**Nonlocal Population Dynamics**

To date I have have had two papers published that focus on models of population dynamics governed by nonlocal species interactions, with an emphasis on migratory traveling wave behaviors. In this context, nonlocality refers to interactions between individuals that occur over a distance. To model this we considered the weighted spatial average of the population, which is represented by a convolution of the population against a kernel function that describes the weighting and controls the extent of the nonlocality. The addition of these integral terms leads to nonlinear integro-partial differential equations that can be analytically intractable. Nonlocal models often have parameter ranges that give rise to patterned states. My research focused on the formation of these patterned states and how they interact with the traveling wave, species migration problem. I sought to approach these problems through both analytical and numerical means.

My project which led to the paper `

*Traveling Waves in a Nonlocal, Piecewise Linear Reaction-Diffiusion Population Model*,’ dealt with a nonlocal reaction-diffusion equation with a piecewise linear source term describing the evolution of a normalized population u. In this model, small populations are governed by local natural growth or decay. When the population increases beyond a threshold, growth is instead controlled by nonlocal competition. The system has two equilibrium states,

*u*º 0 and

*u*º 1, and we considered two cases: the monostable case where only the

*u*º 1 state is stable; and the bistable case where both are stable. Our interest was in examining traveling waves connecting these two states. With a speciffic choice of kernel function, this piecewise linear assumption allowed us to reduce the integro-partial differential equation to a system of algebraic equations, which enabled an analytic characterization of traveling waves in the presence of nonlocality.

My project involving a three-species food chain system in the context of biological control led to the paper `

*Biological Control with Nonlocal Interactions*.’ The ecological system in question consisted of a valuable crop u, a pest v infesting the crop, and an artificially introduced superpredator w designed to devour the pest and help restore the crop. The model incorporates ratio dependent functional responses for predation, which leads to singularities when control is achieved, i.e., when the populations of v and w approach zero. In my research I extended this model from a system of ODEs to a system of integro-partial differential equations to allow for species mobility and superpredator nonlocality, and I developed a pseudo-spectral method to numerically solve this system. Due to the singularity, numerical errors (and even round-off errors) can lead to spurious solutions, i.e., spurious resurgences of the pest v. Real resurgences, however, can occur in certain parameter regimes, i.e., when there is enhanced mobility of the pest relative to the superpredator. I identified parameter regimes where robust control is attainable as well as parameter regimes where control is sensitive to small perturbations and is in effect an unstable equilibrium.

Women in Professional Hierarchies

After graduation, I worked on a collaboration attempting to model how the distribution of women at each level of a professional hierarchy changes over time, which led to a paper `

*Mathematical Model of Gender Bias and Homophily in Professional Hierarchies*.’ In our model, we examined the role that cultural and psychological factors, such as promotion bias and homophily (self-seeking), play in the ascension of women through these hierarchies. Incorporating these factors into a model that accounts for the proportion of women at each level in a hierarchy resulted in a system of nonlinear differential equations. This system exhibits a rich range of dynamics, and, importantly, indicates that gender parity is not inevitable. We collected data on the gender fractionation over time of a number of different professional hierarchies, mainly examining the progression of women through different fields in academia. We used this data to verify our model, and quantify the degree of homophily and bias in each hierarchy.

**Quantifying Gerrymandering**

I recently began working with the Quantifying Gerrymandering group at Duke. The long term goal of this group is to develop `the ensemble method for outlier analysis,’ which is used to generate a representative sample of non-partisan maps from a distribution on redistricting. These samples are then used as a comparison against potentially partisan proposals.

The current ensemble method works by first reading precinct level data of the state or county in question, and creating a planar graph where the vertices correspond to individual precincts and the edges indicate physical adjacencies. It then creates a random initial districting plan, and proceeds to generate new plans by randomly sampling from possible alterations to this initial plan, ensuring that all new proposal comply to specified demographic and geographic criteria. Currently, this sampling is done by selecting a random vertex/precinct in the graph and reassigning it to a random neighboring district. Although this single node flip method works for sampling, the group has been interested in other, more robust approaches.

My work with the group has been focused on the implementation of a merge-split procedure for generating new district proposals. In this method, we impose a random spanning tree on each individual district, and then select two random adjacent districts to be merged. A single merged district is created, and a random spanning tree generated using Wilson's algorithm. We then scan the edges of this merged district seeking `valid cuts’ where the removal of the edge would result in two split districts that each comply to the given demographic and geographic data. We randomly select one of these `valid cuts,’ and this creates the two new districts for updating the districting plan. To determine the probability of our sampler accepting this new proposal, we are required to count the total number of possible spanning trees we could have generated, as well as a rather complicated sum, over the number of alternative edges we could have removed to generate this same cut of the number, of other potential cuts that could instead have been made. We ultimately plan to extend this merge-split method into a multi-scale framework, where each precinct can be broken into census blocks, which would allow for more complex splitting and more flexible districting.

**Des Moines Traffic Stops**

I have recently been in contact with several members of the ACLU in my home state of Iowa, who have gone to a tremendous amount of effort to collect data on traffic citations, arrests, and dismissed cases in the capital city, my hometown, Des Moines. Their analysis so far indicates the presence of police bias in both traffic citations as compared to the rest of the populace, as well as in the rates at which traffic citations lead to arrests. So far, this analysis has been entirely performed through the volunteer work of a retired statistician. My roll in this project will be to perform my own, separate, analysis of the data to verify his results. Ultimately, we would like to connect all of the collected data sets to determine rates at which traffic citations lead to arrests or dismissals for minority populations compared with those rates for the white population. The end goal of this project will be to publish a paper of our results in a well regarded journal. This is a critical step in the litigation process, as it adds credibility to the expert testimony, and this process will hopefully force the city to enact reasonable restrictions on bias in traffic stops.

**Future Research**

For the short term, my research is centered on the gerrymandering and traffic stop projects. In the past, the gerrymandering project has employed a number of undergraduates over the summer, and I would hope to help mentor any students brought in over this upcoming summer. I believe that this project is well suited for undergraduate research as it would give them the ability to work with cleaning and connecting large data sets and the analysis of those data sets, all while working on a socially meaningful and impactful problem.

One project in the area of dynamical systems and differential equations I am planning is collecting data on migratory species and using it for determining traveling wave speeds and profiles. This in turn could allow for estimating the relevant biological parameters of the species examined, something which can be hard to find in this field. A related area of interest to me is the study of these migratory problems in higher dimensions. While a Fourier spectral method with sufficiently distant boundaries could be used to examine the behavior of a single `herd’ moving through an otherwise empty domain, this would be insufficient to study larger migration patterns. The problem is that these larger migrations would necessitate interactions with the artificially imposed domain boundary. Boundary interactions are known to give rise to patterned states in these ecological models, and so it would be necessary to ensure that these do not result the artificially imposed domain boundary. One approach might be to introduce artificial damping in an extended layer around the boundary. This problem would require creating the code base to simulate this system along with its numerical complexities, verifying the code base through a grid refinement study, and then examining the behavior of the system through simulation. I believe that this could provide a number of undergraduate research opportunities, while allowing me to teach them about the model involved, the numerical techniques, and the common practices for verification.