Erin Ellefsen and Lindsey Wong, Department of Applied Mathematics, University of Colorado Boulder
Lyndsey’s Title: Mathematical Models of Wealth Distribution Through an Amenities-Based Theory
Erin’s Title: Efficiently finding Equilibrium Solutions of Nonlocal Models in Ecology
The dynamics of wealth are not fully understood. In order to gain insight on these dynamics, we can use mathematical models. One application of modeling wealth distribution is gentrification. Gentrification refers to the influx of income into a community leading to the improvement of an area through renovation or the introduction of local amenities. It is often accompanied by an increase in the cost of living, which displaces lower income populations. Gentrification is a core issue that affects many urban areas. In this talk, we will present an overview of the work done with our amenities-based approach to modeling wealth distribution.
First, we present a PDE model derived from transport theory and prove the existence and stability of spatially heterogeneous solutions through a global bifurcation result. Next, we improve this PDE model by instead deriving from first principles and then perform a sensitivity analysis to see which parameters create the most change in solutions when perturbed. We then begin to work with data for Baltimore, MD via the Baltimore Neighborhood Indicators Alliance-Jacob France Institute Vital Signs report. To better understand this data, we use machine learning to determine what factors are most important in predicting neighborhood change. Lastly, we discuss some preliminary results of our current project in which we are incorporating this data into metapopulation models which are based on our previous PDE models.
Understanding the factors that drive species to move and develop territorial patterns is at the heart of spatial ecology. We introduce a nonlocal system of reaction-advection-diffusion equations that incorporate the use of nonlocal information to influence the movement of species. One benefit of this model is that groups are able to maintain coherence without the inclusion of artificial dynamics. As incorporating nonlocal mechanisms comes with analytical and computational costs, we explore the potential of using long-wave approximations of the nonlocal model to determine if they are suitable alternatives that are more computationally efficient. We use the gradient flow-structure of the both local and nonlocal models to compute the equilibrium solutions of the mechanistic models via energy minimizers. In an effort to incorporate data into our model, we turn to spectral methods to more efficiently find equilibrium solutions of the nonlocal model and test the model against synthetic data.