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In 2017, Ilya Prigogine would have been 100 years of age. As for any human being, this centenary is a notable event. For him, as a scientist, 2017 was also above all the 50th anniversary of dissipative structures It was indeed in 1967 that for the first time he used this denomination at the occasion of an important scientific event and in publications. The attribution of this qualification for self-organized behaviours of matter only possible far from equilibrium coincided with the outcome of a research effort of more than 25 years. Centred in thermodynamics and statistical physics on the role played by irreversible processes in the physical evolution of matter, the aim of this research is clear from the outset of his scientific career. With visionary personal intuition and iron-willed determination, it was pursued. The road to success had been long and sinuous, but finally it led to what he called the rehabilitation of irreversible processes The progresses that stand out as major landmarks of this endeavour that imposed a U-turn with respect to conceptions of classical physics deeply rooted since the nineteenth century will be described. This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (Part 1)'.

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Desertification due to climate change and increasing drought periods is a worldwide problem for both ecology and economy. Our ability to understand how vegetation manages to survive and propagate through arid and semiarid ecosystems may be useful in the development of future strategies to prevent desertification, preserve flora-and fauna within-or even make use of scarce resources soils. In this paper, we study a robust phenomena observed in semi-arid ecosystems, by which localized vegetation patches split in a process called self-replication. Localized patches of vegetation are visible in nature at various spatial scales. Even though they have been described in literature, their growth mechanisms remain largely unexplored. Here, we develop an innovative statistical analysis based on real field observations to show that patches may exhibit deformation and splitting. This growth mechanism is opposite to the desertification since it allows to repopulate territories devoid of vegetation. We investigate these aspects by characterizing quantitatively, with a simple mathematical model, a new class of instabilities that lead to the self-replication phenomenon observed.

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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.

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Spatially periodic vegetation patterns, forming gaps, bands, labyrinths, or spots, are characteristic of arid and semiarid landscapes. Self-organization models can explain this variety of structures within a unified conceptual framework. All these models are based on the interplay of positive and negative effects of plants on soil water, but they can be divided according to whether they assume the interactions to be mediated by water redistribution through runoff/diffusion or by plants' organs. We carried out a multi-proxy approach of the processes operating in a gapped pattern in southwest Niger dominated by a shrub species. Soil moisture within the root layer was monitored in time and space over one month of the rainy season. Soil water recharge displayed no spatial variation with respect to vegetation cover, but the stock half-life under cover was twice that of bare areas. A kernel of facilitation by the aboveground parts of shrubs was parameterized, and soil water half-life was significantly correlated to the cumulated facilitative effects of shrubs. The kernel range was found to be smaller than the canopy radius (81%). This effect of plants on soil water dynamics, probably through a reduction of evaporation by shading, is shown to be a better explanatory variable than potentially relevant soil and topography parameters. The root systems of five individuals of Combretum micranthum G. Don were excavated. Root density data were used as a proxy to parameterize a kernel function of interplant competition. The range of this kernel was larger than the canopy radius (125%). The facilitation-to-competition range ratio, reflecting the above-to-belowground ratio of plant lateral extent, was smaller than 1 (0.64), a result supporting models assuming that patterning may emerge from an adaptation of plant morphology to aridity and shallow soils by means of an extended lateral root system. Moreover, observed soil water gradients had directions opposite to those assumed by alternative mathematical models based on underground water diffusion. This study contributes to the growing awareness that combined facilitative and competitive plant interactions can induce landscape-scale patterns and shape the two-way feedback loops between environment and vegetation.

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