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Ben-Gurion University of the Negev (2006)

Mathematical models for vegetation patterns and biodiversity

Gilad Erez

Titre : Mathematical models for vegetation patterns and biodiversity

Auteur : Gilad Erez

Université de soutenance : Ben-Gurion University of the Negev

Grade : DOCTOR OF PHILOSOPHY (PhD) 2006

Résumé
In this work we present and study a mathematical model for water-biomass interactions in arid and semi-arid ecosystems. These interactions often involve positive feedbacks between vegetation biomass and water : the larger the biomass the more water available to the vegetation and the faster the vegetation grows. The increase in water availability that comes with biomass can be attributed to reduced evaporation by shading (”shading feedback”), increased infiltration rates of surface water at vegetation patches (”infiltration feedback”), and water uptake by plants’ roots, which lengthen as the plants grow (”uptake feedback”). The model captures all three positive feedback processes, which are the crucial factors affecting water-biomass interactions in water limited systems. The role of these feedbacks in the formation of vegetation pattern have already been studied but their implications for resource distribution and consequently for ecosystem engineering, changes in plant interactions, and species diversity have not been addressed using a modeling approach. By implementing concepts and tools from the theory of pattern formation (e.g. instabilities, bifurcation diagrams, stability analysis), we use the mathematical model to study a variety of currently important questions, including the self-organization of plant communities in arid and semi-arid regions to form vegetation patterns, sudden responses of vegetation to environmental changes, ecosystem engineering, changes in plant interactions along gradients of environmental stresses, and mechanisms of species diversity change. We reproduce and study a sequence of vegetation patterns across a rainfall gradient (a widely observed natural phenomenon), identify ranges of pattern coexistence, and study possible transitions between different stable states of the system (catastrophic shifts). On hill slopes, we identified wide parameter ranges where multistability of vegetation bands occur (i.e. bands patterns characterized by different wavenumbers), and we predict a tradeoff between biological productivity and resilience of the system. We study the different soil-water distributions induced by different vegetation patterns and demonstrate how diversity in vegetation patterns may lead to habitat diversity. Furthermore, at the single-patch scale we predict a facilitation-resilience tradeoff by studying the relative strengths of the infiltration and uptake feedbacks. By generalizing the model to describe the dynamics of n-interacting species populations, we successfully reproduce the widely observed trend of the transition from competition to facilitation as aridity stress increases and we suggest a possible mechanism. In addition, we suggest two novel mechanisms responsible for changes in species diversity, (i) interspecific facilitation induced by intraspecific competition and (ii) en- hanced facilitation at the single-patch scale induced by dynamic pattern changes at the landscape scale (cross-scale processes). Finally, we present a fast algorithm for convolution integrals with space and time variant kernels. This new algorithm was developed in order to overcome several major difficulties in the numerical integration of the model equations

Mots clés : Vegetation patterns, pattern formation, collective dynamics and emergent properties, ecosystem engineer, habitats dynamics, landscape and species diversity, plant interaction, competition and exclusion, facilitation and coexistence, cross-scale processes, convolution, space and time variant kernels, nonlocal terms, integro-differential equations.

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