Deterministic and Stochastic Models of Water Limited Ecosystems: Implications of Pattern Formation, Bifurcations, Model Reduction, and Data

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The investigator and her students will contribute to mathematical modeling efforts for desert ecosystems, which may be vulnerable to desertification under climate change. These models address the underlying mechanisms behind the spatial patterns of vegetation biomass that occur when certain semi-arid ecosystems are stressed by decreased precipitation. Interfacing self-organizing spatial effects for vegetation with the enormous variability of precipitation that define drylands suggests new directions for fundamental pattern formation research. The research will also contribute a mathematical framework for testing the robustness of the ecological proposals that the spatial patterns may serve as early warning signs of tipping points associated with climate change.The research will be developed within a framework of deterministic and stochastic mathematical ecological models, with its objectives enhanced by the availability of rich satellite data that can be used to test model predictions. The first objective is to compare, qualitatively and quantitatively, a class of pattern-forming reaction-diffusion vegetation models to determine the robust transitions between distinct vegetation pattern states that may occur as the model system approaches its trivial desert state. This analysis will be based in bifurcation theory. The second objective is to develop and analyze vegetation models with temporally variable precipitation inputs, and spatially variable drainage networks. This will lead to the development of stochastic models with non-Gaussian noise. Model reduction methods will be used to further determine the consistency between model frameworks. Satellite image data of dry-land ecosystems will inform and verify the modeling effort. The project also has an education component with the goal of training undergraduate and graduate students in interdisciplinary applied mathematics research.