Nonlinear and Non-local Models in Social and Ecological Systems

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

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.