Integrating theory and experiments to link local mechanisms and ecosystem-level consequences of vegetation patterns in drylands
Ricardo Martinez-Garcia, Ciro Cabal, Justin M. Calabrese, Emilio Hernández-García, Corina E. Tarnita, Cristóbal López, Juan A. Bonachela
AArticle
Spatial self-organization of vegetation in water-limitedsystems: mechanistic causes, empirical tests, andecosystem-level consequences
Ricardo Martinez-Garcia ∗ , Ciro Cabal , Juan A. Bonachela , Justin M. Calabrese , EmilioHernández-García , Cristóbal López , Corina E. Tarnita ICTP South American Institute for Fundamental Research & Instituto de Física Teórica - Universidade EstadualPaulista, Rua Dr. Bento Teobaldo Ferraz 271, Bloco 2 - Barra Funda 01140-070 São Paulo, SP Brazil; Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey, United States ofAmerica; Department of Ecology, Evolution, and Natural Resources, Rutgers University, New Brunswick, 08901 NJ (USA) Center for Advanced Systems Understanding (CASUS), Görlitz, Germany; Helmholtz-Zentrum Dresden Rossendorf (HZDR), Dresden, Germany; Department of Ecological Modelling, Helmholtz Centre for Environmental Research – UFZ, Leipzig, Germany; IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), E-07122 Palma de Mallorca, Spain. * Correspondence: [email protected]† These authors contributed equally to this workReceived: date; Accepted: date; Published: date
Abstract:
Self-organized spatial patterns of vegetation are frequent in water-limited regions and havebeen suggested as important ecosystem health indicators. However, the mechanisms underlying theirformation remain unclear. It has been hypothesized that patterns could emerge from a water-mediatedscale-dependent feedback (SDF), whereby interactions favoring plant growth dominate at short distanceswhile growth-inhibitory interactions dominate in the long range. As precipitation declines, thisframework predicts a sequential change from gapped to labyrinthine to spotted spatial patterns. However,we know little about how net plant-to-plant interactions may shift from positive to negative as a functionof inter-individual distance, and in the absence of strong empirical support, the relevance of SDF forvegetation pattern formation remains disputed. Alternative theories show that the same sequence ofpatterns could emerge even if net interactions between plants are always inhibitory, provided thattheir intensity decays sharply enough with inter-individual distance. Importantly, although thesealternative hypotheses lead to visually indistinguishable spatial distributions of plants, the two differentframeworks predict different ecosystem-level consequences for these resulting patterns, thus limiting theirpotential use as ecosystem-state indicators. Moreover, the interaction of vegetation with other ecosystemcomponents can alter the dynamics of the pattern or even introduce additional spatio-temporal scales.Therefore, to make reliable ecological predictions, models need to accurately capture the mechanismsat play in the systems of interest. Here, we review existing theories for vegetation self-organizationand their conflicting ecosystem-level predictions. We further discuss possible ways for reconcilingthese predictions. We focus on the mechanistic differences among models, which can provide valuableinformation to help researchers decide which model to use for a particular system and/or whether itrequires modification.
Keywords:
Water-limited ecosystems, partial differential equations, ecological patterns,reaction-diffusion systems, nonlocal models, scale-dependent feedback, Turing patterns. a r X i v : . [ q - b i o . P E ] J a n of 27 Contents1 Introduction 22 Ecological rationale behind models for vegetation spatial self-organization 43 Review of models for vegetation self-organization 5
Self-organized patterns are ubiquitous in complex biological systems. These regular structures,which can cover large portions of the system, emerge due to many nonlinear interactions among systemcomponents. Examples can be found at any spatiotemporal scale, from microbial colonies [1–3], toentire landscapes [4,5], and both in motile and in sessile organisms [6–8]. Importantly, because harshenvironmental conditions provide a context in which self-organization becomes important, self-organizedpatterns contain important information about physical and biological processes that occur in the systemsin which they form [9].A well-known example of self-organization in ecology is vegetation pattern formation in water-limitedregions [10,11]. Despite forming in very different conditions, both biotic (vegetation species, presence ofdifferent types of fauna) and abiotic (soil type, seasonality, levels of rainfall), these patterns consistentlyshow the same shapes: vegetation spots overdispersed on a matrix of bare soil, soil-vegetation labyrinths,and gaps of bare soil overdispersed on a homogeneous layer of vegetation (see [4,10,12] for a globalcompilation of pattern locations). Importantly, ecosystem water availability strongly determines thespecific shape of the pattern. A Fourier-based analysis of satellite imagery covering extensive areas ofSudan revealed that more humid regions are dominated by gapped patterns, whereas spotted patternsdominate in more arid conditions [13], in agreement with model predictions [14,15]. However, imagery of 27 time series are not long enough to observe whether vegetation cover in a specific region undergoes thesetransitions between patterns in response to growing aridity.After the spotted pattern, models predict that patterned ecosystems undergo a transition to adesert state if precipitation continues to decrease. The observed correlation between pattern shapeand water availability suggests that the spotted pattern could serve as a reliable and easy-to-identifyearly-warning indicator of this ecosystem shift [16–20]. This has reinforced the motivation to developseveral models aiming to explain both the formation of spatial patterns of vegetation and their dependenceon environmental variables [12,14,21–24]. Although Bastiaansen et al. [25] has recently tested some modelpredictions using satellite imagery, theoretical studies using models remain the dominant approach tostudy this hypothesized transition.Spatially-explicit models of vegetation dynamics fall into two main categories. Individual-basedmodels (IBM) describe each plant as a discrete entity whose attributes change in time following a stochasticupdating rule [26–28]. Continuum models describe vegetation biomass and water concentration ascontinuous fields that change in space and time following a system of deterministic partial differentialequations (PDEMs) [29,30]. Because they incorporate much more detail than PDEMs, IBMs requirecomputationally intensive numerical simulations, which makes it difficult to extract general conclusionsabout the fundamental mechanisms that drive the emergence of population-level behaviors and patterns(but see [31–39] for examples of discrete models and analytical tools to solve them). PDEMs, incontrast, neglect most of the details incorporated by IBMs, which makes them analytically tractableas spatially-extended dynamical systems [29,40]. IBMs and PDEMs thus constitute complementaryapproaches to study spatial vegetation dynamics: the former allow for more quantitative, system-specificpredictions, whereas the latter provide more general insights into vegetation pattern formation and theirecological implications if they include the right set of mechanisms. Because here we are interested in thegeneral features of self-organized vegetation dynamics, we focus on PDEMs and discuss how IBMs mayinform improved PDEMs.We review different families of models, discussing how spatial patterns of vegetation emerge andtheir predictions for the ecosystem-level consequences of the patterns. From a mathematical point ofview, we can group PDEMs vegetation models into two main classes: (i) Turing-like models that use asystem of PDEs [41] to describe the coupled dynamics of water and plants, and (ii) kernel-based modelsthat describe the dynamics of the vegetation using a single partial integrodifferential equation in whichthe net interaction between plants is coded in a kernel function [12]. Regardless of their mathematicalstructure, we will refer to models accounting for both positive and negative feedbacks as scale-dependentfeedback (SDF) models. On the other hand, we will refer to all models in which only negative feedbacksare considered as purely competitive (PC). Models within each of these two classes will range from thesimplest ones that capture the two different mechanisms, to the more complex, which include additionalprocesses such as two competing species of plants [42], interactions between vegetation and fauna [43,44],soil-vegetation feedbacks [45–48], landscape topography [49], and different sources of variability, includingboth environmental [50–56] and demographic [57,58].Significantly, although all these models successfully reproduce the sequence of gapped, labyrinthine,and spotted patterns found in satellite imagery, they disagree in their predictions regarding the nature ofthe desertification transition that follows the spotted pattern. Rietkerk et al. [21], for instance, developedan SDF model for vegetation biomass, soil moisture, and surface water and showed that ecosystems mayundergo abrupt desertification, including a hysteresis loop, following the spotted pattern. von Hardenberg et al. [14] used a different SDF model that only accounts for groundwater and vegetation biomass dynamicsand predicted abrupt desertification following the spotted pattern. However, they also found multistabilitybetween patterned states, i.e., for fixed environmental conditions, the shape of the stationary patterndepends on the initial state. Finally, Martínez-García et al. [59] developed a family of purely competitive of 27 models in which desertification occurs gradually with progressive loss of vegetation biomass. The natureof this transition has significant ecological consequences. Abrupt transitions like those predicted byRietkerk et al. [21] and von Hardenberg et al. [14] are almost irreversible, entail hysteresis, and due to theircatastrophic and abrupt character, are difficult to prevent. Continuous transitions, however, are mucheasier to predict and, therefore, to manage. Determining whether ecosystems will respond abruptly orgradually to aridification is critical both from an ecosystem-management and socio-economic point ofview because water-limited ecosystems cover 40% of Earth’s land surface and are home to 35% of theworld population [60].Active lines of theoretical research have focused on understanding how different components ofthe ecosystem may interact with each other to determine an ecosystem’s response to aridification [46],as well as on designing synthetic feedbacks (in the form of artificial microbiomes) that could prevent orsmooth ecosystems collapses [61–63]. The question has also attracted considerable attention from empiricalresearchers [64]. Whether desertification is more likely to occur gradually or abruptly remains largelyunknown, despite evidence suggesting that certain structural and functional ecosystem attributes respondabruptly to aridity [65].Here, we outline and rank strategies to answer this question. In section 2, we discuss the ecologicalrationale behind PDEMs for vegetation self-organization. In section 3, we review different families ofPDEMs for vegetation self-organization. Next, in section 4, we show that, although all lead to seeminglyidentical patterns, different models predict very different transitions into the desert state, limiting thereliability of model predictions regarding how or when the transition will occur and the underlyingmechanisms. In section 5, we discuss possible manipulative experiments and empirical measures thatcould support or discard each of the previously scrutinized models. Finally, in section 6, we envisiondifferent research lines that build on these results and discuss how to apply lessons learned from studyingself-organized vegetation patterns to other self-organizing biological and physical systems.
2. Ecological rationale behind models for vegetation spatial self-organization
Models of spatial self-organization of vegetation rely on simple ecological assumptions about thescale-dependence of the net biotic interaction among individual plants. That is, about the effect thatthe presence of one individual has on the growth and survival of its neighbors as a function of theinter-individual distance. However, this net effect is a simplification, and the mechanisms underpinning thenet interaction between individuals can be very complex [66]. In the case of vegetation, such mechanismsare based on the biophysical effects of the plant canopy on the microclimate underneath and of the rootsystem on the soil conditions (Fig. 1a). While some of these mechanisms are well studied by ecologists,we know little about how they scale with the distance between individual (or clumps of) plants, makingexisting models hard to parameterize using empirical observations.The rationale behind scale-dependent feedbacks is diverse and based on different empiricalobservations. For example, in semiarid and arid open-canopy systems, where the range of the rootsystem is larger than the canopy cover, the positive effects of shade can overcome competition for lightand even be stronger than the effects of root competition, thereby leading to under-canopy facilitation[67]. In this context, focal plants have an overall facilitative effect in the area of most intense shade at thecenter of the crown, which progressively loses intensity and vanishes as shading disappears and givesrise to simple below-ground competition in areas farther from the plant (Fig. 1b). A different rationale isnecessary for models in which the net biotic interaction emerges from the competition between plants forwater or, more specifically, from the capacity that plants have to modify soil structure and porosity, andtherefore enhance soil water infiltration [68]. Enhanced water infiltration has a direct positive effect nearthe plant because it increases soil water content but, as a by-product, it has negative consequences farther of 27 away from its insertion point because, by increasing local infiltration, plants also reduce the amount ofwater that can infiltrate further away in plant-less bare soil locations [69,70]. Spatial heterogeneity in wateravailability due to plant-enhanced infiltration is higher in sloped terrains where runoff water happensexclusively down-slope (Fig. 1c), but it can be assumed in flat landscapes as well [14,41]. If runoff wateris very fast and plants facilitate infiltration substantially, plants will deplete water in their surroundingbare soil areas, even beyond the range of their root system [21]. Slope-mediated dynamics result in aSDF similar to the one emerging from the interplay between canopy shading effects and root-mediatedcompetition for resources, but at a larger scale (Fig. 1c).However, these assumed complex combinations of biophysical mechanisms often lack reliableempirical support and might vary from system to system. For example, Trautz et al. [71] measuredan SDF with short-range competition and long-range facilitation. Moreover, no empirical study hasyet shown that a specific SDF leads to vegetation patterns. In contrast, competition is a ubiquitousinteraction mechanism that affects the relation between any two plants that are located in sufficientproximity. Above-ground, plants compete for light through their canopies; below-ground, they competefor several soil resources, including water and nitrogen, through their roots [72]. If we assume, as PCmodels do, that only competitive mechanisms occur, we should expect plants to compete within a specificspatial range set by their physical reach, either the span of the roots or the extent of the canopy (Fig. 1d).Long but finite-range competition is the only interaction required by PC models to explain vegetationself-organization. PC models are hence the most parsimonious class of models that generates observedvegetation patterns, which makes them easier to test empirically than SDF models (see section 5).In the next section, we review the mathematical basis of SDF and PC models. We start with modelsfor water-mediated SDFs in section 3.1. Then, we move to kernel-based models, starting with SDF modelsin section 3.2) and continuing with PC models in section 3.3.
3. Review of models for vegetation self-organization
In 1952, Turing showed that differences in the diffusion coefficients of two reacting substances canlead to the formation of stable spatial heterogeneities in their concentration [73]. In Turing’s original model,one of the chemicals acts as an activator and produces both the second chemical and more of itself via anautocatalytic reaction. The second substance inhibits the production of the activator and therefore balancesits concentration (see Fig. 2a for a diagram of this reaction). Spatial heterogeneities can emerge arounda stationary balance of chemical concentrations if it is stable to non-spatial perturbations but unstableagainst spatial perturbations. This means that the homogeneous equilibrium reached in the absence ofdiffusion is locally stable, but destabilizes in the presence of diffusion. For this to occur, the differencebetween the diffusion coefficients of each substance is key. Specifically, the inhibitor must diffuse muchfaster than the activator, so that it inhibits the production of the activator at a long range and confines theconcentration of the activator locally (see Fig. 2b for a one-dimensional sketch of how patterns emergein a Turing activation-inhibition principle). The activation-inhibition principle responsible for patternformation thus relies on a scale-dependent feedback: positive feedbacks (autocatalysis) dominate on shortscales and negative, inhibitory feedbacks dominate on larger scales.In the context of vegetation pattern formation, plant biomass acts as the self-replicating activator.Several positive feedbacks have been hypothesized to act as autocatalizers of vegetation growth, such asenhanced water infiltration in the presence of plants [21,41,74] or the attraction of water towards patchesof vegetation by laterally extended roots [43,75]. Water is a limiting resource and, hence, water scarcitywould act as an inhibitor of vegetation growth. Negative feedbacks appear due to the lack of water far of 27
Canopy rangeRoot system range N e t b i o t i c i n t e r a c t i on Negative feedbackPositivefeedback slope addcb Figure 1. (a) Schematic of a plant canopy and spatial distribution of the root system. b-d) Three assumptionsfor the spatial variation of the net biotic interaction between a focal plant and its neighbors. b) Ascale-dependent feedback with short-range facilitation and long-range competition, c) scale-dependentfeedback similar to b) but in a sloped terrain, and d) a purely competitive interaction dominates the netinteraction at all spatial scales. from vegetation patches as a result of the effect of the positive feedbacks. Because plant dispersal occursover much shorter spatial scales than water diffusion, the negative feedback has a much longer range thanthe positive one. In the long-term, fwater-vegetation models including these hypothesized mechanismsrecover the set of gapped, labyrinthine and spotted patterns characteristic of Turing’s activation-inhibitionprinciple (Fig. 3). Importantly, in these models, the transition between each type of pattern is controlled byprecipitation intensity, a proxy for environmental conditions. Gapped patterns emerge for more humidsystems and spotted patterns for more arid ones [14]. More complex transient structures, such as rings ofvegetation, can be observed for certain initial conditions [22].To discuss water-vegetation models, we will first focus on an extension of the seminal work byKlausmeier [41] that describes the interaction between water and vegetation (with densities w ( r , t ) and v ( r , t ) , respectively) in a two-dimensional flat environment. Then, we will study a more complex model,introduced in Rietkerk et al. [21] that distinguishes between soil and surface water, and thus includesadditional feedbacks. of 27 Activator
Inhibitor b a D a D i k aa Inhibitor
Activator T i m e Spatial coordinate C on c en t r a t i on k ai k ia Figure 2. 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 autocatalyticreaction. The inhibitor degrades the activator at rate K ia and diffuses at rate D i > D a . b) Schematicof the pattern-forming process in a one-dimensional system. Average annual rainfall V ege t a t i onb i o m a ss More arid Less arid
Figure 3.
Schematic representation of the patterns of vegetation predicted by Turing-like models along agradient of average annual rainfall. of 27 ∂ w ( r , t ) ∂ t = R − a g ( w ) f ( v ) v ( r , t ) − l w ( r , t ) + D w ∇ w ( r , t ) , (1) ∂ v ( r , t ) ∂ t = a q g ( w ) f ( v ) v ( r , t ) − m v ( r , t ) + D v ∇ v ( r , t ) , (2)where w ( r , t ) and v ( r , t ) represent water concentration and density of vegetation biomass, respectively.In Eq. (1), water is continuously supplied at a precipitation rate R , and its concentration decreases dueto physical losses such as evaporation, occurring at rate l , and local uptake by plants. Water uptake ismodeled by the term a g ( w ) f ( v ) v , in which a is the plant absorption rate, g ( w ) describes the dependenceof vegetation growth on water availability, and f ( v ) is an increasing function of vegetation density thatrepresents the positive effect that the presence of plants has on water infiltration. Finally, water diffuseswith a diffusion coefficient D w . Similarly, Eq. (2) accounts for vegetation growth due to water uptake, plantmortality at rate m , and plant dispersal. In the plant growth term, the parameter q represents the yieldof plant biomass per unit of consumed water. In the original model, the plant absorption rate and theresponse of plants to water are linear ( g ( w ) = w ( r , t ) and f ( v ) = v ( r , t ) ) which facilitates the analyticaltractability of the model. However, other biologically-plausible choices can be made for these functions inorder to account for processes such as saturation in plant growth due to intraspecific competition [42].The generalized Klausmeier model has three spatially-homogeneous equilibria, obtained from thefixed points of Eqs. (1)-(2): an unvegetated state ( R / l ) , stable for any value of the rainfall parameter;and two states in which vegetation and water coexist at non-zero values. Of these two, only one is stableagainst non-spatial perturbations, which guarantees bistability, that is, the presence of alternative stablestates and hysteresis. For spatial perturbations, however, the vegetated state becomes unstable within arange of R , and the system develops spatial patterns, indicating that patterns in this model originate froma Turing instability.3.1.2. Three-equation water-vegetation dynamics: the Rietkerk modelThe Rietkerk model extends the generalized Klausmeier model by splitting Eq. (1) for waterconcentration in two equations: one for surface water, and another one for soil water, and including a termthat represents water infiltration. Moreover, the functions that represent water uptake and infiltration arenonlinear, which makes the model mechanistic, but also more complex, with more feedbacks betweenvegetation, soil moisture and surface water. The model equations are as follows: ∂ u ( r , t ) ∂ t = R − α v ( r , t ) + k w v ( r , t ) + k u ( r , t ) + D u ∇ u ( r , t ) (3) ∂ w ( r , t ) ∂ t = α v ( r , t ) + k w v ( r , t ) + k u ( r , t ) − g m v ( r , t ) w ( r , t ) k + w ( r , t ) − δ w w ( r , t ) + D w ∇ w ( r , t ) (4) ∂ v ( r , t ) ∂ t = c g m v ( r , t ) w ( r , t ) k + w ( r , t ) − δ v v ( r , t ) + D v ∇ v ( r , t ) (5)where u ( r , t ) , w ( r , t ) , and v ( r , t ) are the density of surface water, soil water, and vegetation, respectively. InEq. (3), R is the mean annual rainfall, providing a constant supply of water to the system; the second termaccounts infiltration; and the diffusion term accounts for the lateral circulation of water on the surface. In of 27 Eq. (4), the first term represents the infiltration of surface water into the soil, which is enhanced by thepresence of plants; the second term represents water uptake; the third one accounts for physical lossesof soil water, such as evaporation; and the diffusion term describes the lateral circulation of water in thesoil. Finally, the first term in Eq. (5) represents vegetation growth due to the uptake of soil water, whichis a function that saturates for high water concentrations; the second term accounts for biomass loss atconstant 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. [21] fortheir numerical analysis, are provided in Table 1.In the spatially homogeneous case, this model allows for two different steady states: a nontrivial onein 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. The stability of each of these statesswitches at R =
1. For R <
1, only the plantless equilibrium is stable against non-spatial perturbationswhereas for R > R =
1, both homogeneous equilibria are unstable against spatial perturbations. Whenallowing for spatial heterogeneities, numerical simulations using the parameterization in Table 1 showthe existence of spatial patterns within the interval 0.7 (cid:46) R (cid:46) R ≈ Parameter Symbol Value c Water-biomass conversion factor 10 (g mm − m − ) α Maximum infiltration rate 0.2 (day − ) g m Maximum uptake rate 0.05 (mm g − m − day − ) w Water infiltration in the absence of plants 0.2 (-) k Water uptake half-saturation constant 5 (mm) k Saturation constant of water infiltration 5 (g m − ) δ w Soil water loss rate 0.2 (day − ) δ v Plant mortality 0.25 (day − ) D w Soil water lateral diffusivity 0.1 (m day − ) D v Vegetation dispersal 0.1 (m day − ) D u Surface water lateral diffusivity 100 (m day − ) Table 1.
Typical parameterization of the Rietkerk model [21].
The Rietkerk model assumes constant rainfall, homogeneous soil properties, and only local andshort-range processes. Therefore, all the parameters are constant in space and time, and patternsemerge from scale-dependent feedbacks between vegetation biomass and water availability alone. Thissimplification of the conditions in which patterns form is, however, not valid for most ecosystems.Arid and semi-arid regions feature seasonal variability in rainfall [78]. Kletter et al. [79] showedthat, depending on the functional dependence between water uptake and soil moisture, stochasticrainfall might increase the amount of vegetation biomass in the ecosystem compared to a constantrainfall scenario. Moreover, the properties of the soil often change in space. A widespread cause of thisheterogeneity is soil-dwelling macrofauna, such as ants, earthworms, and termites [5]. Bonachela et al. [46]found that heterogeneity in substrate properties induced by soil-dwelling macrofauna, and modeled byspace-dependent parameters, might interact with SDFs between water and vegetation. This coupling bothintroduces new characteristic spatial scales in the pattern and reduces the abruptness of the transition intoa desert state and its hysteresis loop, which makes the ecosystem more resistant to aridification and easierto restore. Finally, researchers have also extended the Rietkerk model to account for long-range, nonlocal processes. For example, Gilad et al. [43] introduced a nonlocal mechanism in the vegetation density growthof Eqs. (3)-(5) that mimics the long-range of plant root systems. Specifically, they considered that vegetationgrowth at each location depends on the average density of water available within a neighbor region ofthe location rather than by water availability at the focal location. Similarly, they considered that wateruptake at each location depends on the average density of vegetation biomass within a neighborhoodcentered at the location. The size of this neighborhood is a model proxy for root system extension andthe averages are weighted by a kernel function that represents how the influence of each point within theneighborhood decays with distance to the focal location. It is important to note, however, that althoughmodels like the one developed in Gilad et al. [43] contain kernel functions, they do not rely on the shape ofthe kernel for the emergence of patterns, and the pattern-forming instability is still given by difference inwater and vegetation diffusion rates. Therefore, we will consider the Gilad model (and modifications to it)as a Turing-like model instead of a kernel-based one.
Kernel-based models are those in which all the water-vegetation feedbacks are encapsulated in asingle nonlocal net interaction between plants. The nonlocality in the net plant interaction accounts forthe 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 atlocations within the interaction range. The specifics of this coupling, such as whether it enhances or inhibitsplant growth as well as its spatial range, are contained in a kernel function whose mathematical propertiesdetermine the conditions for pattern formation. Moreover, because all water-vegetation feedbacks arecollapsed into a net interaction between plants, kernel-based models do not describe the dynamics of anytype of water and use a single partial integro-differential equation for the spatiotemporal dynamics of thevegetation.Next, we discuss different families of kernel-based models, depending on how the kernel function isintroduced in the equation (linearly or nonlinearly) and the nature of the net interaction it accounts for(scale-dependent feedback or purely competitive).3.2.1. Models with linear nonlocal interactionsThe first family of kernel-based models that we will discuss assumes that plants promote theproliferation of more individuals within their near neighborhood, and they inhibit the establishmentof new plants in their far neighborhood. This distance-dependent switch in the sign of the interactionrepresents a scale-dependent feedback [12]. As explained in Section 2, the facilitation range is usuallyassumed to be determined by the plant crown, while the competition range is related to the lateral rootlength (Fig. 1a). 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[80] (Fig. 1c). Given the analogy between these kernels and the ones used to model processes such aspatterns of activity in neural populations, these models are also termed neural models [81,82].Within kernel-based SDF models, we distinguish between those in which the spatial coupling(nonlocal interactions) enters in the equations linearly [80], and those in which it enters nonlinearly[83]. In the simpler linear case, the spatial coupling is added to the local dynamics, ∂ v ( r , t ) ∂ t = h ( v ) + (cid:90) d r (cid:48) G (cid:0) r (cid:48) ; r (cid:1) (cid:2) v (cid:0) r (cid:48) , t (cid:1) − v (cid:3) , (6)The first term describes the local dynamics of the vegetation, i.e., the temporal changes in vegetationdensity at a location r due to processes in which neighboring vegetation does not play any role. The integral term describes the spatial coupling, i.e., changes in vegetation density at r due to vegetationdensity at neighbor locations r ’. v represents the spatially homogeneous steady state, solution of h ( v ) = G ( r , r (cid:48) ) decays radially with the distance from the focallocation, | r (cid:48) − r | , and it can be written as G ( r (cid:48) , r ) = G ( | r (cid:48) − r | ) . Therefore the dynamics of vegetationdensity is governed by two main contributions: first, if spatial coupling is neglected, vegetation densityincreases or decreases locally depending on the sign of h ( v ) ; second, the spatial coupling enhances ordiminishes vegetation growth depending on the sign of the kernel function and the difference between thelocal vegetation density and the spatially homogeneous steady state v .Assuming kernels that are positive close to the focal location and negative far from it, localperturbations in the vegetation density around v are locally enhanced if they are larger than v andattenuated 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 nonlinearterms in the local-growth function h ( v ) , avoid the unbounded growth of perturbations and stabilize thepattern. However, although this mechanism imposes an upper bound to vegetation density, nothingprevents v from taking unrealistic, negative values. To avoid this issue, numerical integrations of Eq. (6)always include an artificial bound at v = ∂ v ( r , t ) ∂ t = β ( ω ∗ v ) ( r , t ) (cid:20) − ( ω ∗ v ) ( r , t ) K (cid:21) − η ( ω ∗ v ) ( r , t ) (7)where β is the rate at which seeds are produced (a proxy for the number of seeds produced by each plant)and η is the rate at which vegetation biomass is lost due to spontaneous death and external hazards suchas grazing, fires, or anthropogenic factors. The model assumes spatial isotropy, and the symbol ∗ indicatesa linear convolution operation: ( ω i ∗ v ) ( r , t ) = (cid:90) d r (cid:48) ω i ( r − r (cid:48) ; (cid:96) i ) v ( r (cid:48) , t ) (8)in which each ω i is a weighting function with a characteristic spatial scale (cid:96) i that defines the size ofthe neighborhood contributing to the focal process. For instance, ω ( r − r (cid:48) ; (cid:96) ) defines the size of theneighborhood that contributes to the growth of vegetation biomass at r . Similarly, (cid:96) defines the scaleover which plants inhibit the growth of their neighbors, and (cid:96) the scale over which vegetation densityinfluences the spontaneous death rate of vegetation at the focal location (called toxicity length in Lefeverand Lejeune [83]). Because the sign of the interaction is explicit in each term of Eq. (7), the convolutionsonly represent weighted averages of vegetation biomass and the weighting functions must be defined tobe positive. Finally, Lefever and Lejeune [83] set the scale of the inhibitory interactions larger than thescale of the positive interactions ( (cid:96) > (cid:96) ), and thus the model includes a SDF with short-range facilitationand long-range competition. Expanding upon this work, several other models have introduced non-linearspatial couplings via integral terms [84–86], and others have expanded the integral terms and studied theformation of localized structures of vegetation [87]. In previous sections, we invoked the existence of SDFs in the interactions among plants to explain theemergence of self-organized spatial patterns of vegetation. However, competition and facilitation usuallyact simultaneously and are hard to disentangle [88]. This intricate coupling between positive and negativeplant-to-plant interactions, together the various biophysical processes that may underlie each of them,makes it difficult to understand how the net interaction between two neighbors may shift from positiveto negative with the distance between them. For example, Trautz et al. [71] reported a scale-dependentfeedback between neighboring plants in which negative interactions dominate on the short range andpositive interactions dominate on the long range. Moreover, some studies have highlighted the importanceof long-range negative feedbacks on pattern formation, suggesting that short-range positive feedbacksmight be secondary actors that sharpen the boundaries of clusters rather than being key for the instabilitiesthat lead to the patterns [11,89,90]. Following these arguments, Martinez-Garcia et al. [23,59] proposed afamily of purely competitive models with the goal of identifying the smallest set of mechanisms neededfor self-organized vegetation patterns to form. Specifically, the goal of these studies was to determinewhether SDFs are necessary for self-organized patterns to form or if, instead, one of these two feedbacksacting alone can drive the emergence of spatial patterns of vegetation in water-limited ecosystems.3.3.1. Models with linear nonlocal interactionsInspired by the neural models with short-range facilitation and long-range inhibition described byEq. (6), the simplest purely competitive models consider linear nonlocal interactions. Models in this familycan be written as: ∂ v ( r , t ) ∂ t = D ∇ v ( r , t ) + β v ( r , t ) (cid:18) − v ( r , t ) K (cid:19) + λ (cid:90) d r (cid:48) G (cid:0) | r (cid:48) − r | (cid:1) v ( r (cid:48) , t ) (9)where the first term on the right side represents seed dispersal; the second term is a growth term in whichthe logistic-like growth-limiting factor ( − v / K ) represents local competition for space, β is the seedproduction rate, and K the local carrying capacity; the third term accounts for long-range interactionsbetween individuals at r and their neighbors at r (cid:48) . λ > G ( | r (cid:48) − r | ) is necessarily negative to account for a competitive net interaction that inhibits vegetationgrowth.As in Eq. (6), there is no lower bound for v , which can take negative values due to the linear nonlocalterm, Thus, an artificial bound at v = G is a top-hat function of | r − r (cid:48) | , buta linear stability analysis of the model equation reveals that patterns may form for many other kernelshapes [23]. More specifically, a necessary condition for pattern formation is that the Fourier transform ofthe kernel function takes negative values for certain wavenumbers, which indicates a sharp decay in thestrength of the nonlocal interactions [23,59]. Importantly, the Fourier transform of any kernel functionwith a discontinuity at a distance | r − r (cid:48) | takes negative values for a finite range of wavenumbers and canpotentially lead to patterns. Provided that the kernel function meets this condition, the intensity of thenonlocal competition λ controls a transition to patterns, and for large values of λ , the model develops asequence of labyrinthine and spotted patterns similar to those observed in Turing-like. Gapped patterns,however, have not been found in models in which nonlocal interactions are inhibitory and linear. et al. [59] although very similar results are obtained when theymodulate the death term [23]: ∂ v ( r , t ) ∂ t = P E ( (cid:101) v , δ ) β v ( r , t ) (cid:18) − v ( r , t ) K (cid:19) − η v ( r , t ) , (10)where β and K are the seed production rate and the local carrying capacity as defined in Eq. (9), δ is thecompetition-strength parameter, and (cid:101) v ( r , t ) is the average density of vegetation around the focal position r ,termed ‘nonlocal vegetation density’ in the following. Assuming spatial isotropy, this nonlocal vegetationdensity can be calculated as (cid:101) v ( r , t ) = (cid:90) d r (cid:48) G (cid:0) | r (cid:48) − r | (cid:1) v ( r , t ) . (11)where the kernel function G weighs the contribution of vegetation at a location r (cid:48) to the nonlocal vegetationdensity at location r and is necessarily defined positive. Because it is a weighting function, G only definesa range of influence of a focal plant, typically determined by the characteristic scale of the function, q , andhow this influence changes with the distance from the plant [like ω i functions do in Eq. (7)]. The modelsfurther assumes that vegetation losses occur at constant rate η and vegetation grows through a three-stepsequence of seed production, local dispersal, and establishment [36]. Mathematically, this sequence isrepresented by the three factors that contribute to the first term in Eq. (10). First, plants produce seeds at aconstant rate β , which leads to the a growth term β v ( r , t ) . Second, seeds disperse locally and compete forspace which defines a local carrying capacity K . Third, plants compete for resources with other plants,which is modeled using a plant establishment probability, P E . Because the only long-range interaction inthe model is root-mediated interference and competition for resources is more intense in more crowdedenvironments, P E is a monotonically decreasing function of the nonlocal vegetation density ˜ v ( r , t ) definedin Eq. (11). Moreover, P E also depends on a competition-strength parameter, δ , that represents the limitationof resources. In the limit δ =
0, resources are abundant, competition is weak and P E =
1. Conversely, inthe limit δ → ∞ , resources are very scarce, competition is very strong and therefore P E → G and the functional form of the probability of establishment, P E . However, evenwithout fixing these two functions, one can prove the existence of patterns in Eq. (10) from generalproperties of P E . As for models with linear nonlocal interactions, a necessary condition for patterns todevelop is that the Fourier transform of G becomes negative for at least one wavenumber. Once the kernelmeets this condition, the parameter ranges for which pattern formation occurs can be derived via linearstability analysis of the homogeneous solutions of the equation [40]. This analysis was conducted inMartínez-García et al. [59]. For low values of the competition strength δ , a homogeneous state with v (cid:54) = δ increases, the homogeneous state becomes unstableand a sequential series of gapped, labyrinthine and spotted patterns develops. A desert state, howeveris never reached because vegetation density tends asymptotically to zero. Using the seed productionrate β as control parameter, this same sequence of gapped-labyrinthine-spotted patterns develops as β decreases. When seed production rate becomes too low, vegetated patterns cannot be sustained and thesystem collapses into a desert-like, unvegetated state. y / L N o r m a li z ed den s i t y o f v ege t a t i on b i o m a ss a b c Seed production rate, β +_ Figure 4.
As seed production rate increases, which can be seen as resulting from improving environmentalconditions, self-organized patterns from the purely competitive model introduced in Martínez-García et al. [59] transition from spotted to labyrinthine to gapped. The model is integrated on a 2 D square lattice withperiodic boundary conditions and using a exponential function for the seed-establishment probability P E = exp ( − δ ˜ ρ ) . Simulations are started from an uncorrelated random initial condition in which the valueof the vegetation density at each node of the lattice is drawn from a uniform distribution between 0 and 1.Parameterization: δ = η = β = β = β =
15 (panel c). q but smaller than2 q , then there is not inter-patch competition because plants are separated by a distance larger than theinteraction range (Fig. 5a, b). However, because the distance between patches is shorter than 2 q , there is aregion halfway between both clusters in which plants compete with both patches and are thus subject tostronger competition than in each of the patches (Fig. 5c). As result, vegetation tends to disappear fromthese interpatch regions. Moreover, as vegetation dies in the region between patches, individuals withineach of the patches experience weaker competition for resources, which effectively leads to a positivefeedback that increases the biomass inside the patch and enhances the structure of the pattern [23,59].This same mechanism has been suggested to drive the formation of clusters of competing species in theniche space [91–95], and explains why spectral analyses of the patterns developed by purely competitivenonlocal models identify a characteristic wavelength between q and 2 q . a b c Figure 5.
In kernel-based PC models, patchy distributions of vegetation in which the distance betweenpatches is between one and two times the range of the nonlocal interactions are stable. Individuals withineach patch only compete with the individuals in that patch (a,b), whereas individuals in between patchescompete with individuals from both patches (c). Color code: green trees are focal individuals, and dashedcircles limit the range of interaction of the focal individual. Dark grey is used for individuals that interactwith the focal one, whereas light gray indicates individuals that are out of the range of interaction of thefocal individual.
4. Self-organized patterns as indicators of ecological transitions
Models assuming different shapes for the net biotic interaction between neighbor plants havesuccessfully reproduced qualitatively the spatial patterns of vegetation observed in water-limitedecosystems [13]. These different models also predict that the spotted pattern precedes a transition to anunvegetated state and thus could be used as early-warning indicators of ecological transitions [18,19].However, models invoking different mechanisms to explain the formation of the same pattern can leadto very different desertification processes. As an example, we next revisit three different models forvegetation self-organization from previous sections and focus on their contradictory predictions abouthow ecosystems respond to aridification.The Rietkerk model [21] (section 3.1.2) predicts that, if aridity keeps increasing after the system is inthe spotted pattern, the ecosystem eventually collapses into a desert state following an abrupt transitionthat includes a hysteresis loop (Fig. 6a). Abrupt transitions such as this one are typical of bistable systemsin which the stationary state depends on the environmental and the initial conditions. Bistability is apersistent feature of models for vegetation pattern formation, sometimes occurring also in transitionsbetween patterned states [14], and it denotes thresholds in the system that trigger sudden, abrupt responsesin its dynamics. These thresholds are often created by positive feedbacks or quorum-regulated behaviorsas is the case in populations subject to strong Allee effects [96]. In the Rietkerk model, as rainfall decreases,the spatial distribution of vegetation moves through the gapped-labyrinthine-spotted sequence of patterns(Fig. 6a). However, when the rainfall crosses a threshold value ( R ≈ − for parameter valuesin Table 1 and using the initial condition in the caption of Fig. 6), the system responds abruptly, and allvegetation dies. Once the system reaches this unvegetated state, increasing water availability does notallow vegetation recovery until R ≈ − , which results in a hysteresis loop and a region ofbistability ( R ∈ [ ] in Fig. 6a). Bistability and hysteresis loops make abrupt, sudden transitions likethis one extremely hard to revert. Hence, anticipating such abrupt transitions is critical from a conservationand ecosystem-management point of view [18,19].Extended versions of the Rietkerk model have suggested that the interaction between vegetationand other biotic components of the ecosystem may change the transition to the unvegetated state (seesection 3.1.2). Specifically, Bonachela et al. [46] 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 abruptdesertification into a two-step process (Fig. 6b). At a certain precipitation level ( R ≈ − 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. 6b) but persists on the mounds due to improvedproperties for plant growth created by the termites. On-mound vegetation survives even if precipitationcontinues to decline, and is finally lost at a rainfall threshold R ≈ − (T2 in Fig. 6b). As aconsequence of the two-step transition, the ecosystem collapse is easier to prevent because a bare soilmatrix with vegetation only on mounds serves as an early-warning signal of desertification, and it is easierto revert since termite-induced heterogeneity breaks the large hysteresis loop of the original model intotwo smaller ones (compare the hysteresis loops in Fig. 6a and Fig. 6b). -2 -1 N o r m a li z ed v ege t a t i on den s i t y -1 ) V ege t a t i on b i o m a ss ( g m - ) a V ege t a t i on b i o m a ss ( g m - ) Rainfall (mm day -1 )10 -2 -1 c β (T1) b (T2) Increasing R Decreasing R Figure 6.
Although models for vegetation pattern formation may recover the same sequence ofgapped-labyrinthine-spotted patterns from different mechanism, the type of desertification transitionthat follows the spotted pattern strongly depends on the model ingredients. a) Abrupt desertificationas predicted by the Rietkerk model [21]. Simulations were conducted on a squared environment oflateral length 200m with discretization ∆ x = ∆ y =
2m and using the model parameterization in Table1. Simulations were started by introducing peaks of vegetation in 1% of the grid elements, which wereall set in the unvegetated equilibrium. b) Two-step desertification process as predicted in Bonachela et al. [46] simulations were conducted using the same parameterization and initial condition used in panel a. c)Progressive desertification as predicted by the purely competitive model introduced in Martínez-García et al. [59]. Numerical simulations were conducted using the same setup described in Fig. (4).
Finally, the PC model with nonlinear nonlocal interactions of section 3.3.1 [59] predicts a smoothdesertification in which vegetation biomass decreases continuously in response to decreasing seedproduction rate (a proxy for worsening environmental conditions). According to this model, the spotted pattern would persist as precipitation declines, with vegetation biomass decreasing until it eventuallydisappears (Fig. 6c). As opposed to catastrophic shifts, smooth transitions such as the one depicted bythis model do not show bistability and do not feature hysteresis loops. This difference has importantsocio-ecological implications because it enables easier and more affordable management strategies torestore the ecosystem after the collapse [61]. Moreover, continuous transitions are also more predictablebecause the density of vegetation is univocally determined by the control parameter (seed production rate β in Fig. 6c).Therefore, patterns have tremendous potential for ecosystem management as an inexpensive andreliable early indicator of ecological transitions [18,19]. However, predictability requires the developmentof tailored models that reproduce observed patterns from the mechanisms relevant to the focal system. Wehave shown that widespread spotted patterns can form in models accounting for very different mechanisms(Fig. 6). Crucially, however, each of these models predicts a very different type of desertification transition.Because ecosystems are highly complex, it is very likely that spotted patterns observed in different regionsemerge from very different mechanisms (or combinations of them) and thus anticipate transitions of verydifferent natures. Therefore, a reliable use of spotted patterns as early warning indicators of ecosystemcollapse requires a mix of (a) mechanistic models that are parameterized and validated by empiricalobservations of both mechanisms and patterns, (b) quantitative analyses of field observations, and (c)manipulative experiments.
5. Testing models for vegetation self-organization in the field
In this section, we discuss possible experimental approaches to test whether and which of thepreviously reviewed types of models is at play in a specific patterned ecosystem, which would helpdetermine whether an eventual desertification transition is more likely to be abrupt or continuous.The first step that we propose is to test the spatial distribution of the sign of the net interaction betweenpattern-forming plants and an experimental individual plant. Only two net-interaction distributionshave been theoretically predicted to produce spatial vegetation patterns. A PC distribution allowspatterns to emerge from negative net interactions being ubiquitous but stronger in bare-soil areas thanunder the canopy. The SDF distribution generates similar patterns from positive interactions dominatingunder-canopy areas and negative interactions dominating bare-soil areas. A simple experimental setup,based on mainstream plant biotic interaction methodologies[97]], would allow one to discern whether thePC or the SDF distribution of net interactions predominates in the focal ecosystem.Our proposed experiment would compare a fitness proxy (e.g., growth, survival) of experimentalplants planted in the system under study, where we observe a regular vegetation pattern and we assumethat vegetation is in equilibrium with environmental conditions. Each experimental block would consist ina plant growing under-canopy (Fig. 7a), a plant growing in bare soil (i.e., between two vegetation patches)(Fig. 7b), and a control plant growing in the same ecosystem but artificially isolated from the interactionwith pattern-forming individuals (Fig. 7c). To isolate control plants from canopy interaction they need tobe 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 [98]. ]. Comparing the fitness of the control plant withthe fitness of plants growing in vegetation patches or bare soil gives us the sign and strength of the netbiotic interaction. By replicating this experimental block we can statistically determine whether the patternformation results from a SDF, a PC, or whether it involves a different process. The SDF hypothesis wouldbe validated if a predominantly positive net interaction is observed under the canopy, and a negativeinteraction is observed in bare soils. Conversely, the PC hypothesis would be proved if a negative netinteraction is observed in bare soils and under canopy (see Table 2). Any other outcome in the spatialdistribution of the sign of the net interaction between plants would suggest that other mechanisms are at
Canopy range Bare soilRootbarrier (a) (b)(c)
Bare soil
Figure 7.
Schematic representation of a simple experimental setup to test in the field whether the mechanismof spatial patterning is purely competitive (PC) or a classic scale-dependent feedback (SDF). Plant (a) is anexperimental plant growing under-canopy, (b) is growing in bare soil, and (c) is a control plant growingin artificial conditions, free from the biotic interaction using root barriers in bare soil areas of the sameenvironment. play, which could include the action of different ecosystem components, such as soil-dwelling macrofauna[44], or abiotic factors, such as micro-topology.
Under canopy vs control Bare soil vs control Outcome0/ − − −
Purely competitive + − Scale-dependent feedback
Table 2.
Testing the PC versus SDF hypotheses in the experimental setup introduced in Fig. 7. Indexes tocalculate the sign of the net interaction can be taken from Armas et al. [97].
After discriminating between the PC and SDF hypotheses, a second experimental step would beto further explore the biophysical mechanisms responsible for the measured interaction (e.g., aboveand below-ground competition, soil or climate amelioration..) and driving the spatial pattern. Thesebiophysical mechanisms can be complex, and some have been proposed as potential major drivers ofvegetation self-organization [66]. For example, PC models hypothesize that spatial patterns are driven bylong-range below-ground competition for a limiting resource through the formation of exclusion regions.As discussed in section 3.3.2, these exclusion regions are territories between patches of vegetation inwhich the intensity of competition is higher than within the patch [89], possibly because they present a higher density of roots (Fig. 5) [23,59]. To test for the existence of exclusion regions and confirm whetherbelow-ground competition is driving the spatial pattern, researchers could measure root density acrosstransects between two vegetated patches and through the bare soil.Field tests and manipulative experiments to confirm that SDFs are responsible for vegetation patternsare not easy to perform. However, there are still a handful of analyses that researchers could do. Forexample, the Rietkerk SDF model [21] assumes that (i) water infiltration is significantly faster in vegetationpatches than in bare soil areas and (ii) that surface water diffusion (i.e., runoff speed) is several ordersof magnitude larger than vegetation diffusion (i.e., patch growth speed). To test the first assumption,researchers can use infiltrometers to quantify water infiltration rates in both vegetated patches and bare-soilareas [99,100]. This difference in water infiltration due to the presence of vegetation should also resultin higher densities of water in the soil underneath vegetation patches than in the bare soil, which canbe quantified using field moisture sensors [101]. To test the second assumption, field researchers needto measure the intensity of the water runoff and compare it with a measure of the lateral growth rateof vegetation patches. Water runoff is very challenging to measure directly, but reliable estimates canbe calculated using infiltration rates [102]. Note, however, that infiltration rates might be very hard tomeasure due to small-scale soil heterogeneities and expect water runoff estimates derived from themto be reliable only for a subset of ecosystems with more homogeneous soils. The lateral growth rate ofvegetation patches can be estimated based on drone or satellite images repeated over time. Combiningmeasures of both water runoff and expansion rates of vegetation patches, one can estimate approximatedvalues 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, wehave reviewed different modeling approaches employed to understand the mathematical origin and thepredicted consequences of those patterns. We have shown that different models, relying on differentmechanisms, can successfully reproduce the patterns observed in natural systems. However, each of thesemodels predicts very different ecosystem-level consequences of the emergent pattern, which limits theutility of the patterns alone to be used as applied ecological tools in the absence of explicit knowledgeof underlying mechanisms. To solve this issue, we claim that models need to move from their currentuniversal but phenomenological formulation towards a more system-specific but mechanistic one, focusedon isolating the system-specific, key feedbacks for vegetation self-organization. To this endl, we identifyseveral directions for future research.First, biologically-grounded studies should aim to combine system-specific models with empiricalmeasures of vegetation-mediated feedbacks. Existing models for vegetation self-organization are mostlyphenomenological and are only validated qualitatively via the visual comparison of simulated andobserved (macroscopic) patterns. Experimental measures of the (microscopic) processes and feedbackscentral 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-validatedvalues of water diffusion and plant dispersal rates, despite the crucial role of these parameters in theemergence of patterns. Instead, these models fine-tune such values to obtain patterns similar in, forexample, their wavelength, to the natural pattern. Similarly we are only beginning to understandhow plants rearrange their root system in the presence of competing individuals [103], and hencekernel-based models do not incorporate realistic functional forms for the kernels. Instead, these models usephenomenological functions to test potential mechanisms for pattern formation by qualitatively comparingmodel output and target pattern, thus limiting the potential of the models to make quantitative predictions.
PDEMs are analytically more tractable than IBMs and enable the identification of processes that triggerthe instabilities responsible for the patterns [9]. However, such PDEMs only have true predictive power ifderived from the correct microscopic dynamics and properly parameterized via system-specific measures.Thus, in order to establish a dialogue between experiments and theory, models should develop from amicroscopic description of the system [27,28] that allows for a more realistic and accurate description of theplant-to-plant and plant-water interactions, as well as for a better reconciliation between model parametersand system-specific empirical measures. Subsequently, existing tools from mathematics, statistical physics,and/or computer science can be used to reach a macroscopic PDEM that captures the key ingredients ofthe microscopic dynamics. Statistical physics, which was conceived to describe how observed macroscopicproperties of physical systems emerge from the underlying microscopic processes, provides a compellingand well-developed framework to make such a micro-macro connection.Second, recent developments in remotely sensed imagery have enabled the measurement of anecosystem’s state indicators, which will allow researchers to compare observed and simulated patternsquantitatively [25]. On the one hand, using existing databases of ecosystem responses to aridity [65]and satellite imagery of vegetation coverage [13], researchers could conduct a model selection analysisand classify existing models from more to less realistic depending on whether (and how many) featuresof the focal ecosystem the model manages to reproduce in the correct environmental conditions. Forexample, models could be classified depending on whether, after proper parameterization, they canpredict 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 naturalpatterns, geostatistics for quantifying spatial correlations, and time series analysis for tracking changesin the ecosystem properties through time will be essential. On the other hand, once we accumulate along-term database of satellite images of the Earth’s surface, researchers will be able to calculate thecorrelation between pattern shape and mean annual rainfall for a fixed location through time. This analysiswill provide a more robust test for model predictions on the correlation between water availability andpattern type than existing ones using satellite images taken at different locations at the same time [13]because they will ensure that all model parameters except the mean annual rainfall are constant.Finally, theoretical research should try to reconcile reaction-diffusion and kernel-based models.Despite recent efforts [23], the link between the two approaches is still lacking, making it hard to buildbiologically-meaningful kernels. To the best of our knowledge, any attempt to derive a kernel-based modelstarting from a water-vegetation reaction-diffusion model has been unsuccessful in reproducing a kernelshape that generates patterns. Only very few exceptions exist for certain approximations of kernel-basedmodels with SDFs in which the nonlocal term is expanded into a series of differential operators [12]. Wepropose that the micro-macro scaling techniques discussed above can also help shed light on this question.Beyond water-limited ecosystems, both SDF and competition/repulsion alone have been reportedas drivers of spatial self-organization in many other biological and physical systems. A combinationof attractive and repulsive forces acting on different scales is, for instance, believed to be responsiblefor the formation of regular stripes in mussel beds [11]. Other models that investigate the formation ofdifferent structures in animal groupings also rely on similar attraction-repulsion or activation-inhibitionprinciples [104–109]. On the other hand, several biological systems also self-organize only as a consequenceof repulsive or growth-inhibitory interactions alone. For instance, territorial species and central-placeforagers often create a hexagonal, overdispersed pattern of territory packing [44,110–112] (see [5] for acomprehensive review). Species in communities driven by competition have also been predicted to formclumps through the niche space [91,92,94,113] and long-range competition has been recently suggested asa potentially stabilizing mechanism in two-species communities [114]. In physical systems, cluster crystalsform in some molecules and colloids that interact via effective repulsive forces [115–118]. Patterningin these disparate systems shares common properties: competition induces a hexagonal distribution of the clusters, and the transition to patterns is mathematically controlled by the sign of the Fouriertransform of the kernel function, which indicates how quickly the intensity of the competition decayswith the distance between individuals [59,117,119]. Understanding the conditions under which repulsiondominates attraction (or inhibition dominates activation) and finding the key features that distinguish thepatterns that emerge in each of these scenarios across physical systems and different levels of biologicalorganization constitutes another important line for future research.
Acknowledgments:
We acknowledge Robert M. Pringle, Rubén Juanes, and Ignacio Rodríguez-Iturbe for variousdiscussions at different stages of the development of this work
Author Contributions: “Conceptualization, R.M.G., C.C., J.A.B., J.M.C., E.H.G., C.L., C.E.T.; writing–original draftpreparation, 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.”
Funding: “RMG: FAPESP through grants ICTP-SAIFR 2016/01343-7, and Programa Jovens Pesquisadores em CentrosEmergentes 2019/24433-0 and 2019/05523-8, Instituto Serrapilheira through grant Serra-1911-31200, and the SimonsFoundation. CC: the Princeton University May Fellowship in the department of Ecology and Evolutionary Biology.JMC: Center of Advanced Systems Understanding (CASUS) which is financed by Germany’s Federal Ministry ofEducation and Research (BMBF) and by the Saxon Ministry for Science, Culture and Tourism (SMWK) with tax fundson the basis of the budget approved by the Saxon State Parliament. EHG and CL: MINECO/AEI/FEDER through theMaría de Maeztu Program for Units of Excellence in R&D (MDM-2017-0711, Spain). CET & JAB acknowledge supportfrom the Gordon and Betty Moore Foundation, grant
Conflicts of Interest:
The authors declare no conflict of interest.
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