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

The research is aimed at identifying robust features and mechanisms governing the spatio-temporal evolution of deterministic and stochastic models describing vulnerable dry-land ecosystems. This is an important applied direction for pattern formation research, and it o�ffers the challenge that there are no �first principles equations that model these complex systems. Consequently the focus is on robust mechanisms for transitions between diff�erent types of ecological states. The project is motivated by current ecological modeling research of desert ecosystems that describe shifts from vegetated to barren landscapes as "tipping points" or "critical transitions", and which are characterized mathematically as bifurcations. The project addresses the underlying mathematical mechanisms and associated characteristics of 'tipping points' in deterministic and stochastic models of deserti�cation in dry land ecosystems. In the models a patterned state is created through a Turing bifurcations prior to deserti�cation. Bifurcation theory provides a framework for identifying possible robust behavior as a function of system parameters in the models, which will inform stochastic models, and numerical investigations that include appropriate noise, weak spatial heterogeneity, annual variability, and/or temporal drift of key control parameters. Specifi�c objectives of the project are as follows. 1. To compare, qualitatively and quantitatively, a class of pattern-forming reaction-di�ffusion vegetation models to determine the robust transitions between di�fferent vegetation states that may occur as the model system approaches its trivial desert state. 2. To develop model reduction methods for the class of deterministic models of vegetation to further determine the consistency between these models. 3. To develop a simple, flexibly pattern-forming partial diff�erential equation to explore, through global analysis and numerical simulation, the range of behaviors associated with the key, generic modeling assumptions. 4. To develop and analyze vegetation models with spatio-termporally variable precipitation inputs. 5. To examine available satellite image data of dry-land ecosystems to further inform the modeling e�ffort. 6. To educate and train PhD students in interdisciplinary applied mathematics research. The intellectual merit of the proposed research includes contributions to our understanding of ecological tipping points. The project will contribute to deterministic and stochastic modeling of dry-land ecosystems, and meet the challenges of investigating bifurcation problems and complex pattern-forming systems in the setting of a complex system. A broader impact of the proposed activity is that it may help us to predict and plan for changes to vegetation biomass density in semi-arid regions. The project will leverage the infrastructure of the recently established \Mathematics and Climate Research Network" to enhance the training of the students on the project. Interdisciplinary training of the graduate students involved with the project is an integral part of its goals.