René Lefever

Nicolas Barbier

Pierre Couteron

Olivier Lejeune

Deeply gapped vegetation patterns: On crown/root allometry, criticality and desertification

Keywords: vegetation gapped patterns, competition, self-organization, diffusion instability in ecosystems, Swift-Hohenberg equation

The dynamics of vegetation is formulated in terms of the allometric and structural properties of plants. Within the framework of a general and yet parsimonious approach, we focus on the relationship between the morphology of individual plants and the spatial organization of vegetation populations. So far, in theoretical as well as in field studies, this relationship has received only scant attention. The results reported remedy to this shortcoming. They highlight the importance of the crown/root ratio and demonstrate that the allometric relationship between this ratio and plant development plays an essential part in all matters regarding ecosystems stability under conditions of limited soil (water) resources. This allometry determines the coordinates in parameter space of a critical point that controls the conditions in which the emergence of self-organized biomass distributions is possible. We have quantified this relationship in terms of parameters that are accessible by measurement of individual plant characteristics. It is further demonstrated that, close to criticality, the dynamics of plant populations is given by a variational Swift-Hohenberg equation. The evolution of vegetation in response to increasing aridity, the conditions of gapped pattern formation and the conditions under which desertification takes place are investigated more specifically. It is shown that desertification may occur either as a local desertification process that does not affect pattern morphology in the course of its unfolding or as a gap coarsening process after the emergence of a transitory, deeply gapped pattern regime. Our results amend the commonly held interpretation associating vegetation patterns with a Turing instability. They provide a more unified understanding of vegetation self-organization within the broad context of matter order-disorder transitions.

Introduction

Many regions exhibit landscapes whose vegetation cover consists of a regular distribution of densely vegetated areas and zones of bare soil (see e.g., [START_REF] Valentin | The significance of Soil, Water and Landscape Processes in Banded Vegetation Patterning[END_REF][START_REF] Tongway | Banded vegetation patterning in arid and semiarid environments[END_REF][START_REF] Couteron | Quantifying change in patterned semi-arid vegetation by Fourier analysis of digitized aerial photographs[END_REF][START_REF] Deblauwe | The global biogeography of semi-arid periodic vegetation patterns[END_REF]. These landscapes, of which Fig. 1a is an example, are mainly found in climates where the potential evapotranspiration substantially exceeds the mean annual precipitation. The natural formation of these large ecological organizations can be explained by this hydric stress combined with redistribution and conservation mechanisms that allocate more limiting resources, water in particular, to vegetated patches than to bare soil [START_REF] Schlesinger | Biological feedback in global desertification[END_REF][START_REF] Wilcox | Ecohydrology of a resource-conserving semiarid woodland: effects of scale and disturbance[END_REF][START_REF] Ludwig | Vegetation patches and runoff-erosion as interacting ecohydrological processes in semiarid landscapes[END_REF][START_REF] Breshears | The grassland-forest continuum: trends in ecosystem properties for woody plant mosaics[END_REF]. Several mathematical formulations of this explanation have been developed. In general, vegetation patterning is interpreted as a process of self-organization that takes shape via a nonequilibrium instability. This instability may either be a symmetry-breaking phenomenon in the strict sense, likely to take place even if the environment is isotropic [START_REF] Lefever | On the origin of tiger bush[END_REF]HilleRis-Lambers et al., 2001;[START_REF] Von Hardenberg | Diversity of vegetation patterns and desertification[END_REF][START_REF] Shnerb | Reactive glass and vegetation patterns[END_REF][START_REF] D'odorico | Patterns as indicators of productivity enhancement by facilitation and competition in dryland vegetation[END_REF][START_REF] Kéfi | Reading the signs -Spatial vegetation patterns, arid ecosystems and desertification[END_REF], or it may be an advection-induced transition that requires the pre-existence of some environmental (usually topographical) anisotropy [START_REF] Klausmeier | Regular and irregular patterns in semiarid vegetation[END_REF][START_REF] Okayasu | Systematic analysis of periodic vegetation patterns[END_REF][START_REF] Sherratt | An analysis of vegetation stripe formation in semi-arid landscapes[END_REF]; The rhizospheres of five shrubs were excavated. The normalized biovolume b of each shrub was computed as the ratio of its biovolume (i.e., crown area times total height) to that of the largest individual. Values ranged between 0.1 and 1. Roots penetrated down to ca. 40-cm depth but their spread could exceed the crown radius by one order of magnitude. The best fits of Φc were obtained for an exponential function with L 0 c = 1.27 and p = 1/3. Facilitation consisted of a drastic reduction of direct soil evaporation. Soil water sensors were used to parameterize Φ f like Φc by an exponential. Attempts to fit with other functions did not yield substantially better fits. A clear optimum was found for L 0 f = 0.81 [START_REF] Barbier | Self-organised vegetation patterning as fingerprint of climate and human impact on semiarid ecosystems[END_REF][START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF]. [START_REF] Ursino | The influence of soil properties on the formation of unstable vegetation patterns on hillsides of semiarid catchments[END_REF]. Recent campaigns of measurement in the field, notably in the sub-Sahelian territories of Niger, strongly support the hypothesis that the instability underlying the formation of gapped patterns (cf. Fig. 1a) is symmetry-breaking in the strict sense [START_REF] Barbier | Interactions spatiales et autoorganisation des végétations semi-arides[END_REF][START_REF] Barbier | Self-organised vegetation patterning as fingerprint of climate and human impact on semiarid ecosystems[END_REF][START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF].

As such, this instability can occur in the absence of slope-induced water redistribution, a mechanism that has long been thought to be primordial for vegetation pattern formation [START_REF] Tongway | Banded vegetation patterning in arid and semiarid environments[END_REF]. Gapped patterns are observed from Burkina Faso [START_REF] Couteron | Periodic spotted patterns in semi-arid vegetation explained by a propagator-inhibition model[END_REF] to Sudan [START_REF] Deblauwe | The global biogeography of semi-arid periodic vegetation patterns[END_REF]. Besides their periodicity, they display a strong contrast (see Fig. 1a) between the gaps, where the biomass is close to zero, and the vegetated matrix where the vegetation may reach values of 12t/ha in Southern Niger (Hiernaux and Gérard, 1999). So far, in theoretical studies as well as in field studies, the relationship between the elementary structure of individual plants and the collective organization of plant populations has received only scant attention. Our study is intended to clarify this relationship in light of the results obtained in recent campaigns of field measurement [START_REF] Barbier | Interactions spatiales et autoorganisation des végétations semi-arides[END_REF][START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF]. Using a similar approach to that adopted in our earlier works [START_REF] Lefever | On the origin of tiger bush[END_REF]Lejeune et al., 1999;[START_REF] Lefever | Generic modelling of vegetation patterns. A case study of tiger bush in sub-saharian sahel[END_REF][START_REF] Couteron | Periodic spotted patterns in semi-arid vegetation explained by a propagator-inhibition model[END_REF][START_REF] Lejeune | Vegetation spots and stripes: dissipative structures in arid landscapes[END_REF][START_REF] Tlidi | On vegetation clustering, localized bare soil spots and fairy circles[END_REF], we aimed to achieve a general and yet parsimonious theoretical formulation explaining how the structure of individual plants shapes vegetation dynamics and how, as a consequence, vegetation may either adapt to varying environments or conversely collapse and become extinct (desertification).

It is well known that plants adapt their structures, especially their root structures (rhizosphere), to combat water scarcity [START_REF] Schenk | Rooting depths, lateral root spreads and below-ground/above-ground allometries of plants in water-limited ecosystems[END_REF][START_REF] Barbier | Interactions spatiales et autoorganisation des végétations semi-arides[END_REF]. When confronted with drought, they strive to increase their water resources by spreading their roots over a greater territory. Fig. 1b shows that in a shallow soil context, this lateral spread may extend beyond the radius of the aerial structure (crown) by an order of magnitude. In terms of morphological plasticity, this is an extraordinary achievement. However, whereas the lateral expansion of the roots allows for greater water uptake, it significantly increases the degree of competition between neighboring plants. As a consequence, it modifies the balance between competition and the positive feedback called plant-plant facilitation, which favors vegetation development in drylands(e.g. [START_REF] Callaway | Positive interactions among plants[END_REF][START_REF] Callaway | Phenotypic plasticity and interactions among plants[END_REF]. Understanding how such interactions operate at the scale of plant crown (for facilitation) and rhizosphere (for competition) to produce biomass organizations of landscape size is the main biological question studied in sections 2 to 4. The results highlight the role of the crown/root ratio which gives the relative size of the above-ground and below-ground structures. Furthermore, our results demonstrate that the allometric parameter linking plant development to this ratio plays an essential part in all matters regarding stability at the community level. Notably, this allometric parameter determines the coordinates in parameter space of a critical point that controls the feedback conditions for which an organization in the biomass distribution may emerge.

We provide evidence for a fundamental link uniting criticality, vegetation patterning and plant morphology. We further demonstrate that close to their critical point, the dynamics of ecosystems is variational and can be described by a Swift-Hohenberg equation. Our results amend the current interpretation associating vegetation pattern formation with a Turing instability by providing a more unified understanding of vegetation self-organization within the broad context of matter order-disorder transitions. In section 5, we investigate more specifically the pathway of vegetation change in response to increasing aridity. We examine the conditions of gapped pattern formation and the conditions under which desertification takes place either as a local desertification process that does not affect patterns morphology in the course of its unfolding or as a gaps coarsening process that follows the emergence of a transitory deeply gapped pattern regime.

Vegetation interactions and crown-roots allometry

We consider a community within which, as shown in Fig. 1 and as it is often the case in drylands, a single species accounts for most of the biomass. Let L a be the crown radial spread of the community's largest individual, S = π L 2 a be a surface element centered on a point r and

∂ t b(r, t) = b(r, t) [1 -b(r, t)] M f -μ b(r, t)M c + δ Φ D (|r |) [b(r + r , t) -b(r, t) ] dr (1)
be the logistic equation governing the evolution of the biomass density b(r, t) supported by this element at time t, normalized by the biomass density of the surface element S centered on the largest individual. The first two terms of (1) describe the biomass gains and losses; the third term models seed production, dispersion and germination; the logistic factor in the first term, 1b(r, t) , expresses that b(r, t) ∈ [0, 1]. The time constants μ and δ are normalized with respect to the biomass doubling time and are thus dimensionless. They refer to individual (isolated) plants; more precisely, they represent the biomass loss/gain ratio and the ratio of biomass gains due to seeds and to growth. In the third term of (1), integration extends over the entire territory and space is normalized such that its unit be equal to L a . The kernel of the integral is the Gaussian function

Φ D (|r |) = σ π exp -σ |r | 2 , with σ = L a L d 2 ,
(2) weighting the seed flux decrease with distance according to the seed dispersion range L d . M f and M c are state functions that depend on the distribution of biomass around the surface element S centered on point r. Their purpose is to model the facilitative and competitive interactions existing between plants and the dependence of these interactions upon plant development. M f models interactions favoring vegetation development (facilitation) generated by the plant aerial structure. They involve the accumulation of nutrients in the neighborhood of the plants, the reciprocal sheltering of neighboring plants against climatic harshness, and the provision of shade, which improves the water budget in the soil (e.g. [START_REF] Callaway | Positive interactions among plants[END_REF]. The range L f over which facilitation operates depends on the crown size. M c models below-ground interactions involving rhizospheres. Plants confronted with drought develop large superficial root systems relative to their crown size in order to extract enough water from the topsoil [START_REF] Schenk | Rooting depths, lateral root spreads and below-ground/above-ground allometries of plants in water-limited ecosystems[END_REF]. The range L c of the competition generated by root systems depends on this extension [START_REF] Callaway | Competition and facilitation: a synthetic approach to interactions in plant communities[END_REF][START_REF] Callaway | Positive interactions among alpine plants increase with stress[END_REF][START_REF] Callaway | Phenotypic plasticity and interactions among plants[END_REF]. Clearly, M f and M c must be positive definite functions of b(r, t) taking values in the interval [1, ∞) and must become equal to 1 in the limit where plants are too sparsely distributed to interact:

lim b(r+r , t)→0 M ı = 1, (ı = f, c).
(3)

Hence, we assume that they are exponential functions whose argument is a mean field integral taken over the biomass distribution (ı = f, c):

M ı = exp χ ı Φ ı (|r |, L ı ) b(r + r , t) dr . (4)
Their kernels, given by

Φ ı (|r |, L ı ) = N -1 ı exp - |r | L ı , ( 5
)
describe how interactions vary with distance and plant development. To account for these dependencies, we set

L ı = L 0 ı b(r + r , t) p , ( 6
)
and define the norm N ι of (5) as

N ı = exp - |r| L 0 ı dr. ( 7
)
The exponent p and the constants L 0 f and L 0 c can be evaluated by field measurements of the crown and rhizosphere morphology [START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF]. The allometric relation ( 6) allows us to avoid the complication of having to introduce more variables in order to describe the age classes of the community. It expresses that fully grown (mature) plants that produce significant feedback effects are unlikely to be present on a surface element S, centered on r+r , if its biomass density b(r + r , t) tends to zero. The contribution of such elements in the integral argument of ( 4) is negligible if p is a finite (non-zero) positive number. A similar relation, with p = 2, has been used by [START_REF] Gilad | Ecosystems engineers: from pattern formation to habitat creation[END_REF], [START_REF] Meron | Localized structures in dryland vegetation: forms and function[END_REF] and [START_REF] Sheffer | Why do plants in resourcedeprived environments form rings?[END_REF]. These authors however adopt another normalization condition and introduce a finite lower bound below which the range of interactions cannot decrease. We shall not adopt this assumption, which in our view is unnecessary. The parameters χ f , χ c model the interaction's strength; they depend on conditions such as soil nature and moisture index (quantified by the ratio of annual precipitation to potential evapotranspiration). Let us also define the structural ratio

ε ≡ L 0 f L 0 c 2 . ( 8
)
Use of the square power is motivated by the fact that hereafter only the square of L 0 f /L 0 c appears in mathematical expressions. Together with the allometric parameter p, the structural ratio ε plays an important role in estimating the feedback strengths χ f , χ c and in predicting the wavelength of periodic patterns. Finally, with respect to the modeling of plant interactions, let us point out that the field data gathered so far bear out expressions (5-7). For the ecosystem shown in Fig. 1, which is our reference example, the structure and development of the crown and rhizosphere of Combretum micranthum nicely fit with the assumption that interactions decrease exponentially with distance. The values of the parameters are [START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF]:

p ≈ 1/3, L 0 f ≈ 0.81 L a , L 0 c ≈ 1.27 L a . ( 9
)
For the structural ratio estimate, this yields:

ε ≈ 0.41. ( 10
d b(t) d t =e χf b(t) (1+2 p) × b(t)(1 -b(t)) -μ b(t)e Λ b(t) (1+2 p) . ( 13
)
Its stationary states, representing uniform vegetation cover, are the bare soil state b 0 = 0 and the curve b s ∈ [0, 1] solution of

(1 -b s ) -μ exp -Λ b (1+2 p) s = 0. ( 14
)
The feedback difference

Λ ≡ χ f -χ c ( 15
)
measures the community cooperativity or anticooperativity according to whether it is positive or negative. If plants are too sparsely distributed to interact, or if facilitation and competition cancel each other out, Λ = 0. It is noteworthy that Eq. ( 13) is independent from the interaction ranges L f and L c . The normalization condition (7) has been chosen so that this mean field property can be recovered.

The stationary states of (13) are plotted in terms of μ for p = 0 in Fig. 3a, and for p = 1/3 in Fig. 3b. The behavior for p = 1/3 is representative of nonzero (positive) values of the allometric parameter.

It is noteworthy that whatever Λ and p, all curves b s cross the bifurcation point (b = 0, μ = 1) where the bare soil solution b 0 changes stability. At this point, if p = 0, the slope db s /dμ varies with Λ while it is independent from Λ if p = 1/3. In fact, for any p > 0, it is equal to -1, which is the slope of the (green) straight line b s = 1-μ representing communities in which interactions are negligible (Λ = 0). point whose coordinate b c lies above the abscissa axis b = 0. In the 3-dimensional (b, Λ, μ)-space, the coordinates of this critical point are:

b c = 2p 1 + 2p , (Λ -1 ) c = b 2p c , μ c = e bc 1 + 2 p . (16)
Remarkably, the sole knowledge of the allometric parameter p suffices to evaluate these coordinates.

For the field value p = 1/3, they are:

b c = 2 5 , (Λ -1 ) c = 2 5 2 3 ≈ 0.543, μ c = 3 5 exp(2/5) ≈ 0.895. ( 17
)
For p = 0, the slope of the curve b s at its intersection with b = 0 (bare soil) is negative when Λ < 1, infinite for Λ = 1 and positive for Λ > 1. In other words, when p = 0, hysteresis appears on the abscissa axis under conditions where the biomass density is zero, where, as mentioned, interactions are nonexistent and where edaphic factors should not influence dynamics. One can only conclude from this that setting p = 0 is a crude approximation that should be avoided. This is certainly so in the modeling of vegetation at low biomass density and/or undergoing a desertification process.

Stability diagrams

Let f p [b(t)] be the expression between brackets on the right hand side of Eq. ( 13). The function is a potential that completely determines the stability of the stationary states b 0 and b s with respect to uniform perturbations that, in Fourier space, correspond to the mode k = 0. These states are the extrema of (18). They are linearly (locally) stable or unstable according to whether they are a minimum or maximum. The locus of stationary states where the second derivative of F 0,p changes sign is by definition called the spinodal curve [START_REF] Korteweg | Sur les points de plissement[END_REF]. Its expression is given by Eq. (A.6) in appendix A.

F 0,p [b(t)] = -f p [b(t)] db(t) (18)
When, for given values of Λ -1 and μ, the potential F 0,p exhibits more than one minimum, the one that confers the smallest value to F 0,p is said to be stable (more precisely, globally stable). The others are called metastable. The locus of points where, for given Λ -1 and μ, two locally stable stationary states are in equilibrium, i.e., have equal values of F 0,p , is by definition called the connodal curve (also often called the binodal curve). In the absence of a general analytical expression for it, the connodal curve can easily be calculated numerically (cf. green curves in Fig. 4).

The explicit expressions of the potential functions F 0,p for p = 0 and p = 1/3 are given in appendix A.2 (cf. (A.7), (A.8)). Fig. 4 reports the stability diagrams based on these expressions. The (Λ -1 , b)-space consists of three domains where b s is globally stable (above the green connodal curve), metastable (between the red spinodal and green connodal curves) or unstable (in the spinodal domain situated under the red spinodal curve). Quantitatively, as well as qualitatively, Fig. 4a and Fig. 4b are very different. They further illustrate the shortcomings that arise when approximations completely erase the existence of a gradient of plant development: for p = 1/3, as for any p = 0, the spinodal curve crosses the abscissa at b = 0 and b = 1. In between, at the critical point ( 17), it passes through a maximum. In comparison, for p = 0, i.e., when the influence of plant development is neglected, the spinodal abruptly becomes a straight line joining the points (0,1) and (1,0). As a result, the shape and size of the unstable, metastable and stable domains undergo a drastic singular transformation; simultaneously, the critical point (cf. red circle) becomes a transcritical saddle-node point of bifurcation situated on the ordinate axis at Λ -1 = 1. Given these pathologies, we focus in the following sections on the case p = 0, and more specifically on the case p = 1/3 relevant for Fig. 1.

Influence of aridity on interactions and b s pathway of change in (Λ -1 , b)-space

All evidence indicates that climatic aridity due to water scarcity decreases the productivity of plant ecosystems. As most models assume, this decrease can be attributed, at least to some extent, to an increase of the loss/gain ratio μ. It is also known that water scarcity may give rise to pattern formation and ultimately, at unendurable levels of aridity, to vegetation death and desertification [START_REF] Lefever | On the origin of tiger bush[END_REF][START_REF] Hillerislambers | Vegetation pattern formation in semi-arid grazing systems[END_REF][START_REF] Von Hardenberg | Diversity of vegetation patterns and desertification[END_REF]. Before investigating the dynamics underlying these phenomena, it is useful to sort out vegetation kinetic responses to climatic variability and to examine the pathway of stationary curves b s in (Λ -1 , b)space. We therefore consider the combined effects of climate on the loss/gain ratio μ and on the feedback strengths χ f , χ c . Besides the fact that aridity increases μ, it also influences the synergies on which the functioning of ecosystems depends. It enhances or attenuates the interactions between neighboring plants. Shading and increased soil permeability that produce facilitation (embodied by χ f ) and competition for soil water resources (embodied by χ c ) have different, probably weaker, effects in wet regions compared to arid regions. Furthermore, given that χ f and χ c correlate with different structures (crown and rhizosphere), functions (e.g., shading and water uptake) and edaphic factors [START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF], it is likely that the variation of these parameters with aridity is different. This can result into significant variations in cooperativity Λ = χ fχ c , or equivalently of Λ -1 . Increasing values of Λ -1 thus mean that competition increases more rapidly than facilitation.

Writing the stationary state equation ( 14) as

Λ -1 = b (1+2 p) s ln μ 1 -b s , ( 19
)
the dotted curves calculated with ( 19) and plotted in Fig. 4 show how b s evolve in the (Λ -1 ,b)-stability diagram for given μ. Considering these curves, one sees how the transition from uniform vegetation cover, to patterned organization and to desert may take place. Assuming that μ increases monotonously with the deficit of precipitation against potential evapotranspiration (referred to as "aridity") while Λ -1 behaves in a less regular manner, theoretical predictions in regard to increasing levels of climatic aridity can be schematized as follows:

Wet climate: The loss/gain ratio is small and allows for maximum biomass growth (0 μ 1). If water supply is abundant, competition for water is negligible (χ c ≈ 0). Whether facilitation is large or small is a matter of secondary importance: the biomass distribution is uniform, its density is high (b s ≈ 1) and, most likely, corresponds to a system situated in the stable domain on a stationary state curve that remains in this domain, regardless of Λ -1 (cf. e.g., the curve labeled 0.8 in Fig. 4b). The potential F 0,1/3 has two extrema: the first one for b = b 0 is always a maximum and thus unstable, the second one is a minimum whose position evolves along the univalued stationary curve b s .

Dry climate:

The ratio of precipitation against potential evapotranspiration (humidity index) is sufficiently low to causes an increase of μ, which reaches values close to μ c , but is still less than one. Facilitation is now more important because shading by crowns and/or organic material accumulation, which favors water infiltration and conservation, becomes a crucial element that compensates, at least partially, for the increases of μ and of χ c . As a result, b s belongs to a curve passing through or close to the critical point (16) (cf. e.g., the curves labeled μ c or 0.95 in Fig. 4b). Fig. 5 reports the behavior of the potential F 0,1/3 for μ = 0.92, i.e., slightly above the critical value μ c . The ecosystem exhibits a hysteresis loop when 0.485 Λ -1 0.517. Accordingly, F 0,1/3 exhibits two minima. When Λ -1 ≈ 0.5097, the values of F 0,1/3 at these minima are equal (cf. the green circles joined by a horizontal, dotted tie line). In that case, the minima represent equilibrium states of quite different density. Facilitation stabilizes the highly vegetated state (cf. e.g., the curve for Λ -1 = 0.508). Increased competition, on the contrary, makes it metastable (cf. e.g., the curve for Λ -1 = 0.513) and ultimately suppresses it for 0.517 Λ -1 . The low density state is then the only (globally stable) uniform solution possible. The bare soil state b 0 is always unstable and, hence, cannot be reached. Semiarid to arid climate: Potential evapotranspiration is now very large compared to rainfall and μ is greater than one. The vegetation is confronted with rising levels of competition χ c . Facilitation is unable to compensate for this so that Λ -1 increases, which drives the community into the metastable and unstable spinodal domains following curves like those labeled 1 or 2 in Fig. 4b. Hysteresis becomes a more complex phenomenon since now it may also involve the bare soil state b 0 , which may be metastable or stable.

Variational dynamics in dry climate, i.e., for μ ≈ μ c but inferior to 1

Variational behaviors are exceptional in spatially extended non-equilibrium systems. Up to now, they have only been reported in fields distant from ecology like hydrodynamics [START_REF] Swift | Hydrodynamic fluctuations at the convective instability[END_REF][START_REF] Pomeau | Wavelength selection in cellular flows[END_REF], chemistry (see e.g., [START_REF] Hilali | Subharmonic and strong resonances through coupling with a zero mode[END_REF] or non-linear optics (see e.g., [START_REF] Tlidi | Kinetics of localized pattern formation in optical systems[END_REF]. It is remarkable that such behavior can be demonstrated in the modeling of vegetation communities, at least, when they function in the vicinity of the critical point ( 16). This happens under the conditions classified as dry climate in section 3.3. The demonstration is given in appendix B. For the sake of the discussion below, we outline here its main idea and outcome.

The demonstration stems from the observation that at the critical point, by definition, the following two conditions are satisfied:

ω 0 = 0 and ∂ω k ∂k k=0 = 0. ( 20
)
The first condition states that the rate of regression of uniform perturbations, given by ω 0 and corresponding in Fourier space to the zero-mode (k = 0), is equal to zero. It expresses that the critical point is a state of marginal stability. This is easily verified by setting k = 0 and by replacing b s and Λ with their critical values ( 16) in the eigenvalue equation (A.2) describing the time-evolution of Fourier modes. The second condition always holds for k = 0. Indeed, because the environment is isotropic, (A.2) can only depend on even powers of the modulus k, in fact on k 2 . Thus, given (20), if

(∂ 2 ω k /∂k 2 ) k=0 > 0,
then there exists a finite band of non-zero Fourier modes for which the uniform vegetation cover b c , corresponding to the critical point, is unstable. This inequality is interesting. Before we carry on with the investigation of dry climate environments, it merits some general comments. Its explicit expression is of the form:

ε = L f L c 2 < χ c χ f 1 1 + δ σ F (b) , ( 21
)
where F (b) is a positive definite function and

d ≡ δ/σ (22)
is the vegetation transport (dispersion) coefficient. Inequality (21) shows that vegetation patterning is a phenomenon fundamentally due to plant-plant interactions. Vegetation propagation may suppress it, but it does not create it: pattern formation can only take place if the influence of interactions is not overwhelmed by the efficiency of vegetation propagation. If seed production (δ) and/or the dispersion range (L D = L a /σ) increase while ε and χ c /χ f remain constant, then in the strong propagation limit, d → ∞, inequality (21) reduces to the condition ε < 0. Clearly, it is then an unphysical condition that can never be satisfied. In fact, the most favorable conditions for patterning correspond to the weak propagation limit d → 0. From a practical point of view, in regard to the system shown in Fig. 1, the field value ε ≈ 0.41 is a given, measured constant. As such, it sets a lower bound below which the interaction ratio χ c /χ f may not decrease for a symmetry breaking instability to be possible.

In brief, inequality (21) supports the interpretation that vegetation patterns displaying characteristic lengths much larger than the size of individual plants originate from plant-plant interactions that affect biomass development through short range activation (L f ) and long range inhibition (L c ) effects [START_REF] Lefever | On the origin of tiger bush[END_REF]Lejeune et al., 1999). If the efficiency of seed production and/or propagation increases, this can only lead to the suppression of such large-scale structures and the restoration of a uniform vegetation cover. Suppose now that the system is close to the critical point ( 16), that inequality ( 21) is satisfied and thus that there exists a finite band of unstable Fourier modes. Let k f be the fastest growing mode of this band. Owing to the proximity of the critical point, k f can be expanded in terms of a smallness parameter 0 < η 1 measuring the deviation from criticality. Furthermore, k f evolves on a (slow) time scale t * ∝ η 2 . Redefining the units of space and time by the transformation

r → r k f , t → t η 2 t * , ( 23
)
one can then expand Eq. ( 1) in terms of η and show that the excess biomass density β(r, t), defined by the change of variable

b(r, t) = b c [1 + η β(r, t)] , (24)
obeys to the fourth order partial differential equation (PDE) of Swift-Hohenberg form

∂ t β(r, t) = -M + K β(r, t) -β(r, t) 3 + Γ ∇ 2 + 1 2 ∇ 4 β(r, t). ( 25
)
It is well-known that this type of Swift-Hohenberg equation describes a variational dynamics. Eq. ( 25) thus predicts that over the course of time, the community will strive to reach a stationary state, whether it is uniform or not, conferring the smallest possible value compatible with the imposed climatic constraints to the potential function F [β(r, t)] given .9). F [β(r, t)] is therefore called the generator of the system evolution. It can be interpreted as a "spatio-temporal kinetic potential" that governs the global stability of the spatially extended nonequilibrium system considered; it is a generalization of the standard potential (18), which always exists for uniform systems and was used in section 3 to determine their global stability properties.

The dependence of the new loss/gain ratio M, cooperativity K and constant Γ upon the structural parameters p, ε and reproduction parameters δ, σ is given in appendix B. The essential property to keep in mind for the following is that the constant Γ, which as explained in section 4.2, plays both the role of a diffusion coefficient and of a line tension coefficient, is always negative.

An great advantage of the Swift-Hohenberg equation lies in the fact that it permits to obtain analytical expressions for the instability conditions of the density field that are easy to interpret. Notably, the dominant unstable mode k f can be evaluated in terms of the structural parameters (L a , p, ε), the interaction strengths (χ f , χ c ), the kinetic parameter (μ) and the transport coefficient (δ/σ) (see Eq. ( 36) in section 4.

{β sc , K c , M c } = {0, 0, 0}. ( 26
)
Let β s be a uniform stationary solution of (25). One easily finds that a Fourier mode k is stable or unstable depending upon whether its eigenvalue that for

ω k = K -3 β 2 s -Γ k 2 1 - k 2 2 , ( 27
β + ≥ β s ≥ β -, (28)
where

β ± = ± 1 3 (K - Γ 2 ), (29)
there exists a band of unstable modes if

K ≥ Γ/2. ( 30
)
In Fig. 6a, the domain where uniform vegetation covers are unstable (cf. ( 28)) is the part of the stationary state curves β s found between the green dots. One notes that such a domain already exists for the critical stationary state curve, i.e., the curve for K = 0, passing through the critical point. For K = 5.4 (black curve), one notes that the instability occurs on part of the upper and lower branches of the hysteresis loop, which are stable with respect to uniform perturbations. The behavior of the band of unstable modes when β s varies is represented in Fig. 6b. The lower and upper cut-offs of this band are, respectively, k = k -and k u = k + , where

k ± = 1 ± 1 - 2 Γ (K -3 β 2 s ). ( 31
)
Remembering that Γ is negative, one notes that condition (30) may even hold for some negative values of the (excess) cooperativity K. Thus, some modes may already be unstable before the cooperativity Λ is larger than its critical value ( 16). In that case, the uniform states that are unstable belong to stationary state curves situated outside of the spinodal domain (i.e., above the red spinodal curve in Fig. 4b).

In the following, we shall denote by k * the non-zero fastest mode k f when it satisfies the marginality conditions: i.e., we shall set k f = k * when

ω kf = 0 and ∂ω k ∂k k=kf = 0. ( 32
)
For the climatic conditions studied in this section, this happens when β s = β ± (cf. ( 29) and the green curve in Fig. 6b). The red curve in Fig. 6b corresponds to the saddle-node points of the hysteresis loop that exists when K > 0, or to the critical point if K = K sc = 0 and β s = β sc = 0. The black curve in Fig. 6b is typical of systems undergoing an instability that could lead to pattern formation. Fig. 7 shows examples of patterns predicted for fixed K = 0 and Γ = -0.8849. The snapshots are obtained by integrating (34) numerically with zero flux (Neumann) boundary conditions and the average initial densities b(r, 0) indicated. The same noise (less than 1% of β(r, 0) and zero mean value) has been added to all initial conditions. As β s decreases (or equivalently, as the loss/gain ratio increases), the sequence of morphologies displayed by the patterns obtained is similar to that found by previous studies (cf. e.g., [START_REF] Lefever | On the origin of tiger bush[END_REF]; von [START_REF] Von Hardenberg | Diversity of vegetation patterns and desertification[END_REF]): high, intermediate or low initial average biomass densities respectively evolve into a pattern formed of holes distributed throughout a uniform vegetation cover (gapped pattern), into more or less randomly oriented bands (labyrinthine tiger bush), or into a regular distribution of vegetation patches on bare soil (spotted patterns). We shall report on the situations arising when K > 0 elsewhere since this discussion would take us too far away from gapped patterns and desertification, which are our main subjects of interest here.

Diffusional nature of the instability

Eqs (25,A.9) demonstrate that the instability underlying vegetation patterns is a diffusional instability: it is caused by the negative "diffusion coefficient" Γ multiplying the laplacian term in (25). This unusual sign for a diffusion coefficient confirms, as pointed out in the discussion of ( 21), that vegetation patterning is fundamentally caused by plant-to-plant interactions and that vegetation propagation is a process of subordinate importance in this respect.

Inspecting the explicit expression of Γ in terms of the system parameters (cf. (B.15), in which the constant Δ represents the biomass gains due to seeds), one notes that the limited influence of propagation does not take the simple, trivial form of dispersion terms to be negligible in comparison with those due to interactions. More subtly, dispersion manifests itself through the factor (1-Δ) 2 . Obviously, whatever the magnitude of Δ, this factor cannot change the sign of Γ and, in doing so, exert a control upon the origin of vegetation self-organization, i.e., upon the existence of a negative diffusion coefficient. However, in agreement with our comments at the beginning of this section one sees that, through conditions ( 30) and ( 21 21), is a further reason that the reappearance of a band of unstable modes for values Δ 1 becomes impossible. The situation Γ = 0 met when Δ = 1 is not generic and cannot be studied in the framework of the Swift-Hohenberg PDE approximation; its discussion is therefore beyond the scope of this paper. To conclude our examination of Γ properties, it is worth commenting that the finding that Γ cannot change sign is fortunate. If Γ could could change sign, the status of the Swift-Hohenberg PDE (25), as a consistent approximation of (1) would be compromised. Indeed in (25), Γ is not only a diffusion coefficient but also the line-tension coefficient that multiplies the bilaplacian term. When Γ is negative this higher order derivative term is stabilizing, prevents the unbounded growth of unstable modes and, in this manner, insures the establishment of a stable pattern. This property would be lost if Γ could be positive and this loss would jeopardize the expansion procedure that permits to obtain (25). Let us comment now on the similarities between the mechanism producing vegetation patterns and the better known mechanisms of phase separation in fluids. The following situation clarifies this aspect. Taking the critical point (26) as unperturbed state, one sets M = 0, K = 0 in Eqs (25,A.9) which become

∂ t β(r, t) = -β(r, t) 3 + Γ ∇ 2 + 1 2 ∇ 4 β(r, t)
(33) and (A.10). Small deviations u(r, t) from the critical density β sc = 0 obey the linear continuity equation

∂ t u(r, t) = Γ ∇ 2 + 1 2 ∇ 4 u(r, t), ( 34
)
which gives the short-time dynamics of (33). Since Γ is negative, the diffusion process (34) takes on an unusual complexion. To see what is happening, one inspects its eigenvalues

ω k = -Γ k 2 (1 -k 2 ). ( 35
)
One notes that ( 35) is formally identical to the eigenvalue equation associated with generalized diffusion equations encountered in classical first order phase transition theory (see e.g., [START_REF] Langer | An introduction to the kinetics of first-order phase transitions[END_REF]. It reflects the competition between clustering forces, which at the level of the factor (1-k 2 ) favor phase separation (spinodal decomposition) at small k, and the diffusion limitation expressed by the factor k 2 , which tends to slow this separation. The fastest mode k f = 1, as a result, dominates in the system evolution.

In fact, it grows exponentially and determines the wavelength of emerging patterns. However, as the snapshots in Fig. 7 show, in the case of vegetation, little coarsening takes place in the subsequent nonlinear regime: remarkably, in spite of the positive definite term -Γ ∇ 2 + 1 β(r, t) 2 , which represents in (A.10) the extra energy associated with departure from uniformity and which the variational property of the dynamics necessarily strives to minimize, the characteristic length of the stationary pattern finally obtained is in general still strongly dominated by the mode k f . In conclusion, on a short time scale dominated by diffusional instability, vegetation patterning evolves like a phase separation process in a binary mixture composed of bare soil and vegetation. On a long time scale, however, because the dynamics does not conserve the total number of particles (plants), the non-linear scalar term -β(r, t) 3 of (33) comes into play and prevents the complete separation into two "macroscopic phases", i.e., two uniform regions, a vegetated region and a bare soil region separated by a minimal interfacial region.

Comparison of predictions for dry climate with field data

In physical units, at the dominant order of expansion (B.8), the wavelength of the marginal mode k * reads:

λ = πL a √ η 8 5 1+4 p 1+6 p (1 -ε)(1 -Δ) 1/2 . ( 36
)
In agreement with intuition, (B.12) shows that Δ is always positive. Furthermore, biomass gains due to seeds are generally negligible in comparison to those due to growth. The approximation Δ ≈ 0 should often apply; it certainly does for the system of Fig. 1. Hence, the wavelength λ only involves the plant structural parameters L a , p, ε, which are directly accessible by field measurements, and the smallness parameter η, which, by definition for dry edge of these parameters, all critical properties can be predicted: we have already seen that the coordinates of the critical point ( 16) only depend on p; putting Λ = Λ c in (15) and χ 0 = χ c in (B.10) one obtains a very simple system of two equations,

Λ c = χ f -χ c , χ c = Λ c ε 1 -ε ,
whose solutions χ fc and χ cc are the feedback values for which the marginal stability conditions (B.6) are satisfied at the critical point. For the field values p ≈ 1/3 and ε ≈ 0.41, this yields:

χ fc ≈ 3.12, χ cc ≈ 1.28. ( 37
)
The behavior of the wavelength predicted by ( 36) for these values is shown in Fig. 8 as a function of η for three values of L a :

λ ≈ 3.66 L a √ η . ( 38
)
When η < 0.1 and L a > 1, the values obtained for λ are very large, in fact they are much larger than the 50 meters found in the field (represented by the red dashed line). Taking into account the histogram of Fig. 2, which indicates that the field values of L a are close to 3 m, we conclude that for the system of Fig. 1, the predictions of the model are incompatible with the assumption of a system functioning close to the critical point (16).

Dynamics in arid climate for μ ≥ 1

Patterns displaying large variations between a densely vegetated state and bare soil as depicted in Fig. 1 only happen in arid climate where μ ≥ 1. Typically, such conditions correspond to large deviations from criticality that are not covered by the PDE approximation used in section 4. To study arid systems, one must go back to the full integrodifferential version of the model, i.e., Eq. (1). It is not possible through a linear stability analysis of this equation to obtain explicit expressions like (29,31). But fortunately, in practice, as shown in section 5.1, this difficulty can in many cases easily be bypassed thanks to the a priori knowledge of the pattern wavelength. When this essential information, accessible by field measurements, is known, calculation of explicit symmetry breaking conditions is straightforward. General results in this respect are established in section 5.1. They are used in the following sections to study the case μ ≥ 1 of particular interest here.

Stability diagrams of non-zero modes

To avoid cumbersome expressions, we set p = 1/3 and suppose that the approximation δ = 0 is justified. The eigenvalue equation (A.2) then reads

ω k e χf b 5 3 s = -b s + [χ f Ω f (k) -χ c Ω c (k)] (1 -b s ) b 5 3 s . ( 39
)
The expressions of Ω f (k) and Ω c (k) are given in appendix A. For the following, one only needs to keep in mind that they are positive, which makes sense given the assumptions made in the modeling of the feedback functions M f and M c . Inspection of (39) shows that b s is always stable in the absence of facilitation. In order to determine instability conditions with respect to symmetry breaking fluctuations, one proceeds as in section 3.2: in the state space (Λ -1 , b), one determines the spinodal domain in which the mode

k * ≈ 2 π L a λ ( 40
)
is unstable. In writing (40) one identifies λ (expressed in meters) with the wavelength of the fastest mode k f satisfying the marginality conditions (32) and one supposes that this wavelength is given, approximately, by the wavelength measured in the field. This is often a good assumption. Replacing (40) in (32) yields a system of two equations, the solution of which χ

* f = - Ω c (k * ) b 2/3 s (1 -b s ) Ω(k * ) χ * c = Ω f (k * ) b 2/3 s (1 -b s ) Ω(k * )
, where ≡ ∂ ∂k (41)

and

Ω(k * ) = Ω f (k * ) Ω c (k * ) -Ω f (k * ) Ω c (k * ), ( 42
)
gives the feedback levels χ * f and χ * c corresponding to marginal stability for the wavelength λ measured in the field. The expression of the spinodal curve of this wavelength in (Λ -1 , b)-space is then given by

(Λ -1 ) * = 1/(χ * f -χ * c ). ( 43
)
It is represented in Fig. 9a for different values of the crown size L a . The expression of the spinodal curve in (μ, b)-phase space is given by

μ * = (1 -b s ) exp b 5/3 s Λ * . ( 44
)
It is represented in Fig. 9b. The spinodal domains in which vegetation patterns spontaneously form (under (above)the green curves in Fig. 9a (Fig. 9b)) always exceeds the spinodal domain of the mode k = 0 (under or above the red curve); these domains rapidly increase with L a . The coordinates of the critical point for the symmetry breaking mode k are obtained by replacing the solution, b * c , of the equation ∂μ * /∂b = 0 in ( 43) and ( 44); μ * is the smallest value of μ for which k * becomes unstable. The critical coordinates ( 45) are reported in Table 1 for different values of the crown size. The case L a = 0 corresponds to the mean field idealization (plants are points, densities refer to points and the wavelength of the critical mode is infinite (cf. ( 40)). Increasing L a shifts b * c towards higher densities; simultaneously, the critical loss/gain ratio μ * c decreases, both χ * f and χ * c undergo a large increase, but without that the feedback difference Λ * c varies greatly. In Fig. 10, the values of Table 1 for L a = 0 and

* , b = b * c , Λ * c = Λ * (b * c ), μ * c = μ * (b * c ), ( 45
) La k * b * c μ * c χ * f (b * c ) χ * c (b * c ) Λ * c 1/Λ *

Deeply gapped patterns

In the case of deeply gapped patterns, the density switches periodically in space between a highly vegetated state and bare soil. Sufficient conditions for this to be possible are: (i) Bare soil must be locally stable so that it can be reached in the course of the ecosystem evolution; this implies that the inequality μ 1 holds.

(ii) Bare soil must be metastable with respect to some vegetated state b s that is stable globally and situated beyond μ = 1. Conditions (i-ii) are only met in arid climate allowing for the existence of a large hysteresis loop such as shown in Fig. 11. Furthermore, we show below that for b 0 to be metastable with respect to the vegetated state b s , the crown L a must be larger than some lower bound L a .

(iii) The upper states of the hysteresis must exhibit a symmetry breaking instability whose bifurcation point is at μ 1. Hence, μ = 1 is the smallest loss/gain ratio at which deep gapped patterns may appear when competi- tion χ c increases. For this to happen, the symmetry breaking instability must appear in its vicinity. 2) and χc as indicated. For χc = 6.141 (red curve), the symmetry breaking instability appears at μ = 1 (green circle) with λ ≈ 50 m. For χc = 6.15, the ecosystem develops a deeply gapped pattern. At higher levels of competition (red circles at χc = 6.20 and 6.26), the organization transforms into bare bands and spots inside a uniform cover, and into vegetated bands and spots on bare soil (see Fig. 13).

La (m) b * χ * f (b * ) χ * c (b * ) Λ * 1/
respect to the vegetated state b s corresponding to μ = 1 is more difficult to satisfy. For the theory presented, it constitutes a real test of its predictive value and capacity to quantitatively interpret the field data corresponding to the pattern shown in Fig. 1. There is indeed no free adjustable parameter left. The test thus consists of determining whether or not condition (ii) can be satisfied when one combines the results of Table 2 with the data of the histogram in Fig. 2. To investigate this question, one has to compare the value F 0 (b 0 ) of the potential function (A.8) for bare soil with its value F 0 (b * ) for the vegetated state b * and the parameters calculated in Table 2. One easily finds that F 0 (b 0 ) = 0, ∀μ and ∀L a .

In Fig. 12, on the other hand, the behavior of F 0 (b * ) as a function of L a is plotted for the three values of μ indicated. Let L a be the value of L a such that

F 0 (b * ) = 0. Accordingly, b * is stable and b 0 is metastable if L a > L a . ( 46
)
One notes that for μ = 1 and 1 ≤ L a ≤ 3.173, bare soil is stable and, thus, that the vegetated state b * at which the symmetry breaking instability sets in is metastable. On the contrary, for L a > 3.173, b * is stable and bare soil is metastable. Table 3 gives changes sign for values of μ corresponding to arid climate. One sees that a variation of μ over a large interval from 1 up to 3.5 has little effect on the value of L a , which remains approximately equal to 3.3 m. These results show that the allometric and structural parameters p, ε and L a not only control the characteristics of vegetation patterns such as the distance between vegetated bands or patches, but also, through inequality ( 46), they control the very existence of the pattern itself. The other morphological parameters p and ε being given, L a can be viewed as a bench mark dividing plant species into two categories according to whether or not the crown L a of their mature individuals is larger or smaller than L a . In the first case, the upper state b s of the hysteresis is more stable than bare soil. Therefore, the instability, when it appears, cannot trigger desertification but instead induces the formation of deeply gapped patterns. On the contrary, when L a is somewhat smaller than L a , desertification occurs abruptly, as soon as the stationary curve (19) reaches the symmetry-breaking spinodal (43). and3 is satisfying. We find that the field data of [START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF], which fix the maximal crown size of mature plants at L a ≈ 3.25 m (cf. Fig. 2), validate the prediction reached here that, in order to exhibit deep gapped patterns, the maximal L a for mature Combretum micranthum plants should exceed ≈ 3.17 m. A more complete investigation of the case L a < L a is planned. Fig. 13 shows the stationary patterns obtained by numerical simulation of Eq. ( 1) for μ = 1, L a = 3.5 and the green and red circles in Fig. 11 as initial conditions. For χ c = 6.15 (slightly inside the spinodal domain of the symmetry-breaking mode), as expected, the system evolves towards a deep gapped pattern: the vegetation cover exhibits a more or less regular distribution of bare soil spots. When competition increases, for χ c = 6.2, neighboring spots start to coalesce, giving rise to a labyrinthine morphology. Simultaneously, the average pattern density decreases while the maximal density of vegetated domains stays approximately constant, or slightly increases. At the saddle-node point (χ c = 6.26) a stable pattern still forms, indicating that this structured branch extends further, i.e., to competition levels for which uniform solutions b s do not exist: bare soil is then the only uniform solution possible. Under those conditions, development of a labyrinthine pattern of bands or spots distributed over bare soil (not shown) constitutes an adaptive behavior through which a significant average biomass density can still be maintained and extinction can be avoided. Fig. 14 shows stationary patterns obtained for L a = L a = 3.173 m, which is the value at which the exchange of stability and metastability between bare soil and b s takes place. For χ c = 4.230, conditions correspond to the neighborhood of the symmetrybreaking spinodal curve while for χ c = 4.261, they corresponds to the saddle-node point of the meanfield spinodal. When χ c = 0.43, though this value exceeds the hysteresis domain, a stable (subcritical) pattern still exists that exhibits essentially the same morphology as it does for χ c = 4.261, but with a lower average biomass.

μ La (m) b * χ * f (b * ) χ * c (b * ) Λ * 1/

Two mechanisms of desertification: local desertification or gap coarsening

Fig. 15 illustrates the process of desertification from which the ecosystem can no longer escape if χ c increases somewhat further. Taking the third pattern shown in Fig. 14 as the initial condition and setting χ c = 4.5, one sees a rapid decrease over time of the biomass density of the vegetated bands, as well as of the pattern's average biomass. One also notes that the evolution of vegetation towards extinction takes place more or less uniformly. It is a local process of desertification during which the biomass density decreases inside vegetated areas but without much effect on the morphological distribution of vegetated and bare soil areas.

In contrast, desertification may take place much more abruptly when the maximum crown size L a is smaller than the threshold value L a at which F 0 changes sign (The difference L a -L a needs to be slightly positive). Fig. 16 illustrates this second mechanism. Considering that the loss/gain ratio has the value μ = 1 as in Fig. 14 and 15, we suppose that the crown size L a is equal to 2.75 m, i.e., approximately 13% below the threshold value L a = 3.173. As a result, the highly vegetated states b s of the hysteresis are now metastable with respect to bare soil. The sequence of snapshots reported in the figure shows the evolution of an initial condition chosen very close (relative distance: ≈ 0.073%) to the bifurcation point at which the symmetrybreaking instability appears. The gapped pattern organization that appears on a short time scale is now a transitory regime. It is followed by a coarsening process that corresponds to the expansion of bare soil domains that coalesce and invade the entire territory. In the course of this evolution, the pattern progressively undergoes profound morphological modifications. In contrast to what happens in the case of Fig. 15 where vegetation bands tend to disappear by "local desertification", here the area occupied by vegetated bands in the territory progressively decreases but their biomass density stays approximately constant during most of the evolution (it may even increase at some spots); only when the biomass distribution is reduced to a small finite number of isolated spots does the maximum vegetation density finally decrease and rapidly tend to zero everywhere. The numerical integrations carried out suggest that coarse graining desertification starts as soon as the bifurcation point of the symmetry-breaking instability is crossed.

Discussion

To capture the self-organization mechanisms of vegetation under isotropic environmental conditions, two kinds of approach have been developed in the literature. They differ in that they focus either on the vegetation patch level and the processes of water transport by above-ground run off and belowground diffusion [START_REF] Hillerislambers | Vegetation pattern formation in semi-arid grazing systems[END_REF][START_REF] Von Hardenberg | Diversity of vegetation patterns and desertification[END_REF] or on the individual plant level and the properties differentiating above-and below-ground plant structures as mediators of facilitation and competition [START_REF] Lefever | On the origin of tiger bush[END_REF].

It is established [START_REF] Barbier | Interactions spatiales et autoorganisation des végétations semi-arides[END_REF][START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF] that the hypotheses made within the framework of approaches dealing with water redistribution mechanisms do not apply to the site shown in Fig. 1. This site is representative in terms of constitutive species, soil substratum, pattern characteristics and climatic conditions of the vegetation patterns of South West Niger and was therefore used as a reference system in this paper. Gapped patterns with large standing vegetation have also been studied in Burkina Faso [START_REF] Couteron | Periodic spotted patterns in semi-arid vegetation explained by a propagator-inhibition model[END_REF]. On the other hand, in the case of the second kind of approach which emphasizes the characteristics of plant structures, the original modeling, proposed more than ten years ago, needs an upgrade in order to account for the more recent field observations. The appropriate revisions made here are summarized as follows:

(i) In accordance with field results, facilitation and competition are now decoupled structurally and functionally in the modeling. These opposed forms of feedback operate independently and are mediated by plant's clearly separated above-ground and below-ground structures. Functionally, these feedback mechanisms contribute to opposed scale pans of the balance weighting biomass gains against biomass losses. Consistent with this fact, facilitation and competition have been remodeled through the feedback functions M f and M c , and have been associated in Eq. ( 1) with the growth and loss terms respectively. It is noteworthy that since the approach is set up in terms of the characteristics of individual plants and not of vegetation patches, the biomass density b(r, t) refers to the surface element S occupied by a mature plant. This introduces the structural length L a into the description, which is then shown to be a determining parameter controlling the stability of the vegetation cover. In this modeling procedure, the notion of feedback is spatialized; it strictly refers to interactions between plants. Processes through which a plant sustains its own biomass evolution, e.g., the development of its own crown or of its own rhizosphere, do not qualify as feedback effects: self-facilitation or selfcompetition effects do not make sense here and are not postulated. Hence, the denomination of the propagator-inhibitor model as it has been called (Lejeune et al., 1999;[START_REF] Couteron | Periodic spotted patterns in semi-arid vegetation explained by a propagator-inhibition model[END_REF]. These processes are now modeled separately. To describe propagation an integro-differential representation more appropriate and general than the usual laplacian approximation has been introduced. This has allowed us to qualitatively as well as quantitatively assess the influence of vegetation propagation on pattern formation. With regard to the pattern of reference, we conclude that there is no reason to interpret seed dispersal as a cause of deeply gapped pattern organization. Furthermore, the results clearly establish that when the range of seed dispersion L D increases beyond some limit, pattern formation becomes impossible. Intuitively, one expects this prediction to be general, at least within the framework of a time continuous and space continuous approach like the one we adopted. It is however interesting to note that approaches describing biological invasions by processes that are discrete in time and discrete in space (cellular automata) predict the occurrence of fractal fronts when propagation is governed by long range power laws (see e.g., [START_REF] Cannas | Long range dispersal and spatial pattern formation in biological invasions[END_REF]. We shall not venture here into the dangerous territory of comparing continuous and discrete modeling approaches. This is a difficult, non-trivial mathematical problem that is not our subject and lies beyond our biological objectives. On the practical, empirical side, testable expressions have been obtained predicting the conditions of appearance of patterns and their characteristics in terms of a restricted set of parameters. This set is summarized as follows: (i) three structural parameters, L a , p, ε, can be determined with precision by measurements carried out on the individual plants found in the field; (ii) two kinetic parameters, μ, δ/σ (the effective dispersion coefficient), describing biomass growth and propagation can be estimated for some species on the basis of existing literature; (iii) two feedback parameters, χ f , χ c , describe the strength of interactions between plants and the influence of edaphic or climatic factors on these interactions. The instability and desertification conditions of χ f and χ c are predicted by the theory (cf. expressions ( 41) and Tables 123). Using the data established by direct field measurements of the pattern wavelength and plant structural parameters, these expressions have permitted an in depth analysis of the ecosystem shown in Fig. 1.

A striking outcome of this analysis concerns the structural requisite regarding crown size, L a > L a , which insures that stable, deeply gapped patterns (that is bare patches within a vegetation matrix of rather high biomass density) can form under the arid climatic conditions as defined in section 3.3 and studied in section 5.2. This result demonstrates remarkable agreement between theoretical predictions and field observations. Given the measured field values p = 1/3 and ε ≈ 0.41, the calculated value of L a for μ = 1 (the smallest value of μ for which deeply gapped patterns are possible) is L a = 3.173. On the other hand, independently from this theoretical calculation, field measurements yield L a ≈ 3.25 (cf. Fig. 2). Comparing these two values shows that the above inequality is satisfied for the ecosystem of Fig. 1. In other words, the theory predicts that deeply gapped patterns are indeed the organization that one should expect in the case of this system. One also sees in Table 3 that for μ > 1, the values of L a rapidly increase with μ and thus become rapidly much greater than the field value of L a estimated from Fig. 2. Considering that the reference pattern is a stationary, stable organization as multi-temporal remote sensing data indicate [START_REF] Barbier | Self-organised vegetation patterning as fingerprint of climate and human impact on semiarid ecosystems[END_REF], this suggests that in reality this ecosystem functions under loss/gain conditions corresponding to μ close (superior or equal) to one. Another original outcome of the modeling, which the simulations in Figs. et al., 2001;[START_REF] Rietkerk | Self-organized patchiness and catastrophic shifts in ecosystems[END_REF]). An even more catastrophic alternative is put into evidence here, namely, that the symmetry-breaking instability may trigger a transition that short-cuts the pattern organization stage. This causes the ecosystem to switch abruptly, without stable pattern formation, from a uniform, highly vegetated cover directly to the bare soil (desert) state. This mechanism is likely to take place much sooner, i.e., well before the saddle-node point of the hysteresis is reached. It is the fate which, as aridity increases, awaits those species whose crown size L a is significantly below L a .

On the conceptual side, the prediction that the organization of vegetation communities is ruled by a critical point is all the more remarkable since the coordinates ( 16) of this point in {b, Λ, μ}-space depend on the sole knowledge of the allometric parameter1 . This prediction clarifies the nature of the instability underlying vegetation patterns. It also remedies a shortcoming displayed by all earlier self-organization models of vegetation patterns. To conclude, let us comment on these two aspects. First, we note that ( 16) is a true ecological analogue of the physico-chemical conditions that govern the onset of immiscibility in regular solutions where unmixing is associated with the minimization of the energy rather than with the maximization of the entropy in the free energy. Self-organized landscape patterning obeys a mechanism that, phenomenologically speaking, has much in common with a phase transition in which vegetation and bare soil "phase separate" e.g., like some polymer mixtures [START_REF] Leibler | Theory of microphase separation in block copolymers[END_REF][START_REF] Bates | Polymer-Polymer phase behavior[END_REF] that display 3d-organizations whose morphology in 2-dimensions recalls those of 2d-vegetation patterns. Given the fact that close to the critical point vegetation patterning dynamics can be captured by a variational principle, generating a Swift-Hohenberg equation whose "diffusion coefficient" Γ is negative is the demonstration of this phase transition type of phenomenology. Seed dispersion is not a basic factor in this respect. Vegetation self-organizations originate from a diffusional instability due to interactions between plants. It would be appropriate to classify these organizations as such, rather than as the result of a Turing type of instability. The latter instability is a "chemical" phenomenon in the description of which interactions other than chemical reactions, e.g., physical forces like the Van der Waals intermolecular interactions, play no role and can in general be completely neglected. A similar non-Turing interpretation also applies to the non-periodic (so called glassy) vegetation patterns of frozen disordered plant spots found in some arid regions [START_REF] Shnerb | Reactive glass and vegetation patterns[END_REF][START_REF] Maruvka | Nonlocal competition and logistic growth: Patterns, defects, and fronts[END_REF]. From a theoretical modeling point of view, the introduction of the allometric parameter p not only crucially influences the order-disorder transitions involved in patterning and desertification, it also improves a weak point of previous self-organization models. In general, they either do not take into account the variation of interaction range with plant development, or they underestimate its influence [START_REF] Lefever | On the origin of tiger bush[END_REF][START_REF] Hillerislambers | Vegetation pattern formation in semi-arid grazing systems[END_REF][START_REF] Von Hardenberg | Diversity of vegetation patterns and desertification[END_REF][START_REF] Zeng | Multiple equilibrium states and the abrupt transitions in a dynamical system of soil water interacting with vegetation[END_REF][START_REF] Kéfi | Local facilitation, bistability and transitions in arid ecosystems[END_REF]. As Fig. 4 illustrates, setting p = 0 is a convenient approximation but quite unrealistic because it suggests that hysteresis, a phenomenon that requires large feedback effects, remains possible even when the biomass tends to zero, i.e., under conditions where plants are so sparsely distributed that they cannot interact. This problem is automatically avoided when the allometric constant p is different from zero (positive) and by the modeling of M f and M c , which insures that these functions tend to one (cf. ( 3)) when the biomass density tends to zero. Biologically speaking, our approach provides a reliable quantitative description of phytosocietal interactions for ground-limiting resources in terms of plant size and development at low and high biomass densities. It highlights the importance of the crown/root ratio, which reflects the plasticity of plant structures in the face of water stress and demonstrates that vegetation patterning is conceptually an order-disorder transition whose dynamics has properties similar to those of some phase separation processes.

B.2. Swift-Hohenberg equation of vegetation

To find a PDE approximation of Eq. ( 1), we expand this integro-differential logistic equation in terms of η as follows.

To begin with, let k * be a finite (non-zero) Fourier mode that satisfies the conditions of marginal stability Order η 2 :

ω k * =
β s = ± 1 3 K - Γ 2 , (B.13
)

k 1 = C 1 + C 2 β + C 3 κ, (B.14) with Γ = - 9 5 (1 -ε) 2 p εΛ 2 c 1 + 4 p 1 + 2 p 2 (1 -Δ) 2 (1 + 6 p) , C1 = b c Λ c Δ 1 -Δ - (1 + 2 p)(1 -ε 2 ) 2 p ε + Δ 10 (1 -ε)(1 + 4 p) σ (1 + 6 p) , C2 = -2 p + Δ 1 -Δ 1 + 2 p 2 -ε 1 -ε C3 = - Λ c ε (1 -ε)(1 -Δ)
.

(B.15)

Next, before expanding Eq. (1) in terms of η, we apply the transformation (23) to rescale its space and time units, we put k f = k * and take expansion (B.8) into account. We then replace in (1) the expressions (B.1-B.5), (B.9-B.15) and the explicit expressions of M f , M c and of the seed dispersion term. As a result, we find that up to terms of order η 2 , Eq. ( 1) is automatically satisfied. At the order η 3 ,we find that

  Fig. 1. (a) Sub-Sahelian gapped landscape dominated by the shrub species Combretum micranthum G. Don in South-West Niger. Shrub crown radius and bare spot distances are approximately 1.75 m and 50 m. (b)The rhizospheres of five shrubs were excavated. The normalized biovolume b of each shrub was computed as the ratio of its biovolume (i.e., crown area times total height) to that of the largest individual. Values ranged between 0.1 and 1. Roots penetrated down to ca. 40-cm depth but their spread could exceed the crown radius by one order of magnitude. The best fits of Φc were obtained for an exponential function with L 0 c = 1.27 and p = 1/3. Facilitation consisted of a drastic reduction of direct soil evaporation. Soil water sensors were used to parameterize Φ f like Φc by an exponential. Attempts to fit with other functions did not yield substantially better fits. A clear optimum was found for L 0 f = 0.81[START_REF] Barbier | Self-organised vegetation patterning as fingerprint of climate and human impact on semiarid ecosystems[END_REF][START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF].

FrequencyFig. 2 .

 2 Fig. 2. Crown radius histogram: number of plants as a function of crown size for 750 individuals of C. micranthum above 1.5 m height measured in a plot of 0.4 ha representative of the Niger site seen in Fig. 1.

)Fig. 2

 2 Fig. 2 reports a histogram of the crown radii established on the basis of measurements made for 750 individual plants constitutive of the pattern shown in Fig. 1a. Note that while the wavelength of this pattern (mean distance between bare spots) is λ ≈ 50 m, (11) the maximum crown radius measured in the sampling for the most developed trees is L a ≈ 3.25 m. (12)3. Uniform stationary vegetation covers

Fig. 4 .

 4 Fig. 4. Stability domains in (Λ -1 , b) space: in (a) for p = 0, in (b) for p = 1/3. The arrows indicate the direction of bs's pathway of change when competition increases faster with aridity than facilitation.The red circle indicates the position of the critical point. The green circles joined by a dashed line represent two locally stable stationary states corresponding to the same values of μ and Λ (bistability), which are in equilibrium, i.e., whose potential value F 0,p as given by (A.7) (p = 0) or by (A.8) (p = 1/3) is the same.

Fig. 5 .

 5 Fig. 5. Behavior of the potential function F 0 under dry climatic conditions for the values of Λ -1 indicated.

  ) in appendix B). These analytical expressions are studied and compared to field data in sections 4.1-4.3 below. They clarify the kinetic mechanism underlying vegetation patterns, its properties in relation to plant structure and exogenous factors, the similarities between selforganization in plant ecology and other domains.4.1. Local stability with respect to fluctuations in the PDE approximationWe have shown in Fig.4that the stationary states b 0 and b s are locally unstable with respect to uniform perturbations when these states fall into the spinodal domain of stability diagrams. Let us now study the local stability of uniform vegetated covers with respect to arbitrary, non-uniform fluctuations. Using (25), the treatment of this problem is quite straightforward. The coordinates of the critical point are now

  Fig. 6. (a) Excess stationary density βs as a function of M for Γ = -0.8849, K = 0 (in red: stationary curve when cooperativity equals its critical value) and K = 5.4 (in black: a stationary curve displaying hysteresis). The intersections (green dots for βs = β ± ) with the green line are the bifurcation points of the symmetry-breaking instability. (b) Linear stability of the excess stationary densities βs. The black curve displays a finite band of unstable modes; the black dots are its lower and upper cut-offs k and ku.

Fig. 7 .

 7 Fig. 7. Stationary patterns in the vicinity of the critical point for K = 0. Parameters have been determined so that the wavelength is approximately equal to 50 meters: p = 1/3, Γ = -0.8849, χ f ≈ 3.408, ε = 0.4096 ≈ χc/χ f , La ≈ 1.080 m and δ = 0. The average initial conditions b(r, 0) = bc [1 + η β(r, 0) ] indicated are calculated for η = 0.1 and bc = 2/5.

  ), dispersion (Δ) may control the existence of a band of unstable modes. For example, with the field values p ≈ 1/3 and ε ≈ 0.41, (B.15) predicts that 0.883 (1 -Δ) 2 , and thus that there exists a band of unstable modes if Δ ≈ 0, K > -0.441 and the competition strength χ c is big enough for inequality (21) to be satisfied with ε = 0.41. If all other parameters stay constant while dispersion increases, this band is suppressed when Δ = 1 at the latest, or possibly before that if inequality (21) no longer holds for ε = 0.41. On the other hand, for Δ > 1, the expression of the marginal mode k * becomes unphysical (see, (36) below), which, besides the violation of condition (

Fig. 8 .

 8 Fig.8. Plot of the critical wavelength in meters predicted by Eq. (36) vs η for p = 1/3, K = 0, ε = 0.4096, δ = 0 and the values of La indicated.

Fig. 9 .

 9 Fig. 9. (a) Spinodal curves in (Λ -1 , b)-phase space for p = 1/3, ε = 0.4096, λ = 50 m and the values of La indicated. Each curve divides the phase space into a domain (above the curve) where uniform biomass distributions are stable and a domain (under the curve) in which vegetation clustering can take place. (b) Spinodal curves in (μ, b)-phase space. The same values of parameters as in (a) are used. Above the curves, uniform biomass distributions are unstable.

Fig. 10 .

 10 Fig. 10. Spinodal curves corresponding to k * = 0 (mean field case, in red, La = 0) and k * = 0.503 (symmetry breaking case, in green, La = 4 m, λ = 50 m, ε = 0.4096). The uniform stationary curves are plotted for μ * = 0.667, μc = 0.895 and μ = 1. The position of the mean field and symmetry breaking critical points are indicated by the red and green circles, respectively.L a = 4 have been used to represent the mean field (in red, L a = 0) and symmetry breaking spinodal (in green, L a = 4) together with the critical stationary

b

  Fig. 11. Stationary curves bs for p = 1/3, ε = 0.4096, La = 3.5 m, χ f = 8.405 (cf. Table2) and χc as indicated. For χc = 6.141 (red curve), the symmetry breaking instability appears at μ = 1 (green circle) with λ ≈ 50 m. For χc = 6.15, the ecosystem develops a deeply gapped pattern. At higher levels of competition (red circles at χc = 6.20 and 6.26), the organization transforms into bare bands and spots inside a uniform cover, and into vegetated bands and spots on bare soil (see Fig.13).

  Fig. 12. Potential F 0 (b * ) vs La for p = 1/3, ε = 0.4096, λ = 50 m and μ as indicated. For μ = 1, F 0 (b * ) = 0 when La = La ≈ 3.173.

  Fig. 13. Stationary patterns for the conditions depicted in Fig. 11 (green and red circles), i.e., La = 3.5 m, χ f = 8.405 and increasing competition levels. With regard to the question raised earlier concerning the predictive merits of the present theory, the discussion of Fig 12, and of the results of Tables2 and 3is satisfying. We find that the field data of[START_REF] Barbier | Spatial decoupling of facilitation and competition at the origin of gap vegetation patterns in SW Niger[END_REF], which fix the maximal crown size of mature plants at L a ≈ 3.25 m (cf. Fig.2), validate the prediction reached here that, in order to exhibit deep gapped patterns, the maximal L a for mature Combretum micranthum plants should exceed ≈ 3.17 m. A more complete investigation of the case L a < L a is planned. Fig.13shows the stationary patterns obtained by numerical simulation of Eq. (1) for μ = 1, L a = 3.5 and the green and red circles in Fig.11as initial conditions. For χ c = 6.15 (slightly inside the spinodal domain of the symmetry-breaking mode), as expected, the system evolves towards a deep gapped pattern: the vegetation cover exhibits a more or less regular distribution of bare soil spots. When competition increases, for χ c = 6.2, neighboring spots start to coalesce, giving rise to a labyrinthine morphology. Simultaneously, the average pattern density decreases while the maximal density of vegetated domains stays approximately constant, or slightly increases. At the saddle-node point (χ c = 6.26) a stable pattern still forms, indicating that this structured branch extends further, i.e., to competition levels for which uniform solutions b s do not exist: bare soil is then the only uniform solution possible. Under those conditions, development of a labyrinthine pattern of bands or spots distributed over bare soil (not shown) constitutes an adaptive behavior through which a significant

Fig. 14 .

 14 Fig. 14. Stationary patterns obtained for the value La = 3.173 m, at which b 0 and bs exchange stability and metastability, for χ f = 6.408 and increasing competition levels χc.

  Fig.15. Desertification: transition from stable pattern to bare soil when La = La = 3.173 m and Λ -1 significantly exceeds the saddle-node value of the mean field spinodal. The sequence of snapshots is taken after increasing the competition level from χc = 0.43 to χc = 0.45. The level of facilitation χ f = 6.408 is constant (see initial condition in Fig.14).

Fig. 16 .

 16 Fig. 16. Metastability of uniform vegetation cover: snapshots at different times showing the formation of a gapped patterns triggered by the symmetry-breaking instability and the subsequent transformation of this pattern via the gap coarsening process into uniform bare soil (desert) state. Values of parameters: p = 1/3, ε = 0.4096, La = 2.75 m, λ = 50 m, μ = 1, χ f = 5.224, χc = 3.070

  (ii) In order to take into account the influence of plant development on the range over which feedback operates, the educated guess (ansatz) consisting to let the facilitation and competition ranges L f and L c depend upon the biomass density has been introduced. From the modeling point of view, this has the advantage that it avoids the complication of increasing the number of variables that the consideration of several classes of age would otherwise impose. From a quantitative point of view, this guess can straightforwardly be calibrated to of a particular plant by using the measured values of the allometric parameter p and structural ratio ε. So far, field values for these parameters are only known in the case of the reference site of Fig.1. Hence, for the purposes of quantitatively illustrating our modeling predictions and of comparing their outcome with a concrete, well defined system, we have used the values p = 1/3 and ε ≈ 0.41 corresponding to the shrub species Combretum micranthum G. Don, which predominates at this site. Whether similar values apply to other species is an open, fundamental and motivating question. It is hoped that the approach we report will stimulate further field work yielding progress and clarifications in this respect. (iii) In the original modeling, both processes of biomass growth and biomass propagation (by seeds or vegetative) are lumped together in the gain term.

  15 and 16 illustrate, is the prediction that vegetation extinction (desertification) can take place following the gap coarsening pathway which so far had not yet been described. Strikingly, this mechanism of extinction is met for smaller values of the loss/gain ratio μ than the lo-, which takes place when the loss/gain ratio exceeds the (upper) turning point of the hysteresis loop. So far, it is this kind of situation that has been predicted as potentially catastrophic because it implies an abrupt transition of the ecosystem from a vegetated state (typically a more or less periodic distribution of vegetated spots) towards the bare soil state (von Hardenberg

  χ c = χ 0 + η, χ f = χ c + Λ c (1 + η 2 κ), μ = μ c 1 + i η i μ i , b s = b c (1 + η β s ) , (B.1)replace these expressions in (14) and solve the equation obtained at each order in η. This yields: is the equation of state for the excess stationary biomass density β s .

  k * to exist, the biomass "diffusion coefficient" must be small (in fact of order η or smaller)close to the critical point, the wavelength of a mode that satisfies (B.6) is much greater than L a , we expand k * 2 in terms of η:k * 2 = η k 0 t) = b c (1 + η β(r, t)) , (B.9)replace the expressions (B.1-B.5) and (B.7-B.9) in (B.6). Up to O(η 2 ) the solution of the system of two equations obtained in this manner reads:

  variational. F [β(r, t)] can only decreases in time till it reaches some minimum value compatible with the constraints. For K > 0, the

Table 2

 2

	reports for increasing values of L a the val-
	ues of the other parameters calculated in imposing
	the condition that μ = 1 is the bifurcation point of
	the symmetry breaking instability. This insures that
	conditions (i) and (iii) above are satisfied.
	Condition (ii) requiring b 0 to be metastable with

Table 3 Minimal

 3

	Λ *

size La the crown of mature plants should exceed to satisfy condition (ii) vs μ for p = 1/3, ε = 0.4096, λ = 50 m.

It is also interesting to note that species for which the inequality p 1 would hold, reach criticality and thus patterning conditions for average biomass densities very close to one.

Acknowledgments

We thank John Wm Turner, Mustapha Tlidi and Vincent Deblauwe for helpful discussions. This work has benefited from a support granted by the European project ECOPAS.

 1), only retain the terms linear in A(k, 0). Hence, we obtain that the Fourier mode k is unstable if its eigenvalue

is positive. Inspection of (A.2) and of the expressions

shows that: (i) For p < 1, both Ω f (k) and Ω c (k) are positive whatever the biomass density b or the structural ratio ε may be. It is interesting to note that facilitation rather than competition is likely to destabilize uniform states.

(ii) The third term of ω k , contributed by plant propagation, is negative and thus always stabilizing. Furthermore, close to the critical point, this term is negligible for Fourier modes whose wave number is close to zero.

(iii) For k → ∞, ω ∞ < 0. Hence, the Fourier modes k 1 are always stable. This is consistent with the requirement that the biomass distribution of the surface element S with respect to which the density b(r, t) is defined must be uniform. (iv) For bare soil, the eigenvalue ω 0 is simply:

Thus, b 0 is linearly unstable with respect to uniform perturbations for 0 ≤ μ < 1 and linearly stable otherwise. On the other hand, using ( 14) to eliminate μ in (A.2) shows that for vegetated states

s . (A.5) Accordingly, the spinodal curve (see Fig. 4)

separates the (Λ -1 , b)-space into two domains where vegetated states are either stable (above the spinodal) or unstable (below the spinodal).

A.2. Potential functions

The potentials governing the global stability of uniform states when p = 0 and p = 1/3 read:

is the Whittaker M function. The potential from which the Swift-Hohenberg PDE derives reads (p = 1/3):

(A.9)

At the critical point, it reduces to:

(A.10)

Appendix B. PDE approximation of vegetation spatio-temporal dynamics

B.1. Approximation of uniform stationary states in the vicinity of the critical point

Let 0 < η 1 be a smallness parameter that fixes the deviation of Λ from its critical value Λ c given by ( 16). We set