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导致人们认识到，要对涉及的非常复杂的系统和现象进行定性和定量的理解，需要将这些科学中开发的许多概念模型数学化。该项目的总体目标是为这类概念模型制定数学框架，并有助于对这些复杂系统的定性和定量理解。其中一个项目将侧重于人口流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流动策略是最优的，以及为什么竞争种群的聚集有效。随着环境和经济的快速变化，研究不同迁徙策略的相对成功程度至关重要，以便能够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骚乱活动的模型。这将使我们能够梳理出不同骚乱动态背后最重要的因素。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犯罪的多尺度模型，该模型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量化框架和工具，供非技术科学家用来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帮助量化其影响的政策战略的方法。这些项目将与一个教育计划相结合，该计划重点是通过与CU Boulder的科学发现合作的一系列活动来增强拉美裔社区的能力。首先，PI将通过青少年科学咖啡馆项目与青少年会面，讨论她的研究。其次，PI将为代表不足的少数族裔的高中生提供研究经验机会，目的是鼓励他们上大学并主修STEM领域。最后，国际和平协会将开展实践活动，帮助拉丁裔社区了解在科罗拉多州新冠肺炎大流行期间实施的社交距离和居家政策所使用的模型。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-扩散框架来了解不同运动策略的最佳性，特别是在存在Allee效应的情况下。第二个项目将侧重于暴动活动模型的开发和研究。特别是，PI将研究行波解决方案，这种解决方案已经在许多现实生活中观察到。最后一个项目将借鉴数学流行病学方面的工作，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还将促进对空间不均匀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。

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.