Nancy Rodriguez, Ph.D. - US grants

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.

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.