CAREER: Mathematical Frameworks and Theory for Conceptual Models in Economics, Ecology and Criminology

犯罪学、经济学和生态学的最新进展使人们认识到，要对非常复杂的系统和现象进行定性和定量的理解，就需要将这些科学中发展起来的许多概念模型数学化。该项目的首要目标是为这些概念模型开发数学框架，并有助于对这些复杂系统的定性和定量理解。一个项目将侧重于运动在生态系统中的作用或再分配在经济系统中的作用。我们的目标是了解什么样的运动策略是最优的，以及为什么竞争种群的聚类是有效的。随着环境和经济的快速变化，研究不同迁移策略的相对成功是至关重要的，这样才能预测哪些种群将在竞争中生存下来。另一个项目将开发和研究暴乱活动的模式。这将使我们能够梳理出不同暴乱动态背后的最重要因素。最后一个项目将借鉴数学流行病学的工作，从公共卫生的角度开发一个犯罪的多尺度模型，这可以帮助我们测试不同政策对暴力犯罪动态的影响。这项研究将提供一个定量的框架和工具，非技术科学家可以利用这些框架和工具在他们的工作中取得重大进展。这里开发的模型将提供测试干预和政策策略的方法，有助于量化其效果。这些项目将与一项教育计划相结合，重点是通过与科罗拉多大学博尔德分校的科学发现合作开展一系列活动，增强拉丁裔社区的能力。首先，PI将与青少年见面，通过青少年科学咖啡馆项目讨论她的研究。其次，该项目将为少数族裔高中生提供研究经验机会，鼓励他们上大学，主修STEM领域。最后，PI将开展实践活动，帮助拉丁裔社区了解用于为科罗拉多州COVID-19大流行期间实施的社交距离和居家政策提供信息的模型。本研究由三个项目组成。第一个项目旨在使用局部和非局部反应-平流-扩散框架来理解不同运动策略的最优性，特别是在存在Allee效应的情况下。第二个项目将侧重于发展和研究暴乱活动的模式。特别是，PI将研究行波解，这已经在许多现实生活中的骚乱中被观察到。最后的项目将借鉴数学流行病学的工作，利用动力学方程框架从公共卫生的角度开发犯罪的多尺度模型。这些项目的成功完成也将促进对空间异构环境中局部和非局部反应-平流-扩散方程的行波解和广义前沿的定性理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Recent advances in criminology, economics, and ecology have led to the realization that to derive qualitative and quantitative understanding of very complex systems and phenomena involved one needs to mathematize many conceptual models developed in these sciences. The overarching objective of this project is to develop mathematical frameworks for such conceptual models and contribute to the qualitative and quantitative understanding of these complex systems. One project will focus on the role that movement has in ecological systems or that redistribution has in economic systems. The objectives are to understand what movement strategies are optimal and why clustering of competing populations work. With rapid shifts in the environment and in the economy, it is crucial to investigate the relative success of different movement strategies to be able to predict which populations will survive under competition. Another project will develop and study models for rioting activity. This will enable us to tease out the most important factors behind the dynamics of different riots. A final project will draw from work in mathematical epidemiology to develop a multi-scale model for crime from a public health perspective, which can help us to test the effect of different policies on the dynamics of violent crimes. This research will provide a quantitative framework and tools that non-technical scientists can use to make significant advances in their work. The models developed here will provide ways to test intervention and policy strategies helping to quantify their effects.These projects will be integrated with an educational plan focused on empowering the Latinx community through a series of activities in collaboration with CU Boulder’s Science Discovery. First, the PI will meet with teenagers to discuss her research through the Teen Science Cafe program. Second, the PI will provide research experience opportunities for high school students who are underrepresented minorities, with the aim of encouraging them to attend college and major in a STEM field. Finally, the PI will develop hands-on activities to help the Latinx community understand the models used to inform the social distancing and stay-at-home policies implemented during the COVID-19 pandemic in Colorado. This research consists of three projects. The first project aims to use a local and non-local reaction-advection-diffusion framework to understand the optimality of different movement strategies, particularly, in the presence of an Allee effect. The second project will focus on development and study of models for rioting activity. In particular, the PI will study traveling wave solutions, which have been observed in many real-life riots. The final project will draw from work in mathematical epidemiology to develop a multi-scale model for crime from a public health perspective using a kinetic equations framework. The successful completion of these projects will also advance the qualitative understanding of traveling wave solutions and generalized fronts for local and non-local reaction-advection-diffusion equations in spatially heterogeneous environments.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.