Towards a Qualitative Theory on Non-Linear Geographic Systems: Regular and Stochastic Evolution in Fluvial Landforms

以地理实体和现象为特征的空间模式和过程，如流域、城市系统和信息流，往往表现出组织和复杂性的显著混合。这些相互作用是由大量空间分布的相互作用引起的，这些相互作用的特征是涉及随机效应和不稳定性的非线性过程。这项研究的主要目标是改进和扩展一系列基于物理的模型，以模拟流域演化的关键方面。该项目将结合理论、计算和实地调查，以完成以下工作：(1)应用新兴的方法来描述无限维、非线性动力系统的定性行为，以解释流域演变中出现的空间模式；(2)证明这些方法必须与传统方法结合使用，以充分描述河流陆面的演变和对各种尺度规律的理解；(3)开发更多的分析工具，并将其应用于其他复杂地理现象的例子；以及(4)传播对这些方法及其价值的理解。数学和建模结果将为定性和定量预测提供基础，这些预测可以与对真实景观的观察进行比较，最终可以基于这种理解开发类似地理现象的模型，如人口迁移、创新扩散和信息流。

Spatial patterns and processes characterizing geographical entities and phenomena, such as drainage basins, systems of cities, and flows of information, often show a remarkable mix of organization and complexity. These arise from a large number of interactions that are spatially distributed and characterized by non-linear processes involving stochastic effects and instabilities. The dominant objective of this research is to refine and extend a family of physically based models for simulating key aspects of drainage basin evolution. The project will incorporate theoretical, computational, and field-based investigations to accomplish the following: (1) apply emerging methodologies for characterizing the qualitative behavior of infinite-dimensional, non-linear dynamical systems in explaining the emergence of spatial patterns in the evolution of drainage basins; (2) demonstrate that such methods must be used in conjunction with traditional methods in developing a full characterization of the evolution of fluvial land surfaces and an understanding of various scaling laws; (3) develop additional tools for such analyses and apply them to other examples of complex geographic phenomena; and (4) disseminate an understanding of these methods and their value. The mathematical and modeling results will provide a basis for qualitative and quantitative predictions that may be compared with observations on real landscapes, and ultimately, models of analogous geographic phenomena, such as population movements, diffusion of innovations, and information flows can be developed based on this understanding.