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Mathematical Models for Vegetation Patternsand Biodiversity

Thesis submitted in partial fulfillmentof the requirements for the degree of

“DOCTOR OF PHILOSOPHY”

by

Erez Gilad

Submitted to the Senate of Ben-Gurion Universityof the Negev

December 10, 2006

Beer-Sheva

Mathematical Models for Vegetation Patternsand Biodiversity

Thesis submitted in partial fulfillmentof the requirements for the degree of

“DOCTOR OF PHILOSOPHY”

by

Erez Gilad

Submitted to the Senate of Ben-Gurion Universityof the Negev

Approved by the advisorApproved by the Dean of the Kreitman School of Advanced Graduate Studies

December 10, 2006

Beer-Sheva

This work was carried out under the supervision of Prof. Ehud MeronIn the Department of PhysicsFaculty of Natural Sciences.

i

Acknowledgements

I am grateful for the opportunity to work under the supervision of Prof. Ehud Meron.His creative thinking and deep understanding, along with a unique perception of scienceand the courage to step into new scientific territories were an inspiration to me.

I would also like to thank Dr. Jost von Hardenberg, a brilliant scientist and agood friend, from whom I learned so much. Many ideas were conceived and a consider-able amount of work was done during my visits to Jost in Italy and during Jost’s visitsto Israel.

I would like to express my great appreciation to Prof. Moshe Shachak and Prof.Antonello Provenzale for extremely interesting and helpful discussions and for upmostimportant comments and ideas regarding my work.

Special thanks to other members of our research group, Dr. Hezy Yizhaq andEfrat Sheffer, for an enjoyable collaboration, and many thanks to Amos Yarom (myoffice mate), Ran Salem, Gidi ”the cat” Sarusi and the entire ”Bobo” gang forhaving a good time.

The financial support by the Department of Solar Energy and Environmental Physics,The Blaustein Institutes for Desert Research in Sede-Boqer and the Department ofPhysics, Faculty of Natural Sciences is gratefully acknowledged.

Finally, I am most grateful to Mazal, my precious wife, for her endless love andunconditional support.

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To my parents and Shelly

Contents

Abstract 1

1 Introduction 3

2 Model for dryland water-vegetation systems 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Modeling positive feedback processes . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Infiltration feedback . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Uptake feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Non-dimensional form of the model . . . . . . . . . . . . . . . . . . . . . 102.5 Aridity parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Landscape states 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Mapping the landscape states along aridity gradients . . . . . . . . . . . 133.3 Coexistence of landscape states and state transitions . . . . . . . . . . . 153.4 Aridity parameter and stability of uniform states . . . . . . . . . . . . . 18

3.4.1 Aridity classification . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.2 Bare state stability threshold . . . . . . . . . . . . . . . . . . . . 193.4.3 Uniform vegetation stability threshold . . . . . . . . . . . . . . . 20

3.5 Structural and dynamical aspects of banded vegetation states . . . . . . 213.5.1 Uniform and pattern solutions . . . . . . . . . . . . . . . . . . . . 213.5.2 Multiple band states and biological productivity . . . . . . . . . . 223.5.3 The productivity-resilience tradeoff of band states . . . . . . . . . 24

4 Plants as ecosystem engineers 254.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 The facilitation-resilience tradeoff . . . . . . . . . . . . . . . . . . . . . . 264.3 Facilitation vs. competition . . . . . . . . . . . . . . . . . . . . . . . . . 284.4 Engineering niches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5 Local engineering by global pattern changes . . . . . . . . . . . . . . . . 304.6 Landscape diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.6.1 The niche concept and the niche map . . . . . . . . . . . . . . . . 324.6.2 Bistability as a mechanism for landscape diversity . . . . . . . . . 32

iv

5 Dynamics and spatial organization of plant communities 355.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 A model for n-interacting species populations . . . . . . . . . . . . . . . 365.3 Modeling positive feedback processes . . . . . . . . . . . . . . . . . . . . 365.4 Non-dimensional form of the model . . . . . . . . . . . . . . . . . . . . . 385.5 Plant interactions along a stress gradient . . . . . . . . . . . . . . . . . . 39

5.5.1 The woody-herbaceous ecosystem . . . . . . . . . . . . . . . . . . 395.5.2 From competition to facilitation along an aridity gradient . . . . . 405.5.3 Back to competition . . . . . . . . . . . . . . . . . . . . . . . . . 425.5.4 Interspecific facilitation induced by intraspecific competition . . . 43

5.6 Mechanisms of species diversity change in stressed environments . . . . . 455.6.1 Linking plant interactions and species composition . . . . . . . . 455.6.2 Spatial patterning effects . . . . . . . . . . . . . . . . . . . . . . . 47

6 A fast algorithm for convolution integrals with space and time variantkernels 486.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.3 Computing the approximation coefficients . . . . . . . . . . . . . . . . . 516.4 Examples for some common kernel functions . . . . . . . . . . . . . . . . 52

6.4.1 One dimensional decaying exponential . . . . . . . . . . . . . . . 536.4.2 Two dimensional Gaussian . . . . . . . . . . . . . . . . . . . . . . 546.4.3 Three dimensional Lorentzian . . . . . . . . . . . . . . . . . . . . 55

6.5 Accuracy of the approximation . . . . . . . . . . . . . . . . . . . . . . . 556.5.1 The error in the kernel approximation . . . . . . . . . . . . . . . 556.5.2 The error in the integral approximation . . . . . . . . . . . . . . . 57

6.6 Numerical performances of the algorithm . . . . . . . . . . . . . . . . . . 59

7 Conclusions 607.1 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.1.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607.1.2 Landscape states . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.1.3 Ecosystem engineering . . . . . . . . . . . . . . . . . . . . . . . . 627.1.4 Dynamics of plant interactions and species diversity . . . . . . . . 637.1.5 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.2 Model limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.3 Future studies and model developments . . . . . . . . . . . . . . . . . . . 64

A Derivation of the surface water equation 66

B Linear stability analysis 68B.1 Analysis of the n-interacting species model . . . . . . . . . . . . . . . . . 68B.2 Results for the n = 1 case . . . . . . . . . . . . . . . . . . . . . . . . . . 71B.3 Results for the n = 2 case . . . . . . . . . . . . . . . . . . . . . . . . . . 72

v

C Numerical methods 74C.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74C.2 Time integration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 75C.3 Solving the surface water equation . . . . . . . . . . . . . . . . . . . . . . 76

Bibliography 78

vi

List of Tables

2.1 Model parameters, their units, and numerical values . . . . . . . . . . . . 112.2 Scaling relations between non-dimensional and dimensional variables and

parameters of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.1 Scaling relations between non-dimensional and dimensional variables andparameters of the n-interacting species populations model . . . . . . . . . 38

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List of Figures

2.1 The infiltration rate I as a function of biomass density B . . . . . . . . . 9

3.1 Bifurcation diagram for homogeneous stationary solutions of the modelequations 2.5–2.8 showing the biomass vs. precipitation and basic vege-tation patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Linear growth-rate curves of non-homogeneous spatial perturbations ap-plied to the bare state and uniform vegetation solutions . . . . . . . . . . 15

3.3 Development of vegetation bands from an unstable uniform vegetationstate on hill slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Bifurcation diagram for homogeneous stationary solutions of the modelequations 2.5–2.8 showing the biomass vs. aridity parameter a . . . . . . 16

3.5 Coexistence of different stable landscape states along the rainfall or ariditygradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 Bifurcation diagram showing a wide coexistence range of stable bare soiland stable spots pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.7 Local disturbance leading to a global state transition (e.g. bands to spotspattern) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.8 A local clearcut, similar to that shown in Fig. 3.7, but in a precipitationrange where spots pattern is unstable . . . . . . . . . . . . . . . . . . . . 18

3.9 A local clearcut, similar to that shown in Fig. 3.7, but in plane topography 183.10 Neutral stability curves for the bare soil solution obtained from the linear

stability result pc = ΛP/MN = 1 . . . . . . . . . . . . . . . . . . . . . . 193.11 Stability balloon describing the stability range of the uniform vegetation

solution to spatial perturbations along an aridity gradient . . . . . . . . . 213.12 A bifurcation diagram showing biomass vs. precipitation for a uniform

slope topography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.13 The effect of precipitation on the widths of vegetation bands for a pattern

with a given wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.14 The dependence of biological productivity on the pattern’s wavenumber k 233.15 Low resilience of vegetation bands to precipitation downshifts . . . . . . 24

4.1 Spatial profiles of the variables b, w, and h as affected by the parametersthat control the main positive biomass-water feedbacks, f and η . . . . . 27

4.2 Maximal soil-water densities under ecosystem engineer patches for twodifferent engineer species as functions of the infiltration contrast . . . . . 28

viii

4.3 Competition changes to facilitation as aridity increases . . . . . . . . . . 294.4 Hump and ring shaped spatial distributions of soil-water at the scale of a

single patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5 A pattern transition at the landscape scale (from bands to spots) affects

engineering at the single patch scale . . . . . . . . . . . . . . . . . . . . . 314.6 Bistability as a mechanism for landscape diversity (i) . . . . . . . . . . . 334.7 Bistability as a mechanism for landscape diversity (ii) . . . . . . . . . . . 34

5.1 Bifurcation diagram showing homogeneous and pattern solutions of thewoody-herbaceous system . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Model solutions showing a transition from competition to facilitation asprecipitation decreases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Water uptake Gw and infiltration I rates per unit area of a b1 patch asprecipitation decreases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.4 Transition back to competition at extreme aridity . . . . . . . . . . . . . 425.5 Model solutions showing a transition from competition to facilitation as

a result of biotic stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.6 Interspecific facilitation induced by intraspecific competition as a transient 445.7 Implications of the competition to facilitation transition for species com-

position changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.8 Implications of interspecific facilitation (induced by intraspecific compe-

tition) for species diversity . . . . . . . . . . . . . . . . . . . . . . . . . . 465.9 Facilitation induced by a pattern shift at the landscape level . . . . . . . 47

6.1 Typical shapes of the approximation coefficients αl(φ) as functions of φfor three different kernels and three different distributions of the series φl 53

6.2 Plot of the error in the kernel approximation as a function of φ usingthree different distributions of φl . . . . . . . . . . . . . . . . . . . . . 56

6.3 Plots of the average error in the kernel approximation as a function of Nl

using three different distributions of φl . . . . . . . . . . . . . . . . . . 576.4 Three different spatial patterns of the field ψ and the corresponding er-

ror in the integral approximation as a function of the number of basisfunctions Nl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.5 Performance comparison of the approximation and direct brute force al-gorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

ix

Abstract

In this work we present and study a mathematical model for water-biomass interactionsin arid and semi-arid ecosystems. These interactions often involve positive feedbacksbetween vegetation biomass and water: the larger the biomass the more water availableto the vegetation and the faster the vegetation grows. The increase in water availabilitythat comes with biomass can be attributed to reduced evaporation by shading (”shadingfeedback”), increased infiltration rates of surface water at vegetation patches (”infiltra-tion feedback”), and water uptake by plants’ roots, which lengthen as the plants grow(”uptake feedback”). The model captures all three positive feedback processes, whichare the crucial factors affecting water-biomass interactions in water limited systems.

The role of these feedbacks in the formation of vegetation pattern have already beenstudied but their implications for resource distribution and consequently for ecosystemengineering, changes in plant interactions, and species diversity have not been addressedusing a modeling approach.

By implementing concepts and tools from the theory of pattern formation (e.g. in-stabilities, bifurcation diagrams, stability analysis), we use the mathematical model tostudy a variety of currently important questions, including the self-organization of plantcommunities in arid and semi-arid regions to form vegetation patterns, sudden responsesof vegetation to environmental changes, ecosystem engineering, changes in plant inter-actions along gradients of environmental stresses, and mechanisms of species diversitychange.

We reproduce and study a sequence of vegetation patterns across a rainfall gradient(a widely observed natural phenomenon), identify ranges of pattern coexistence, andstudy possible transitions between different stable states of the system (catastrophicshifts). On hill slopes, we identified wide parameter ranges where multistability ofvegetation bands occur (i.e. bands patterns characterized by different wavenumbers),and we predict a tradeoff between biological productivity and resilience of the system.

We study the different soil-water distributions induced by different vegetation pat-terns and demonstrate how diversity in vegetation patterns may lead to habitat diver-sity. Furthermore, at the single-patch scale we predict a facilitation-resilience tradeoffby studying the relative strengths of the infiltration and uptake feedbacks.

By generalizing the model to describe the dynamics of n-interacting species pop-ulations, we successfully reproduce the widely observed trend of the transition fromcompetition to facilitation as aridity stress increases and we suggest a possible mecha-nism. In addition, we suggest two novel mechanisms responsible for changes in speciesdiversity, (i) interspecific facilitation induced by intraspecific competition and (ii) en-

1

Abstract

hanced facilitation at the single-patch scale induced by dynamic pattern changes at thelandscape scale (cross-scale processes).

Finally, we present a fast algorithm for convolution integrals with space and timevariant kernels. This new algorithm was developed in order to overcome several majordifficulties in the numerical integration of the model equations.

Keywords: Vegetation patterns, pattern formation, collective dynamics and emergentproperties, ecosystem engineer, habitats dynamics, landscape and species diversity, plantinteraction, competition and exclusion, facilitation and coexistence, cross-scale pro-cesses, convolution, space and time variant kernels, nonlocal terms, integro-differentialequations.

2

Chapter 1

Introduction

Plants, as primary producers, constitute the fundamental components of most ecosys-tems. Significant questions related to ecosystem function and stability are thereforeaddressed at the level of plant communities. Currently important research topics in-clude self-organization of plant communities in arid and semiarid regions to form vege-tation patterns [1], the sudden responses of vegetation to environmental changes [2, 3],ecosystem engineering [4–6], changes in plant interactions along gradients of environmen-tal stress [7–9], maintenance of species diversity [10, 11], and the impacts of diversitychanges on ecosystem function and stability [12,13].

A fundamental problem in understanding global ecosystem characteristics, such asbiodiversity, productivity, and stability, is how local properties at the individual andsingle plant-patch level, where laboratory and field experiments can be carried out, areup-scaled to the ecosystem or landscape level [12]. This problem has two facets, concep-tual and technical. Conceptually, at any higher level of organization, emergent phenom-ena may appear. Emergent properties involve collective dynamics of species individualsand therefore cannot be understood by studying individuals’ properties alone. The con-cept of emergent properties has a long history in ecology [14], and it has been adoptedin many fields of physics, including superconductivity, pattern formation, and networkdynamics [15]. A striking example in water-limited ecosystems is banded vegetationon hill-slopes, where positive water-biomass feedbacks and intraspecific competition atthe level of the individual patch lead to self-organizing band patterns at the landscapelevel [16].

Vegetation patterns, such as bands on hill slopes [16, 17], have been observed inmany arid and semiarid regions worldwide. The characteristic length scales associatedwith these patterns suggest the existence of intrinsic pattern formation mechanisms [18,19], independent of the heterogeneity of the physical environment. According to thisapproach vegetation patterns follow from spatial instabilities that reflect intraspecific 1

plant competition over a scarce water resource. The mechanisms responsible for theseinstabilities most often involve positive feedbacks [20] between vegetation biomass andwater: the larger the biomass the more water available to the vegetation and the fasterthe vegetation grows. The increase in water availability that comes with biomass can be

1Interspecific (intraspecific) interactions are interactions between individuals of different (the same)species.

3

Chapter 1. Introduction

attributed to reduced evaporation by shading (”shading feedback”), increased infiltrationrates of surface water at vegetation patches (”infiltration feedback”), and water uptakeby plants’ roots, which lengthen as the plants grow (”uptake feedback”).

Progress in understanding self-organized patchiness has largely been due to the de-velopment of mathematical models of vegetation growth [21–26]. The input informationused in formulating these models is at the level of a single plant-patch, while the pre-dictive power extends to the patchwork created at the landscape level. These modelsreproduce various biomass patterns observed in the field [1], predict the possible coex-istence of different stable patterns under given environmental conditions, and identify ageneric sequence of basic pattern states along the rainfall gradient [23,27].

Another phenomenon that has attracted considerable interest recently is the possibleoccurrence of sudden vegetation responses to small gradual environmental changes [1–3].Sudden responses, or ”catastrophic shifts”, have been interpreted as transitions betweentwo contrasting stable states taking place at the verge of a range of coexistence of thetwo states [28]. Examples of catastrophic shifts include sudden loss of transparencyand vegetation in shallow lakes subjected to human-induced eutrophication [29,30], andthe regeneration of woodlands as a result of low herbivore activity due to epidemic andhunting [31, 32]. Desertification in drylands [33] may also be viewed as a catastrophicshift involving a sudden transition from a patchy perennial vegetation state to a state ofbare soil, possibly with ephemeral plants, induced by climatic events or overgrazing [2].This view has recently been supported by mathematical models that demonstrated co-existence ranges of stable uniform vegetation with stable bare soil [34] and of patchyvegetation and bare soil [23].

The dynamics and spatial structures of plant communities strongly depend on inter-specific plant interactions [35]. These interactions can be negative, implying competition,or positive, implying facilitation. Recent studies have identified changes from negativeto positive interactions in conjunction with increases in abiotic stresses or consumerpressures [7, 9, 36–38]. In water-limited systems such changes have been observed withshrubs under conditions of increasing aridity. Facilitation in this case is manifested bythe appearance of annuals under the shrub canopies and has been attributed to theamelioration of micro-environmental conditions (reduced evaporation, nutrient accrue-ment, etc.) by the shrub [8, 39]. Recent experimental observations [40] however, areinconsistent with the reported shift in plant interactions from negative to positive asabiotic stresses increase [41].

Other studies addressing similar phenomena emphasized the importance of abioticlandscape modulations and resource redistributions by plant species. Shrubs modifythe landscape by forming patches of biomass with soil mounds and litter underneath.The soil mounds and the litter increase the water infiltration rate and form patches richwith soil-water and organic nutrients [42]. Contributing to this process are biologicalcrusts [43,44] (e.g. cyanobacteria crusts) that cover the bare soil, reduce the water infil-tration rate, and increase the runoff that is trapped at the soil mounds. The overall effectis the creation of favorable conditions for the growth of other species, such as annuals,under shrub canopies. Species facilitating the growth of other species by modulating thelandscape and concentrating resources have been termed ”ecosystem engineers” [4–6].

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Despite their relative success in reproducing biomass patchiness, current models [1]provide very limited information about the dynamics and the spatial distributions of thewater resource and thus, about engineering. In general, the water resource is coupledto the plant biomass through various positive-feedback processes, of which shading,infiltration, and uptake feedbacks are the dominant ones in water-limited systems.

Shading and infiltration feedbacks contribute to positive plant interactions, or fa-cilitation, by increasing the water resource at vegetation patches, whereas the uptakefeedback contributes to negative plant interactions, or competition, by reducing thesoil-water available to other plant species. Engineering is therefore highly sensitive tothe relative strength of these counteracting processes. Current models of self-organizedpatchiness do capture the water concentration processes but not the water uptake pro-cess, and therefore are not suitable for studying ecosystem engineering.

The impacts of facilitation or ecosystem engineering on plant communities andspecies diversity have largely been ignored in ecological theories [7, 38]. The ”real-ized niche” concept in niche theory [45] is a good example; it has traditionally beenconceived as a subset of the niche occupied by an isolated species (the ”fundamentalniche”), because of competitive interactions and exclusion by other species. As empha-sized recently [38], the realized niche of a given species can increase in the presence ofother species due to positive interactions, and as a result, species diversity can increaseas well [46, 47].

The mathematical models introduced in this work and used to study the above topicsare systems of nonlinear partial differential equations containing nonlocal terms in theform of spatial integrals. These integrals, which are not convolutions, model the uptakefeedback by capturing the root length dependence on plant biomass (the larger theplant’s biomass, the longer its roots). Numerical integration of the model’s equations iscomplicated by two inherent difficulties. First, a need to compute the nonlocal integralterms at each time step and second, the different time scales of the various processes inthe model. These time scales range between seconds-hours (e.g. flow of surface water)and months-years (e.g. plant’s growth rate). The immense computational complexityof these models requires the use of High Performance Computing system along with thedevelopment of fast numerical algorithms [48].

This dissertation is organized as follows: In Chapter 2 we describe a mathematicalmodel for vegetation growth in water-limited systems [49] applicable to many of theaspects discussed above. It is the first model of its kind to include all three positivefeedbacks between biomass and water: reduced evaporation by shading, increased in-filtration at vegetation patches, and water uptake by plants’ roots. In Chapter 3 weuse the model to study vegetation patterns along aridity gradients, we identify rangesof environmental conditions where different patterns coexist and study possible tran-sitions between them, and we suggest new insights regarding aridity classification ofecosystems. Furthermore, biological productivity and resilience properties of bandedvegetation are studied [50]. In Chapter 4 we study ecosystem engineering, addressingaspects such as resilience to disturbances, engineering niches, and the effects of spatialpatterning at the landscape level [51]. In Chapter 5 we generalize the model to includen-interacting species populations and use it to study changes in plant interactions along

5


different stress gradients, niche dynamics, and the implications for changes in speciesdiversity [52]. The numerical algorithm that was developed and used for the fast nu-merical integration of the models [48] is described in chapter 6. Finally, in chapter 7we summarize the main results of this work, discuss their validity in the context of themodel’s limitations, and conclude by suggesting possible future research directions andextensions of the model for the study of new problems.

6

Chapter 2

Model for dryland water-vegetationsystems

2.1 Introduction

A few mathematical models have been introduced to describe vegetation pattern for-mation in water-limited systems. Earlier models include cellular automata models de-scribing the vegetation biomass state in the system [53, 54] and continuous models inthe form of partial differential equations for a single dynamical biomass variable [21].Later on, models containing two dynamical variables representing biomass and soil-waterwere developed [22, 23], followed by three variable models where a distinction betweensoil-water and surface water is made [24, 25]. The three variable models are the mostappropriate for studying water-vegetation interactions, but they do not account for thewater uptake feedback. We introduce a mathematical model by Gilad et al. [49] thatcaptures all three feedbacks between biomass and water, including infiltration, uptake,and shading.

2.2 Model equations

We introduce a model for a single species population where the limiting resource is water.A ”patch” in the context of the model is defined to be an area covered by the plant, whichgenerally differs in its water content from the surrounding bare soil. The three dynamicalvariables of the model are: (a) the biomass density, B(X, T ), representing the plant’sbiomass above ground level in units of [kg/m2], (b) the soil-water density, W (X, T ),describing the amount of soil-water available to the plants per unit area of groundsurface in units of [kg/m2], and (c) the surface water variable, H(X, T ), describing theheight of a thin water layer above ground level in units of [mm]. Rainfall and topographyare introduced parametrically; thus, vegetation feedbacks on climate and soil erosion are

7

Chapter 2. Model for dryland water-vegetation systems

assumed to be negligible. The model equations are:

BT = GBB (1−B/K)−MB +DB∇2B

WT = IH −N (1−RB/K)W −GWW +DW∇2W

HT = P − IH +DH∇2(H2

)+ 2DH∇H · ∇Z + 2DHH∇2Z , (2.1)

where the subscript T denotes the partial time derivative, X = (X, Y ), and ∇2 =∂2

X + ∂2Y .

The quantity GB [yr−1] represents the biomass growth rate, while K [kg/m2] is themaximum standing biomass. The quantity GW [yr−1] represents the soil-water con-sumption rate, the quantity I [yr−1] represents the infiltration rate of surface water intothe soil, and the parameter P [mm/yr] stands for the precipitation rate. The param-eter N [yr−1] represents soil-water evaporation rate, while R describes the reductionin soil-water evaporation rate due to shading. The parameter M [yr−1] describes therate of biomass loss due to mortality and different kinds of continuous disturbances(e.g. grazing). The term DB∇2B represents seed dispersal while the term DW∇2Wdescribes soil-water transport in non-saturated soil [55]. Finally, the non-flat groundsurface height [mm] is described by the topography function Z(X) while the parame-ter DH [m2/yr (kg/m2)−1] represents the phenomenological bottom friction coefficientbetween the surface water and the ground surface.

While the equations for B and W are purely phenomenological (resulting from mod-eling processes at the single patch scale), the equation for H was motivated by shallowwater theory. The shallow water approximation is based on the assumptions of a thinlayer of water where pressure variations are very small and the motion becomes almosttwo-dimensional [56]. The detailed derivation of the equation for H is described inAppendix A.

2.3 Modeling positive feedback processes

The shading positive feedback is modeled by the parameter R, which measures thereduction in evaporation rate due to the presence of biomass. The two remaining positivefeedbacks are modeled through the explicit forms of the infiltration rate term I and thegrowth rate term GB. The infiltration feedback is modeled by assuming a monotonouslyincreasing dependence of I on biomass: the larger the biomass, the higher the infiltrationrate and the more soil-water is available to the plants. This form of the infiltrationfeedback mirrors the fact that in many arid regions infiltration is low far from vegetationpatches due to the presence of the biogenic crust. Conversely, the presence of shrubsdestroys the cyanobacterial crust and favors water infiltration. The uptake feedback ismodeled by assuming a monotonously increasing dependence of root length on biomass:the larger the biomass, the longer the roots and the more soil-water is taken up by theroots.

8


2.3.1 Infiltration feedback

The explicit dependence of the infiltration rate of surface water into the soil on thebiomass density is chosen as [25,57,58]:

I(X, T ) = AB(X, T ) +Qf

B(X, T ) +Q, (2.2)

where A [yr−1], Q [kg/m2] and f are constant parameters. Two distinct limits of thisterm (see Fig. 2.1) are noteworthy. When B → 0, this term represents the infiltrationrate in bare soil, I = Af . When B À Q, it represents the infiltration rate in fullyvegetated soil, I = A. The parameter Q represents a reference biomass beyond whichthe plant approaches its full capacity to increase the infiltration rate. It is a plantproperty reflecting, for example, leaf width. The difference between the infiltrationrates in bare and vegetated soil (hereafter the ”infiltration contrast”) is quantified bythe parameter f , defined to span the range 0 < f < 1. When f ¿ 1 the infiltration ratein bare soil is much smaller than that in vegetated soil. Such values can model bare soilscovered by biological crusts [43, 44]. As f gets closer to 1, the infiltration rate becomesindependent of the biomass density B, representing non-crusted soil where infiltration ishigh everywhere. The parameter f measures the strength of the positive feedback dueto increased infiltration at vegetation patches. The smaller f , the stronger the feedbackeffect.

10.80.60.40.20

infil

trat

ion

rate

[yr-1

] A

Af

B/K

Q/K

Figure 2.1: The infiltration rate I = A(B + Qf)/(B + Q) as a function of biomass density B. Whenthe biomass is diminishingly small (B ¿ Q) the infiltration rate approaches the value of Af . Whenthe biomass is large (B À Q) the infiltration rate approaches A. The infiltration contrast between bareand vegetated soil is quantified by the parameter f , where 0 < f < 1; when f = 1 the contrast is zeroand when f = 0 the contrast is maximal. Small f values can model biological crusts which significantlyreduce the infiltration rates in bare soils. Disturbances involving crust removal can be modeled byrelatively high f values. The parameters used are given in Table 2.1.

9


2.3.2 Uptake feedback

The growth rate GB at a point X at time T has the form:

GB(X, T ) = Λ

∫

Ω

G(X,X′, T )W (X′, T )dX′ ,

G(X,X′, T ) =1

2πS20

exp

[− |X−X′|2

2[S0(1 + EB(X, T ))]2

], (2.3)

where the integration is over the entire domain Ω and the kernel G (X,X′, T ) is normal-ized such that for B = 0 the integration over the entire domain equals unity. Accordingto this form the biomass growth rate depends not only on the amount of soil-water atthe plant location, but also on the amount of soil-water in the neighborhood spannedby the plant’s roots. A measure of the root-system size [m] is given by the width of theGaussian function in Eq. (2.3), S0(1+EB(X, T )), where E [(kg/m2)−1] is root extensionper unit biomass, beyond some minimal root length S0. The parameter Λ has units of[(kg/m2)−1 yr−1] and represents the plant’s growth rate per unit amount of soil-water.The parameter E measures the strength of the positive feedback due to water uptakeby the roots. The larger E, the stronger the feedback effect.

The soil-water consumption rate at a point X at time T is similarly given by

GW (X, T ) = Γ

∫

Ω

G(X′,X, T )B(X′, T )dX′ . (2.4)

Note that G(X′,X, T ) 6= G(X,X′, T ). The soil-water consumption rate at a givenpoint is due to all plants whose roots extend to this point. The parameter Γ has units[(kg/m2)−1yr−1] and measures the soil-water consumption rate per unit biomass.

The parameter values used in this work (unless stated otherwise) are summarizedin Table 2.1. They are chosen to describe shrub species and are taken or deducedfrom [22,25,55,59]. The model solutions described here are robust and do not depend onthe delicate tuning of any particular parameter. The precipitation parameter representsmean annual rainfall in this work and assumes constant values. This approximation isjustified for species such as shrubs, whose growth time scales are much longer than thetime scale of rainfall variability.

2.4 Non-dimensional form of the model

It is advantageous to study the model equations using non-dimensional variables andparameters, for it eliminates dependent parameters and reveals the possible equivalenceof different parameters in controlling the states of the system. Rescaling the modelvariables and parameters as in Table 2.2, we obtain the following non-dimensional formof the model equations:

bt = Gbb(1− b)− b+ δb∇2b

wt = Ih− ν(1− ρb)w −Gww + δw∇2w

ht = p− Ih+ δh∇2(h2) + 2δh∇h · ∇ζ + 2δhh∇2ζ , (2.5)

10


Parameter Units Description Value/Range

K kg/m2 Maximum standing biomass 1Q kg/m2 Biomass reference value beyond 0.05

which infiltration rate under a patchapproaches its maximum

M yr−1 Rate of biomass loss due to mortality 1.2and disturbances

A yr−1 Infiltration rate in fully vegetated soil 40N yr−1 Soil-water evaporation rate 4E (kg/m2)−1 Root’s extension per unit biomass 3.5Λ (kg/m2)−1yr−1 Biomass growth rate 0.032

per unit soil-waterΓ (kg/m2)−1yr−1 Soil-water consumption rate 20

per unit biomassDB m2/yr Seed dispersal coefficient 6.25× 10−4

DW m2/yr Transport coefficient for soil-water 6.25× 10−2

DH m2/yr (kg/m2)−1 Bottom friction coefficient between 0.05surface water and ground surface

S0 m Minimal root length 0.125Z(X) mm Topography functionP kg/m2 yr−1 Precipitation rate [0, 1000]R – Evaporation reduction due to shading 0.95f – Infiltration contrast between bare soil 0.1

and vegetated soil

Table 2.1: A list of parameters of the model, their units, and their numerical values. The parametervalues are set to represent shrubs, using [22,25,55,59].

where t and x = (x, y) are the non-dimensional time and spatial coordinates, ∇2 =∂2

x + ∂2y , and x′ = (x′, y′).

The infiltration term has the non-dimensional form

I(x, t) = αb(x, t) + qf

b(x, t) + q, (2.6)

the growth rate term Gb is written as

Gb(x, t) = ν

∫

Ω

g(x,x′, t)w(x′, t)dx′ , (2.7)

g(x,x′, t) =1

2πexp

[− |x− x′|2

2(1 + ηb(x, t))2

],

and the soil-water consumption rate can be similarly written as

Gw(x, t) = γ

∫

Ω

g(x′,x, t)b(x′, t)dx′ . (2.8)

11


Quantity Scaling Quantity Scaling

b B/K p ΛP/MNw ΛW/N δb DB/MS2

0

h ΛH/N δw DW/MS20

q Q/K δh DHN/MΛS20

ν N/M ζ ΛZ/Nα A/M ρ Rη EK t MTγ ΓK/M x X/S0

Table 2.2: Relations between non-dimensional variables and parameters and the dimensional onesappearing in the dimensional form of the model equations 2.5–2.8.

2.5 Aridity parameter

In obtaining equations 2.5–2.8, we eliminated four dependent parameters (K, M, Λ, S0).The non-dimensional form of the precipitation parameter

p =ΛP

MN, (2.9)

proves the equivalence of decreasing the precipitation rate, P , to increasing the mortality(grazing) rate, M , or the evaporation rate, N , in traversing the basic instabilities of thesystem. The non-dimensional precipitation p can be used to define an aridity parameter,a = p−1. This form extends an earlier definition [21] by adding the two parameters Pand M .

12

Chapter 3

Landscape states

3.1 Introduction

The landscape of a dryland ecosystem is a patchwork of biomass and resources. Thispatchwork changes with rainfall conditions, grazing stress, soil properties, ground topog-raphy, plant species traits, etc. The effects of these factors on the system’s landscape canbe studied by solving the model equations for various parameter values. In this chapterwe (a) map the basic landscape states that appear along aridity gradients, (b) studycoexistence ranges of stable states and state transitions (catastrophic shifts), (c) usethe aridity parameter definition suggested in section 2.5 to gain new insights regardingthe degree of aridity of the system, and (d) study structural and dynamical aspects ofbanded vegetation states.

3.2 Mapping the landscape states along aridity gra-

dients

The model has two homogeneous stationary solutions representing bare soil and uniformsoil coverage by vegetation. The two solutions and their linear stability ranges for planetopography are shown in the bifurcation diagram displayed in Fig. 3.1. The bifurcationparameter is p, the dimensionless form of the precipitation parameter, P . The natureof the different bifurcations is demonstrated in Fig. 3.2. The linear stability analysisleading to this bifurcation diagram and the linear stability of the solutions is describedin Appendix B.

The bare soil solution, denoted in Fig. 3.1 by B, is given by b = 0, w = p/ν, andh = p/αf . It is linearly stable for p < pc = 1 and loses stability at p = 1 to uniformperturbations 1, as shown in Fig. 3.2a. The uniform vegetation solution, denoted by V ,exists for p > 1 in the case of a supercritical bifurcation and for p > p1 (where p1 < 1) inthe case of a subcritical bifurcation. It is stable, however, only beyond another threshold,p = p2 > p1. As p is decreased below p2 the uniform vegetation solution loses stability

1The bifurcation is subcritical (supercritical) depending whether the expression 2ην/[ν(1 − ρ) + γ]is greater (lower) than unity.

13

Chapter 3. Landscape states

0

0.2

0.4

0.6

biom

ass

[kg/

m2 ]

d)

precipitation pp

cp1 p2

V

B

a) b) c)

Figure 3.1: Bifurcation diagram for homogeneous stationary solutions of the model equations 2.5–2.8showing the biomass B vs. precipitation p (panel d) for plane topography. The solution branches Band V represent, respectively, bare-soil and uniform-vegetation solutions. Solid lines represent stablesolutions, and dashed and dotted lines represent solutions unstable to uniform and non-uniform pertur-bations, respectively. Also shown are the basic vegetation patterns, spots (a), stripes (b), and gaps (c),along the precipitation gradient, obtained by numerical integration of the model equations. Dark shadesof gray represent high biomass density. The domain size in panels a-c is 10 × 10 m2. The bifurcationpoints in panel d are: pc = 1, p1 = 0.65, p2 = 1.91. The parameters used are given in Table 2.1 andcorrespond to a woody shrub.

to non-uniform perturbations in a finite wavenumber (Turing-like) instability [18], asshown in Fig. 3.2b. These perturbations grow to form large amplitude patterns. Asequence of basic patterns (gaps, stripes, and spots) is found for plane topography withsteadily decreasing precipitation values (see panels c, b, and a in Fig. 3.1).

The basic landscape states persist on slopes but with two major differences: stripes,which form labyrinthine patterns on a plane, reorient perpendicular to the slope directionto form parallel bands, and the patterns travel uphill. (Typical speeds for the parametersused in this study are of the order of centimeters per year.). Fig. 3.3 shows the devel-opment of bands traveling uphill from an unstable uniform vegetation state. Travelingbands on a slope have been found in earlier models as well [21–25, 49, 53, 54]. Anotherdifference is the coexistence of multiple band solutions with different wavenumbers inwide precipitation ranges (see section 3.5).

The bifurcation diagram displayed in Fig. 3.1 can be expressed in terms of the aridityparameter a = p−1 as the bifurcation parameter, rather than p. The resulting diagram isshown in Fig. 3.4. It shows how the homogeneous stationary solutions and their stabilityproperties change by increasing the grazing stress, M , or the evaporation rate, N , as ais proportional to M and N .

The sequence of basic landscape states (uniform vegetation, gaps, stripes or bands,spots, and bare soil) as the system’s aridity is increased has been found in earlier vege-

14


-0.1

0

0.1

0 1 2

a)

-0.1

0

0.1

210

b)

kk

ℜ[λ

(k)]

ℜ[λ

(k)]

p > pc

p = pc

p < pc

p > p2

p = p2

p < p2

kc

Figure 3.2: Linear growth-rate curves of non-homogeneous spatial perturbations obtained from theeigenvalues of the Jacobian given by Eq. (B.16), and applied to (a) the bare state and (b) the uniformvegetation solutions of the model equations 2.5–2.8, around the points p = pc and p = p2, respectively(see Fig 3.1). The bare state becomes unstable at p = pc to homogeneous perturbations, as the fastestgrowing mode has a wavenumber k = 0 (panel a). The uniform vegetation solution undergoes a finitewavenumber instability at p = p2 and becomes unstable to non-homogeneous perturbations, as thefastest growing mode has a wavenumber k = kc 6= 0 (panel b).

c) t=55 d) t=106b) t=20a) t=0

downhill

Figure 3.3: Development of vegetation bands traveling uphill from an unstable uniform vegetation state,obtained by numerical integration of the model equations 2.5–2.8 at P = 600 mm/yr. The differentpanels a–d show snapshots of this process at different times (t in years). Dark shades of gray representhigh biomass density. The bands are oriented perpendicular to the slope gradient and travel uphill withtypical speed of a few centimeters per year. The parameters used are given in Table 2.1, the domainsize is 5× 5 m2, and the slope angle is 15o.

tation models [21, 23–25, 27] and is consistent with field observations [1, 16, 21, 27]. Thepresent model, however, contains information on the water resource as well. Two land-scape states that appear to have the same spatial vegetation pattern, e.g. spots, maydiffer in their soil-water distributions due to different relative strengths of the infiltra-tion and the roots feedbacks. We will discuss this difference in chapters 4 and 5 in thecontext of ecosystem engineering and plant competition and facilitation.

3.3 Coexistence of landscape states and state tran-

sitions

Any pair of consecutive landscape states along the rainfall or aridity gradient has a rangeof bistability (coexistence of two stable states): bistability of bare soil with spots, spots

15


0

0.2

0.4

0.6

0.8

1

biom

ass

[kg/

m2 ]

aridity parameter aa

ca1a2

V

B

Figure 3.4: Bifurcation diagram for homogeneous stationary solutions of the model equations 2.5–2.8similar to that shown in Fig. 3.1d except that the bifurcation parameter is the aridity parameter a. Thebifurcation points are: ac = 1.00, a1 = 1.10, a2 = 0.12. The parameters used are given in Table 2.1.

with stripes, stripes with gaps, and gaps with uniform vegetation as demonstrated inFig. 3.5. On a slope, tristability of bare soil, spots, and bands has been found. In addi-tion, multiple band solutions with different wavenumbers coexist in wide precipitationranges (see section 3.5 for detailed results).

precipitation

x x x x

y

Figure 3.5: Coexistence of different stable landscape states along the rainfall or aridity gradient. Shownare examples, obtained by numerical integration of the model equations 2.5–2.8 at increasing precipi-tation values, of coexistence of bare soil with spots, spots with stripes, stripes with gaps and gaps withuniform vegetation. The parameters used are given in Table 2.1 and the domain size is 7.5× 7.5 m2.

Bistability of different landscape states has at least three important implications: (i)it implies hysteresis, which has been used to elucidate the irreversibility of desertificationphenomena [2,23], (ii) it implies vulnerability to environmental stresses by disturbancesinvolving biomass removal, and (iii) it increases landscape diversity as patterns involvingspatial mixtures of two distinct landscape states become feasible. This latter issue willbe addressed in chapter 4.

A climatic fluctuation, such as drought, that drives the system beyond the bistabilityrange can result in an irreversible transition to a less biologically productive state, aphenomenon known as ”desertification” [33]. Fig. 3.6 illustrates such a scenario. Theinitial state corresponds to stable spots (the S branch) that coexists in the range p0 <

16


0

0.2

0.4

0.6

0.8

biom

ass

[kg/

m2 ]

precipitation

pc

p1p0

V

B

S

Figure 3.6: Bifurcation diagram similar to that shown in Fig. 3.1d except that in addition to the uniformstates, B and V, it displays a solution branch S representing a stable spots pattern. Note the widecoexistence range, p0 < p < pc, of stable bare soil and stable spots. The parameters used are given inTable 2.1, p0 = 0.33 (50 mm/yr) and pc = 1 (150 mm/yr).

p < pc with bare soil (the B branch). A precipitation downshift below p0 results ina transition (catastrophic shift) to the bare soil state (downward arrow). When thedrought is over and the original precipitation resumes the vegetation does not recoverbecause of the stability of the bare soil state. A particularly rainy period with annualprecipitation rates exceeding pc is needed for the vegetation to recover. This type ofprocess is known in other contexts (e.g. magnetism) as hysteresis.

State transitions can also be induced by temporal disturbances such as clearcutting,crust removal, or fires. Of particular interest are circumstances where local disturbancesinduce global transitions. Banded patterns on slopes provide nice examples for globaltransitions. Fig. 3.7 shows model simulations of a transition from a stable band patternto a stable spot pattern induced by a local biomass removal (through initial conditionsfor the biomass variable) that mimics the effect of clearcutting. The initial cut of theuppermost band allows for more runoff to accumulate at the band section just below it(panel a). As a result this section grows faster, draws more water from its surroundingsand induces vegetation decay at the nearby band sections. The entire process continuesrepeatedly until the whole pattern is transformed into stable spots pattern (panel d).

a) t=0 b) t=6 c) t=11 d) t=25

downhill

Figure 3.7: Local disturbance leading to a global state transition. A local clearcut along the uppermostband of a linearly stable band pattern on a slope (panel a) induces a chain reaction that culminatesin a stable spot pattern (panel d). The driving forces of the process are runoff flow and intraspecificcompetition as explained in the text. The parameters used are given in Table 2.1 with P = 225 mm/yr.The domain size is 5× 5 m2 and t in years.

17


Figure 3.8 shows an example of a different type of global transition induced by a localclearcutting disturbance, where precipitation value (P=500 mm/yr) is high enough sothat a spots pattern is not stable. In this case, the landscape state retains a bandedvegetation state, but a transition to a lower wavenumber occurs. The coexistence ofbands patterns with different wavenumbers and possible transitions between them affectthe biological productivity and resilience of the ecosystem and is further studied insection 3.5.

time

downhill

Figure 3.8: A local clearcut, similar to that shown in Fig. 3.7, but in a precipitation range where spotspattern is unstable. As a result, a transition to a banded vegetation pattern with lower wavenumberoccurs. The parameters used are given in Table 2.1 with P = 500 mm/yr. The domain size is 5× 5 m2.

On a plane topography similar local disturbances have no global effects, as Fig. 3.9demonstrates. Shown in this figure is a stable stripe pattern which coexists with a stablespots pattern. The impact of a local clearcut in the initial stripe pattern remains local.

a) b)

Figure 3.9: A local clearcut, similar to that shown in Fig. 3.7, but in plane topography, has no globaleffect, as indicated by the initial and asymptotic states shown in panels a and b respectively. Theparameters used are given in Table 2.1 and the domain size is 7.5× 7.5 m2.

3.4 Aridity parameter and stability of uniform states

3.4.1 Aridity classification

The term aridity refers to a permanent pluviometric deficit whose strength bears onthe degree of vegetation the system can support. Aridity classes are introduced toreflect different landscape states at different pluviometric conditions, defined by theannual rainfall or by an aridity index (such as the ratio of the annual rainfall to theevapotranspiration rate) [60].

18


To circumvent this difficulty, von Hardenberg et al. [23] proposed using the inherentlandscape states of the system as a basis for classifying drylands. In this context, the factthat the model reproduces the basic sequence of landscape states and their coexistence,as predicted in [23], corroborates their proposed classification of aridity.

3.4.2 Bare state stability threshold

The point along the aridity gradient at which the bare state loses stability, i.e. pc, mayprovide some indication regarding the degree of aridity of the system, since it designatesthe point beyond which the system can spontaneously recover from desertification. Ac-cording to a linear stability analysis, the bare state instability threshold (the neutralstability curve) is pc = PΛ/MN = 1. This implies that four parameters alone affectthe bare state stability. The effects of the four parameters Λ, P , M , and N on the barestate stability threshold are shown in Fig. 3.10.

0

100

200

0 1 2

stable

unstable

a)

P

M

A BB′

0

5

10

0 0.1

unstable

stable

b)

Λ

N

0

100

200

0 0.1

unstable

stable

c)

P

Λ

∆Λ

∆Λ

∆P

∆P

0

10

20

0 1 2

stable

unstable

d)

M

N

Figure 3.10: Neutral stability curves for the bare soil solution obtained from the linear stability resultpc = ΛP/MN = 1, by plotting P = (NΛ−1)M (a), N = (PM−1)Λ at P = 100 mm/yr (b), P =(MN)Λ−1 (c), and N = (ΛP )M−1 at P = 150 mm/yr (d). All other parameters are given in Table 2.1.The bare soil solution loses stability as any of these curves is traversed. The points A and B in panel adenote the state of two systems that differ only in grazing stress, leading to different stability propertiesof the bare state (see text for more details). The rectangles in panel c show two upshifts in growth ratesthat are required to cross the recovery threshold, pc, of a desertified system. The dryer the system, thelarger the required upshift.

Consider an ecosystem characterized by a precipitation rate between P and P +∆P ,a grazing stress between M and M+∆M , an evaporation rate between N and N+∆N ,and a biomass growth rate per unit soil-water between Λ and Λ + ∆Λ. We define thestate of the system as the 4-dimensional hyper-volume it occupies in the 4-dimensionalparameter space spanned by the four parameters Λ, P , M , and N . The dynamics of thestate of the system in this parameter space correspond to changes in environmental or

19


biological conditions of the system (e.g. changes in precipitation P or in grazing stressM). The bare state stability threshold is a 3-dimensional surface in the 4-dimensionalparameter space defined by PΛ/MN = 1. When the system state crosses this threshold,the bare state can become stable (implying an increase in aridity) or unstable (implyinga decrease in aridity) depending on the direction of the crossing.

We illustrate the above by considering two ecosystems in two different states (denotedby A and B) and their dynamics in the subspace P −M (for a given Λ and N) as shownin Fig. 3.10a. The two systems, A and B, differ only in their M values (MA < MB),i.e., the grazing stress in system A is lower than that in system B. Clearly for the givenM values, the bare state loses stability at a lower precipitation value in system A thanin system B, implying the stronger aridity of system B. If we consider variations alongthe P axis, a downshift in P may cause system A to cross the bare state threshold fromthe unstable to the stable regime (see downward arrow), thus becoming more arid. Onthe other hand, in order for the bare state in system B to become unstable, an upshiftin P is required (see upward arrow). The closer system B is to the threshold (e.g., lowergrazing stress), the smaller this required upshift in P (see point B′).

Since similar arguments hold for additional subspaces that include other parameters(see panels b–d), we may conclude that as the state of an ecosystem moves away fromthe bare state threshold into the stable (unstable) regime, the degree of aridity of theecosystem increases (decreases).

Noteworthy is the hyperbolic relationship between P and Λ (Fig. 3.10c). A regionthat has gone through desertification due to a precipitation drop can spontaneouslyrecover by seedling with species that have Λ values sufficiently high to cross the stabilitythreshold. This practice, however, becomes increasingly hard to implement as the systembecomes dryer because, as the rectangles in Fig. 3.10c illustrate, the upshift required inΛ is getting larger.

3.4.3 Uniform vegetation stability threshold

The stability range of the uniform vegetation solution to spatial perturbations alongan aridity gradient can be described by a stability balloon, analogous to the Busseballoon for the Rayleigh-Benard convection [61], with the aridity parameter a = p−1 asthe control parameter. Figure 3.11 shows the neutral stability curve that denotes thethreshold for a finite wavenumber instability of the uniform vegetation solution as thecontrol parameter a is varied. At low values of a (a < a2), the uniform vegetation solutionis stable for spatial perturbations. As a increases beyond some critical value a = a2, theuniform vegetation solution becomes unstable to spatial perturbations, where a bandof wavenumbers have positive growth rates. The uniform vegetation solution remainsunstable until another threshold is crossed, a = a1, beyond which this solution no longerexists.

20


1

043210

unstable

stable

k

a=

p−1

kc

a1

a2

Figure 3.11: Stability balloon describing the stability range of the uniform vegetation solution to spatialperturbations along an aridity gradient, with the aridity parameter a = p−1 as the control parameter.Shown is the neutral stability curve, obtained from Eq. (B.16), that denotes the threshold for a finitewavenumber instability as the control parameter a is varied. For a < a2, the uniform vegetationsolution is stable for spatial perturbations. As a increases beyond some critical value a = a2, theuniform vegetation solution becomes unstable to spatial perturbations. The uniform vegetation solutionremains unstable until another threshold is crossed, a = a1, beyond which this solution no longer exists.The thresholds a1 and a2 correspond to p1 and p2 in Fig. 3.1.

3.5 Structural and dynamical aspects of banded veg-

etation states

One striking example of vegetation patterns resulting from self-organization is bandedvegetation. Vegetation bands on hill slopes, consisting primarily of trees and shrubs,have been observed in arid and semi-arid regions throughout the world [16, 21, 22, 24].The bands are normally oriented perpendicular to the slope direction and are a few tensof meters apart. Some field observations suggest that bands migrate uphill at a rate ofa few tens of centimeters per year [62].

In this section we use the model to study the biological productivity and the resilienceof banded vegetation states by considering structural and dynamical aspects. Under-standing these aspects is significant for the conservation and rehabilitation of desertifiedregions [3].

3.5.1 Uniform and pattern solutions

The two stationary uniform states representing bare soil and uniform vegetation, as wellas patterned states, still hold on a uniform slope topography, as the bifurcation diagramin Fig. 3.12 shows. The insets show two basic patterned states, spot patterns at low

21


precipitation rates and band patterns at higher precipitation rates. Stable spot patternsappear at precipitation values where the bare soil state is still stable. Likewise, stableband patterns appear at precipitation values where spot patterns are still stable. Withappropriate parameter choices a tristability of bare state, spots, and bands is found.These ranges of multistability have significant implications on desertification [23] andhabitat diversity [49].

0

0.2

0.4

0.6

0

biom

ass

[kg/

m2 ]

precipitation

pc

p1 p2

V

B

Figure 3.12: A bifurcation diagram showing biomass B vs. precipitation p for a uniform slope topogra-phy. The branch B represents a uniform bare soil solution and the branch V a uniform vegetation state.The bare state becomes unstable at pc = 1, whereas the uniform vegetation state becomes unstableat p2 = 4.146. The insets show typical patterns along the precipitation gradient, spots at low p (200mm/yr) and bands at higher p (200, 250 and 500 mm/yr for a single band and five and nine bands,respectively). Dark shades of grey represent high biomass density. pc corresponds to 250 mm/yr andp2 to 1036.5 mm/yr. The parameters used are: ν = 2, η = 5, α = 160, f = 0.1, q = 0.05, γ = 5, ρ = 1,δb = 0.02, δw = 2, δh = 200, slope = 3 degrees.

3.5.2 Multiple band states and biological productivity

Multistability of stable states also exists within the precipitation range of stable bandswhere the bare state and spot patterns are unstable. The coexisting stable states cor-respond to band patterns with different wavenumbers. At relatively low precipitationvalues, where the bare state is still stable, single-band patterns are possible (secondinset from left in Fig. 3.12). At higher precipitation values stable patterns with higherwavenumbers appear but many of the lower wavenumber patterns remain stable. Thesestates are likely related to the discrete families of Eckhaus stable solutions that have beenfound in simpler models of finite systems, such as the Ginzburg-Landau model [63] andthe complex Swift-Hohenberg model [64,65]. The existence of a discrete family of stablebanded patterns suggests two principal responses to a gradual precipitation downshift:(i) an increase in the ratio of the interband distance to the band width (interband-band-ratio, or IBR index [66]) without a change in the pattern’s wavenumber, and (ii) adecrease in the pattern’s wavenumber. Fig. 3.13 shows how the IBR index changes with

22


precipitation. The numerical data fits an exponential function nicely in agreement withfield observations [66].

Precipitation [mm/year]

x [m

]

200 300 400 5000

2

4

6

8

10

100 200 300 400 5001

2

3

4

Precipitation [mm/year]100 200 300 400 500

0.5

1

1.5

2

bto

tal[ k

g]

(a) (b)

IBR

Figure 3.13: The effect of precipitation on the widths of vegetation bands for a pattern with a givenwavenumber. The numerically computed Interband-band-ratio (IBR) (circles in panel b) fits an expo-nentially decreasing function (thick solid line) well. The total biomass grows linearly with precipitation(thin solid line). The parameters used are the same as for Fig. 3.12.

0 2 4 60

0.25

0.5

0.75

1

bto

tal

0 2 4 60

500

1000

1500

Wto

tal

0 2 4 61500

2000

2500

k k k

R

(a) (b) (c)

Figure 3.14: The dependence of biological productivity on the pattern’s wavenumber k. Both totalbiomass, btotal, and total amount of water consumed by the vegetation, wtotal, increase with k, but theratio R = wtotal/btotal decreases. Thus, high wavenumber patterns consume less water and are thereforemore productive. p = 0.8 (200 mm/year) and all other parameters are the same as in Fig. 3.12.

The multistability of band states affects the rate at which vegetation consumes water,and as a result, the biological productivity of the ecosystem: the smaller this rate thehigher the productivity. We evaluated the water consumption rate per unit of biomassfor all band solutions that coexist for given precipitation and slope values by calculatingthe total water consumption rate wtotal =

∫Gwwdx, the total biomass, btotal =

∫bdx,

and the ratio wtotal/btotal. The results are shown in Fig. 3.14. Both wtotal and btotal

increase with the pattern’s wavenumber (or number of bands) but wtotal increases moreslowly. As a consequence the water consumption per unit biomass decreases as the pat-tern’s wavenumber increases. This result suggests that establishing higher wavenumberpatterns in rehabilitating desertified regions will increase the biological productivity ofthese regions.

23


3.5.3 The productivity-resilience tradeoff of band states

Banded patterns with higher wavenumbers are more biologically productive but areless resilient to environmental changes. We demonstrate this property by studying theresponse of banded vegetation to a precipitation downshift. Fig. 3.15 shows the responseof a relatively high wavenumber pattern (k0 = 8.8) to a precipitation downshift withinthe range of multistability of band states. The downshift leads to an Eckhaus-typeinstability [18] that culminates in a significantly lower wavenumber (k = k0/2). Anidentical downshift of a band pattern with an intermediate wavenumber (k0 < 8) leavesthe wavenumber unchanged. Thus, lower wavenumber patterns which are less productiveare more resilient to precipitation downshifts such as prolonged droughts.

a) t=0 b) t=47 c) t=60 e) t=80

downhill

d) t=72 f) t=130

Figure 3.15: Low resilience of vegetation bands to precipitation downshifts. A high wavenumber pattern(k0 = 8.8) which is stable at p = 2 (500 mm/yr) responds to a precipitation downshift to p = 1.2 (300mm/yr) by gradual downhill elimination of every second band. The parameters used are the same asin Fig. 3.12 and t is in years.

24

Chapter 4

Plants as ecosystem engineers

4.1 Introduction

Ecosystem dynamics at the landscape level involve a multitude of interacting species withmany traits with respect to resource acquisition and distribution. Some species, how-ever, have greater impacts than others in the sense that their introduction or removal candramatically alter the ecosystem’s behavior. One example are ”keystone species” [67],whose roles in energy flow and nutrient cycling can affect the trophic web structure ofthe system. Recently, another type of key species, named ”ecosystem engineers” [4–6],has been identified. Unlike keystone species, which directly affect the biotic compo-nent, ecosystem engineers affect the physical environment, thereby changing resourcedistributions and, as a result, species diversity and ecosystem functioning [4–6,68–72].

Processes of resource distribution are normally combinations of biotic factors, asso-ciated with ecosystem engineers, and abiotic factors. One abiotic factor contributingto water redistribution in drylands is dust and sediment deposition in rocky water-sheds [73]. The higher water infiltration rates in deposition patches induce source-sinkwater flow relationships and enrich the water content in these patches. Biotic factorsinclude cyanobacteria that form partially impermeable biogenic crusts, limiting infiltra-tion of rainfall and favoring surface runoff generation, and plants, particularly shrubs,that act as sinks for the surface-water flow. The accumulation of litter and dust underthe shrub forms a mound with high infiltration capacity, which not only absorbs directrainfall but also intercepts the runoff water generated by the soil crust. Reduced evap-oration under the shrub canopy further contributes to local water accumulation. Thewater-enriched patch and the accumulated dust and litter create an island of fertility [74]under the shrub canopy, characterized by higher concentrations of water, nutrients, andorganic matter relative to the surrounding crusted soil.

At the landscape level self-organization processes generating spatial patterns maytake place. Depending on environmental conditions (rainfall, topography, grazing stress,etc.) and species traits, different vegetation patterns can emerge (see chapter 3). Oneimportant aspect of vegetation patterns that escaped attention in earlier theoreticalstudies is how pattern dynamics at the landscape level affect water distribution at thesingle patch level. We refer to this interplay between processes at the single patch scale

25

Chapter 4. Plants as ecosystem engineers

and processes at the landscape scale as cross-scale processes (e.g. competition at thepatch scale leads to pattern formation at the landscape scale, which in turn feeds backon processes at the patch scale). Such cross-scale processes are studied in the contextof micro-habitat dynamics (see below for micro-habitat definition).

Both positive feedbacks, increased infiltration at vegetation patches and water uptakeby roots, act to deplete soil-water from the patch surroundings and induce intraspecificcompetition over the water resource. As a consequence, both feedbacks, independentlyof one another, can induce instabilities that lead to spatial patterns. While the twofeedbacks give rise to similar biomass patterns, they differ in the resource distributionsthey induce, a difference which reflects on the function of plants as ecosystem engineers.We discuss this difference in two contexts: (a) facilitation vs. resilience to disturbancesand (b) facilitation vs. competition along aridity gradients.

Throughout this chapter we define engineering as the capacity of a plant speciesto concentrate soil-water beyond the level pertaining to bare soil. The concentrationprocess can be either increased infiltration rate at the plant patch, or runoff interceptionby the soil mound the plant forms. To simulate the former process we need to choosesmall f values. To simulate the latter process a topography function ζ(x) that mimicssoil mounds should be chosen. For simplicity we study the former process.

We define a ”micro-habitat” of a given species as a small area in the physical do-main, at the scale of a single patch of the ecosystem engineer, that is mapped into thefundamental niche of that species in niche space [45]. The micro-habitats created by anecosystem engineer are characterized by a higher soil-water density than what would ex-ist in the absence of the ecosystem engineer. The definition of habitats generally includeadditional resources beside soil-water, such as light, nutrients, soil pH, etc., which arenot explicitly modeled in equations 2.5–2.8. In drylands, however, the water resourcecan often be considered the limiting factor, justifying the use of the model equations forstudying habitat creation.

4.2 The facilitation-resilience tradeoff

The infiltration feedback concentrates soil-water under vegetation patches of the ecosys-tem engineer, leading to facilitation (or ”engineering”) by creating favorable conditionsfor the growth of other species. The uptake feedback, on the other hand, leads toresilience as it increases the water uptake capability of the ecosystem engineer.

We study engineering by varying the parameters f and η (the non-dimensional formof E) that control the strength of the infiltration and the uptake feedbacks, respectively.We look for conditions that maximize the engineering capacity and ask what price thesystem has to pay for attaining strong engineering (facilitation). The results are sum-marized in Fig. 4.1 and indicate the existence of a tradeoff between the engineeringcapacity of a plant and its resilience to disturbances. Conditions that favor ecosystemengineering, resulting in water enriched patches or micro-habitats, imply low resilience,and conditions that favor high resilience imply weak or no engineering.

Shown in Fig. 4.1 are spatial profiles of b, w, and h for a single patch of the ecosystemengineer at decreasing values of η, representing species with different root extension

26


η = 12 η = 3.5 η = 2

f=

0.1

a) b

w

h

b) b

w

h

c) b

w

h

f=

0.9

d) b

w

h

e) b

w

h

f) b

w

h

Figure 4.1: Spatial profiles of the variables b, w, and h as affected by the parameters that controlthe main positive biomass-water feedbacks, f (infiltration feedback) and η (uptake feedback). Theprofiles are cross sections of two-dimensional simulations of the model equations 2.5–2.8. In all panels,the horizontal dotted lines denote the soil-water level in bare soil. Strong infiltration feedback andweek uptake feedback (panel c) leads to high soil-water concentration reflecting strong engineering.Strong uptake feedback results in soil-water depletion and no engineering, irrespective of the infiltration-feedback strength (panels a,d). While the species characterized by η = 2 is the best engineer underconditions of strong infiltration contrast (panel c), it leads to low system resilience; the engineer alongwith the micro-habitat it forms completely disappear when the infiltration contrast is strongly reduced,e.g. by crust removal (panel f). A species with somewhat stronger uptake feedback (η = 3.5) still actsas an ecosystem engineer (panel b) and also survives disturbances that reduce the infiltration contrast(panel e), thereby retaining the system’s resilience. The Parameters used are given in Table 2.1 withP = 75 mm/yr. Panels a and d span a horizontal range of 14 m while all other panels span 3.5 m.

properties, and for two extreme values of f . The value f = 0.1 models high infiltrationrates under an engineer’s patches and low infiltration rates in bare soil, which mayresult from a biological crust covering the bare soil. The value f = 0.9 models highinfiltration rates everywhere. This case may describe, for example, un-crusted sandysoil. Engineering effects resulting in soil-water concentration (facilitation) appear onlyin the cases of (i) low infiltration in bare soil and (ii) an engineer species with limitedroot extension capabilities, η = 3.5, η = 2 (panels b and c in Fig. 4.1). The soil-waterdensity under an engineer’s patch in this case exceeds the soil-water density level in baresoil (shown by the dotted lines), thus creating opportunities for species that require thisextra amount of soil-water to colonize the water-enriched patch.

While a weak uptake feedback enhances facilitation, it reduces the resilience of theecosystem engineer (and all dependent species) to disturbances. Fig. 4.1f shows theresponse of an engineer species with the highest engineering ability to concentrate water(η = 2, Fig. 4.1c) to a disturbance that strongly reduces the infiltration contrast (f =0.9). In the following we refer to crust removal, but other disturbances that reducethe infiltration contrast, such as erosion of bare soil, will have similar effects. Theengineer, and consequently the micro-habitat it forms, disappear altogether for tworeasons: (i) surface water infiltrates equally well everywhere and the plant patch is nolonger effective in trapping water, (ii) the engineer’s roots are too short to collect waterfrom the surrounding area.

27


Resilient ecosystem engineers are obtained with strong infiltration feedbacks andmoderate uptake feedbacks (η = 3.5) as Fig. 4.1e shows. Removal of the crust (by in-creasing f) destroys the micro-habitats (soil-water density is smaller than the bare soil’svalue) but the engineer persists. Once the crust recovers the ecosystem engineer regainsits capability to concentrate water and the micro-habitats recover as well. Interestingly,when the uptake feedback is too strong, the plant persists but it no longer functions asan ecosystem engineer (Fig. 4.1a,d).

Fig. 4.2 provides another view of the facilitation-resilience tradeoff. Shown in thisfigure are two graphs of the maximum soil-water density under an engineer patch asa function of the surrounding crust’s coverage, which is parameterized by f . A plantwith high engineering strength (η = 2) has low resilience to disturbances that reducethe infiltration contrast (increase f), e.g. crust removal. As f increases past a criticalvalue fc the engineer patch disappears altogether (catastrophic shift), leaving behindbare soil with no water enriched patches and no ability for the system to recover (arrowpointing downwards). In contrast, a plant with lower engineering strength (η = 3.5)survives severe reduction of the infiltration contrast. The engineering effect no longerexists for f > fe, but once the disturbances disappear and the infiltration contrast buildsup again, engineering resumes.

45

30

15

0.450.30.20.10

Max

. soi

l wat

er [k

g/m

2 ]

η=2η=3.5

fc

fe

f

Figure 4.2: Maximal soil-water densities under ecosystem-engineer patches for two different engineerspecies, η = 2 (dashed curve) and η = 3.5 (solid curve), as functions of the infiltration contrast,quantified by f . At low values of f (e.g. in the presence of a soil crust) both species concentrate thesoil-water resource under their patches beyond the soil-water level of bare soil (horizontal dotted line),thereby creating water enriched micro-habitats for other species (see Fig. 4.1). The engineer specieswith η = 2 outperforms the η = 3.5 species, but does not survive low infiltration contrasts (resultingfrom crust removal); as f exceeds a threshold value, fc = 0.15, the engineer’s patch dries out and themicro-habitat it forms is irreversibly destroyed. The η = 3.5 species, on the other hand, survives lowinfiltration contrasts and while its engineering capacity disappears above a second threshold fe = 0.38,its engineering capacity is regained as the infiltration contrast builds up (e.g. by crust recovery). TheParameters used are given in Table 2.1 with P = 75 mm/yr.

4.3 Facilitation vs. competition

For a given plant ecosystem engineer (fixed η) and water infiltration rate (fixed f) fa-cilitation effects can develop as the environment becomes more arid. Figure 4.3 shows

28


biomass distributions at decreasing precipitation rates (panels a,b,c) and the correspond-ing soil-water distributions (panels d,e,f). At high precipitation values the soil-waterdensity under the engineer’s patch is lower than the density in bare soil, implying com-petitive relations with other plant species (”negative engineering”). At low precipitationvalues the soil-water density under the engineer’s patch is higher than the density in baresoil, implying facilitation. This model-based prediction is consistent with field observa-tions [7, 9, 36–38].

The model offers the following explanation. As the systems becomes more arid thepatch area becomes smaller and the water consumption decreases significantly. Theinfiltration rate, on the other hand, does not change much since the biomass densityremains high. As a result a unit patch area in the more arid environment traps nearlythe same amount of surface water, but a significantly smaller amount of soil-water isconsumed due to fewer plant individuals in the surrounding region. A detailed study ofthese field observations using the model is given in chapter 5.

P =103.5 mm/yr P =67.5 mm/yr P =45 mm/yr

Bio

mas

s

a) b) c)

Soil

wat

er

x

d)

x

e)

x

f)

Figure 4.3: Competition changes to facilitation as aridity increases. Plant-biomass distributions (panelsa,b,c) and the corresponding soil-water distributions (panels d,e,f) along a precipitation gradient. Athigh precipitation values (panels a,d) the soil-water density under the plant patch is lower than in thesurrounding bare soil, implying competitive interactions with other plant species. At low precipitationvalues the soil-water density under the plant patch is higher than in the surrounding bare soil, implyingfacilitation. The parameters used are given in Table 2.1 and the domain size is 5× 5 m2.

4.4 Engineering niches

The next question we address is what forms of engineering at the single patch levelcan be expected, and what types of niches or micro-habitats do they create? Modelsimulations suggest the existence of two basic engineering forms as shown in Fig. 4.4:a hump-shaped form, where the maximum of the soil-water distribution occurs at thecenter of the patch, and a ring-shaped form, where the soil-water maximum is alongthe circumference of the patch. The ring-shaped form is associated with bigger biomasspatches and can be obtained from a hump-shaped form by increasing the precipitationrate or the infiltration contrast.

29


y

w

x x

x x

a) b)

Figure 4.4: Two typical spatial distributions of soil-water at the scale of a single patch, obtained withdifferent infiltration rates in bare soil. The upper panels show two-dimensional soil-water distributions(dark shades of gray represent high soil-water density), while the lower panels show the correspondingone-dimensional cross-sections along the patch center. In the range of intermediate to low infiltrationrates (panel a, f = 0.1) the soil-water density is maximal at the center of the patch, whereas at verylow infiltration rates (panel b, f = 0.01) the maximum is shifted to the edge of the patch. Similarsoil-water distributions can be obtained at different precipitation values. The soil-water distributionswere obtained by numerical integration of the model equations 2.5–2.8 at P = 75 mm/yr, with a domainsize of 5× 5 m2. All other parameters are given in Table 2.1.

The two soil-water distributions represent different niches for herbaceous species.Species in need of abundant sunlight will favor ring-shaped distributions along whichthe shading is weak. Species sensitive to grazing, on the other hand, will prefer hump-shaped distributions where the water-rich regions are protected by the engineer canopies(see chapter 5).

4.5 Local engineering by global pattern changes

The water-biomass feedbacks induce intraspecific competition at the single patch scalewhich leads to spatial patterning (or self-organized patchiness [1]) at the landscape scale.Are there cross-scale feedbacks, i.e. processes at the landscape scale that feed back tothe level of a single patch? Studies of the model indeed suggest that such processescan exist; global landscape transitions from one vegetation pattern to another can affectengineering at the single patch level.

Transitions from one vegetation pattern to another can result from climatic orhuman-induced environmental changes, such as alternation in rainfall regime or biomassharvesting. These transitions affect the patterns formed by the ecosystem engineer andconsequently, the micro-habitats they create for other organisms.

We illustrate such cross-scale feedback process by considering a micro-habitat changeas a result of transition from a banded ecosystem engineer pattern to a spotted patternon a uniform slope as precipitation decreases. Snapshots of the pattern transition andthe associated soil-water distributions are shown in Fig. 4.5. The upper panels showthe response of a banded pattern on a slope to a small precipitation downshift from

30


P = 225 mm/yr to P = 200 mm/yr. The lower panels show the soil-water distributionsalong the transects denoted by the dashed lines in the corresponding upper panels. Theinitial pattern (panel a) loses stability and evolves into a stable spot pattern (panel d).

Surprisingly, the transition to spots is accompanied by increased engineering; the soil-water density under a spot is significantly higher than the density under a band. Themechanism of this counter-intuitive behavior can be understood as follows. The spotspattern self-organizes to form a hexagonal pattern. As a result each spot experiences abare area uphill which is twice as large as the bare area between successive bands, andtherefore absorbs more runoff. In addition, the higher biomass density of spots (due towater uptake from all directions) increases the infiltration contrast.

w

y

x x x x

downhill

d) t=50a) t=0 b) t=15 c) t=20

Figure 4.5: A transition from bands to spots in response to a precipitation downshift, leading toenriched soil-water patches. The upper panels show the response of a banded pattern on a slope toa small precipitation downshift from P = 225 mm/yr to P = 200 mm/yr. The lower panels showsoil-water distributions along the transects denoted by the dashed lines in the corresponding upperpanels. The initial pattern (panel a) loses stability and evolves into a stable spot pattern (panel d).The transition is accompanied by increased soil-water densities under vegetation patches despite thedrier conditions. The panels were obtained by numerical integration of the model equations 2.5–2.8with domain size of 5× 5 m2, slope angle of 15o and all other parameters are given in Table 2.1.

A transition from bands to spots involving soil-water gain can also be induced by alocal disturbance (e.g. clearcutting) at a given precipitation value corresponding to acoexistence range of stable bands and stable spots, as shown in Fig. 3.7.

We have already discussed, in the context of a single patch, the ability of ecosystemengineers to create micro-habitats richer in soil-water as the system becomes more arid(see section 4.3). Here we see the same trend but at the landscape or multi patch scale.The soil-water gain in this case is an emergent property resulting from the collectivedynamics of many individuals that respond to environmental changes by self-organizinginto different landscape patterns [51].

31


4.6 Landscape diversity

Species richness in drylands is expected to be strongly affected by the presence of plantecosystem engineers because of the facilitation effects of the latter and the vegetationpatterns they form. A useful concept in relating species richness to the spatial patternsof ecosystem engineers is the ”niche”. First we consider this concept in the context ofthe model, and then we use the model to study the conditions that give rise to highspecies richness. The effect of landscape diversity on species richness is further studiedin chapter 5 using a two species model.

4.6.1 The niche concept and the niche map

A niche of a given species can be defined as the ranges of environmental variables withinwhich that species survives and reproduces [45, 75–77]1. Following Hutchinson [45], weconsider a niche as a hyper-volume in a multi-dimensional ”niche space” spanned by therelevant environmental variables. The latter form the ”niche axes” and can representresources, such as soil-water, and consumer pressures, such as grazing stress.

Given a physical environment, one may conceive a niche map that associates anarea element in physical space with a volume element in niche space [45]. This isa many to one map in general as many different physical domains can lead to thesame niche. The physical domains which are mapped into the niche of a given speciesare defined here as the micro-habitats of that species. Competition with other speciesmay reduce the micro-habitat of a given species2. The niche map may not be easilymeasurable in the field but can be calculated using mathematical models. In the presentcontext, niche maps are simply solutions of the model equations 2.5–2.8. SolutionsW = W (X, Y, T ) of these equations define niche maps from the physical 2-dimensionalplane (X,Y ) ∈ R2 to a 1-dimensional niche space spanned by the soil-water resourceW ∈ R+. The niche space can be extended to two dimensions by including a second nicheaxis representing grazing and specifying a map M = M(X,Y ). The other component ofthe map, W = W (X, Y, T ), will then be obtained by solving 2.5–2.8 with the specifiedform M = M(X,Y ).

Note that niche maps obtained as solutions of the model equations account for theimpacts of ecosystem engineers, which, by concentrating the soil-water resource, increasethe micro-habitats or realized niches of other species [38].

4.6.2 Bistability as a mechanism for landscape diversity

Mechanisms for stable coexistence of plant species based on the niche concept containseveral ingredients including [10,77,78]: (a) differentiation of species in niche space (dif-ferent species occupy different volumes), (b) landscape diversity giving rise to spatial

1If the species has no competitors or enemies the niche is often called the ”fundamental niche”.2The micro-habitats in the physical space are sometimes referred to as the ”realized niches” [38]

although originally the realized niche has been defined as a volume in niche space that takes intoaccount species competition effects [45].

32


heterogeneity, and (c) tradeoffs in species traits (none of the species is a superior com-petitor with respect to all niche axes). These elements allow for the stable coexistenceof different species in different physical locations (see chapter 5). Moreover, a strongpositive correlation is expected between landscape diversity and species richness.

In the context of the model, landscape diversity is determined by the diversity ofpattern solutions that dictate the instantaneous spatial distributions of the ecosystemengineer’s biomass and of the soil-water resource. A high diversity of pattern solutionscan be realized in parameter regimes giving rise to irregular solutions. One possiblemechanism leading to irregular solutions is spatial chaos. So far, however, we havenot identified chaotic solutions and therefore, we will not address this possibility here.Irregular patterns can also result from the coexistence of stable states (e.g. bistability),where spatial mixtures of the coexisting states form stationary or long lived patterns.

t=0

t=0.5 t=1.25 t=45

t=110t=42.5t=20

t=0

P=

75 m

m/y

rP

=30

0 m

m/y

r

Figure 4.6: Bistability as a mechanism for landscape diversity. Snapshots of the time evolution (t inyears) of the same initial state (leftmost panels) when spotted patterns are the only stable state (upperpanels, P = 300 mm/yr), and when spotted patterns stably coexist with bare soil (lower panels, P = 75mm/yr). In the former case the asymptotic state is independent of the initial state; any initial statewill converge to a spotted pattern as this is the only stable state of the system. In the latter casethe asymptotic state is highly sensitive to the initial one; although the spot size changes the patternremains invariant. The domain size is 7.5× 7.5 m2, and the parameters used are given in Table 2.1..

We demonstrate two aspects of these patterns; the first pertains to the dominant rolesarbitrary initial conditions have in shaping the asymptotic patterns, and the secondto the irregular soil-water distributions that can result from these patterns. Fig. 4.6shows the time evolution of two identical initial conditions (leftmost panels), one in aparameter range where spots are the only stable state (upper panels) and the other ina coexistence range of spots and bare soil (lower panels). In the former case the systemevolves towards a spots pattern and the initial pattern has little effect. In the lattercase the initial pattern has a strong imprint on the asymptotic pattern; the systembecomes sensitive to random factors and the asymptotic patterns will generally showhigh landscape diversity. Fig. 4.7a,b show biomass and soil-water distributions in acoexistence range of stripes and gaps. As the one-dimensional cross-section in Fig. 4.7c

33


shows, the soil-water distribution is irregular, creating a diversity of micro-habitats ascompared with uniform or regular periodic patterns.

a) b) c)

2

y

x x

y

soil

wat

er [k

g/m

]

x

Figure 4.7: Biomass (a) and soil-water distributions (b,c) of an asymptotic pattern in a coexistencerange of stripes and gaps. Panel c shows the soil-water profile along the transect denoted by the dashedline in panel b. The soil-water distribution is irregular as is evident by the variable grey shades inpanel b and by the profile in panel c. Such irregular distributions create a diversity of micro-habitats ascompared with uniform or regular periodic patterns. The domain size is 7.5×7.5 m2 and the parametersused are given in Table 2.1 with P = 950 mm/yr.

34

Chapter 5

Dynamics and spatial organizationof plant communities

5.1 Introduction

The dynamics and spatial organization of plant communities strongly depend on in-traspecific and interspecific interactions. Intraspecific interactions can lead to spatialself-organization resulting in vegetation patterns of various forms [21–25, 49]. Interspe-cific interactions can induce species competition or facilitation resulting in local exclusionor coexistence of species. Field studies of plant interactions along environmental gradi-ents reveal a change from competition to facilitation (or from ”negative” to ”positive”plant interactions) as abiotic stresses or consumer pressures increase [7–9,36,79,80]. Inwater-limited systems such changes have been observed in woody-herbaceous commu-nities under conditions of increased aridity. Facilitation in this case is manifested bythe growth of annuals, grasses, and other species under the canopy of woody plants [8].Recent experimental observations [40] are inconsistent with the reported shift in plantinteractions from negative to positive as abiotic stresses increase [41].

Despite the progress that has been made in understanding vegetation patterns [1]and plant interactions along environmental gradients [7], the spatio-temporal responseof plant communities to environmental stresses and the underlying mechanisms are stillpoorly understood [81]. Mathematical modeling of vegetation pattern formation has sofar been limited to a single species population [21–25,49], whereas most models of inter-acting species populations, beginning with the Lotka-Volterra model [19], have focusedon competitive interactions [82–84] while overlooking facilitation [7, 38]. In addition,cross-scale effects, such as the interplay between spatial patterning at the landscapelevel and species composition at the single-patch level, have hardly been addressed.

In this chapter we present a study of a nonlinear mathematical model for n-interactingspecies populations in water-limited systems. This model extends an earlier model fora single species population (presented in chapter 2) that has been used to study vege-tation patterns (chapter 3) and ecosystem engineering (chapter 4). Using a two speciesmodel we elucidate mechanisms governing the change in plant interactions as a resultof aridity stresses and disturbances, and we highlight the interplay between intraspe-

35

Chapter 5. Dynamics and spatial organization of plant communities

cific and interspecific interactions and between landscape and single-patch dynamics.We further study the relationship between changes in species-interaction and changes inspecies composition by explicitly modeling herbaceous-species traits (such as tolerancesto shading and grazing), and we suggest novel mechanisms for species-diversity changein stressed environments.

5.2 A model for n-interacting species populations

We consider a spatially extended system where the limiting resource is water, and assumethe system contains n life forms of vegetation. Depending on the particular context, alife form can represent a single species, or a community of species, whose traits fallin a narrow range of values relative to species belonging to other communities. Themodel contains n+ 2 dynamical variables: n biomass variables, Bi(X, T ) (i = 1, . . . , n),representing biomass densities above ground level of the n life forms in units of [kg/m2],a soil-water variable, W (X, T ), describing the amount of soil-water available to theplants per unit area of ground surface in units of [kg/m2], and a surface water variable,H(X, T ), describing the height of a thin water layer above ground level in units of [mm].The model equations are:

∂Bi

∂T= Gi

BBi (1−Bi/Ki)−MiBi +DBi∇2Bi i = 1, . . . , n

∂W

∂T= IH −N

(1−

n∑i=1

RiBi/Ki

)W −W

n∑i=1

GiW +DW∇2W

∂H

∂T= P − IH +DH∇2

(H2

)+ 2DH∇H · ∇Z + 2DHH∇2Z , (5.1)

where ∇2 = ∂2X + ∂2

Y . The quantity GiB [yr−1] represents the growth rate of the ith

life form, GiW [yr−1] represents its soil-water consumption rate, and Ki [kg/m2] is its

maximum standing biomass. The quantity I [yr−1] represents the infiltration rate ofsurface water into the soil, the parameter P [mm/yr] stands for the precipitation rate,N [yr−1] represents the soil-water evaporation rate, and Ri > 0 describes the reduction inevaporation rate due to shading by the ith life form. The parameter Mi [yr−1] describesthe biomass loss rate of the ith life form due to mortality and various disturbances(grazing, fire, etc.). The terms DBi

∇2Bi and DW∇2W represent, respectively, local seeddispersal of the ith life form, and soil-water transport in non-saturated soil [55]. Finally,the non-flat ground surface height [mm] is described by the topography function Z(X)where the parameter DH [m2/yr (kg/m2)−1] represents the phenomenological bottomfriction coefficient between the surface water and the ground surface.

5.3 Modeling positive feedback processes

The shading feedback is captured by the reduced evaporation terms in the equationfor W (assuming Ri > 0), whereas the infiltration and uptake feedbacks are modeledthrough the explicit forms of the infiltration rate I and the growth rate Gi

B.

36


The explicit dependence of the infiltration rate of surface water into the soil onbiomass density is a generalized form of Eq. (2.2)

I(X, T ) = A

∑i YiBi(X, T ) +Qf∑i YiBi(X, T ) +Q

, (5.2)

where A [yr−1], Q [kg/m2], Yi, and f are constant parameters and Y1 = 1. Two dis-tinct limits of this term are noteworthy. When

∑i YiBi → 0, this term represents the

infiltration rate in bare soil, I = Af . When∑

i YiBi À Q it represents the infiltrationrate in fully vegetated soil, I = A. The parameter Q represents a reference biomassdensity, beyond which the vegetation density approaches its full capacity to increase theinfiltration rate. The infiltration contrast (between bare and vegetated soil) is quantifiedby the parameter f , defined to span the range 0 < f < 1. When f ¿ 1 the infiltrationrate in bare soil is much smaller than the rate in vegetated soil. As f gets closer to 1,the infiltration rate becomes independent of the biomass densities Bi. The parameter fmeasures the strength of the positive feedback due to increased infiltration at vegetationpatches. The smaller f , the stronger the feedback effect.

The growth rate GiB at a point X at time T has the form:

GiB(X, T ) = Λi

∫

Ω

Gi(X,X′, T )W (X′, T )dX′ , (5.3)

Gi(X,X′, T ) =

1

2πS2i

exp

[− |X−X′|2

2[Si(1 + EiBi(X, T ))]2

],

where the integration is over the entire physical domain Ω and the kernel Gi (X,X′, T )

is normalized such that for Bi = 0 the integration over the entire domain equals unity.According to this form, the biomass growth rate depends not only on the amount ofsoil-water at the plant location, but also on the amount of soil-water in the neigh-borhood spanned by the plant’s roots. A measure for root length [m] is given bySi(1 + EiBi(X, T )), where Ei [(kg/m2)−1] is root extension per unit biomass of theith life form, beyond some minimal root length Si. The parameter Λi has units of[(kg/m2)−1 yr−1] and represents plant growth rate per unit of soil-water. The parameterEi measures the strength of the positive feedback due to water uptake by the roots. Thelarger Ei, the stronger the feedback effect of species i.

The soil-water consumption rate at a point X at time T is similarly given by

GiW (X, T ) = Γi

∫

Ω

Gi(X′,X, T )Bi(X

′, T )dX′ . (5.4)

Note that Gi(X′,X, T ) 6= Gi(X,X

′, T ). The soil-water consumption rate at a givenpoint is due to all plants whose roots extend to this point. The parameter Γi has units[(kg/m2)−1yr−1] and it measures the soil-water consumption rate per unit biomass ofthe ith life form.

37


5.4 Non-dimensional form of the model

Using non-dimensional variables and parameters defined in Table 5.1, the non-dimensionalform of the model equations is:

∂bi∂t

= Gibbi(1− bi)− µibi + δbi

∇2bi i = 1, . . . , n

∂w

∂t= Ih− ν

(1−

n∑i=1

ρibi

)w − w

n∑i=1

Giw + δw∇2w

∂h

∂t= p− Ih+ δh∇2(h2) + 2δh∇h · ∇ζ + 2δhh∇2ζ , (5.5)

where ∇2 = ∂2x + ∂2

y and t and x = (x, y) are the non-dimensional time and spatialcoordinates. The infiltration term now reads

I(x, t) = α

∑i ψibi(x, t) + qf∑i ψibi(x, t) + q

, (5.6)

the growth rate term Gib is

Gib(x, t) = νλi

∫

Ω

gi(x,x′, t)w(x′, t)dx′ , (5.7)

gi(x,x′, t) =

1

2πσ2i

exp

[− |x− x′|2

2 [σi(1 + ηibi(x, t))]2

],

and the soil-water consumption rate is

Giw(x, t) = γi

∫

Ω

gi(x′,x, t)bi(x′, t)dx′ . (5.8)

Quantity Scaling Quantity Scaling

bi Bi/Ki p Λ1P/NM1

w Λ1W/N γi ΓiKi/M1

h Λ1H/N ηi EiKi

ν N/M1 ρi Ri

λi Λi/Λ1 σi Si/S1

µi Mi/M1 δbiDBi

/M1S21

α A/M1 δw DW/M1S21

q Q/K1 δh DHN/M1Λ1S21

x X/S1 ψi YiKi/K1

t M1T ζ Λ1Z/N

Table 5.1: Relations between non-dimensional variables and parameters and the dimensional onesappearing in the dimensional form of the model equations 5.1–5.4.

38


5.5 Plant interactions along a stress gradient

5.5.1 The woody-herbaceous ecosystem

Motivated by recent field studies of plant interactions in woody-herbaceous communitiesalong aridity gradients [8], we consider a system of two species (n = 2), representingwoody vegetation (b1) and herbaceous vegetation (b2). Accordingly, we choose the max-imum standing biomass (or carrying capacity) of the woody species to be an orderof magnitude higher than that of the herbaceous species while its growth and mor-tality rates are taken to be significantly slower. We confine ourselves to the case ofstrong infiltration feedback (f ¿ 1) and moderate uptake feedback of the woody species(η1 ∼ O(1)). These conditions are often realized in drylands where biological soil crustsincrease the infiltration contrast and the woody vegetation consists of shrubs [42].

Three types of abiotic and biotic stresses are examined. The first two are abioticor environmental stresses in the forms of an aridity stress and local disturbances in-volving woody-biomass removal such as clearcutting and over-grazing. The third typeinvolves biotic stresses resulting from intraspecific competition of nearby woody vegeta-tion patches for water. The parameter values used in this chapter, except where definedotherwise, are: ν = δw = 1.667, α = 16.667, q = 0.05, f = 0.1, δh = 416.667, η1 = 3.5,η2 = 0.35, γ1 = 2.083, γ2 = 0.208, ρ1 = 0.95, ρ2 = ψ2 = 0.005, δb1 = δb2 = 0.167,λ1 = µ1 = σ1 = σ2 = 1, λ2 = 10, and µ2 = 4.1. In all simulations the domain sizecorresponds to 7.5× 7.5 m2.

b1

b2

p

pb1

pb2

B

V1

V2

S

Figure 5.1: Bifurcation diagram showing homogeneous and pattern solutions of the woody-herbaceoussystem. The solution branches B, V1, and V2 represent, respectively, uniform bare-soil, uniform woodyvegetation, and uniform herbaceous vegetation. The branch S represents the amplitudes of spotspatterns. Solid lines represent stable solutions, and dashed and dotted lines represent solutions unstableto uniform and non-uniform perturbations, respectively. The thresholds pb2 = µ2/λ2 and pb1 = 1correspond to 307.5 mm/yr and 750 mm/yr, respectively.

The stationary uniform solutions for the woody-herbaceous system and their linearstability are described by the bifurcation diagram in Fig. 5.1. These solutions representbare soil (B), uniform woody vegetation (V1), and uniform herbaceous vegetation (V2).An additional uniform solution, not shown in Fig. 5.1 and representing mixed vegetation,

39


is unstable in this parameter range. Nonuniform solutions in the low precipitation rangedescribe spot patterns. These solutions are described by the branch S in Fig. 5.1.Throughout this chapter we will confine ourselves to a parameter range where stableuniform herbaceous vegetation coexists with stable woody spots.

5.5.2 From competition to facilitation along an aridity gradient

First we reproduce an interaction trend observed in field studies, whereby competi-tive interspecific plant interactions change to facilitative as environmental stresses in-crease [7–9,36,79,80]. We associate plant competition (facilitation) with cases where thesoil-water density under a patch of woody vegetation is lower (higher) than the densityin the surrounding bare soil. Figure 5.2a shows the results of simulating the model fordecreasing precipitation values and starting the simulations with initial states includingno herbaceous (b2) vegetation. The line B shows the soil-water content in bare soil whilethe line S shows the maximal water density under a b1 patch. The two lines intersectat p = pf suggesting a crossover from competition at high precipitation (p > pf ), wherethe soil-water density under a b1 patch is lower than in bare soil, to facilitation at lowprecipitation (p < pf ), where the soil-water density under a patch exceeds that of baresoil.

150

100

50

00.60.40.2

soil

wat

er [k

g/m

2 ]

precipitation

a)

0

b) c)

0

d) e)

BS

pf

xx

b1b1

b1b1

b2b2

ww

p < pf

pf

< p

Figure 5.2: Model solutions showing a transition from competition to facilitation as precipitation de-creases. The lines B and S in panel (a) show, respectively, the soil-water density in bare soil and undera b1 patch (in the absence of b2) as functions of precipitation. Above (below) p = pf the water contentunder the b1 patch is lower (higher) than in bare soil, implying competition (facilitation). Panels b–eshow spatial profiles of b1, b2 and w in the competition range p > pf (c,e) where an herbaceous speciesb2 is excluded by the woody species b1, and in the facilitation range p < pf (b,d) where b2 grows underthe b1 canopy. Precipitation values are: p = 0.25 (187.5 mm/yr) for b,d, p = 0.6 (450 mm/yr) for c,e,and pf = 0.5 (378 mm/yr).

40


Figures 5.2b,c show examples of spatial profiles of b1 and w in the competition range(c) and in the facilitation range (b). Note that the line S terminates at some lowprecipitation value. Below that value the woody vegetation (b1) no longer survives thedry conditions and a catastrophic shift [2] to bare soil occurs.

The model offers the following explanation for this crossover. As the system becomesmore arid, the b1 patch area becomes smaller and the water uptake decreases significantly.The infiltration rate at the reduced patch area, however, decreases only slightly sincethe biomass density remains high, i.e. b1 À q, implying a weak dependence of theinfiltration rate I on the biomass density b1 (see Fig. 2.1). As a result, a unit area of ab1 patch in a more arid environment traps nearly the same amount of surface water, buta significantly smaller amount of soil-water is consumed in that unit area due to fewerb1 individuals in the surrounding region (see Fig. 5.3). The outcome is an increased soil-water density at the b1 patch area which the b2 species can benefit from. Two factorsprevent the b1 species from exhausting the soil-water for its own growth: its carryingcapacity, which limits the local growth, and the depletion of soil-water in the immediatevicinity of the b1 patch which prevents its expansion.

1

0.8

0.6

0.4

0.2

00.560.40.30.2

precipitation

pf

Gw

I

Figure 5.3: Water uptake Gw and infiltration I rates per unit area of a b1 patch as precipitationdecreases. As the system becomes more arid, the b1 patch area becomes smaller (see upper panels)and the water uptake decreases significantly. The infiltration rate at the reduced patch area, however,decreases only slightly since the biomass density remains high. The values of Gw and I are calculatedper unit area which is marked with a circle in the upper panels. These values are normalized with respectto their maximal value and correspond to the model solutions denoted by the S curve in Fig. 5.2.

Simulations of the model equations with small, randomly distributed initial b2 biomass,indeed show a transition from competition at p > pf , where the b1 species excludes theb2 species from its patches and their immediate neighborhoods (Fig. 5.2e), to facilitationat p < pf , where the b2 species cannot survive the aridity stress and can only grow inpatches of the b1 species (Fig. 5.2d). In this range the woody vegetation acts as anecosystem engineer [4–6].

41


5.5.3 Back to competition

The transition from competition to facilitation as precipitation decreases (Fig. 5.2) wasobtained assuming that the parameter η1, measuring the capacity of the woody vegeta-tion to extend its roots as it grows, is independent of the precipitation p. Most drylandplants, however, have developed the capability to extend their roots further in responseto aridity stresses [85]. Figure 5.4a shows a schematic dependence of η1 on p that ac-counts for such a capability. The increased value of η1 at low p improves the resilienceof b1 plants to aridity stresses as they are now capable of uptaking more soil-water fromtheir surroundings. The improved resilience is accompanied, however, by a decrease inthe soil-water density under a b1 patch (dashed part of the line S in Fig. 5.4b), andwhen the increase in η1 at low p is steep enough, another crossing point, pc, of the B andS lines may appear. Thus, depending on the particular η1(p) dependence, some woodyspecies may show a transition back to competition as the aridity stress further increases.

15

10

5

00.60.40.20

precipitation

a)

100

50

00.60.40

soil

wat

er [k

g/m

2 ]

precipitation

b)BS

S ′

pfpc ≈ p′

f

η 1(p

)

Figure 5.4: Transition back to competition at extreme aridity, obtained by introducing a functionaldependence of the root extension parameter η1 on the precipitation rate p (panel a). The increase in η1at low p (dashed line) improves the resilience of b1 plants to aridity stresses, but decreases the soil-watercontent under their patches (dashed part of the S line in panel b). Transition back to competition occursbelow the second crossing point of the B and S lines, pc < pf (pc corresponds to 140 mm/yr). The S ′ lineshows the soil-water content under the patches of another b1 species for which the upper crossing point,p′f (competition to facilitation), is close to the lower crossing point, pc (facilitation to competition), ofthe first species. Under such circumstances both directions of plant interaction change, competition tofacilitation and facilitation to competition, can be realized along the same rainfall gradient.

The specific form of the S line may depend on other species traits as well. A specieswith a higher growth rate, for example, is characterized by an S ′ line that is shifted tolower precipitation values. Such a species may have a crossing point p′f (competitionto facilitation) close to the crossing point pc (facilitation to competition) of another

42


species, as shown in Fig. 5.4b. In such cases both directions of plant interaction change,competition to facilitation and facilitation to competition, can be realized along thesame environmental gradient, as field observations suggest [41].

5.5.4 Interspecific facilitation induced by intraspecific compe-tition

So far we considered the effects of abiotic stresses on interspecific interactions at thescale of a single patch. However, biotic stresses, resulting from intraspecific competi-tion of nearby patches of woody vegetation for water, may also affect woody-herbaceousinterspecific interactions. Figure 5.5 shows a possible biotic stress effect predicted bythe model. When the patches are far from one another, and for given environment andspecies traits, the b1 species competes with the b2 species and excludes it (Fig. 5.5a).However, for the same environment and species, when the patches are close enough,coexistence of the two species within the patches becomes possible (Fig. 5.5b). Thecompetition for water reduces the b1 patch size and, consequently, the soil-water con-sumption (see Fig. 5.3). As a result, more soil-water is left for the b2 species, thusallowing for its coexistence with the b1 species.

a) b)

1

2

b

b

Figure 5.5: Model solutions showing a transition from competition to facilitation as a result of bioticstress. Shown are the spatial distributions of the two biomass densities, b1 and b2, for sparsely scatteredb1 patches (a) and for a closely packed hexagonal pattern of b1 patches (b). The biotic stress resultsfrom intraspecific competition among b1 patches over the water resource. The smaller b1 patch sizein case (b) reflects a stronger stress. For the chosen environment and species traits, the b2 species isexcluded from b1 patches and their close neighborhoods when the patches are sparsely scattered, butcoexists with b1 (within its patches) when the patches are closely packed. The biotic stress, associatedwith intraspecific b1 patches competition, leads to interspecific facilitation. The parameters used aregiven in the text except for: ν = δw = 3.333, α = 33.333, δh = 333.333, γ1 = 8.333, γ2 = 0.833,δb1 = δb2 = 0.033, µ2 = 4.3, and p = 0.55 (82.5 mm/yr).

In the case of an isolated group of b1 patches, this effect can be transient [80], asdemonstrated in Fig. 5.6. Due to intraspecific competition among b1 patches within

43


the group they tend to grow toward the surrounding open space, thus increasing thedistances between them and reducing the biotic stress they exert on one another. Asa consequence, the b1 patches grow in size, consume more water, and exclude the b2species. Thus, species coexistence gradually turns into species exclusion.

b1

b2

t=0 t=12.5 t=125 t=250 t=375

Figure 5.6: Interspecific facilitation induced by intraspecific competition as a transient. Shown aresnapshots of the evolution in time of an isolated group of b1 patches. Due to intraspecific competitionamong b1 patches within the group they tend to grow toward the surrounding open space, thus increasingthe distances between them and reducing the biotic stress they exert on one another. As a consequence,the b1 patches grow in size, consume more water, and exclude the b2 species. Parameters used are asin Fig. 5.5 and t is in years.

44


5.6 Mechanisms of species diversity change in stressed

environments

5.6.1 Linking plant interactions and species composition

Transitions from competition to facilitation as described above are likely to involvechanges in species composition. The results shown in Fig. 5.2d,e apply to herbaceousspecies whose growth is solely limited by water availability. Consider two additionalfactors that can limit the growth of herbaceous species: grazing and shading. A speciessensitive to grazing can be modeled by assigning it a relatively fast biomass decay rateexcept in woody patches, where it is protected by the woody canopies. For simplicity wechoose a linear relationship, µ2 = µ20 − µ21b1. Similarly, a species sensitive to shading(i.e. requiring abundant sunlight) can be modeled by assigning it a relatively fast growthrate except in woody patches, λ2 = λ20 − λ21b1

1.

0

a) b)

0

c) d)

0

e) f)

xx

b1b1

b1b1

b1b1

b2b2

b2Sb2S

b2Gb2G

p < pf pf < p

Figure 5.7: Implications of the competition to facilitation transition for species composition changes.Panels a–f show spatial profiles of b1 and b2 above pf (b,d,f) and below it (a,c,e). An herbaceous speciesb2, whose growth is limited solely by water, is excluded by the woody species b1 for p > pf but growsunder its canopy for p < pf (a,b). An herbaceous species b2S , whose growth is also limited by shading,survives away from b1 for p > pf , but disappears from under the b1 canopy for p < pf (c,d). A speciesb2G, whose growth is limited by grazing, cannot survive the open areas away from b1 for p > pf , butcan grow under its canopy for p < pf (f,g). Precipitation values are: p = 0.25 (187.5 mm/yr) for a,c,e,p = 0.6 (450 mm/yr) for b,d,f, and pf = 0.5 (378 mm/yr).

We repeated the model simulation shown in Fig. 5.2d,e, once with a shading-sensitiveherbaceous species whose biomass density is denoted by b2S, and once with a grazing-sensitive species whose biomass density is denoted by b2G. The results are shown in

1We chose µ20 = 5.5, µ21 = 2.25 and λ20 = λ21 = 10.

45


Fig. 5.7. The b2 species, whose growth is limited solely by water, is excluded by thewoody species b1 for p > pf but grows under its canopy for p < pf (Fig. 5.7a,b). The b2S

species, that grows in the competition range (p > pf ) away from the woody patches, maynot survive the shading under the woody canopies in the facilitation range (p < pf ), andmay disappear as a result of the transition from competition to facilitation (Fig. 5.7c,d).By contrast, the b2G species that cannot survive the open areas in the competition rangemay appear and grow under the canopies of the woody patches in the facilitation range(Fig. 5.7e,f).

Interspecific facilitation of herbaceous species may also result from intraspecific com-petition between woody patches under fixed environmental conditions (see section 5.5.4).When the patches are far from one another, the woody species competes with the herba-ceous species and excludes it. In this case, only species exhibiting high tolerance tograzing can survive the open areas away from the b1 patches, as Fig. 5.8a shows (b2S

marked in yellow). When the patches are close enough, the intraspecific competition ofthe woody species for water reduces the sizes of its patches, and consequently, its soil-water consumption, giving rise to facilitation of the herbaceous species in these patches.In this case, only herbaceous species exhibiting high tolerance to shading can survive,as Fig. 5.8b shows (b2G marked in purple). Herbaceous species having high tolerances toshading and grazing (b2) are likely to appear both in sparse and dense b1 patch patterns(see Fig. 5.5).

1

2Sb

b2G

b

a) b) c)

Figure 5.8: Implications of interspecific facilitation (induced by intraspecific competition) for speciesdiversity. The green, yellow, and purple shades represent, respectively, the spatial biomass distributionsof the woody species, b1, the shading-sensitive herbaceous species, b2S , and the grazing-sensitive herba-ceous species, b2G. When the woody patches are sparse (a) the woody species competes and excludesthe herbaceous species. In the open space away from the woody patches the grazing-sensitive speciescannot cope with the high grazing stress and the shading-sensitive species prevails. When the b1 patchesare dense (b) the woody species facilitates the growth of the herbaceous species under its canopies. Inthis case the shading-sensitive species cannot survive and the grazing-sensitive species prevails. Whenthe pattern involves regions of sparse and dense patches (c), the two herbaceous species coexist, eachin its own niche. The precipitation is p = 0.55 (82.5 mm/yr) and all other parameters are as Fig. 5.5.

46


Irregular patterns, involving regions of sparse and dense patches, can accommodateboth types of species (b2S and b2G) as Fig. 5.8c shows, thus increasing the diversity ofherbaceous species. This finding is consistent with the niche approach to biodiversitytheory [77]. Irregular patch patterns exist in bistability ranges of uniform herbaceousvegetation and mixed spot patterns, and can be maintained in the field by appropriateclearcutting and grazing management.

5.6.2 Spatial patterning effects

At the landscape level, symmetry breaking vegetation patterns can appear [1,49]. At thisscale environmental stresses or consumer pressures may affect interspecific interactionsby shifting the system from one pattern state to another (see section 4.5). Figure 5.9shows a woody species transition on a slope from vegetation bands to vegetation spotsas a result of a local clearcut along one of the bands and the response of the herbaceousspecies to such a transition.

Simulations of the model equations 5.5–5.8 show exclusion of the b2 species by the b1species in the banded pattern and facilitation of the b2 by the b1 species in the spottedpattern. This predicted behavior is consistent with the results presented in section 4.5,where the transition from bands to spots pattern is accompanied by increased soil-waterdensities under vegetation spots (see Fig. 4.5).

This is an example of a cross-scale effect where pattern transitions at the landscapescale affect interspecific interactions at the single-patch scale [49], thereby allowing thelocal coexistence of b1 and b2. The fact that the transition from bands to spots takesplace under constant environmental conditions (including precipitation) indicates it isa pure spatial patterning effect rather than a single-patch facilitation induced by anaridity stress as discussed earlier.

1

2

t=25

b

b

t=0 t=58 t=104

downhill

Figure 5.9: Facilitation induced by a pattern shift at the landscape level. Shown is a sequence ofsnapshots at different times (t in years) describing a transition from vegetation bands to vegetationspots on a slope induced by a local clearcut along one of the bands (b2 is randomly distributed att = 0). In the banded pattern the b1 species excludes the b2 species, but in the spotted pattern theycoexist due to enhanced runoff concentration. The slope angle is 15o, the precipitation is p = 1.6 (240mm/yr), and all other parameters are as in Fig. 5.5 except for γ1 = 16.667, γ2 = 1.667, and µ2 = 4.8.

47

Chapter 6

A fast algorithm for convolutionintegrals with space and timevariant kernels

6.1 Introduction

The mathematical models presented in this work were studied mainly numerically. Whennumerically solving the model equations 2.5–2.8 and 5.5–5.8, we face two major difficul-ties. First, different time scales exist for the various processes in the dynamical model.These time scales range between seconds-hours (e.g. flow of surface water) and months-years (e.g. plant’s growth rate). Second, and most crucial, is the need to compute thenonlocal integral terms at each time step. The resulting computational complexity isof the order O(N2), where N is the total number of grid points. The immense com-putational complexity of these models on the one hand, and the need to carry out acomprehensive numerical study on the other hand, have led to the development of a fastand accurate numerical code that overcomes these difficulties.

The numerical methods by which the time scales problem is solved are described inAppendix C, and are rather standard techniques for the numerical integration of PDE’s.However, for the evaluation of the nonlocal terms (i.e. the spatial integrals) we havedeveloped a novel algorithm for the fast approximation of convolution integrals withspace and time variant kernels [48].

The numerical integration of convolution integrals was made very efficient by theintroduction of the Fast Fourier Transform algorithm [86], allowing for the evaluationof convolution integrals as O(N logN) processes using the Convolution Theorem [87].This has led, since the mid-1960s, to a large number of applications in various areas ofthe physical sciences. In particular, the integration of partial differential equations con-taining nonlocal integral terms (i.e. integro-differential equations) has profited from thistechnique. Examples include mathematical biology [88], population dynamics [89], non-linear optics [90], epidemics [91], and superfluidity [92]. Note that in all these cases, thenonlocal terms appearing in the problem are in the form of convolutions, thus allowingfor this efficient approach.

48

Chapter 6. A fast algorithm for convolution integrals with variant kernels

One particular example of such a situation, which motivated this work, is given bythe integral terms appearing in the model equations 2.5–2.8 and 5.5–5.8. The nonlocalterms in these models are of the general form

I1 (x) =

∫

D

K [x− x′;φ(x′)]ψ(x′)dx′ (6.1)

or

I2 (x) =

∫

D

K [x− x′;φ(x)]ψ(x′)dx′ , (6.2)

where the integration is taken over a domain D and the time dependence of the fieldsis omitted for clarity. In this example, the kernel function K represents the effect ofthe root system and was chosen as a two dimensional Gaussian with a characteristicwidth φ, which is a linear function of the biomass density. Both integrals I1(x) andI2(x) cannot be written as convolutions due to the kernel dependence on φ, which im-plies a dependence of the kernel on either x or x′. This fact is crucial, since it preventstheir evaluation as O(N logN) processes using the Convolution Theorem. Similar in-tegrals also arise in the fields of image processing [93] and nonstationary linear signalfiltering [94].

In this paper we present a fast algorithm for approximating nonlocal terms of theform of equations (6.1) and (6.2) for different kernel functions, including the examplesof exponential, Gaussian, and Lorentzian kernels, in one, two, and three dimensions.The chapter is organized as follows: in section 6.2 we schematically present the algo-rithm with some remarks on its computational cost. Due to the heuristic approach ofthis section, some aspects of the algorithm are discussed briefly, and a more detaileddescription is deferred to subsequent sections. In sections 6.3 and 6.4 we show in detailhow the approximation coefficients required by the algorithm can be computed, with afew specific examples. The accuracy achievable using this algorithm and the relevantfactors controlling the error in the approximation are discussed in section 6.5. Finally,in section 6.6 we show some examples of numerical performances.

6.2 The algorithm

Our goal is to evaluate integral terms of the form of equations (6.1) or (6.2), where bothφ and ψ are known scalar fields over a D-dimensional domain D. We assume that thatD allows for the computation of Fourier Transforms and that the kernel function K iswell defined in the sense that its integral over D converges.

Clearly, unless the dependence on φ can be factored out of the kernel, in general theintegrals in equations (6.1) or (6.2) are not convolutions since, due to the dependence ofthe field φ on x or x′, K is not a function of x− x′ alone. This prevents us from usingthe Convolution Theorem to evaluate the terms I1(x) and I2(x) trivially as products oftwo fields in Fourier space. The main idea described in this paper is to approximatethe kernel, for any value of φ, as a linear superposition of instances of the same kernel,evaluated at fixed values of φ. This shifts the implicit kernel dependence on space tothe linear approximation coefficients, thus enabling the evaluation of the integral as alinear combination of convolutions.

49


To be more specific, we write the following approximation to the kernel function K:

K (s;φ) ≈ K (s;φ) ≡Nl∑

l=1

αl(φ)K (s;φl) , (6.3)

where Nl is the number of terms in the approximation, φl is a series of constants andαl(φ) are unknown functions of φ. Note that the basis functions in this expansion aregiven by the same kernel function we wish to approximate, computed at fixed valuesof φ. This is a natural choice, as it allows for an exact reproduction of the kernelwhenever φ = φl (see section 6.5 for more details). Since the field φ depends on space,the approximation coefficients, being functions of φ, have an implicit dependence onspace, αl(φ) = αl [φ(x)] or αl(φ) = αl [φ(x′)]. As we will show in section 6.3, optimalcoefficients αl(φ) can be easily computed by minimization of an appropriately definederror, and, as we will show in section 6.4, in some cases exact closed form formulas canbe derived. The series φl and the number of terms used in the approximation, Nl,represent free parameters of the approximation, which should be chosen according to thedistribution of φ values in the specific problem. We will discuss this issue in section 6.5.

For simplicity we start our illustration with the case of Eq. (6.1), when φ is a functionof x′. Substituting Eq. (6.3) into Eq. (6.1) gives

I1(x) ≈∫

D

[Nl∑

l=1

αl(φ)K (x− x′;φl)

]ψ(x′)dx′ =

Nl∑

l=1

∫

D

K (x− x′;φl)αl(φ)ψ(x′)dx′ ,

(6.4)which we rewrite as

I1(x) ≈Nl∑

l=1

∫

D

fl(x− x′)gl(x′)dx′ =

Nl∑

l=1

fl ∗ gl , (6.5)

wherefl(x− x′) = K (x− x′;φl) , gl(x

′) = αl [φ(x′)]ψ(x′) , (6.6)

and the operator ”∗” denotes a convolution of two fields over the domain D. As shownby Eq. (6.5), the field I1(x) is now approximated by a sum of Nl convolutions which canbe efficiently evaluated in Fourier space thanks to the Convolution Theorem.

In case the field φ in the kernel depends on x instead on x′, as in Eq. (6.2), theresulting approximation can be similarly written as

I2(x) ≈Nl∑

l=1

αl (φ)

∫

D

fl(x− x′)g′l(x′)dx′ =

Nl∑

l=1

αl [φ(x)] (fl ∗ g′l) , (6.7)

where nowfl(x− x′) = K (x− x′;φl) , g′l(x

′) = ψ(x′) . (6.8)

The algorithm’s different steps can be summarized as follows, using the ConvolutionTheorem for evaluating the integrals:

50


1. Choose Nl constants φl;2. Compute the approximation coefficients αl(φ) as described in section 6.3;

3. Compute

I1(x) ≈ F−1

[Nl∑

l=1

F (fl)F (gl)

]or (6.9)

I2(x) ≈ F−1

[Nl∑

l=1

αl(φ)F (fl)F (g′l)

], (6.10)

where the operators F and F−1 indicate a direct and an inverse Fourier transform,respectively, and the fields fl, gl, and g′l are defined above.

In case the evaluation of I1(x) or I2(x) needs to be iterated, as is usually the casewhen the problem is time dependent, some of the terms above need to be computedonly once. In particular, since fl is simply the kernel function evaluated at a fixed φl,its Fourier transform F(fl) can be evaluated only once at the beginning of the code.Additionally, if the evaluation of the coefficients αl(φ) is time consuming, they can alsobe evaluated only once at the beginning for discretized values of φ.

The computational complexity of this algorithm is of the orderO [NlN log(N)], whereN is the number of grid points in the spatial domain of interest. This has to be comparedwith a complexity of O(N2) if a direct, brute force, approach is used. Since a fairlysmall number of terms in the approximation (Nl ¿ N) is enough to obtain a goodapproximation, as is shown in the following, the advantage for large values of N isobvious.

6.3 Computing the approximation coefficients

We compute optimal approximation coefficients αl (φ) for a kernel function K (s;φ), byapplying a standard minimization procedure to a measure of the approximation error.For any given, fixed, φ, we write a measure of the error in the kernel approximation,K (s;φ), as a functional of αl (φ) and of Nl:

F [αl (φ) , Nl] ≡∫

Φ

∫

D

[K (s;φ,Nl)−K (s;φ)

]2

dsdφ , (6.11)

where D is the physical domain and Φ is the domain spanned by the values of φ. Havingfixed Nl, we wish to minimize the functional F [αl (φ)] with respect to all αl. We usethe method of variations [95], and define a small variation in αl (φ) as αl (φ) + δαl (φ).Substituting this into equations (6.3) and (6.11) and expanding up to first order in

51


δαl (φ) gives

F + δF =

∫

Φ

∫

D

∑

l

[αl (φ) + δαl (φ)]K (s;φl) − K (s;φ)

2

dsdφ

=

∫

Φ

∫

D

[∑

l

αl (φ)K (s;φl)−K (s;φ) +∑

l

δαl (φ)K (s;φl)

]2

dsdφ

=

∫

Φ

∫

D

[∑

l

αl (φ)K (s;φl)−K (s;φ)

]2

dsdφ+

+ 2

∫

Φ

∫

D

[∑

l

αl (φ)K (s;φl)−K (s;φ)

][∑j

δαj (φ)K (s;φj)

]dsdφ+

+ O (δα2

). (6.12)

Hence, we obtain an expression for the variation in F with respect to variations in αl:

δF = 2

∫

Φ

∑j

∫

D

[∑

l

αl (φ)K (s;φl)K (s;φj)−K (s;φ)K (s;φj)

]ds

δαj (φ) dφ

= 2

∫

Φ

∑j

[∑

l

Mjlαl (φ)− bj (φ)

]δαj (φ) dφ , (6.13)

where we defined

Mjl ≡∫

D

K (s;φj)K (s;φl)ds , bj (φ) ≡∫

D

K (s;φ)K (s;φj)ds . (6.14)

The requirement δF = 0 for any δαl (φ) yields a set of Nl linear equations for theapproximation coefficients

Nl∑

l=1

Mjlαl (φ) = bj (φ) , j = 1, . . . , Nl , (6.15)

which in some cases can be solved analytically, providing an exact, compact expressionfor αl as a function of φ. Alternatively, this linear system can be solved numerically fordiscrete values of φ.

6.4 Examples for some common kernel functions

In this section we will demonstrate the method described above to compute the approxi-mation coefficients, applying it to some common kernel functions and deriving analyticalexpressions for the dependence of αl on φ.

Typical shapes of the resulting approximation coefficients αl(φ) as a function of φare summarized in Fig. 6.1, with different choices of the series φl. Notice the propertythat whenever φ = φl, then αl(φl) = 1 while αj 6=l(φl) = 0. This results from havingused instances of the same kernel we wish to approximate (computed at φl) as basisfunctions for the approximation.

52


0

1

a)

PSfrag repla ements

1 2 3 4 5 l()

1 2 3 4 5

0

1

b)

PSfrag repla ements

1 2 3 4 5 l()

1 2 3 4 5

0

1

c)PSfrag repla ements

1 2 3 4 5 l()

1 2 3 4 5

Figure 6.1: Typical shapes of the approximation coefficients αl(φ) as functions of φ for three differentkernels and three different distributions of the series φl, with Nl = 5. The coefficients αl(φ) areshown for a) one dimensional decaying exponential kernel (Eq. 6.19), with φl distributed accordingto φl ∼ l2, b) two dimensional Gaussian kernel, (Eq. 6.23), with φl distributed according to φl ∼ 2l/2,and c) three dimensional Lorentzian kernel (Eq. 6.27), with φl distributed according to φl ∼ l. Noticethat whenever φ = φl, αl(φl) = 1 while αj 6=l(φl) = 0. All φl series are distributed over the rangeΦ = [0, 1] and are normalized such that φNl

= 1.

6.4.1 One dimensional decaying exponential

We consider a decaying exponential kernel in an infinite one dimensional domain, witha characteristic length determined by φ:

K (s;φ) = e−|s|φ . (6.16)

We assume that the range of possible φ values in the problems is given by φ ∈ Φ (φ > 0).We define a series φlNl

l=1 ∈ Φ and approximate the kernel according to Eq. (6.3):

e−|s|φ ≈

Nl∑

l=1

αl(φ)e− |s|

φl . (6.17)

53


We calculate the entries Mjl and bj(φ) of the linear system in Eq. (6.15) using Eq. (6.14)as follows:

Mjl =

∞∫

−∞

e− |s|

φj e− |s|

φl ds = 2φjφl

φj + φl

bj(φ) =

∞∫

−∞

e−|s|φ e

− |s|φl ds = 2

φφj

φ+ φj

. (6.18)

The solution of the linear system Eq. (6.15), using these entries, yields the followinganalytic expression for the approximation coefficients (see Fig. 6.1a):

αl (φ) =2φ

φ+ φl

∏

j 6=l

φ− φj

φl − φj

φl + φj

φ+ φj

. (6.19)

6.4.2 Two dimensional Gaussian

We consider a Gaussian kernel in an infinite two dimensional domain, with a character-istic width determined by φ:

K (s;φ) = e− |s|2

2φ2 . (6.20)

We assume that φ ∈ Φ (φ > 0) and define the series φlNll=1 ∈ Φ. The kernel can be

approximated using Eq. (6.3):

e− |s|2

2φ2 ≈Nl∑

l=1

αl(φ)e− |s|2

2φ2l . (6.21)


Mjl =

∫

D

e− |s|2

2φ2j e− |s|2

2φ2l ds = 2π

∞∫

0

s ds e− s2

2

1

φ2j

+ 1

φ2l

!

= 2πφ2

jφ2l

φ2j + φ2

l

bj(φ) =

∫

D

e− |s|2

2φ2 e− |s|2

2φ2j ds = 2π

φ2φ2j

φ2 + φ2j

. (6.22)

The solution for the linear system given in Eq. (6.15), using these entries, yields thefollowing analytic expression for the approximation coefficients (see Fig. 6.1b):

αl (φ) =2φ2

φ2 + φ2l

∏

j 6=l

φ2 − φ2j

φ2l − φ2

j

φ2l + φ2

j

φ2 + φ2j

. (6.23)

54


6.4.3 Three dimensional Lorentzian

We consider a Lorentzian kernel in an infinite three dimensional domain, with a charac-teristic width determined by φ:

K (s;φ) =1

1 + |s|2φ2

. (6.24)

We assume that φ ∈ Φ (φ > 0) and define the series φlNll=1 ∈ Φ. We approximate the

kernel according to Eq. (6.3):

1

1 + |s|2φ2

≈Nl∑

l=1

αl(φ)1

1 + |s|2φ2

l

. (6.25)


Mjl =

∫

D

1

1 + |s|2φ2

j

1

1 + |s|2φ2

l

ds = 4π

∞∫

0

s2ds1

1 + s2

φ2j

1

1 + s2

φ2l

=π

2

φ2jφ

2l

φj + φl

bj(φ) =

∫

D

1

1 + |s|2φ2

1

1 + |s|2φ2

j

ds =π

2

φ2φ2j

φ+ φj

. (6.26)

The solution for the linear system given in Eq. (6.15), using the entries calculated inEq. (6.26), yields the following analytic expression for the approximation coefficients(see Fig. 6.1c):

αl (φ) =2φ2

φl (φ+ φl)

∏

j 6=l

φ− φj

φl − φj

φl + φj

φ+ φj

. (6.27)

6.5 Accuracy of the approximation

The accuracy which can be achieved with the algorithm described above depends onan appropriate choice of the distribution of φl over Φ and of the number of basisfunctions Nl. We illustrate this dependence by separately considering (i), the error inapproximating the kernel alone using Eq. (6.3), and (ii), the error in approximatingintegrals similar in form to Eq. (6.1), compared with a direct, brute force, algorithm ina practical application example.

6.5.1 The error in the kernel approximation

We write the error in the kernel approximation (i.e. Eq. 6.3) for an arbitrary φ as

e2 (φ;Nl) =

∫D

[K (s, φ;Nl)−K (s, φ)

]2

ds∫

DK (s, φ)2 ds

, (6.28)

55


where φ is regarded as an argument of the error (not a parameter), and the dependenceof the error on the parameter Nl is explicitly noted. We illustrate the dependence of theerror in the kernel approximation on the particular choice of φl by fixing Nl = 12 andplotting this error as a function of φ ∈ Φ = (0, 1] (see Fig. 6.2). The errors are shown fora) a one dimensional exponential kernel (Eq. 6.17) and b) a two dimensional Gaussiankernel (Eq. 6.21) using three different distributions of φl over Φ.

100

10-5

10-10

10-15

10-20

0 0.2 0.4 0.6 0.8 1

a)

PSfrag repla ementsflg1flg2flg3

e2 ()

100

10-5

10-10

10-15

0 0.2 0.4 0.6 0.8 1

b)


e2 ()

Figure 6.2: Plot of the error in the kernel approximation e2(φ) (Eq. 6.28) as a function of φ for a) a onedimensional exponential kernel (Eq. 6.17) and b) a two dimensional Gaussian kernel (Eq. 6.21) usingthree different distributions of φl over the range φ ∈ (0, 1]. In both cases Nl = 12 and the seriesφl is distributed according to φl1 ∼ 2l/2, φl2 ∼ l, φl3 ∼ l2 and normalized such that φNl

= 1.Notice that the error drops to 0 whenever φ = φl.

Figure 6.2 illustrates how, for a fixed number of basis functions, an appropriatechoice of the φl series can lead to a highly accurate approximation of the kernel overa desired range of φ values in a given problem. Obviously, no universally valid choice ofthe φl distribution over Φ can be given as this depends on the particular distributionof values of φ in the problem at hand. In theory, if the distribution of φ in the problemwere known in advance, it would be possible to define an optimal set of φl for a givenNl, but we will not attempt this exercise here.

In order to study the dependence of the kernel approximation accuracy on the num-ber of basis functions Nl, we define an average error in the approximation, averagingEq. (6.28) over Φ:

e2(Nl) =

∫Φe2 (φ;Nl) dφ∫

Φdφ

. (6.29)

Figure 6.3 shows plots of this average error as a function of Nl for the cases of aone dimensional exponential kernel (Eq. 6.17) and a two dimensional Gaussian kernel(Eq. 6.21), using the three distributions of φl described in Fig. 6.2. This plot demon-strates the performance of the algorithm, but also the importance of appropriately choos-ing the distribution φl: Two of the choices of φl, i.e. φl2 ∼ l and φl3 ∼ l2, leadto an error which decreases exponentially in a wide range of Nl whereas in the case of

56


φl1 ∼ 2l/2 initially the error decreases more rapidly, but it reaches a plateau early. 1

10-4

10-8

10-12

10-16

10-20

4 8 12 16 20

a)


Nle2 (N l)

10-2

10-4

10-6

10-8

10-10

10-12

4 8 12 16 20

b)


Nle2 (N l)

Figure 6.3: Plots of the average error in the kernel approximation (Eq. 6.29) as a function of Nl, for thecases of a) a one dimensional exponential kernel (Eq. 6.17) and b) a two dimensional Gaussian kernel(Eq. 6.21), using the same three distributions of φl over Φ as described in Fig. 6.2. In this figure weused Φ = [0.2, 1] for the evaluation of Eq. (6.29).

6.5.2 The error in the integral approximation

By numerical exploration and a practical application example, we analyze the differencebetween an approximated integral, I(x), computed using the approximation proceduredescribe above, and an integral, I(x), computed using an alternative, standard, bruteforce approach. Accordingly, we define an error between the two as

e2I(Nl) =

∫D

[I(x;Nl)− I(x)

]2

dx∫

DI(x)2dx

. (6.30)

This error is calculated for the case of Eq. (6.1) with a two dimensional Gaussiankernel and for the three different distributions of φl defined above. The field ψ(x) istaken from [51] where it represents a spatial distribution of biomass density and assumesvalues in the range ψ ∈ [0, 1]. The field φ(x) in this example represents the root lengthand is a linear function of the biomass according to φ(x) = 1 + ψ(x), so that φ ∈ [1, 2].We calculate the error defined in Eq. (6.30) for three different spatial patterns of biomassdensity shown in the upper panels of Fig. 6.4, spots, stripes, and gaps (see also [23]).

1For this particular distribution of φl, the number of values in the range Φ saturates for Nl > 5,with all new values in the sequence being added below the lowest limit of the range Φ. While in generalwe found that in order to have a good approximation it is also necessary to include a few φl valuesoutside the range of φ, in this case these new values have a negligible effect on the accuracy of theapproximation.

57


The domain size of these patterns is L = 6 and the series φl are chosen in the rangeΦ = (0, L], with φNl

= L.All continuous fields are discretized over a 128×128 points grid with periodic bound-

ary conditions. The field I(x) is calculated according to the approximation algorithmdescribed in section 6.2, while the field I(x) is calculated using a direct brute forcemethod of integration, specifically the trapezoidal rule [96], as follows:

I(xi, yj) = ∆x∆y

N/2∑

k,l=−N/2

e− [(i−k)∆x]2+[(j−l)∆y]2

2φ(xk,yl)2 ψ(xk, yl) , i, j = 1, . . . , N , (6.31)

where N is the number of grid points in each direction and ∆x = ∆y = L/N . Thisexpression can also be considered simply a discrete version, on a finite grid, of Eq. (6.1)with a Gaussian kernel.

(r)e2 I(N l) 1

10-4

10-8

10-12

4 8 12 16 20

d)


Nl

1

10-4

10-8

10-12

4 8 12 16 20

e)


Nl

1

10-4

10-8

10-12

4 8 12 16 20

f)


NlFigure 6.4: Three different spatial patterns of the field ψ: a) spots, b) stripes, and c) gaps and thecorresponding error in the integral approximation e2I(Nl) (Eq. 6.30) as a function of the number of basisfunctions Nl (panels d–f). The field ψ(x) is taken from [51], where it represents the spatial distributionof biomass density and assumes values over the range ψ ∈ [0, 1] (darker shades of gray represent higherbiomass density) and φ(x) = 1 + ψ(x). The error e2I(Nl) is calculated for each spatial pattern of ψusing a two dimensional Gaussian kernel and three different distributions of φl, as previously definedin Fig. 6.2.

The resulting error is plotted in panels d–f in Fig. 6.4 as a function of the number ofbasis functions used in the approximation Nl. Note that for the choices φl2 and φl3

the error can be made arbitrarily small by choosing a small, finite value of Nl. The errorfor the φl1 case reaches an asymptotic value as Nl ∼ 8, for reasons similar to thosealready discussed in footnote 1 for Figure 6.3.

58


In summary, Figures 6.2–6.4 show that by correctly tuning both the number of basisfunctions Nl and the distribution of the series φl, a highly accurate approximationcan be achieved for both the kernel function and for the integral itself over the desiredrange of φ values in the problem. Furthermore, only a small number of basis functionsNl ¿ N are required to achieve good accuracy.

6.6 Numerical performances of the algorithm

We demonstrate the advantage of using the approximation described in this paper bycomparing the speed of a numerical implementation of the algorithm with a direct bruteforce approach (see Eq. 6.31). We computed the single processor CPU time needed toperform one integration of integral I1 on two dimensional grids ranging in resolutionfrom 64 × 64 points up to 1024 × 1024 points, using a Gaussian kernel function withperiodic boundary conditions and setting Nl = 16.

The numerical simulations were run using Fortran implementations of the algorithms,on an SGI-Altix 3700 SuperCluster with Intel Itanium2 1.3GHz processors and 32GBmemory, running 64-bit Linux OS [97]. Intel Compilers [98] were used for compilationand the FFTW library [99] was used for the Fourier transforms. All simulations useddouble precision arithmetic.

Figure 6.5a shows the CPU user time required to complete a single calculation ofthe integral as a function of the grid size, using both a direct brute force algorithm(dashed line) and the approximation algorithm (solid line). Figure 6.5b shows the ratiobetween these two curves. Note that the speed-up obtained by using the approximationalgorithm with respect to the direct brute force algorithm ranges between 103 and 105,whereas the error in the approximation remains small.

10-3

10-2

10-1

100

101

102

103

104

105

106

12 13 14 15 16 17 18 19 20

CP

U u

ser

time

(sec

)

a) approximationdirect

PSfrag repla ements log2(N) 103

104

105

12 13 14 15 16 17 18 19 20

t dire

ct/t a

ppro

x

b)

PSfrag repla ements log2(N)Figure 6.5: a) CPU user time required to complete a single integral calculation as a function of gridsize, using two different algorithms: a direct brute force algorithm (dashed line) and the approximationalgorithm (solid line). b) CPU user time ratio between the two algorithms obtained as a ratio of thetwo curves in panel a). The number of basis functions in these tests is Nl = 16.

59

Chapter 7

Conclusions

7.1 Summary and discussion

7.1.1 The model

We presented a mathematical model for the dynamics of a single species population andfor n-interacting species populations in water-limited systems. Two ingredients of themodel are particularly significant for understanding self organized vegetation patterns,ecosystem engineering and the dynamics of plant interactions, i.e., the infiltration anduptake feedbacks. Each feedback, independently of the other, induces intraspecific in-teractions that lead to similar spatial patterns at the landscape level. The two feedbacksdiffer, however, in the interspecific interactions they induce. The infiltration feedback in-duces positive interactions (facilitation) by trapping surface water in vegetation patches,thereby increasing the soil-water resource in these patches that provide habitats for othervegetation species. The uptake feedback induces negative interactions (competition) byexploiting the soil-water resource and depleting it in and around vegetation patches.Thus, dominance of the uptake feedback leads to competition and exclusion, whereasdominance of the infiltration feedback results in facilitation and coexistence.

Another significant aspect of the model is that it allows for the study of how processesat the level of a single patch (e.g. biomass-water feedbacks) affect patterning processesat the landscape level. Such studies can reveal emergent landscape processes that feedback to the single patch level, thereby illuminating mechanisms that would be hardto deduce by studying that level alone [100]. An example of such an emergent processinvolves the pattern shift from banded to spotted vegetation on a slope and the resultingfacilitation of herbaceous species growth in woody patches (Figures 4.5 and 5.9).

The explicit modeling of the infiltration and uptake feedbacks represents a ”first-principle” approach whereby community and landscape properties, such as spatial struc-tures, habitat diversity, plant interactions, and species composition, emerge as solutionsof the model equations rather than being preset in formulating the model. For exam-ple, positive or negative interspecific interactions are not predetermined by the signs ofcoupling terms in the model equations, as is the case with the Lotka-Volterra modeland with some other models [101], but rather follow from the relative strength of thewater-biomass feedbacks that, in turn, are affected by a variety of basic, measurable

60

Chapter 7. Conclusions

parameters describing species traits and environmental conditions.

7.1.2 Landscape states

The model predicts the possible development of vegetation patterns at the landscapescale due to intraspecific competition over the limited water resource. The patternformation mechanism is the positive feedback between biomass and water that acts atthe single patch scale. Vegetation patterns, which appear at the landscape scale, reflectcollective dynamics of many individuals involving long spatial correlations of biomassand soil-water.

The model has proved successful in reproducing several observations and in illumi-nating mechanisms of ecosystem processes. The model reproduces various vegetationpatterns that have been observed in arid and semiarid regions, including specific char-acteristics of banded vegetation on hill slopes [16], such as migration uphill and thedependence of inter-band to band ratios on rainfall [50]. Vegetation patterns have beenproduced by earlier models as well that captured the biomass-water feedback.

Most of the earlier models and the models introduced here predict the same sequenceof patterns along aridity gradients: uniform vegetation, gaps, stripes, spots and bare soil(Fig. 3.1). Moreover, most models predict coexistence ranges of different stable states:uniform vegetation with gaps, gaps with stripes, stripes with spots and spots with baresoil (Fig. 3.5). This apparent agreement among most models is due to the universalnature of the instabilities that are responsible for the pattern formation phenomena.

The coexistence of different stable vegetation states, predicted by the model, impliesa vulnerability to desertification. Desertification may be induced either by varyingenvironmental conditions, thereby driving the system out of a state coexistence range,or by disturbances that induce transitions between the coexisting states. In the formercontext the model explains the irreversible character of desertification by relating itto hysteresis (Fig. 3.6). In the latter context the model highlights the dramatic rolesregional topography may play. Local disturbances, such as clearcutting, remain local ona plane, but can induce global state transitions on a slope (Figures 3.7 and 3.9).

On a slope we find a multistability of stable states within the precipitation rangeof stable bands. The coexisting stable states correspond to band patterns with dif-ferent wavenumbers (Fig. 3.12). Two principal responses to a gradual precipitationdownshift were identified: (i) an increase in the ratio of the interband-band-ratio (IBRindex) without a change in the pattern’s wavenumber, and (ii) a decrease in the pat-tern’s wavenumber (Figures 3.8 and 3.15). The model also predicts a tradeoff betweenbiological productivity and resilience; banded patterns with higher wavenumbers aremore biologically productive but are less resilient to environmental changes (Figures 3.8and 3.15). These results are significant for rehabilitation of desertified regions. In choos-ing the inter-band distances in rehabilitation projects [102], two conflicting factors mustbe carefully considered: biological productivity and system vulnerability to environ-mental changes, such as a precipitation downshift, both of which increase in response toshort inter-band distances.

61


7.1.3 Ecosystem engineering

In the context of ecosystem engineering, the coexistence of different stable vegetationpatterns (Fig. 3.5) implies higher landscape and species diversity. Periodic patterns,such as hexagonal spot patterns, probe small volumes in niche space, as the samemicro-habitats repeat themselves, and therefore are associated with low species diversity.Strictly periodic patterns are never expected to be observed in nature because of the longtime scales needed for approaching these asymptotic states. According to the model,however, under environmental conditions where spot patterns constitute the only stablestate of the system, any initial biomass distribution will evolve on a relatively short timescale into a space-filling spot pattern, as Fig. 4.6 (top) demonstrates. These patterns,even though not periodic, are still expected to give rise to rather low species diversity.

Higher species diversity can be realized under conditions where the coexistence ofdifferent stable states occurs. Under these conditions engineer patterns can be highlyirregular, and therefore can probe larger volumes in niche space. Fig. 4.6 (bottom) showsthe time evolution of the same initial state as in the top part. The initial state remains”frozen” apart from patch size adjustments because of the stability of the bare soil stateand the depletion of soil-water in the patch vicinity. This implies that highly irregularinitial biomass distributions, dictated by random factors, will also remain highly irregularasymptotically. Another example is shown in Fig. 4.7, where the stripes pattern coexistswith gaps pattern, giving rise to an asymptotically irregular soil-water distribution.

A significant prediction of the model is the tradeoff between facilitation and resilienceas the relative strength of the two feedbacks changes (Figures 4.1 and 4.2). This tradeoffis significant for understanding the stability and functioning of water-limited ecosystemson micro (patch) and macro (watershed) spatial scales [42]. The soil moisture accumu-lation at an engineer patch accelerates litter decomposition and nutrient production andculminates in the formation of fertility islands. Watershed-scale disturbances that areincompatible with the resilience of dominant engineer species can destroy the engineerpatches and the fertility islands associated with them, thereby damaging the stabilityand functioning of the ecosystem.

We also found that environmental conditions affecting the engineer patch size canmodify the soil-water distribution under the engineer’s patch and thus the micro-habitatit creates. We demonstrated a transition from a hump-shaped distribution to a ring-shaped distribution (Fig. 4.4) by increasing the infiltration contrast (decreasing f), butsimilar transitions can be obtained in the model by changing other parameters, e.g.increasing the precipitation rate, P , decreasing the evaporation rate, N , or decreasingthe grazing stress, M .

We also studied the possible effects of pattern transitions at the landscape levelon engineering at the single patch level (i.e. cross-scale processes). We found that aglobal transition from vegetation bands to spots on a slope can result in stronger localengineering, suggesting that studies of single-patch dynamics should not be confined tothe patch level alone. This effect is tightly related to the hexagonal structure of the spotpattern (any spot has 6 neighbor spots), where the distance between any adjacent spotsalong the slope direction is about twice as large as the distance between two adjacentbands. In heterogeneous systems (containing e.g. rocky or eroded soil parts) the spot

62


pattern will generally appear disordered, but the basic principle leading to enhancedengineering still holds because of the reduced patch connectivity in spotted patterns ascompared with banded patterns and the resulting higher runoff accumulation at spottedvegetation patches.

7.1.4 Dynamics of plant interactions and species diversity

Our study of plant interactions in a mixed woody-herbaceous ecosystem suggests twonew mechanisms of plant interaction change in water stressed environments. The first in-volves a change in the balance between the infiltration feedback and the uptake feedback(Fig. 5.3) that results in a transition from interspecific competition to facilitation [8] andvice versa [41]. The transition from competition to facilitation can be induced by anaridity stress (Fig. 5.2), or by intraspecific patch competition over the water resource(Fig. 5.5). Transitions back to competition as the aridity stress further increases canresult from the capability of a plant species to further extend its roots in response to anaridity stress (Fig. 5.4). We also demonstrate how both directions of plant interactionchange, competition to facilitation and facilitation to competition, can be realized alongthe same environmental gradient. The results shed new light on conflicting field obser-vations [41] and provide a possible explanation for this inconsistency. This mechanismacts at the level of a single patch or a few interacting patches.

An important prediction of the model, associated with the fact that a transition fromcompetition to facilitation may result from a biotic stress induced by nearby patches,is that irregular vegetation patterns involving regions of sparse and dense patches cansupport higher species diversity than regular patterns. We demonstrated this by con-sidering the tradeoffs between different herbaceous species traits, such as tolerances toshading or grazing (Fig. 5.8).

The second mechanism involves cross-scale processes, in which pattern transition atthe landscape level feeds back to the level of a single patch by increasing the soil-waterdensity and facilitating herbaceous vegetation growth. This transition can be inducedby local disturbances, such as biomass removal (Fig. 5.9) or by increased aridity stress(Fig. 4.5). Figures 5.7, 5.8, and 5.9 demonstrate how the mechanisms of plant interactionchange bear on species composition.

7.1.5 Numerical methods

We presented a new algorithm for the fast approximation of convolution integrals withspace and time variant kernels in the form of Eqs. (6.1) and (6.2). The main idea isto approximate the integrals as a linear combination of a small number of convolutions.Good accuracy can be achieved with this algorithm, with pronounced advantages interms of computational complexity and, thus, speed of integration.

The coefficients required for the approximation are easily obtained either analyticallyor numerically. We have developed analytical solutions for the coefficients and testedthe algorithm for decaying exponential, Gaussian, and Lorentzian kernels in one, two,and three dimensions, respectively; other kernels, in arbitrary D-dimensional domains,could also be explored.

63


We believe that given its efficient tradeoff between speed and accuracy, the methodwe present in this work can be a useful technique for the numerical integration of nonlocalPDE’s and of integro-differential problems containing convolution integrals with spaceand time variant kernels.

7.2 Model limitations

We restricted our analysis to the rather artificial circumstances of homogeneous systemswith respect to soil properties, topography, rainfall, evaporation, mortality, etc. We didso to highlight the roles of self-organizing patterns of biomass and soil-water in breakingthe spatial symmetry of the system and its implications for ecosystem engineering andplant interaction dynamics. Heterogeneous factors, as well as temporal rainfall vari-ability, can be incorporated into the model by introducing time- and space-dependentparameters.

The model in its present form (equations 2.5–2.8 and 5.5–5.8) accounts for the majorfeedbacks between vegetation and water but leaves out a few other feedbacks. The at-mosphere affects vegetation via precipitation and evaporation rate parameters, but thevegetation is assumed to have no feedback on the atmosphere. Organic nutrient effectsare parameterized by the biomass growth rate, but litter decomposition [103] that feed-backs on nutrient concentrations is not considered. Furthermore, ground topography,parameterized by the function ζ, affects runoff and water concentration, but topogra-phy changes due to soil erosion by water flow are neglected. In drylands, where thevegetation is sparse and the limiting resource is water, the vegetation feedbacks on theatmosphere and on organic nutrients are often of secondary importance. Soil erosionprocesses, however, can play important roles, e.g. in desertification, and restrict themodel’s applicability.

Another limitation of the model is the elimination of the soil depth dimension, whichrestricts the variety of root effects the model can capture. Thus, engineering by hydrauliclifts [104] is not captured by the model. Finally, plant recruitment and growth arelumped together in a single biomass equation with a diffusion term modeling local seeddispersal. This modeling form rules out long-distance dispersion, e.g. by wind, waterflow, or animals, and seed predation, e.g. by insects or birds [105].

7.3 Future studies and model developments

In chapter 5 we focused on a system consisting of two life forms (woody and herbaceousplants) growing in planes and uniform slopes under uniform and stationary environ-mental conditions. The model, however, is much more widely applicable: the two lifeforms can represent other functional groups, additional life forms can be included, andthe effects of non-uniform time-dependent environments can be studied. Two examplesare noteworthy. First, the inclusion of two herbaceous species and one engineer speciescan be used to study tradeoffs and species-coexistence mechanisms associated with self-organized heterogeneities induced by patterns of the ecosystem engineer. Second, the

64


inclusion of additional engineer species can be used to study the effects of the dynamicsand mutual interactions of two engineer species on landscape diversity. Furthermore,the model can be used to test new rehabilitation approaches and their effectiveness interms of productivity, species diversity, and resilience [50].

Extensions of the model to include soil erosion, nonlocal seed dispersal, and theimpacts of vegetation on the atmosphere and on nutrient concentrations are possible.Soil erosion, for example, can be included by writing a dynamical equation for thetopography function ζ(x, t)

∂ζ

∂t= f (h, b;R) , (7.1)

where the dynamics of the soil surface depend on the surface runoff h, the biomass b,and other relevant parameters R. Non-local seed dispersal can be considered by split-ting the biomass equation into two dynamical equations, one for mature plants andone for seeds. This will allow for the modeling of seed dynamics separately from thedynamics of mature plants. In that case, convective terms (using some wind field, forexample) can model nonlocal seed dispersal. Obviously, such developments require con-siderable modifications of the model. There are, however, simpler extensions which arenot less significant. One extension is the introduction of environmental heterogeneitiesand temporal fluctuations by means of space- and time-dependent model parameters.This extension is significant in studying tradeoffs between species, species richness, andecosystem response to extreme variations in climatic conditions.

Extended models as described above may provide powerful tools for testing the nicheconcept and developing a niche theory based on a pattern formation approach. A fewelements of this theory can already be outlined: (i) Model solutions provide the mapsthat associate hyper-volumes in niche space (the fundamental niches) with domains inphysical space (the micro-habitats) and determine where in physical space a given speciescan exist. (ii) These maps include the effects of species interactions and therefore elim-inate the need to define ”realized niches” [45,75–77] in niche space. The micro-habitatsare already ”realized” in the sense that their sizes include the effects of competitionor facilitation. (iii) Landscape diversity is not merely a result of heterogeneous en-vironmental factors but can also follow from spatial instabilities leading to symmetrybreaking patterns (e.g. vegetation patterns). (iv) Species diversity responses to environ-mental changes may be driven by collective species dynamics, e.g. transitions betweenecosystem-engineer patterns that involve micro-habitat creation or destruction.

65

Appendix A

Derivation of the surface waterequation

In order to derive a dynamical equation for the surface water depth above the ground,defined by d(x, y, t), we will use the continuity and momentum equations from the shal-low water approximation [56]. Neglecting vertical accelerations and internal friction inthe case of a shallow fluid layer (as it is the case for surface runoff), the shallow-watermomentum equations are written on the horizontal plane (x, y) and as a function of timet as

Du

Dt= −g∇ (ζ + d) +

1

ρF , (A.1)

where u = (u, v) is the two-dimensional, depth-averaged fluid velocity [m/s] and is afunction of x, y, and t, d(x, y, t) is the depth of the surface water layer, ζ(x, y) is theheight of the soil surface, g is the acceleration of gravity [m s−2], ρ is the (constant) fluiddensity [kg/m3], and F represents bottom friction [kg/m2 s−2].

The shallow-water continuity equation is written as

ρ

[∂d

∂t+

∂

∂x(ud) +

∂

∂y(vd)

]= P − Iρd , (A.2)

where P is the average precipitation measured in units of [kg/m2 s−1] and I is theinfiltration rate measured in units of [s−1]. Since ρ is constant, we scale it out using d,and define the new quantity H = ρd, which has units of [kg/m2] (Note that for water[kg/m2]=[mm]). Thus, we can write

∂H

∂t+

∂

∂x(Hu) +

∂

∂y(Hv) = P − IH . (A.3)

We can simplify the above equations by noticing that on the time scales of inter-est here the fluid velocity adjusts rapidly to changes in inputs and outputs, and fluidaccelerations (Du/Dt = 0) can be neglected. If we further assume a linear (Rayleigh)bottom friction term of the form F = −ku, where k is a constant coefficient, we obtain

66

Appendix A. Derivation of the surface water equation

the following expressions from the momentum equation

u = −κ(∂Z

∂x+∂H

∂x

)

v = −κ(∂Z

∂y+∂H

∂y

)(A.4)

where we have defined κ = g/k and we have introduced the rescaled soil-height variableZ = ρζ, in analogy to what has been done for the water depth.

The final equation for H is obtained by substituting Eq. (A.4) into Eq. (A.3) andcollecting terms1:

∂H

∂t= P − IH +

κ

2∇2(H2) + κ∇H · ∇Z + κH∇2Z (A.5)

Different forms for the bottom friction term F lead to different forms of the dynamicalequation for H. For example, bottom friction of the form F = −ku/dp, where p is aninteger, would lead to the following equation for H:

∂H

∂t= P − IH +

κ

(p+ 2)∇2Hp+2 + κ∇ (

Hp+1) · ∇Z + κHp+1∇2Z . (A.6)

Although quantitatively different, these alternate forms for the H equation do not bringin any qualitatively new behavior.

1In Eqs. 2.1 and 5.1 we define DH = κ/2

67

Appendix B

Linear stability analysis

B.1 Analysis of the n-interacting species model

A Linear stability analysis of the model homogeneous stationary solutions for planetopography is performed by adding an infinitesimal spatial perturbation and lineariz-ing around the solution to calculate the linear growth rate of the perturbation. Wedenote the homogeneous stationary solutions of the model equations 5.5–5.8 by U0 =(b10, . . . , bn0, w0, h0)

T and define the perturbed solution as

U(x, t) = U0 + δU(x, t) , (B.1)

where

U = (b1, . . . , bn, w, h)T

δU = (δb1, . . . , δbn, δw, δh)T . (B.2)

The perturbation dependence on time and space is given by

δU(x, t) = a(t)eik·x + c.c. , (B.3)

wherea(t) = [ab1(t), . . . , abn(t), aw(t), ah(t)]

T . (B.4)

Substitution of the perturbed solution (B.1) into the model equations 5.5–5.8 gives(time and space dependence of the perturbation is omitted for convenience)

(δbi)t = Gib

∣∣U0+δU

(bi0 + δbi) [1− (bi0 + δbi)]− µi(bi0 + δbi) + δbi∇2(bi0 + δbi)

(δw)t = I|U0+δU (h0 + δh)− ν

[1−

n∑i=1

ρi(bi0 + δbi)

](w0 + δw)

−(w0 + δw)n∑

i=1

Giw

∣∣U0+δU

+ δw∇2(w0 + δw)

(δh)t = p− I|U0+δU (h0 + δh) + δh∇2[(h0 + δh)2

], (B.5)

68

Appendix B. Linear stability analysis

where i = 1, . . . , n. The infiltration term is expanded up to the first order in δbi accordingto Eq. (5.6) and reads

I|U0+δU = I0 +n∑

i=1

∂I∂bi

∣∣∣∣bi=bi0

δbi

= α

∑i ψibi0 + qf∑i ψibi0 + q

+ α

n∑i=1

ψiq (1− f)

(∑

k ψkbk0 + q)2 δbi +O(δbi2) . (B.6)

To evaluate the terms Gib|U0+δU and Gi

w|U0+δU, we expand the kernels, as defined inEqs. (5.7) and (5.8), up to the first order in δbi:

gi (x,x′) = g0

i (x,x′) + g1i (x,x′) δbi (x) +O (

δbi2)

gi (x′,x) = g0

i (x′,x) + g1i (x′,x) δbi (x

′) +O (δbi

2), (B.7)

where

g0i (x,x′) = g0

i (x′,x) = gi (x,x′)|bi=bi0

=1

2πσ2i

e−|x−x′|2/2ξ2i

g1i (x,x′) = g1

i (x′,x) =∂

∂bi[gi (x,x

′)]

∣∣∣∣bi=bi0

=ηi

2πσiξ3i

|x− x′|2 e−|x−x′|2/2ξ2i ,(B.8)

where ξi ≡ σi(1 + ηibi0) and |x− x′|2 = (x− x′)2 + (y − y′)2.After expanding the kernels and substituting the perturbed solution, we get the fol-

lowing expressions up to the first order in the perturbation (time dependence is omittedfor clarity):

Gib

∣∣U0+δU

= νλi

∫gi (x,x

′)w (x′) dx′

≈ νλi

∫ [g0

i (x,x′) + g1i (x,x′) δbi (x)

][w0 + δw (x′)] dx′

= νλiw0

∫g0

i (x,x′) dx′ + νλi

∫g0

i (x,x′) δw (x′) dx′

+νλiw0δbi (x)

∫g1

i (x,x′) dx′

Giw

∣∣U0+δU

= γi

∫gi (x

′,x) bi (x′) dx′

≈ γi

∫ [g0

i (x′,x) + g1i (x′,x) δbi (x

′)][bi0 + δbi (x

′)] dx′

= γibi0

∫g0

i (x′,x) dx′ + γi

∫g0

i (x′,x) δbi (x′) dx′

+γibi0

∫g1

i (x′,x) δbi (x′) dx′ , (B.9)

where the integration is over the entire domain. Solving the integrals in Eq. (B.9) gives

Gib

∣∣U0+δU

= νλiw0(ξi/σi)2 + νλi(ξi/σi)

2e−k2ξ2i /2δw (x) + 2νλiηi(ξi/σi)w0δbi (x)

Giw

∣∣U0+δU

= γibi0(ξi/σi)2 + γi(ξi/σi)e

−k2ξ2i /2

[1 + ηibi0

(3− k2ξ2

i

)]δbi(x) . (B.10)

69


Substitution of Eqs. (B.6) and (B.10) into Eq. (B.5) and linearization around U0

can now be performed in a straightforward manner. The equations for the unperturbedhomogeneous stationary solution U0 = (b10, . . . , bn0, w0, h0)

T are

0 = νλibi0 (1− bi0) (ξi/σi)2w0 − µibi0 i = 1, . . . , n

0 = αh0


− ν

(1−

n∑i=1

ρibi0

)w0 − w0

n∑i=1

γibi0 (ξi/σi)2

0 = p− αh0


. (B.11)

At the first order in the perturbation δU we get the governing equations for thelinear spatio-temporal dynamics of the perturbation:

(δbi)t = νλiw0 (ξi/σi)[1− 2bi0 + ηibi0 (3− 4bi0)− µi − δbi

k2]δbi

+νλibi0 (1− bi0) (ξi/σi)2 ek2ξ2

i /2δw i = 1, . . . , n

(δw)t =n∑

i=1

αh0ψiq (1− f)

(∑

k ψkbk0 + q)2 + νw0ρi − w0γi (ξi/σi) ek2ξ2

i /2[1 + ηibi0

(3− k2ξ2

i

)]δbi

−[ν

(1−

n∑i=1

ρibi0

)+

n∑i=1

γibi0 (ξi/σi)2 + δwk

2

]δw + α


δh

(δh)t = −n∑

i=1

[αh0ψiq (1− f)

(∑

k ψkbk0 + q)2

]δbi +

[α


− 2h0δhk2

]δh . (B.12)

These equations can be written in vector form as

∂

∂t[δU(x, t)] = J (k) δU(x, t) , (B.13)

where J (k) ∈ Rn+2×n+2 is the Jacobian matrix. After eliminating the spatial part of theperturbation (i.e. eik·x) and assuming exponential time dependence for the perturbation(i.e. ai(t) ∼ eλt) we get the following eigenvalue problem:

λ a(t) = J (k) a(t) , (B.14)

where the entries of the Jacobian matrix J (k) are given by Eq. (B.12). By solving thiseigenvalue problem (Eq. B.14) we obtain the dispersion relation λ(k) for the perturba-tion.

70


B.2 Results for the n = 1 case

According to Eq. (B.11), the equations for the homogeneous stationary solutions (b0, w0, h0)T

of the model equations 2.5–2.8 are1

0 = νb0 (1− b0) (1 + ηb0)2w0 − b0

0 = αh0b0 + qf

b0 + q− ν (1− ρb0)w0 − γw0b0 (1 + ηb0)

2

0 = p− αh0b0 + qf

b0 + q, (B.15)

and the Jacobian entries, given by Eq. (B.12), are

J11 = (1 + ηb0)νw0 [1− 2b0 + ηb0 (3− 4b0)]− 1− δbk2

J12 = νb0(1− b0)(1 + ηb0)2e−k2(1+ηb0)2/2

J13 = 0

J21 = αh0q(1− f)

(b0 + q)2+ ρνw0 − γw0(1 + ηb0)e

−k2(1+ηb0)2/2[1 + ηb0

(3− k2 (1 + ηb0)

2)]

J22 = −ν(1− ρb0)− γb0 (1 + ηb0)2 − δwk

2

J23 = αb0 + qf

b0 + q

J31 = −αh0q(1− f)

(b0 + q)2

J32 = 0

J33 = −αb0 + qf

b0 + q− 2δhh0k

2 . (B.16)

For the bare state solution, b0 = 0, w0 = p/ν, and h = p/αν, the Jacobian takes thefollowing form:

J (k) =

p− 1− δbk2 0 0

p(1−f)qf

− γpek2/2

ν+ ρp −ν − δwk

2 αf

−p(1−f)qf

0 −αf − 2pδh

αfk2

(B.17)

and its eigenvalues are given by the diagonal terms

λ1(k) = p− 1− δbk2

λ2(k) = −ν − δwk2

λ3(k) = −αf − 2pδhαf

k2 . (B.18)

Obviously, λ2, λ3 < 0 for any k; however λ1 becomes zero for k = 0 at p = 1. Thus thebare state solution becomes unstable to homogeneous perturbations at p = pc = 1, i.e.

1Note that ξi/σi = 1 + ηibi0 and according to Table 5.1 we set λ1 = µ1 = σ1 = ψ1 = 1.

71


the fastest growing perturbation is the one with zero wavenumber k = kc = 0, as shownin Fig. 3.2a. For the uniform vegetation solution, the analytic expressions are too longto be written here, but the same procedure leads to a Turing-like instability [18] of theuniform cover as p decreases below some critical value (e.g. p2 in Fig. 3.1). In that case,the uniform cover solution becomes unstable to perturbations with finite wavenumber(i.e. k = kc 6= 0), which has the fastest growth rate, as shown in Fig. 3.2b.

B.3 Results for the n = 2 case

According to Eq. (B.11), the equations for the homogeneous stationary solutions (b10, b20, w0, h0)T

of the model equations 5.5–5.8 are

0 = νb10 (1− b10) (1 + η1b10)2w0 − b10

0 = νλ2b20 (1− b20) (1 + η2b20)2w0 − µ2b20

0 = αh0b10 + ψ2b20 + qf

b10 + ψ2b20 + q− ν (1− ρ1b10 − ρ2b20)w0

−γ1w0b10 (1 + η1b10)2 − γ2w0b20 (1 + η2b20)

2

0 = p− αh0b10 + ψ2b20 + qf

b10 + ψ2b20 + q, (B.19)

72


and the Jacobian entries, given by Eq. (B.12), are

J11 = ν(1 + η1b10)w0 [1− 2b10 + η1b10 (3− 4b10)]− 1− δb1k2

J12 = 0

J13 = νb10(1− b10)(1 + η1b10)2e−k2(1+η1b10)2/2

J14 = 0

J21 = 0

J22 = νλ2(1 + η2b20)w0 [1− 2b20 + η2b20 (3− 4b20)]− µ2 − δb2k2

J23 = νλ2b20(1− b20)(1 + η2b20)2e−k2[σ2(1+η2b20)]

2/2

J24 = 0

J31 =αh0q (1− f)

(b10 + ψ2b20 + q)2 + νρ1w0

−w0γ1(1 + η1b10)ek2(1+η1b10)2/2

[1 + η1b10

(3− k2(1 + η1b10)

2)]

J32 =αh0ψ2q (1− f)

(b10 + ψ2b20 + q)2 + νρ2w0

−w0γ2(1 + η2b20)ek2[σ2(1+η2b20]

2/2[1 + η2b20

(3− k2 [σ2(1 + η2b20)]

2)]

J33 = −ν(1− ρ1b10 − ρ2b20)− γ1b10 (1 + η1b10)2 − γ2b20 (1 + η2b20)

2 − δwk2

J34 = αh0b10 + ψ2b20 + qf

b10 + ψ2b20 + q

J41 = − αh0q (1− f)

(b10 + ψ2b20 + q)2

J42 = − αh0ψ2q (1− f)

(b10 + ψ2b20 + q)2

J43 = 0

J44 = −αb10 + ψ2b20 + qf

b10 + ψ2b20 + q− 2h0δhk

2 . (B.20)

73

Appendix C

Numerical methods

The earlier numerical simulations were carried out on various machines that were at ourdisposal, ranging from standard Linux-based PC’s with Intel processors to Unix serverswith multiple alpha EV67 processors. However, starting from November 2004 the maincomputational platform we used was an SGI Altix 3700 64-bit Linux SuperCluster with24 1.3GHz Itanium2 processors and 32GB memory. The programming languages weused were Fortran and C along with the appropriate Intel and GNU compilers, whichare the most suitable for building high performance applications in a Linux environment.

The numerical solutions were computed using the method of lines for partial differ-ential equations [106]. The method of lines decouples the discretization of the spatialand temporal operators into independent problems. The spatial derivatives were repre-sented on centered, uniform grids with finite difference approximations and the resultingset of coupled ordinary differential equations were solved by a suitable time integrationscheme.

Most of the numerical simulations were done using a two-dimensional grid withperiodic boundary conditions and a resolution of either 128×128 or 256×256 grid points.However, a resolution of 512×512 and even 512×256 grid points were occasionally used(see Fig. 3.15).

C.1 Spatial discretization

We write the model equation in the form

Ut = N (U;x, t) +DLG (U) , (C.1)

where U represents the dynamical variables like bi and w, N and G are nonlinear func-tions (see Eqs. 2.5 and 5.5), D is a matrix of coefficients (e.g. δbi

,δw), and L stands fordifferential operators like ∇2. All continuous functions and operators are descretizedover a uniform grid of N ×N grid points. Space location x = (x, y) is now denoted asxi,j = (xi, yj) where i, j = 1, . . . , N . Given the domain size Lx × Ly, the grid spacingis ∆x = Lx/N and ∆y = Ly/N . Any continuous function is represented by its value ateach grid point

f(x) = f(xi, yj) ≡ fi,j , (C.2)

74

Appendix C. Numerical methods

and any continuous operator is replaced with a discrete finite differences approximation

∇f(x) = ∇f(xi,j) =f(xi+1,j)− f(xi−1,j)

2∆xx+

f(xi,j+1)− f(xi,j−1)

2∆yy +O(∆x2,∆y2) ,

or simply

∇fi,j =fi+1,j − fi−1,j

2∆xx+

fi,j+1 − fi,j−1

2∆yy +O(∆x2,∆y2) , (C.3)

and

∇2fi,j =fi+1,j − 2fi,j + fi−1,j

∆x2+fi,j+1 − 2fi,j + fi,j−1

∆y2+O(∆x2,∆y2) . (C.4)

The periodic boundary conditions in this second order finite differences approximationare given by

x : f0,j = fN,j, fN+1,j = f1,j j = 1, . . . , N

y : fi,0 = fi,N , fi,N+1 = fi,1 i = 1, . . . , N (C.5)

C.2 Time integration scheme

After the spatial operators are discretized we are left with a set of coupled ordinarydifferential equations of the form

Ut = F(U;x; t) , (C.6)

where F represents the spatially discretized right-hand-side of Eq. (C.1) . We maythen use numerical methods for ordinary differential equations to advance the solutionforward in time. We use the Adams-Bashforth-Moulton predictor-corrector scheme withan explicit third-order Adams-Bashforth predictor and an implicit fourth-order Adams-Moulton corrector with a fixed time step [107]. For a time step of size ∆t the solutionof Eq. (C.6) at time step n + 1 is given in terms of the solution at the previous threetime steps, n, n− 1, and n− 2 according to the following scheme:

Predict : Upn+1 = Un +

∆t

12(23Fn − 16Fn−1 + 5Fn−2) +O(∆t4)

Evaluate RHS : Fn+1 = F(Upn+1)

Correct : Un+1 = Un +∆t

24(9Fn+1 + 19Fn − 5Fn−1 + Fn−2) +O(∆t5)

Evaluate RHS : Fn+1 = F(Un+1) . (C.7)

To start the integration from the initial conditions we make two time steps, one with afirst-order Adams-Bashforth predictor and a second-order Adams-Moulton corrector

Upn+1 = Un + ∆tFn +O(∆t2)

Un+1 = Un +∆t

2(Fn+1 + Fn) +O(∆t3) , (C.8)

75


and the second with a second-order Adams-Bashforth predictor and a third-order Adams-Moulton corrector

Upn+1 = Un +

∆t

2(3Fn − Fn−1) +O(∆t3)

Un+1 = Un +∆t

12(5Fn+1 + 8Fn − Fn−1) +O(∆t4) . (C.9)

C.3 Solving the surface water equation

A conspicuous phenomenon in the models is the different time scales according to whichthe dynamical variables change significantly. Specifically, the time scale of surface waterdynamics (minutes–hours) is faster by orders of magnitude than that of the biomass(years) or the soil-water (days–weeks). Consequently, the biomass and soil-water vari-ables may be considered ”frozen” with respect to the time scale of the surface water,leading to the well justified approximation ∂h/∂t = 0 in Eqs. (2.5) and (5.5). Based onthis approximation we may write the equation for the surface water as

0 = p− Ih+ δh∇2(h2) + 2δh∇h · ∇ζ + 2δhh∇2ζ . (C.10)

The significance of this approximation is that h immediately adjusts itself to thestate of bi and w, thus allowing for Eq. (C.10) to be solved under the assumptionof constant bi and w. This approach enables the elimination of the computationallyexpensive need to advance h in time. For reasons of numerical stability, we make thenonlinear transformation h2 = u in order to avoid the nonlinear term δh∇2(h2). Theresulting set of nonlinear equations is1

0 = p− Iu1/2 + δh∇2u+ 2δh∇u1/2 · ∇ζ + 2δhu1/2∇2ζ ≡ F (u) . (C.11)

We seek an iterative solution to this equation by linearizing F (u) and using Newton’smethod. With the iteration parameter k the linearization is

F (uk) +∂F

∂u

∣∣∣∣uk

(uk+1 − uk

)= 0 . (C.12)

We denote the Jacobian matrix ∂F∂u

∣∣uk by J (uk) and

(uk+1 − uk

)by ∆uk+1. Solving

Eq. (C.12) for ∆uk+1 gives

∆uk+1 = −J −1(uk)F (uk) , (C.13)

and the Newton iteration step is defined as

uk+1 = uk + ∆uk+1 . (C.14)

We iterate on k until the norm of the right-hand-side in Eq. (C.11), ‖ F (uk) ‖, issmaller than some given absolute error tolerance αtol. Typically the maximum number

1We omit the vector notation for clarity.

76


of iterations allowed was 3. The linear system defined by Eq. (C.12) is solved using thediterative routine – a sparse iterative linear system solver, which is a part of the SCSLScientific Library that have been optimized for use on SGI computer systems. We useda Jacobi preconditioner along with a conjugate gradient squared (CGS) solver [108].

The stopping condition for the linear iteration was modified from the one contained inthe SCSL library. This modification was needed because the diterative code terminateswhen the norm of the residual relative to the norm of the right-hand-side is less thanβtol, which in our case means

‖ F (uk)− J (uk)∆uk+1 ‖‖ F (uk) ‖ < βtol . (C.15)

The problem is that we are solving for the difference of the solution, ∆uk+1 = uk+1−uk,and the right-hand-side of the linear system, F (uk), goes to zero. Since we are workingin finite precision, when the solution is close to convergence, the error we are measuringusing this test is mostly due to roundoff. The code was modified to terminate wheneverthe norm of the residual is smaller than some given absolute error tolerance

‖ F (uk)− J (uk)∆uk+1 ‖< αtol , (C.16)

where we chose αtol = 10−2 and limited the number of iterations to a maximum of 20.

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