Models for Social, Ecological, and Biological Systems: Narrowing the Gap Between Theory and Applications

社会、生态和生物系统模型：缩小理论与应用之间的差距

• 批准号: 1516778
• 负责人: Nancy Rodriguez
• 金额: $ 17万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-15 至 2019-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1516778&HistoricalAwards=false
• 关键词: Models; Social; Ecological; Biological; Systems

This research is aimed at development of mathematical tools for understanding and predicting behavior of a variety of important and apparently dissimilar biological phenomena and sociological processes, such as the spread of diseases, ecological changes, or the distribution and prevention of crime. Some salient features of these diverse phenomena can be described by mathematical models based on reaction-advection-diffusion equations that have been used extensively to investigate fundamental and ubiquitous phenomena in several areas of biology, and more recently in the social sciences. While these models are simplified versions of reality, their mathematical analysis has contributed to the understanding of many important phenomena. At the same time, the underlying equations can be extremely interesting and challenging from the mathematical point of view as their solutions can exhibit rich behaviors, e.g., pattern formation and traveling wave structures. The behavior of solutions reflects critical features of the system that mathematics models and have clear and convincing parallels with the evolution of real-world systems. However, there is still a large gap between real world systems and their more tractable mathematical models. One of the goals of this project is bridging this gap through the use of accumulated concrete data and the corresponding calibration and modification of mathematical models. Of particular importance is the analysis of systems which include heterogeneous environments, the understanding of the effects that non-local dispersal has on the behavior of the solutions, and the validation of these models with real-world data. Efforts will be focused on four reaction-advection-diffusion systems where the heterogeneities are due to: climate change in an ecological context, non-local and asymmetrical spread of information in the context of riots, income heterogeneities in the context of social segregation, and environment heterogeneities due to specific concentrated inputs. The primary goal of the project is to understand how heterogeneous environments and non-local dispersal impact solution patterns in both a general class of nonlinear reaction-diffusion equations as well in specific ecological and social contexts. Throughout this project the investigators will maintain and foster contacts with social scientists in order to conduct the much needed discourse between the mathematical theory and the applications. This project will focus on the development of an extensive theory for reaction-advection-diffusion systems in heterogeneous environments with applications in ecology and sociology. In particular, the objective of this research is three-fold: to expand the current mathematical theory for local and non-local reaction-advection-diffusion systems in heterogeneous environments, to gain insight into various (ecological, sociological, and biological) complex systems by modeling them using these types of systems, and to take an initial step toward bridging the gap between basic mathematical models and the complex real-world systems they aim to describe by incorporating the use of data. From the qualitative perspective, this work will be mainly concerned with exploring the effects that various dispersal mechanism have on the propagation (and lack thereof) of a solution in a heterogeneous environment: for example, determining the spreading speed, the existence of traveling wave solutions, pulsating fronts, traveling pulses, or generalized fronts in heterogeneous systems. Additionally, the fundamental issues of the global well-posedness of solutions to these systems, intermediate and long-term asymptotics, and existence and uniqueness of non-trivial steady-state solutions will be rigorously analyzed.

这项研究旨在开发数学工具，以理解和预测各种重要且显然不同的生物学现象和社会学过程的行为，例如疾病的传播，生态变化或犯罪的分布和预防。这些不同现象的某些显着特征可以通过基于反应 - 注入扩散方程的数学模型来描述，这些模型已广泛用于研究生物学几个领域的基本和无处不在现象，以及最近在社会科学中。 尽管这些模型是现实的简化版本，但它们的数学分析促成了对许多重要现象的理解。同时，从数学的角度来看，基础方程可能非常有趣且具有挑战性，因为它们的解决方案可以表现出丰富的行为，例如模式形成和波动波结构。解决方案的行为反映了数学模型的系统的关键特征，并且与现实世界系统的演变具有清晰而令人信服的相似之处。但是，现实世界系统与其更可行的数学模型之间仍然存在很大的差距。 该项目的目标之一是通过使用累积的混凝土数据以及数学模型的相应校准和修改来弥合这一差距。 特别重要的是对系统的分析，这些系统包括异质环境，对非本地分散对解决方案行为的影响的理解以及使用现实世界数据对这些模型进行验证。 努力将集中在四个反应 - 侵蚀扩散系统上，在这些系统中，由于：在骚乱的背景下，在生态环境中的气候变化，非本土和不对称的信息传播，社会隔离的收入异质性，以及由于特定的集中输入而导致的环境异质性。 该项目的主要目的是了解一般一类非线性反应扩散方程中的异质环境和非本地分散影响解决方案模式以及特定的生态和社会环境中。 在整个项目中，调查人员将维持并促进与社会科学家的接触，以进行数学理论与应用之间的急需论述。 该项目将着重于在生态学和社会学中应用的异质环境中的反应 - 接地扩散系统的广泛理论。 特别是，这项研究的目的是三个折叠：在异质环境中扩展当前的数学理论，以扩展本地和非本地反应 - 反转扩散系统，以深入了解各种（生态，社会学和生物学）复杂系统，通过使用这些类型的系统建模它们，并通过这些类型的系统来建立差距，并通过基本的数学模型来构建实际的数学模型之间的差异。从定性的角度来看，这项工作将主要关注探索各种分散机制对异质环境中溶液传播（且缺乏溶液）的影响：例如，确定传播速度，行进浪潮解决方案的存在，脉动脉动，脉动阵线，旅行脉冲，或在异质体系中的普遍前线。 此外，将严格分析解决这些系统的全球解决方案，中间和长期渐近性以及非平凡稳态解决方案的存在和独特性的基本问题。