisadores em Centros Emergentes 2019/24433-0 and
2019/05523-8, Instituto Serrapilheira through grant Serra-1911-31200, and the
Simons Foundation (RMG); the Princeton University May Fellowship in the
department of Ecology and Evolutionary Biology (CC); MINECO/AEI/FEDER through
the María de Maeztu Program for Units of Excellence in R&D (MDM-2017-0711,
Spain) (EHG & CL); The Gordon and Betty Moore Foundation, grant #7800 (CET &
JAB); and the NSF grant DMS-2052616 (JAB). This work was partially funded by
the Center of Advanced Systems Understanding (CASUS) which is financed by
Germany’s Federal Ministry of Education and Research (BMBF) and by the Saxon
Ministry for Science, Culture and Tourism (SMWK) with tax funds on the basis
of the budget approved by the Saxon State Parliament (JMC).

### Author contribution

Conceptualization, R.M.G., C.C., J.A.B., J.M.C., E.H.G., C.L., C.E.T.;
writing–original draft preparation, R.M.G.; writing–review and editing,
R.M.G., C.C., J.A.B., J.M.C., E.H.G., C.L., C.E.T.; visualization, R.M.G.,
C.C. All authors have read and agreed to the published version of the
manuscript.

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