Nonlinear and Non-local Models in Social and Ecological Systems

这个项目的目标是介绍一个理论框架，以了解和预测在许多复杂的社会和生态系统中形成的宏观模式。这项研究的动机主要是两个现象：社会动物中的社会和环境相互作用导致地域模式的发展；城市士绅化模式的发展。尽管在概念上不同，但这些现象将由非常相似的数学模型来模拟，这些模型适合于反应-平流-扩散(RAD)系统的框架。RAD系统是本研究的重点；它们的使用将有助于深入了解需要了解宏观模式的复杂社会和生态系统。在这个框架中，PI将致力于纳入真实世界的数据，以提取客观信息，这些信息将有助于揭示导致在生态学和社会学中观察到的复杂模式的最具影响力的因素。与这一研究项目相关的是一项指导计划，重点是为科罗拉多博尔德大学STEM专业的少数族裔学生提供建议。这将主要通过在科学分会中发起奇卡诺人和美洲原住民协会来实现。其目的是为这些学生提供一个能够帮助他们在STEM取得成功的网络。这项研究的总体目标是基于真实的观测和数据开发、分析和模拟反应-平流-扩散(RAD)系统。例如，数据驱动的RAD系统必须包括异构性(空间和时间)以及非本地操作符，这带来了巨大的数学和计算挑战。RAD系统还提供了一个完美的框架来检验其他领域的研究人员假设的假设，因为他们的解可以作为概率密度函数，用于使用各种方法，如最大似然估计，以将参数与数据相匹配。PI将利用这一框架开发面向生态学家和社会科学家的基础设施(理论、算法和软件)，通过使用适当的统计技术将RAD类型的模型与数据进行匹配来验证RAD类型的模型。这项研究将包含三个相互关联的项目：第一个项目将围绕社会动物的社会和环境相互作用的建模，以便通过使用非本地和不同种类的RAD系统来了解区域格局；第二个项目的重点将是制定一种方法和数字框架，以纳入(社会)数据，以检验社会学家提出的假说，并将士绅化作为第一个案例研究；最后的项目将集中于发展非局部反应扩散方程的理论，它产生于出生跳跃过程，这是非常适合于出生和扩散不能分开的过程的模型。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of this project is the introduction of a theoretical framework to understand and predict macroscopic patterns that are formed in many complex social and ecological systems. This research is motivated primarily be the two phenomena: the social and environmental interactions in social animals that lead to development of territorial patterns; development of urban gentrification patterns. Although conceptually different, these phenomena will be modeled by very similar mathematical models that fit into the framework of reaction-advection-diffusion (RAD) systems. RAD systems are the focus of this research; their use will help to gain insight into complex social and ecological systems where there is a need to understand macroscopic patterns. In this framework the PI will work on incorporating real-world data to extract objective information that will help shed light into what are the most influential factors leading to the complex patterns which are observed in ecology and sociology. Associated to this research project is a mentoring plan focused on advising underrepresented minority students at University of Colorado Boulder majoring in a STEM field. This will mainly be done through the initiation of a Society of Chicanos and Native Americans in the Science chapter. The aim is to provide these students with a network that can help them succeed in STEM. The overarching objective of this research is to develop, analyze, and simulate reaction-advection-diffusion (RAD) systems based on real-life observations and data. For example, RAD systems that are data-driven must include heterogeneities (spatial and temporal) as well as nonlocal operators, posing significant mathematical and computational challenges. RAD systems also provide a perfect framework to test hypothesis postulated by researchers in other fields, since their solutions can serve as a probability density function for the use of various methodologies, such as maximum likelihood estimation, to fit parameters to data. The PI will take advantage of this framework to develop an infrastructure (theory, algorithms, and software) targeted toward ecologists and social scientists that will validate RAD-type models by fitting them to data using appropriate statistical techniques. This research will contain three interrelated projects: the first one will be centered around the modeling of social and environmental interactions in social animals in order to understand territorial patterns through the use of non-local and heterogeneous RAD systems; the focal point of the second project will be on the development of a methodology and numerical framework to incorporate (social) data in order to test hypothesis developed by sociologists, with gentrification as a first case study; the final project will focus on developing the theory for non-local reaction-diffusion equations that arise from birth-jump processes, which are very suitable models for processes for which birth and dispersal cannot be separated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.