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

研究人员和她的学生将为沙漠生态系统的数学建模工作做出贡献，这些生态系统在气候变化下可能容易受到荒漠化的影响。这些模型涉及到某些半干旱生态系统因降水减少而受到压力时出现的植被生物量空间格局背后的基本机制。将植被的自组织空间效应与定义旱地的降水量的巨大变化相结合，为基本格局形成研究提出了新的方向。这项研究还将为检验生态学建议的可靠性提供一个数学框架，即空间格局可作为与气候变化有关的临界点的预警信号，这项研究将在确定性和随机性数学生态学模型的框架内进行，其目标将因可用于检验模型预测的丰富卫星数据的可用性而得到加强。第一个目标是比较，定性和定量，一类图案形成反应扩散植被模型，以确定不同的植被格局状态之间的强大的过渡，可能会发生的模型系统接近其平凡的沙漠状态。这种分析将基于分叉理论。第二个目标是开发和分析植被模型与时间可变的降水输入，空间可变的排水网络。这将导致非高斯噪声的随机模型的发展。模型简化方法将用于进一步确定模型框架之间的一致性。旱地生态系统的卫星图像数据将为建模工作提供信息并加以核实。该项目还有一个教育部分，目标是培训跨学科应用数学研究的本科生和研究生。

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.