Habitat Dynamics in Drylands: The Role of Collective Modes E. Gilad Department of Physics, Ben-Gurion University, Beer Sheva, 84105, Israel Department of Solar Energy and Environmental Physics, BIDR, Ben Gurion University, Sede Boker Campus 84990, Israel
[email protected] J. von Hardenberg ISAC-CNR, Corso Fiume 4, I-10133 Torino, Italy CIMA, Universit`a di Genova e della Basilicata, Via Cadorna 7, 17100 Savona, Italy
[email protected] A. Provenzale ISAC-CNR, Corso Fiume 4, I-10133 Torino, Italy CIMA, Universit`a di Genova e della Basilicata, Via Cadorna 7, 17100 Savona, Italy
[email protected] M. Shachak Mitrani Department of Desert Ecology, BIDR, Ben Gurion University, Sede Boker Campus 84990, Israel
[email protected] E. Meron1 Department of Solar Energy and Environmental Physics, BIDR, Ben Gurion University, Sede Boker Campus 84990, Israel
–2–
Department of Physics, Ben-Gurion University, Beer Sheva, 84105, Israel
[email protected] Received
;
accepted
Elements to appear in the expanded online edition: Appendices A,B, Tables 1, 2, Figs. 1,3, 4,7,10, 12,13. This manuscript was prepared with the AASTEX LATEX macros v5.2.
1
Corresponding author
–3– ABSTRACT
Dryland landscapes are mosaics of patches that differ in resource concentration, biomass production and species richness. Understanding their structure and dynamics often calls for the identification of key species that modulate the abiotic environment, redistribute resources and facilitate the growth of other species. Shrubs in drylands are excellent examples; they self-organize to form patterns of mesic patches which provide habitats for herbaceous species. In this paper we present a mathematical model for studying the dynamics of habitats created by plant ecosystem engineers in drylands. Using a pattern formation approach we study conditions and mechanisms for habitat creation, habitat diversity, and habitat response to environmental changes. We highlight the roles of two collective modes of habitat dynamics, associated with the engineering effects of plant species and the self-organizing spatial patterns they form, and identify emergent mechanisms of habitat dynamics. Subject headings: collective dynamics, pattern formation, ecosystem engineer, habitats dynamics, diversity, drylands
–4– 1.
Introduction
A fundamental problem in understanding global ecosystem aspects, such as biodiversity, productivity and stability, is how local properties at the individual and single plant-patch level, where laboratory and field experiments can be carried out, are up-scaled to the ecosystem or landscape level (Loreau et al. 2001). This problem has two facets, conceptual and technical. Conceptually, at any higher level of organization, emergent phenomena may appear. Emergent properties involve collective dynamics of species individuals and therefore cannot be understood by studying individuals’ properties alone. The concept of emergent properties has a long history in ecology (Odum 1969, 1971) and it has been adopted in many fields of physics ranging from superconductivity through pattern formation to network dynamics (Anderson 1972, 1995). A striking example in water limited ecosystems is banded vegetation on hill-slopes, where positive water-biomass feedbacks and intraspecific competition at the level of the individual patch lead to self-organizing band patterns at the landscape level (Valentin et al. 1999). The self-organized dynamics at the ecosystem or landscape level usually involve interactive activities of many species spanning a diversity of traits in respect to resource acquisition and distribution. There are cases, however, where a few species have greater impacts on the trophic web structure than others, through their roles in energy flow and nutrient cycling. Such species have been named “keystone species” (Power et al. 1996), as their introduction or removal can dramatically alter the ecosystem’s behavior. Recently, a non-trophic organismal trait, termed ecosystem engineering, was suggested as an additional organizational trait that links species and ecosystems (Jones et al. 1994, 1997; Olding-Smee et al. 2003). While keystone species are assumed to directly affect the biotic component, ecosystem engineers affect the physical environment. Plant ecosystem engineers modulate the landscape by creating patches and patterns of biomass and resources, thereby affecting
–5– species diversity and ecosystem functioning (Perry 1998; Gilad et al. 2004; Shachak et al. 2005). From a technical point of view, the attempt of addressing the up-scaling problem from individual patches to the whole ecosystem is plagued with the difficulty of studying large systems characterized by long length and time scales. In the context of water limited ecosystems, significant progress in this direction has recently been made by the introduction of spatially explicit mathematical models of vegetation growth (Lefever & Lejeune 1997; Klausmeier 1999; von Hardenberg et al. 2001; Okayasu & Aizawa 2001; Rietkerk et al. 2002; Shnerb et al. 2003; Gilad et al. 2004). The input information used in formulating these models is at the level of a single plant-patch, while the predictive power extends to the patchwork created at the landscape level. These models have reproduced various vegetation patterns observed in the field (Rietkerk et al. 2004), predicted the possible coexistence of different stable patterns under given environmental conditions, and identified a generic sequence of pattern states along the rainfall gradient (von Hardenberg et al. 2001; Meron et al. 2004). In this paper we use a mathematical modelling approach to study emergent mechanisms of habitat creation and change in drylands, where the limiting resource is water. We focus on plant species that act as ecosystem engineers by trapping soil water under the patches they form. The water enriched patches provide micro-habitats for other species. In modeling plant ecosystem engineers we capture two collective modes representing interspecific and intraspecific dynamics. The interspecific mode is associated with the micro-habitats that are created by the ecosystem engineer, which other species can benefit from. The intraspecific mode is associated with the spatial patterns formed by the ecosystem engineer, which reflect cooperative dynamics of many individuals over long distances. Taking into account these collective modes is important for understanding the impact of abiotic or human
–6– induced environmental disturbances on species diversity. Such disturbances may change the spatio-temporal patterns of the ecosystem engineer and thus the habitats it forms. The implications for species diversity inferred in this way may differ and even contradict those inferred by merely considering the tolerance limits of individual species. A brief account of the model to be presented here has been given in (Gilad et al. 2004). In this paper we give a full account of the model, provide detailed model analysis, and present new results concerning habitat types, habitat diversity and scenarios for habitat change.
2.
Background
Water limited landscapes are a mosaic of patches that differ in resource concentration, biomass production and species richness. Redistribution of the water resource by biotic and abiotic factors is a major factor in the formation of this patchwork. An abiotic factor contributing to water redistribution is dust and sediment deposition in rocky watersheds (Yair & Shachak 1987). The higher water infiltration rates in deposition patches induce source-sink water flow relationships and enrich the water content in these patches. Biotic factors include cyanobacteria that form partially impermeable biogenic crusts, limiting infiltration and favoring surface runoff, and plants, particularly shrubs, that act as sinks for the surface-water flow. The accumulation of litter and dust under the shrub forms a mound with high infiltration capacity, which not only absorbs direct rainfall but also intercepts the runoff water generated by the soil crust. Reduced evaporation under the shrub canopy further contributes to local water accumulation. The water-enriched patch and the accumulated dust and litter create an island of fertility under the shrub canopy, characterized by a concentration of water, nutrient and organic matter that is higher than in the surrounding crusted soil.
–7– Organisms capable of modulating the landscape and redistributing resources, such as shrubs in drylands, have been termed ”ecosystem engineers”. There is an increasing recognition that ecosystem engineers have important implications for biodiversity and for ecosystem function (Olding-Smee 1988; Jones et al. 1994; Shachak & Jones 1995; Gurney & Lawton 1996; Jones et al. 1997; Laland et al. 1999). Studies in the Negev desert, for example, indicate that the concentration of resources, particularly soil water, under shrub patches significantly increases species richness and density of annual plants (Boeken & Shachak 1994). The implications of the shrub patchwork for species richness at the landscape scale, however, is largely unexplored. At this scale collective species dynamics, manifested in the form of symmetry breaking spatial patterns, become important. Depending on environmental conditions (rainfall, topography, grazing stress, etc.) and species traits, different vegetation patterns have been observed, including banded or labyrinthine patterns, spotted patterns, and gaps in uniform vegetation (Valentin et al. 1999; Rietkerk et al. 2002; Meron et al. 2004; Rietkerk et al. 2004). The self organization of vegetation in water limited systems to form spatial patterns has recently been attributed to an instability (Cross & Hohenberg 1993; Murray 1993) of uniform vegetation which has been produced by several independent models (Lefever & Lejeune 1997; von Hardenberg et al. 2001; Okayasu & Aizawa 2001; Rietkerk et al. 2002; Gilad et al. 2004). The mechanism of the instability is based on a positive feedback between biomass and resources. A common realization of such a feedback is the increased infiltration of surface water at vegetation patches discussed above. Another significant realization of a biomass-water feedback is water uptake by plants’ roots; the larger the biomass the longer the roots and the larger amount of soil-water the roots take up. This feedback is not captured by earlier mathematical models. The biomass-water positive feedback is also responsible for the coexistence of different
–8– stable vegetation states, uniform or patterned, under the same environmental conditions (von Hardenberg et al. 2001; Meron et al. 2004; Rietkerk et al. 2004). Coexistence of stable states over limited parameter ranges (precipitation rate, grazing stress, etc.) may explain desertification processes (von Hardenberg et al. 2001; Meron et al. 2004), and catastrophic shifts in general (Scheffer et al. 2001; Scheffer & Carpenter 2003; Rietkerk et al. 2004), as transitions between coexisting states. Coexistence of stable states also gives rise to a high pattern diversity as mixtures of the different stable states in various spatial distributions can be realized (Meron et al. 2004). Spatial vegetation patterns are important agents in resource concentration processes on the landscape level. Vegetation patterns such as spots, bands and gaps differ in the proportion of bare soil to vegetative cover and thus in the resource patches they form. Environmental changes or human disturbances may induce transitions from one pattern state of the ecosystem engineer to another, thereby affecting the habitats it forms. This landscape-scale mechanism of habitat change has not been explored yet.
3.
A model for patch dynamics
We introduce a model for a single plant species where the limiting resource is water. A ”patch” in the context of the model is defined to be an area covered by the plant, which generally differs in its water content from the surrounding bare soil. Two main positive feedback mechanisms between plant biomass and water are explicitly modelled: increased water infiltration rates at vegetation patches (hereafter ”infiltration feedback”) and water uptake by plants’ roots (hereafter ”uptake feedback”). Both feedbacks are responsible for the appearance of symmetry breaking patterns of biomass and soil water at the landscape scale. They control, however, different complementary properties of the ecosystem induced by the plant: habitat creation and resilience to disturbances.
–9– The model contains three dynamical variables: (a) the biomass density variable, B(r, t), representing the plant’s biomass above ground level and has units of [kg/m2 ], (b) the soil-water density variable, W (r, t), describing the amount of soil water available to the plants per unit area of ground surface in units of [kg/m2 ], and (c) the surface water variable, H(r, t), describing the height of a thin water layer above ground level in units of [mm]. Rainfall and topography are introduced parametrically; thus vegetation feedback on climate and soil erosion are assumed to be negligible. The model equations are: BT = GB B (1 − B/K) − M B + DB ∇2 B WT = IH − N (1 − RB/K) W − GW W + DW ∇2 W ¡ ¢ HT = P − IH + DH ∇2 H 2 + 2DH ∇H · ∇Z + 2DH H∇2 Z ,
(1)
2 where the subscript T denotes partial time derivative, r = (X, Y ) and ∇2 = ∂X + ∂Y2 .
The quantity GB [yr−1 ] represents biomass growth rate, while K [kg/m2 ] is the maximum standing biomass. The quantity GW [yr−1 ] represents the soil water consumption rate, the quantity I [yr−1 ] represents the infiltration rate of surface water into the soil and the parameter P [mm/yr] stands for the precipitation rate. The parameter N [yr−1 ] represents soil water evaporation rate, while R describes the reduction in soil-water evaporation rate due to shading 1 . The parameter M [yr−1 ] describes the rate of biomass loss due to mortality and different kinds of disturbances (e.g. grazing, fire). The term DB ∇2 B represents seed dispersal while the term DW ∇2 W describes soil water transport in non-saturated soil (Hillel 1998). Finally, the non-flat ground surface height is described by the topography function Z(r) while the parameter DH [m2 /yr (kg/m2 )−1 ] represents the phenomenological bottom friction coefficient between the surface water and the ground surface. 1
Reduced evaporation by shading is another positive feedback mechanism between
biomass and water that is captured by the model. Throughout this study, however, we keep this feedback constant by fixing the value of R.
– 10 – While the equations for B and W are purely phenomenological (resulting from modelling processes at the single patch scale), the equation for H was motivated by shallow water theory. The shallow water approximation is based on the assumption of a thin layer of water where pressure variations are very small and the motion becomes almost two-dimensional (Weiyan 1992). The derivation of the equation for H is described in online Appendix A. The two positive feedback processes are modelled in the equations through the explicit forms of the infiltration rate term I and the growth rate term GB . The infiltration feedback is modelled by assuming a monotonously increasing dependence of I on biomass; the bigger the biomass the higher the infiltration rate and the more soil-water available to the plants. This form of the infiltration feedback mirrors the fact that in many arid regions infiltration is low far from vegetation patches due to the presence of the biogenic crust. Conversely, the presence of shrubs destroys the cyanobacterial crust and favor water infiltration. The uptake feedback is modelled by assuming a monotonously increasing dependence of root length on biomass; the bigger the biomass the longer the roots and the bigger amount of soil-water the roots take up from the soil. The explicit dependence of the infiltration rate of surface water into the soil on the biomass density is chosen as (HilleRisLambers et al. 2001; Rietkerk et al. 2002): I=A
B(r, t) + Qf , B(r, t) + Q
(2)
where A [yr−1 ], Q [kg/m2 ] and f are constant parameters. Two distinct limits of this term, illustrated in online Fig. 1, are noteworthy. When B → 0, this term represents the infiltration rate in bare soil, I = Af . When B À Q it represents infiltration rate in fully vegetated soil, I = A. The parameter Q represents a reference biomass beyond which the plant approaches its full capacity to increase the infiltration rate. The difference between the infiltration rates in bare and vegetated soil (hereafter the ”infiltration contrast”) is
– 11 – quantifed by the parameter f , defined to span the range 0 < f < 1. When f ¿ 1 the infiltration rate in bare soil is much smaller than the rate in vegetated soil. Such values can model bare soils covered by biological crusts (Campbell et al. 1989; West 1990). As f gets closer to 1, the infiltration rate becomes independent of the biomass density B, representing non-crusted soil where the infiltration is high everywhere. The parameter f measures the strength of the positive feedback due to increased infiltration at vegetation patches. The smaller f the stronger the feedback effect. As we shall see below, this feedback plays a crucial role in determining the engineering behavior of shrubs. The growth rate GB has the form: Z GB (r, t) = Λ G(r, r0 , t)W (r0 , t)dr0 , Ω · ¸ 1 |r − r0 |2 0 G(r, r , t) = exp − , 2πS02 2[S0 (1 + EB(r, t))]2
(3)
where the integration is over the entire domain Ω and the kernel G (r, r0 , t) is normalized such that for B = 0 the integration over the entire domain equals unity. According to this form, the biomass growth rate depends not only on the amount of soil water at the plant location, but also on the amount of soil water in the neighborhood spanned by the plant’s roots. The roots length [m] is given by S0 (1 + EB(r, t)), where E [(kg/m2 )−1 ] is the roots’ extension per unit biomass, beyond some minimal root length S0 . The parameter Λ has units of [(kg/m2 )−1 yr−1 ] and represents the plant’s growth rate per unit amount of soil water. The parameter E measures the strength of the positive feedback due to water uptake by the roots. The larger E the stronger the feedback effect. The soil water consumption rate at a point r is similarly given by Z GW (r, t) = Γ G(r0 , r, t)B(r0 , t)dr0 .
(4)
Ω
Note that G(r0 , r, t) 6= G(r, r0 , t). The soil water consumption rate at a given point is due to all plants whose roots extend to this point. The parameter Γ has units [(kg/m2 )−1 yr−1 ]
– 12 – and it measures the soil water consumption rate per unit biomass. The parameter values used in this paper are summarized in online Table 1. They are chosen to describe shrub species and are taken or deduced from (Hillel 1998; Sternberg & Shoshany 2001; Rietkerk et al. 2002). The model solutions described here are robust and do not depend on delicate tuning of any particular parameter. The precipitation parameter represents mean annual rainfall in this paper and assumes constant values. This approximation is justified for species, such as woody shrubs, whose growth time scales are much larger than the typical rainfall time scale. The effects of temporal rainfall variability will be explored in a future study. It is convenient to study the model equations using non-dimensional variables and parameters as defined in online Table 2. The model equations can now be written in the following non-dimensional form: bt = Gb b(1 − b) − b + δb ∇2 b wt = Ih − ν(1 − ρb)w − Gw w + δw ∇2 w ht = p − Ih + δh ∇2 (h2 ) + 2δh ∇h · ∇ζ + 2δh h∇2 ζ .
(5)
The infiltration term has the non dimensional form: I=α
b(r, t) + qf . b(r, t) + q
(6)
The growth rate term Gb is written as: Z g(r, r0 , t)w(r0 , t)dr0 , Ω · ¸ |r − r0 |2 1 0 exp − , g(r, r , t) = 2π 2(1 + ηb(r, t))2 Gb (r, t) = ν
and the soil water consumption rate can be similarly written as: Z Gw (r, t) = γ g(r0 , r, t)b(r0 , t)dr0 , Ω
(7)
(8)
– 13 – where r = (x, y) and r0 = (x0 , y 0 ). The non-dimensional precipitation parameter p can be used to define an aridity parameter, a, as a = p−1 =
MN . ΛP
(9)
This parameter captures the effects of four factors on aridity: precipitation, evaporation, mortality or grazing, and biomass growth rate per unit amount of soil water. It extends an earlier definition (Lefever & Lejeune 1997) in including the two pluvial parameters, precipitation rate and evaporation rate.
4.
Model solutions describing basic landscape states
The model has two homogeneous stationary solutions representing bare soil and uniform coverage of the soil by vegetation. Their existence and linear stability ranges for plane topography are shown in the bifurcation diagram displayed in Fig. 2. The bifurcation parameter is the inverse of the aridity parameter, p. The linear stability analysis leading to this diagram is described in online Appendix B. The bare soil solution, denoted in Fig. 2 by B, is given by b = 0, w = p/ν and h = p/αf . It is linearly stable for p < pc = 1 and it loses stability at p = 1 to uniform perturbations 2 . Note that the instability threshold at pc = 1 is independent of all other non-dimensional parameters. We will further discuss the significance of this threshold at the end of the section. The uniform vegetation solution, denoted by E, exists for p > 1 in the case of a supercritical bifurcation and for p > p1 (where p1 < 1) in the case of a subcritical bifurcation. It is stable, however, only beyond another threshold, p = p2 > p1 . 2
The bifurcation is subcritical (supercritical) depending whether the quantity 2ην/[ν(1 −
ρ) + γ] is greater (lower) than unity.
– 14 – As p is decreased below p2 the uniform vegetation solution loses stability to non-uniform perturbations in a finite wavenumber (Turing like) instability (Cross & Hohenberg 1993). These perturbations grow to form large amplitude patterns. The following sequence of basic patterns has been found at decreasing precipitation values for plane topography (see panels A-C in Fig. 2): gaps, stripes and spots. The sequence of basic landscape states, uniform vegetation, gaps, stripes, spots and bare soil, as the precipitation parameter is decreased, has been found in earlier vegetation models (Lefever et al. 2000; von Hardenberg et al. 2001; Okayasu & Aizawa 2001; Rietkerk et al. 2002; Gilad et al. 2004). Any pair of consecutive landscape states along the rainfall (precipitation) gradient has a range of bistability (coexistence of two stable states): bistability of bare soil with spots, spots with stripes, stripes with gaps, and gaps with uniform vegetation (see online Fig. 10). Bistability of different landscape states has at least three important implications: (i) it implies hysteresis, which has been used to elucidate the irreversibility of desertification phenomena (von Hardenberg et al. 2001; Scheffer et al. 2001), (ii) it implies vulnerability to desertification and (iii) it increases landscape diversity as patterns involving spatial mixtures of two distinct landscape states become feasible. The basic landscape states persist on slopes with two major differences: stripes, which form labyrinthine patterns on a plane, reorient perpendicular to the slope direction to form parallel bands, and the patterns travel uphill (typical speeds for the parameters used in this paper are of the order of centimeters per year). Online Fig. 3 shows the development of bands travelling uphill from an unstable uniform vegetation state. Travelling bands on a slope have been found in earlier models as well (Thi´ery et al. 1995; Lefever & Lejeune 1997; Dunkerley 1997; Klausmeier 1999; von Hardenberg et al. 2001; Okayasu & Aizawa 2001; Rietkerk et al. 2002; Gilad et al. 2004). Another difference is the coexistence of multiple band solutions with different wavenumbers in wide precipitation ranges (Yizhaq
– 15 – et al. 2005). The landscape states on plane and slope topographies predicted by this and earlier models are consistent with field observations (Valentin et al. 1999; Rietkerk et al. 2004; Lejeune et al. 2004). The sequence of landscape states described above represents the states that appear along the rainfall gradient as p is proportional to P . We could alternatively use Eq. (9) to express the same sequence of landscape states, but in reverse order, as a function of the grazing (mortality) rate M = (P Λ/N )p−1 , as shown in Fig. 2E. The bare state instability threshold, p = P Λ/M N = 1, implies that only four parameters affect the bare-state stability: the precipitation rate P , the mortality or grazing rate, M , the evaporation rate, N , and the biomass growth rate per unit amount of soil water, Λ. The effects of the four parameters on the bare-state stability threshold are shown in online Fig. 4. Noteworthy is the hyperbolic relationship between P and Λ (online Fig. 4c). The bare-state stability threshold designates the point at which the system can spontaneously recover from desertification. A region that has gone through desertification due to a precipitation drop can spontaneously recover by seedling with species that have Λ values sufficiently high to cross the stability threshold. This practice, however, becomes increasingly hard to implement as the system becomes dryer for the required up shift in Λ is getting larger as the rectangles in online Fig. 4c illustrate.
5.
Habitat creation by ecosystem engineers
The model equations (5) are now used to study micro-habitat creation by plant ecosystem engineers. We define a ”micro-habitat” of a given species as a small area in the physical domain, at the scale of a single patch of the ecosystem engineer, that is mapped into the fundamental niche of that species in niche space (Hutchinson 1957). The micro-habitats
– 16 – created by an ecosystem engineer is characterized by soil-water density higher than the density that would exist in the absence of the ecosystem engineer. We reserve the term ”habitat” for a local area at the landscape (many patches) scale characterized by a given pattern of the ecosystem engineer and its associated micro-habitats. The definition of habitats generally include additional resources besides soil water - light, nutrients, soil pH, etc., which are not explicitly modelled in equations (5). In drylands, however, the water resource can often be considered the limiting factor, justifying the use of the model equations for studying habitat creation. The micro-habitats to be considered here are results of trapping soil water by plant ecosystem engineers. In the following we use the model equations to study conditions for engineering effects and for ecosystem resilience, focusing on the parameters that control the two positive feedbacks, η and f .
5.1.
Engineering effects
The engineering capacity of a plant species is measured by its ability to capture surface water, either by increasing the water infiltration rate under its canopy, or by intercepting runoff water by the soil mounds it forms. Both ways can be simulated using the model but for simplicity we concentrate here on the first. Figure 5 shows spatial profiles of b, w and h for a single patch of the ecosystem engineer at decreasing values of η, representing species with different root extension properties, and for two extreme values of f . The value f = 0.1 models high infiltration rates under engineer’s patches and low infiltration rates in bare soil, which may result from a biological crust covering the bare soil (or from a litter layer formed by the engineer). The value f = 0.9 models high infiltration rates everywhere. This case may describe, for example,
– 17 – un-crusted sandy soil. Engineering effects resulting in soil water concentration appear only in the case of (i) low infiltration in bare soil, (ii) engineer species with rather limited root extension capabilities, η = 3.5, η = 2 (frames B and C in Fig. 5). The soil water density under an engineer’s patch in this case exceeds the soil-water density level of bare soil (shown by the dotted lines), thus creating opportunities for species that require this extra amount of soil water to colonize the water-enriched patch. We conclude that ecosystem engineering, resulting in water enriched patches or micro-habitats, is obtained when the infiltration feedback is strong (small f ) and the uptake feedback is weak (small η).
5.2.
Resilience
While a weak uptake feedback enhances the engineering ability, it reduces the resilience of the ecosystem engineer (and all dependent species) to disturbances. Fig. 5F shows the response of an engineer species with the highest engineering ability to concentrate water (η = 2, Fig. 5C) to a disturbance that strongly reduces the infiltration contrast (f = 0.9). In the following we refer to crust removal, but other disturbances that reduce the infiltration contrast, such as erosion of bare soil, will have similar effects. The engineer, and consequently the micro-habitat it forms, disappear altogether for two reasons: (i) surface water infiltrate equally well everywhere and the plant patch is no longer effective in trapping water, (ii) the engineer’s roots are too short to collect water from the surrounding area. Resilient ecosystem engineers are obtained with uptake feedbacks of intermediate strength (η = 3.5) as Fig. 5E shows. Removal of the crust (by increasing f ) destroys the micro-habitats (soil-water density is smaller than the bare-soil’s value) but the
– 18 – engineer persists. Once the crust recovers the ecosystem engineer resumes its capability to concentrate water and the micro-habitats recover as well. It is also of interest to comment that when the uptake feedback is too strong, the engineer persists but it no longer functions as an ecosystem engineer (Fig. 5A,D). Fig. 6 provides another view of the resilience of the engineering effect. Shown in this figure are two graphs of the maximum soil water density under an engineer patch as a function of the coverage of the surrounding crust, which is parametrized by f . One graph (solid line) describes a resilient engineering effect (η = 3.5) while the other graph (dashed line) describes a non-resilient engineering effect (η = 2). Also shown is the soil water level in bare soil (dotted line). The resilient engineer survives highly damaged crusts but loses the engineering ability to concentrate water at some f = fe (the crossing point of the solid line and the dotted line). The non-resilient engineer has better engineering abilities in the case of weakly damaged crusts (dashed line above solid line) but does not survive crust damages beyond f = fc . Resilience to climatic fluctuations, such as droughts, is achieved by a wide coexistence range of stable spots and stable bare-soil (see online Fig. 10A and the discussion in section 7.1). The larger η (or uptake feedback) the wider this coexistence range becomes. We can conclude that resilient ecosystem engineers call for a compromise in the strength of the uptake feedback; a weak feedback (small η) favors engineering, as the engineer consumes less water, while a strong feedback (large η) favors resilience both to disturbances such as crust removal and to climatic fluctuations such as drought.
– 19 – 6.
Habitat diversity
At the scale of a single patch of the ecosystem engineer we found two types of soil-water distributions, representing different micro-habitats. Online Fig. 7 shows possible infiltration conditions giving rise to the two distributions. For moderately low infiltration rates in bare soil (f = 0.1), the maximum of the soil water density coincides with the maximum of the engineer’s biomass (online Fig. 7A) whereas for very low infiltration rates (f = 0.01) the soil water density attains maximal values at the edge of the patch (online Fig. 7B). The two types of micro-habitats provide different niches with respect to water availability and shading. Species which are sensitive to the absence of sunlight are expected to prefer the micro-habitat of online Fig. 7B whereas species which are not sensitive are expected to be found in both micro-habitats. Habitat diversity on a landscape scale is determined by the diversity of spatial patterns the ecosystem engineers form. A high diversity of pattern solutions can be obtained in parameter regimes giving rise to irregular solutions. One possible mechanism leading to irregular solutions is spatial chaos. So far, however, we have not identified chaotic solutions and therefore we will not address this possibility here. Irregular patterns can also result from coexistence of stable states (e.g. bistability), where spatial mixtures of the coexisting states form stationary or long lived patterns. We demonstrate two aspects of these patterns; the first pertains to the dominant role that arbitrary initial conditions have in shaping the asymptotic patterns, and the second to the irregular soil-water distributions that can result from these patterns. Fig. 8 shows the time evolution of two identical initial conditions (leftmost frames), one in a parameter range where spots are the only stable state (upper frames) and the other in a coexistence range of spots and bare soil (lower frames). In the former case the system evolves towards a spot pattern and the initial pattern has little effect. In the latter case the initial pattern has a strong imprint on the asymptotic pattern;
– 20 – the system becomes sensitive to random factors and the asymptotic patterns will generally show high landscape diversity. Fig. 9A,B show biomass and soil-water distributions in a coexistence range of stripes and gaps. As the one-dimensional cut in Fig. 9C shows the soil-water distribution is rather irregular, creating a diversity of micro-habitats as compared with uniform or regular periodic patterns.
7.
Habitat response to environmental changes
Climatic or human induced environmental changes, such as alternation in rainfall regime or biomass harvesting, can affect the patterns formed by the ecosystem engineer and consequently the habitats they provide for other organisms. The most significant pattern changes are those involving transitions between different biomass pattern states. We focus here on two cases of pattern transitions induced by gradual precipitation decreases or disturbances to biomass patches. The first case is a ”catastrophic shift” (Scheffer et al. 2001; Scheffer & Carpenter 2003; Rietkerk et al. 2004) that involves an abrupt drop in the soil water concentration under the biomass patch. The second case, which we term a ”resource-gain shift”, involves soil-water gain as a result of a pattern transition. The two cases are discussed and illustrated below.
7.1.
Catastrophic shifts
Consider a coexistence range of stable spots and stable bare-soil (online Fig. 10A – coexistence of the stable N and B branches over the precipitation range p0 < p < pc ). The soil-water densities associated with these two states are shown in online Fig. 10B. For a soil initially covered with spots of biomass, a precipitation decrease below p0 will result in the decay of the spotted vegetation and the convergence to the bare soil state (see the arrows
– 21 – in online Figs. 10A,B). Along with this transition, the enriched soil moisture patches under the spots disappear as the soil-water density drops down to the bare soil level. A similar transition, from spot to bare-soil, can result from crust disturbances which increase the water infiltration rate I away from biomass patches. As implied by Fig. 6, an increase in the value of f beyond fc , for an engineer species characterized by short roots (η = 2), results in the decay of the ecosystem engineer and the drop of the soil water density to the bare soil level (see the arrow in Fig. 6).
7.2.
Resource-gain shifts
We illustrate this case by considering a transition from a banded biomass and water pattern induced by the ecosystem engineer to a spotted pattern on a uniform slope as precipitation decreases. Snapshots of the pattern transition and the associated soil-water distributions are shown in Fig. 11. Surprisingly, the transition involves the appearance of spot patches with higher soil-water densities, despite the lower precipitation value. This counter-intuitive result can be explained as follows. The spot pattern self-organizes to form an hexagonal pattern. As a result each spot ”sees” a bare area uphill which is twice as large as the bare area between successive bands, and therefore absorbs more runoff. The higher biomass density of spots (due to water uptake from all directions) also increases the infiltration rate at the patch. A transition from bands to spots involving soil-water gain can also be induced by a local disturbance (e.g. clear-cutting) at a given precipitation value corresponding to a coexistence range of stable bands and stable spots. The transition occurs by expansion of the spot pattern downhill into the area occupied by the banded pattern as illustrated in online Fig. 12.
– 22 – The resource gain shifts described above reflect emergent mechanisms of habitat dynamics resulting from the two collective modes associated with ecosystem engineering and spatial symmetry breaking. Overlooking the roles of these modes, when inferring the impact of environmental changes on habitat dynamics, may result in misleading conclusions.
8.
Discussion
We presented here a mathematical model for the dynamics of dryland vegetation that allows up-scaling processes occuring at the single plant-patch level to system behaviors at the landscape level, and thereby exploring emergent mechanisms of habitat creation, diversity and change. The model captures the biomass-water feedback due to water uptake by plants’ roots (unlike earlier models of this kind), in addition to the two other biomass-water feedbacks, increased infiltration at vegetation patches and reduced evaporation by shading. These feedbacks are tightly related to collective inter and intra-specific species dynamics. In the following we discuss these relations and the implications they bear on ecosystem processes, stability and diversity, as predicted and elucidated by the model. We conclude by discussing the limitations of the model and by delineating further directions of development.
8.1.
Biomass and resource patterns
The model predicts the possible development of vegetation patterns at the landscape scale due to intra-specific competition over the limited water resource. The pattern formation mechanism is the positive feedback between biomass and water that acts at the single patch scale. Vegetation patterns, which appear at the landscape scale, reflect collective individual dynamics involving long spatial correlations of biomass and soil-water. Vegetation patterns have been produced by earlier models that captured the biomass-water
– 23 – feedback. Most of these models, and the model introduced here, predict the same sequence of patterns along aridity gradients; uniform vegetation, gaps, stripes, spots and bare soil, as Fig. 2 shows. Moreover, most models predict coexistence ranges of different stable states; uniform vegetation with gaps, gaps with stripes, stripes with spots and spots with bare soil. This apparent agreement among most models is due to the universal nature of the instabilities that are responsible for the pattern formation phenomena. The difference between the present model and earlier ones lies in the details of the biomass and the water-resource spatial distributions. By capturing the uptake feedback in the model equations and controlling its strength relative to the infiltration feedback, a whole range of interspecific plant interactions, from negative (or competitive) to positive (or facilitative), can be studied. Competitive interactions occur when the uptake feedback dominates and the high water uptake depletes the amount of soil water under the vegetation patch and its surroundings (Fig. 5D). Facilitation or ecosystem engineering occurs when the infiltration feedback dominates, leading to soil-water accumulation under vegetation patches (Fig. 5C). Ecosystem engineering represents an interspecific mode of collective dynamics whereby one species, the ecosystem engineer, strongly affects the dynamics of other species by creating water-rich micro-habitats. Both modes of collective species dynamics, the intraspecific pattern mode and the interspecific engineering mode, are realized in vegetation patterns of ecosystem engineers. The coupling between the two modes offers a mechanism for species-richness change that has not been explored yet. Fig. 11 illustrates an example of this mechanism by showing soil-water enrichment (resource-gain shift) resulting from banded patterns transforming into spotted patterns on a slope in response to a precipitation down shift. There is another aspect to the landscape modulation that ecosystem engineers induce besides resource modulation and habitat creation. The basic patterns, spots, stripes and
– 24 – gaps, represent different forms of landscape fragmentation, which bear on population, community and ecosystem level processes (Fahrig 2003). Spot patterns (Fig. 2A) create open landscapes through which organisms and materials can move or flow with limited hindrance. Gap patterns (Fig. 2C), on the other hand, form closed landscapes through which movements of organisms and materials are highly restricted. The effect of landscape fragmentation on the dynamics of populations, communities and ecosystems is of great concern to ecologists (Debinski & Holt 2000). Most studies, however, refer to fragmentation induced by human factors (Harrison & Bruna 1999). Our model studies suggest that natural processes such as biomass pattern formation are also important agents in landscape fragmentation.
8.2.
Engineering strength and resilience
Ecosystem engineering in the present context refers to the ability of plant species to trap and concentrate the water resource under biomass patches. It can be quantified by the difference between the maximal value of the soil water density under an engineer patch and the soil water density at bare soil, far from the patch (see online Fig. 10B) 3 . Strong engineering is obtained when the infiltration feedback dominates. In this case the infiltration contrast between bare soil and vegetated soil is high (f ¿ 1) and the capacity of the plant to extend its roots as it grows is weak (η ∼ O(1)) (see Fig. 5C). In our model, resilience refers to the ranges of environmental conditions where the ecosystem engineer survives and reproduces, that is, to the size of its fundamental niche (Hutchinson 1957). Fixing all environmental conditions (precipitation, slope, etc.) except 3
When this difference is negative, i.e. in water deprived patches, the engineering can be
regarded as negative (Jones et al. 1997)
– 25 – the infiltration contrast, the resilience of the system can be quantified by the range of f values within the fundamental niche. A significant prediction of the model is a tradeoff between engineering strength and resilience, as Fig. 6 shows. An engineer with high strength (η = 2) has low resilience to disturbances that reduce the infiltration contrast (increase f ), e.g. crust removal. As f increases past a critical value fc the engineer patch disappears altogether, leaving behind a bare soil with no water enriched micro-habitats and no ability for the system to recover (arrow pointing downwards). In contrast, a plant with lower engineering strength (η = 3.5) survives severe reduction of the infiltration contrast (see Fig. 5E which corresponds to f = 0.9). The engineering effect no longer exists for f > fe , but once the disturbances disappear and the infiltration contrast builds up again, positive engineering resumes. This tradeoff is significant for understanding the stability and functioning of water limited ecosystems on micro (patch) and macro (watershed) spatial scales (Shachak et al. 1998). The soil-moisture accumulation at an engineer patch accelerates litter decomposition and nutrient production, and culminates in the formation of ”fertility islands” (Charley & West 1975). Watershed-scale disturbances that are incompatible with the resilience of dominant engineer species can destroy the engineer patches and the fertility islands associated with them, thereby damaging the stability and functioning of the ecosystem.
8.3.
Self-organized heterogeneity and species diversity
Spatial heterogeneity, along with species differentiation in niche space, are considered to be major factors in inducing species coexistence and diversity (Tilman 1990, 1994; Tilman & Kareiva 1997; Chesson et al. 2004; Silvertown 2004). Different spatial locations are then mapped into different fundamental niches (Hutchinson 1957) and provide micro-habitats
– 26 – for different species. The two collective modes, whose overall effect is the formation of self-organized patterns of water enriched biomass patches, induce spatial heterogeneity even under conditions of homogeneous environments. This form of heterogeneity has not been studied so far in the context of species diversity. The heterogeneity induced by the two collective modes create micro-habitats characterized by a few niche variables including soil moisture and grazing stress4 . The model, and further extensions thereof as discussed below, provide means for studying the niche space probed by these micro-habitats and thus species diversity. Periodic patterns, such as hexagonal spot patterns, probe small volumes in niche space, as the same micro-habitats repeat themselves, and therefore are associated with low species diversity. Strictly periodic patterns are never expected to be observed in nature because of the long time scales needed for approaching these asymptotic states. According to the model, however, under environmental conditions where spot patterns constitute the only stable state of the system, any initial biomass distribution will evolve on a relatively short time scale into a space filling spot pattern as Fig. 8 (top) demonstrates. These patterns, even though not periodic, are still expected to give rise to rather low species diversity. Higher species diversity can be realized in more arid conditions where coexistence of stable bare-soil and spots states occurs. Under these conditions engineer patterns can be highly irregular, and therefore can probe larger volumes in niche space. Fig. 8 (bottom) shows the time evolution of the same initial state as in the top part. The initial state remains ”frozen” apart of patch size adjustments because of the stability of the bare-soil 4
Assuming engineer’s biomass patches can reduce grazing of understorey herbaceous
species, soil-water distributions as in online Figs. 7A and 7B imply micro-habitats characterized by different grazing stresses.
– 27 – state and the depletion of soil water in the patch vicinity. This implies that highly irregular initial biomass distributions, dictated by random factors, will remain highly irregular also asymptotically.
8.4.
Further model development
We restricted our analysis to the rather artificial circumstances of homogeneous systems with respect to soil properties, topography, rainfall, evaporation, mortality, etc. We did so to highlight the roles of self-organizing patterns of biomass and soil water in breaking the spatial symmetry of the system and in creating habitats. Heterogeneous factors can easily be incorporated into the model by introducing spatially dependent parameters. The coupling of this heterogeneity to coexistence of stable states is expected to intensify the diversity of possible habitats discussed in Sections 6 and 8.3. The model in its present form takes into account the major feedbacks between vegetation and water but leaves out a few other feedbacks. The atmosphere affects vegetation through the precipitation and evaporation rate parameters, but the vegetation is assumed to have no feedback on the atmosphere. Organic nutrients effects are parametrized by the biomass growth rate, but litter decomposition (Moro et al. 1997) that feedbacks on nutrient concentrations is not considered. Finally, ground topography, parametrized by the function ζ(r), affects runoff and water concentration, but topography changes due to soil erosion by water flow are neglected. In drylands, where the vegetation is sparse and the limiting resource is water, the vegetation feedbacks on the atmosphere and on organic nutrients are often of secondary importance. Soil erosion processes, however, can play important roles, e.g. in desertification, and restrict the circumstances the model applies to. Another limitation of the model is the elimination of the soil depth dimension which
– 28 – restricts the variety of roots effects the model can capture. Thus, engineering by hydraulic lifts (Caldwell et al. 1998) is not captured by the model. Finally, we consider in this model local seed dispersal, ruling out dispersion by wind or animals. While extensions of the model to include the factors discussed above are interesting and significant, they may require considerable modifications of the model (especially the modeling of feedbacks with the atmosphere, and soil-erosion). A more direct extension of the model, that is already underway, is the inclusion of additional biomass variables representing herbaceous species which benefit from the habitats created by the ecosystem engineer. Each additional species is modeled by a biomass equation similar to that of the engineer but differing considerably in the values of various species parameters such as growth rate, mortality, maximum standing biomass, etc. In this context, the introduction of environmental heterogeneities and temporal fluctuations by means of space and time dependent model parameters is particularly significant in studying tradeoffs between species and species richness. Including additional biomass variables will also allow studying systems with competing ecosystem engineers, and the effects of this competition on habitat creation along environmental gradients.
Acknowledgements This research was supported by the James S. McDonnell Foundation (grant No. 220020056) and by the Israel Science Foundation (grant No. 780/01).
– 29 – Online Appendices
A.
Derivation of the surface water equation
In order to derive a dynamical equation for the surface water depth above the ground, d(x, y, t), we will use the continuity and momentum equations in the shallow water approximation (Weiyan 1992). Neglecting vertical accelerations and internal friction and in the case of a shallow fluid layer (as it is the case for surface runoff), the shallow-water momentum equations are written on the horizontal plane (x, y) and as a function of time t as Du 1 = −g∇ (ζ + d) + F , Dt ρ
(A1)
where u = (u, v) is the two-dimensional, depth-averaged fluid velocity [m/s], function of x, y and t, d(x, y, t) is the depth of the surface water layer, ζ(x, y) is the height of the soil surface, g is the acceleration of gravity [m s−2 ], ρ is the (constant) fluid density [kg/m3 ] and F represents bottom friction [kg/m2 s−2 ]. The shallow-water continuity equation is written as ·
¸ ∂d ∂ ∂ ρ + (ud) + (vd) = P − Iρd , ∂t ∂x ∂y
(A2)
where P is the average precipitation measured in units of [kg/m2 s−1 ] and I is infiltration rate measured in units of [s−1 ]. Since ρ is constant, we scale it out using h, and define the new quantity H = ρd, which has units of [kg/m2 ] (Note that for water [kg/m2 ]=[mm]). We can thus write ∂H ∂ ∂ + (Hu) + (Hv) = P − IH . ∂t ∂x ∂y
(A3)
We can simplify the above equations by noticing that on the time scales of interest here the fluid velocity adjusts rapidly to changes in inputs and outputs, and fluid accelerations
– 30 – (Du/Dt = 0) can be neglected. If we further assume a linear (Rayleigh) bottom friction term of the form F = −ku, where k is a constant coefficient, from the momentum equation we obtain µ
¶ ∂Z ∂H u = −κ + ∂x ∂x ¶ µ ∂Z ∂H v = −κ + ∂y ∂y
(A4)
where we have defined κ = g/k and we have introduced the rescaled soil-height variable Z = ρζ, in analogy to what has been done for the water depth. The final equation for H is obtained by substituting Eqs. (A4) into Eq. (A3) and collecting terms: ∂H κ = P − IH + ∇2 H 2 + κ∇H · ∇Z + κH∇2 Z ∂t 2
(A5)
Different forms for the bottom friction term F lead to different forms of the dynamical equation for H. For example, bottom friction of the form F = −ku/dp , where p is an integer, would lead to the following equation for H: ¡ ¢ ∂H κ = P − IH + ∇2 H p+2 + κ∇ H p+1 · ∇Z + κH p+1 ∇2 Z . ∂t (p + 2)
(A6)
Although quantitatively different, these alternate forms for the H equation do not bring in any qualitatively new behavior.
B.
Linear stability analysis
Linear stability analysis of the model homogeneous stationary solutions for plane topography is performed by adding an infinitesimal spatial perturbation and linearizing around the solution in order to calculate the linear growth rate. We denote the homogeneous
– 31 – stationary solution by U0 = (b0 , w0 , h0 )T and define the perturbed solution as U(r, t) = U0 + δU(r, t) ,
(B1)
where U = (b, w, h)T and δU = (δb, δw, δh)T . The perturbation dependence on time and space is: δU(r, t) = a(t)eik·r + c.c ,
(B2)
where a(t) = [ab (t), aw (t), ah (t)]T . Substitution of the perturbed solution (B1) into the model equations Eq. (5) gives (time and space dependence of the perturbation is omitted for convenience): (δb)t = Gb |U0 +δU (b0 + δb) [1 − (b0 + δb)] − (b0 + δb) + δb ∇2 (b0 + δb) (δw)t = I|U0 +δU (h0 + δh) − ν [1 − ρ(b0 + δb)] (w0 + δw) − Gw |U0 +δU (w0 + δw) + δw ∇2 (δw) £ ¤ (δh)t = p − I|U0 +δU (h0 + δh) + δh ∇2 (h0 + δh)2 , (B3) where the infiltration term Eq. (6) reads I|U0 +δU =
b0 + qf + δb b0 + qf q (1 − f ) 2 = + 2 δb + O(δb ) . b0 + q + δb b0 + q (b0 + q)
(B4)
In order to evaluate the terms Gb |U0 +δU and Gw |U0 +δU we expand the kernels, as defined in Eq. (7), up to first order in δb: ¡ ¢ g (r, r0 ) = g 0 (r, r0 ) + g 1 (r, r0 ) δb (r) + O δb2 ¡ ¢ g (r0 , r) = g 0 (r0 , r) + g 1 (r0 , r) δb (r0 ) + O δb2 ,
(B5)
and 1 −|r−r0 |2 /2σ2 e g 0 (r, r0 ) = g 0 (r0 , r) = g (r, r0 )|b=b0 = ¯ 2π ¯ η ∂ 0 2 2 2 = [g (r, r0 )]¯¯ |r − r0 | e−|r−r | /2σ , g 1 (r, r0 ) = g 1 (r0 , r) = 3 ∂b 2πσ b=b0 where σ ≡ 1 + ηb0 and |r − r0 |2 = (x − x0 )2 + (y − y 0 )2 .
(B6)
– 32 – After expanding the kernels and substituting the perturbed solution, we get up to first order Z Gb |U0 +δU = ν
g (r, r0 ) w (r0 ) dr0 Z
¤ g 0 (r, r0 ) + g 1 (r, r0 ) δb (r) [w0 + δw (r0 )] dr0 (B7) Z Z Z 0 0 0 0 0 0 0 = νw0 g (r, r ) dr + ν g (r, r ) δw (r ) dr + νw0 δb (r) g 1 (r, r0 ) dr0 ,
≈ ν
£
where the integration is over the entire domain. Solving the integrals in Eq. (B7) results in the following expression for Gb |U0 +δU Gb |U0 +δU = νw0 σ 2 + νσ 2 e−k
2 σ 2 /2
δw (r) + 2νησw0 δb (r) .
(B8)
Applying the same procedure to Gw |U0 +δU we get Gw |U0 +δU = γb0 σ 2 + γσe−k
2 σ 2 /2
£
¡ ¢¤ 1 + ηb0 3 − σ 2 k 2 δb(r) .
(B9)
Substituting Eqs. (B4), (B8) and (B9) into Eq. (B3) and linearizing around U0 can now be performed in a straightforward manner. The equations for the unperturbed state U0 are 0 = νb0 (1 − b0 ) (1 + ηb0 )2 w0 − b0 b0 + qf − ν (1 − ρb0 ) w0 − γw0 b0 (1 + ηb0 )2 b0 + q b0 + qf 0 = p − αh0 , b0 + q 0 = αh0
(B10) (B11) (B12)
which are the equations for the homogeneous stationary solutions (b0 , w0 , h0 )T . At first order in the perturbation δU we get the governing equations for the linear spatio-temporal dynamics of the perturbation: ∂ [δU(r, t)] = J (k) δU(r, t) , ∂t
(B13)
where J (k) ∈ R3×3 . After eliminating the spatial part of the perturbation (i.e. eik·r ) and assuming exponential time dependence for the perturbation (i.e. aj (t) ∼ eλt ) we get the
– 33 – following eigenvalue problem: λ a(t) = J (k) a(t) ,
(B14)
where the entries of the Jacobian matrix J (k) are given by J11 = σνw0 [1 − 2b0 + ηb0 (3 − 4b0 )] − 1 − δb k 2 J12 = νb0 (1 − b0 )σ 2 e−k
2 σ 2 /2
J13 = 0 £ ¡ ¢¤ q(1 − f ) 2 2 + ρνw0 − γw0 σe−k σ /2 1 + ηb0 3 − σ 2 k 2 2 (b0 + q) = −ν(1 − ρb0 ) − γb0 σ 2 − δw k 2
J21 = αh0 J22
b0 + qf b0 + q q(1 − f ) = −αh0 (b0 + q)2 = 0
J23 = α J31 J32
J33 = −α
b0 + qf − 2δh h0 k 2 . b0 + q
(B15)
By solving this eigenvalue problem we obtain the dispersion relation λ(k) as shown in online Fig. 13.
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This manuscript was prepared with the AAS LATEX macros v5.2.
– 40 –
Table 1. Dimensional parameters and their values/range
Parameter
Value/Range
Units
K
1
kg/m2
Q
0.05
kg/m2
M
1.2
yr−1
A
40
yr−1
N
4
yr−1
E
3.5
(kg/m2 )−1
Λ
0.032
(kg/m2 )−1 yr−1
Γ
20
(kg/m2 )−1 yr−1
DB
6.25 × 10−4
m2 /yr
DW
6.25 × 10−2
m2 /yr
DH
0.05
m2 /yr (kg/m2 )−1
S0
0.125
m
Z
mm
P
[0, 1000]
kg/m2 yr−1
R
[0, 1]
-
f
[0, 1]
-
Note. — (Online) Table 1 contains a list of parameters of the model, their numerical values
– 41 – and their units. The parameters values were set using (Sternberg & Shoshany 2001; Hillel 1998; Rietkerk et al. 2002).
– 42 –
Table 2. Non-dimensional parameters and variables definitions
non-dimensional
definition
quantity
b
B/K
w
ΛW/N
h
ΛH/N
q
Q/K
ν
N/M
α
A/M
η
EK
γ
ΓK/M
p
ΛP/M N
δb
DB /M S02
δw
DW /M S02
δh
DH N/M ΛS02
ζ
ΛZ/N
ρ
R
t
MT
x
X/S0
– 43 –
Note. — (Online) Table 2 contains the definitions of the nondimensional quantities (parameters and variables) used for studying the model equations (Eqs. 5–8).
– 44 –
infiltration rate [yr-1]
A
Af 0
Q/K
0.2
0.4
0.6
0.8
1
B/K
Fig. 1.— (Online) The infiltration rate I = A(B + Qf )/(B + Q) as a function of biomass density B. When the biomass is diminishingly small (B ¿ Q) the infiltration rate approaches the value of Af . When the biomass is large (B À Q) the infiltration rate approaches A. The infiltration contrast between bare and vegetated soil is quantified by the parameter f , where 0 < f < 1; when f = 1 the contrast is zero and when f = 0 the contrast is maximal. Small f values can model biological crusts which significantly reduce the infiltration rates in bare soils. Disturbances involving crust removal can be modeled by relatively high f values. Parameter values used: Q = 0.1, K = 1 and A = 1.
– 45 – A)
biomass [kg/m2]
0.6
B)
C)
D) E
0.4
0.2 B
0 p1 pc
1
precipitation
p2
E)
biomass [kg/m2]
0.8 0.6 0.4
E
0.2 B
0 M2
mortality rate [yr-1]
Mc
M1
Fig. 2.— Bifurcation diagrams for homogeneous stationary solutions of the model equations (Eqs. 5–8) showing the biomass B vs. precipitation p (frame D) and mortality M (frame E) for plane topography. The solution branches B and E denote, respectively, the baresoil and uniform-vegetation solutions. Solid lines represent linearly stable solutions, dashed lines represent solutions which are unstable to homogeneous perturbations, and dotted line represents solutions which are stable to homogeneous perturbations but unstable to nonhomogeneous ones. Also shown are the basic vegetation patterns along the precipitation gradient, spots (frame A), stripes (frame B) and gaps (frame C), obtained by numerical integration of the model equations. Dark shades of gray represent high biomass. The same sequence of patterns but in a reverse order appears along the mortality axis. The parameters used are given in online Table 1 and correspond to a woody shrub.
– 46 –
Fig. 3.— (Online) Development of vegetation bands traveling uphill from an unstable uniform vegetation state, obtained by numerical integration of the model equations (Eqs. 5–8) at P = 600 mm/yr. The different frames A–D shows snapshots of this process at different times (t in years). Dark shades of gray represent high biomass. The bands are oriented perpendicular to the slope gradient and travel uphill with typical speed of a few centimeters per year. The parameters used are given in online Table 1, the domain size is 5 × 5 m2 and the slope angle is 15o .
– 47 –
2
12.5 A)
B) 10
1.5
7.5
1
stable
N
M
stable
5 unstable
unstable
0.5
2.5
0
0 0
50
100
150
200
250
0
0.05
0.1
0.15
Λ
P 250
20 D)
C) 200
15
150
unstable
stable N
P
∆P ∆Λ
100
∆P
50
10
5
unstable
∆Λ
stable 0
0 0
0.05
0.1
0.15
0
Λ
0.4
0.8
1.2
1.6
2
M
Fig. 4.— (Online) Neutral stability curves for the bare soil solution obtained from the linear stability result p = pc =
ΛP MN
= 1, by plotting M =
Λ P N
(frame A), N =
P Λ M
at P = 100
mm/yr (frame B), P = (M N )Λ−1 (frame C), and N = (ΛP )M −1 at P = 150 mm/yr (frame D). All other parameters are given in online Table 1. The bare-soil solution loses tability as any of these curves is traversed. The rectangles in frame C show two upshifts in growth rates that are required to cross the recovery threshold, pc , of a desertified system. The dryer the system, the larger the required upshift.
– 48 –
η = 12
η = 3.5
η=2
b w h
B)
b w h
C)
b w h
D)
b w h
E)
b w h
F)
b w h
f = 0.9
f = 0.1
A)
Fig. 5.— Spatial profiles of the variables b, w and h as affected by the parameters that control the main positive biomass-water feedbacks, f (infiltration feedback) and η (uptake feedback). The profiles are cross sections of two-dimensional simulations of the model equations (Eqs. 5–8). In all frames, the horizontal dotted lines denote the soil-water level at bare soil. Strong infiltration feedback and week uptake feedback (frame C) leads to high soil-water concentration reflecting strong engineering. Strong uptake feedback and weak infiltration feedback (frame D) results in soil-water depeletion and no engineering. While the species characterized by η = 2 is the best engineer under conditions of strong infiltration contrast (frame C), it leads to low system resilience; the engineer along with the microhabitat it forms completely disappear when the infiltration contrast is strongly reduced, e.g. by crust removal (frame F). A species with somewhat stronger uptake feedback (η = 3.5) still acts as an ecosystem engineer (frame B) and also survives disturbances that reduce the infiltration contrast (frame E), thereby retaining the system’s resilience. Parameter values are given in online Table 1 with P = 75 mm/yr. Frames A) and D) span a horizontal range of 14 m while all other frames span 3.5 m.
– 49 –
Max. soil water [kg/m2]
45
η=2 η=3.5
30
15 0
0.1
fc
0.2
0.3
fe 0.4
f
Fig. 6.— Maximal soil-water densities under ecosystem-engineer patches for two different engineer species, η = 2 (dotted curve) and η = 3.5 (solid curve), as functions of the infiltration contrast, quantified by f . At low values of f (e.g. in the presence of a soil crust) both species concentrate the soil-water resource under their patches beyond the soil-water level of bare soil (horizontal dashed line), thereby creating water enriched micro-habitats for other species (see Fig. 5). The engineer species with η = 2 outperforms the η = 3.5 species, but does not survive low infiltration contrasts (resulting e.g. from crust removal); as f exceeds a threshold value, fc = 0.15, the engineer’s patch dries out and the micro-habitat it forms is irreversibly destroyed. The η = 3.5 species, on the other hand, survives low infiltration contrasts and while its engineering capacity disappears above a second threshold fe = 0.38, the engineering capacity is regained as the infiltration contrast builds up (e.g. by crust recovery). Parameter values are given in online Table 1 with P = 75 mm/yr.
– 50 –
Fig. 7.— (Online) Two typical spatial distributions of soil water at the scale of a single patch, obtained with different infiltration rates in bare soil. The upper frames show twodimensional soil water distributions (dark shades of gray represent high soil-water densities), while the lower frames show the corresponding one-dimensional cross-sections along the patch center. In the range of intermediate to low infiltration rates (frame A, f = 0.1) the soil-water density is maximal at the center of the patch, whereas at very low infiltration rates (frame B, f = 0.01) the maximum is shifted to the edge of the patch. Similar soil water distributions can be obtained at different precipitation values. The soil-water distributions were obtained by numerical integration of the model equations (Eqs. 5–8) at P = 75 mm/yr, with domain size of 5 m2 . All other parameters are given in online Table 1.
– 51 –
Fig. 8.— Bistability as a mechanism for pattern diversity. Snapshots of the time evolution of the same initial state (leftmost frames) when a spot pattern is the only stable state (upper frames, P = 300 mm/yr), and when spots stably coexist with bare soil (lower frames, P = 75 mm/yr). In the former case the asymptotic state is independent of the initial state; any initial patch configuration will converge to a spot pattern as this is the only stable state of the system. In the latter case the initial patch configuration remains unchanged as time evolves (appart of patch size adjustments), implying high sensitivity to random factors that affect the initial state, and thus high pattern diversity. The images were obtained by numerical integration of the model equations (Eqs. 5–8). The domain size is 7.5 × 7.5 m2 , t in years and the parameters are given in online Table 1.
– 52 –
Fig. 9.— Biomass (A) and soil-water distributions (B,C) of an asymptotic pattern in a coexistence range of stripes and gaps. Frame C shows the soil-water profile along the transect denoted by the dashed line in B. The soil-water distribution is pretty irregular as is evident by the variable grey shades in frame B and by the profile in frame C. Such irregular distributions create a diversity of micro-habitats as compared with the cases of uniform or regular periodic patterns. The images were obtained by numerical integration of the model equations (Eqs. 5– 8) at P = 950 mm/yr. The domain size is 7.5 × 7.5 m2 and the parameters used are given in online Table 1.
– 53 –
biomass [kg/m2]
0.8
A)
N
0.6 0.4 E
0.2 B
0
p0
p1 pc precipitation
60
soil water [kg/m2]
B)
40 N E
20 B
0
p0
p1 pc precipitation
Fig. 10.— (Online) Solution branches for the biomass density of an ecosystem engineer and the soil-water density it induces along the low end of the precipitation axis. The branches B and E denote, respectively, bare-soil and uniform-vegetation solutions. The branch N describes a spot-pattern solution (see frame A in Fig. 2). In frame A it denotes the amplitude of the biomass density, while in frame B it denotes the soil-water density at the center of a biomass spot. The figure shows the coexistence range, p0 < p < pc , of stable spot-pattern and bare-soil solutions, and illustrates the occurrence of a catastrophic shift (see arrow) as the precipitation drops below p0 . The figure also shows the range of engineering where the value of the soil-water density under a vegetation spot (given by N ) exceeds the value at bare soil (given by B). The N branch was calculated by numerical integration of the model equations (Eqs. 5–8). The parameters used are given in online Table1. With these parameters p0 corresponds to 50 mm/yr and pc to 150 mm/yr.
– 54 –
Fig. 11.— Resource-gain shift induced by a transition from vegetation bands (frame A) to spots (frame D) on a slope, in response to a precipitation downshift from P = 225 mm/yr (A) to P = 200 mm/yr (frames B–D). The transition is shown by displaying snapshots of two-dimensional biomass patterns (frames A–D, with darker gray shades corresponding to higher biomass density) at successive times (t in years), obtained by solving numerically Eqs. 5–8. The lower frames show soil water profiles along the transects denoted by the dashed lines in the corresponding upper frames. The soil-water density under spots is higher than the density under bands despite the precipitation downshift. Domain size is 5 × 5 m2 , slope angle is 15o and all other parameters are given in online Table 1.
– 55 –
Fig. 12.— (Online) Resource-gain shift induced by a transition from vegetation bands (frame A) to spots (frame D) on a slope, in response to a clear-cutting disturbance. The transition is shown in frames A–D by displaying snapshots of biomass patterns (darker gray shades correspond to higher biomass density) at successive times (t in years). The patterns were obtained by solving numerically Eqs. (5)–(8) at a precipitation value where both band and spot patterns are stable. The initial cut of the uppermost band allows for more runoff to accumulate at the band section just below it (frame A). As a result this section grows faster, draws more water from its surrounding and induces vegetation decay at the nearby band sections. The whole process continues repeatedly until the whole pattern transforms into a spot pattern. Domain size is 5 × 5 m2 , P = 225 mm/yr, slope angle is 15o and all other parameters are given in online Table 1.
– 56 –
0.0025
pp2 -0.0025
-0.005 6.5
7
7.5 8 wavenumber k
8.5
9
Fig. 13.— (Online) Linear growth-rate curves of non-homogeneous spatial perturbations applied to the uniform vegetation solution of the model equations Eqs. (5)–(8) (denoted by E in Fig. 2), around the point p = p2 . When p > p2 all wavenumbers have negative growth rates and any perturbation decays. At p = p2 a critical wavenumber becomes marginal; perturbations having this wavenumber neither grow nor decay. When p < p2 there exists a band of wavenumbers with positive growth rates; the uniform solution is unstable and perturbations characterized by the critical wavenumber grow faster than all others.
