| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | June 14, 2019 |
| Latest Amendment Date: | May 7, 2021 |
| Award Number: | 1909638 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Pedro Embid
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | June 15, 2019 |
| End Date: | August 31, 2022 (Estimated) |
| Total Intended Award Amount: | $301,796.00 |
| Total Awarded Amount to Date: | $301,796.00 |
| Funds Obligated to Date: |
FY 2020 = $112,384.00 FY 2021 = $77,008.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
3100 MARINE ST Boulder CO US 80309-0001 (303)492-6221 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
3100 Marine Street, Room 481 Boulder CO US 80303-1058 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | APPLIED MATHEMATICS |
| Primary Program Source: |
01002021DB NSF RESEARCH & RELATED ACTIVIT 01002122DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
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.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The work funded by this project set out to develop a theoretical framework to gain insight into complex social and ecological systems. There were three main projects, two related to ecological systems and one related to social systems. The body of work funded by this grant encompassed the development of models, theory, numerical schemes, and algorithms to incorporate data. We have pushed the theoretical boundaries of new mathematical models and obtained insight into many applications along the way.
One project was centered around the modeling of social and environmental interactions in social animals in order to understand territorial patterns using the non-local nature of their decision making. This work was motivated by the fact that a good wildlife management strategy relies on the understanding of (1) the factors that drive animals to move the way they do and of (2) what good movement strategies are. Thanks to the funding from this grant, we have been able to develop mathematical models that incorporate different factors that are hypothesized to lead to animal movement, including non-local information. These models can be computationally expensive if one considers ecosystems with many groups. We have developed an efficient numerical scheme these types of systems, which are easily adaptable to other fields. Note that an efficient scheme is essential when trying to incorporate data into these models to be able to determine what factors tend to influence animal movement strategy the most. We have also developed algorithms to incorporate data into these models and then do model selection and have tested them on synthetic data. This provides an objective way to determine what factors are important in the movement of different animals once field data is available. With regards to the second point of ?good? movement strategies, some of the work funded by this grant began to explore this issue for species which are subject to a strong Allee effect, for example species who are highly reliable to their social structure and, therefore, decay when the population is small. We determined that species which depend on their social network benefit from aggregation; however, if the aggregation happens via movement towards areas of high resources moving too quickly can be detrimental in a competitive situation.
On a more theoretical side, this work has contributed to the theory of integro-differential equations that arise from birth-jump processes. We have developed fundamental mathematical properties of integro-differential equations that appear in cases when birth and dispersal cannot be separated. We have studied the equations with and without passive diffusion. The theory we developed looked at maximum principles and comparison principles. We have also developed a foundation to analyze the global well-posedness of a general class of population models with cross-diffusion without an overcrowding effect. These models appear in many areas such in ecology and human population dynamics.
Our last emphasis was on social phenomena. We set out to develop of a methodology and numerical framework to incorporate (social) data in order to test hypotheses developed by sociologists, with gentrification as a first case study. We developed a theory that can help policy makers determine which parameter regimes will lead to wealth hotspots. Our work also contributed to the understanding of solutions to models, which have been previously introduced to model urban crime and social unrest. We studied a system of reaction-advection-diffusion equations modelling urban crime and provide parameter conditions that lead to the existence of spatially heterogeneous solutions and time periodic solutions. Moreover, we discover a region where solutions appear to be chaotic. This has significant consequences from the application perspective, essentially stating that predictions with this model are not reliable as small noise in data could lead to a completely incorrect solution. Moreover, we study the existence of traveling wave solutions which are non-monotone, and which resemble data from the French 2005 riots. Our results imply that the French 2005 riots are tension-inhibitive, in the sense that protesting led to a reduction of social tension.
Last Modified: 11/15/2022
Modified by: Nancy Rodriguez
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