Olivier Lejeune

On the origin of tiger bush

We propose a model which describes the dynamics of vast classes of terrestrial plant communities growing in arid or semi-arid regions throughout the world. On the basis of this model, we show that the vegetation stripes (tiger bush) formed by these communities result from an interplay between short-range cooperative interactions controlling plant reproduction and long-range self-inhibitory interactions originating from plant competition for environmental resources. Isotropic as well as anisotropic environmental conditions are discussed. We find that vegetation stripes tend to orient themselves in the direction parallel or perpendicular with respect to a direction of anisotropy depending on whether this anisotropy influences the interactions favouring or inhibiting plant reproduction; furthermore, we show that ground curvature is not a necessary condition for the appearance of arcuate vegetation patterns. In agreement with in situ observations, we find that the width of vegetated bands increases when environmental conditions get more arid and that patterns formed of stripes oriented parallel to the direction of a slope are static, while patterns which are perpendicular to this direction exhibit an upslope motion.

A model for the explanation of vegetation stripes (tiger bush)

Regular vegetation patterns appear on aerial views of plateaux in SW Niger where densely and sparsely popu- lated zones alternate with each other. This spatial organization of the vegetation is an endogenous phenomenon which is not limited to specific plants or soils; it is a characteristic land- scape of many arid regions throughout the world. The phe- nomenon is interpreted as the result of a spatial range differ- ence between two biologically distinct interactions operating at the plant population level. The proposed mechanism is independent of external heterogeneities deriving from soil geomorphology or meteorology. We present a model to simu- late the genesis of vegetation stripes. In addition, the model predicts the occurrence of vegetation hexagons corresponding to higher or lower density spots arranged in a hexagonal lattice. The distinction between the two spatial symmetries is discussed in terms of their Fourier transforms.

Short range co-operativity competing with long range inhibition explains vegetation patterns

We model the non-local dynamics of vegetation communities and interpret the formation of vegetation patterns as a spatial instability of intrinsic origin: the wavelength of the patterns predicted within the framework of this approach is determined by the parameters governing the dynamics rather than by boundary conditions and/or geometrical constraints. The spatial periodicity results from an interplay between short-range co-operative interactions and long-range self-inhibitory interactions inside the vegetation community. The influence of environmental anisotropies on pattern symmetry and orientation is discussed. As a case study, the approach is applied to a system of vegetation bands situated in the north-west of Burkina Faso. The parameters describing the co-operative and inhibitory interactions at the origin of the patterns are evaluated.

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

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.

Generic Modelling of Vegetation Patterns. A Case Study of Tiger Bush in Sub-Saharian Sahel

The conditions underlying the mean field description of vegetation patterns are reviewed. A generic partial differential equation model describing vegetation propagation over space by reproduction under isotropic as well as anisotropic environmental conditions is discussed. An example of Tiger Bush in Burkina Faso is analyzed on the basis of this model.

Kinetics of single stripe formation in intracavity second harmonic generation

Abstract The second harmonic generation of type I with diffraction is considered. When the spatial modulational instability occurs near a saddle node bifurcation associated with bistability, stable stationary single stripes may be generated spontaneously from the unstable connecting branch of homogeneous steady states. The kinetics of this localized structure formation is found to be described by power laws.