undergraduate research
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Dispersal-induced global extinction in two-patch model under the Allee effect
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description
From the summer of 2015 until the spring of 2017, I had the pleasure of working on a population dynamics research project with Drs. Leah Shaw and Junping Shi of the William & Mary mathematics department. I started with an established system of two coupled logistic equations undergoing dispersal, which modeled the dynamics of a marine population (inspired by Chesapeake Bay oysters), and was given free rein to explore the dynamics. From there, I uncovered a rich dynamical architecture, and created diagrams exploring its steady-state and bifurcation structure in a qualitative fashion. Currently, the paper is still in preparation. my contribution The abstract, literature review, non-dimensionalization of the model in Section 2; Lyapunov proof in Section 3; and all results, diagrams, and analyses were authored by myself with guidance from my advisors. The general model presented in Section 2 was devised by Leah Shaw and Junping Shi, and proofs from Section 3, Theorem 2 until the end of that section were constructed by Junping Shi. |
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abstract
"Aquatic currents often create unequal patterns of dispersal from one population of organisms to another. We study two population patches coupled by this asymmetric dispersal, with population conserved during migration. In addition, our coupled patches are subject to the positively density-dependent Allee effect: Initial populations below an inherent threshold decline towards extinction, while those above the threshold persist towards their carrying capacity. Analytically, we show that there are no periodic orbits, find a perpetually locally stable equilibrium at the extinction state (0,0), and define parameter ranges within which positive equilibria must occur. Numerically, we explore one- and two-dimensional steady-state bifurcation structures, varying dispersal rates and the Allee threshold. At high Allee thresholds, we uncover large parameter ranges in which the extinction state is the only fixed point. These regions have not been shown to occur with asymmetric dispersal rates with no mortality during migration; we therefore focus on the bifurcations that lead to this surprising extinction state. It is crucial to understand this behavior in efforts to restore endangered species that exhibit an Allee effect and asymmetric dispersal."
Because the paper is in preparation, I am not able to attach a copy at this time.
"Aquatic currents often create unequal patterns of dispersal from one population of organisms to another. We study two population patches coupled by this asymmetric dispersal, with population conserved during migration. In addition, our coupled patches are subject to the positively density-dependent Allee effect: Initial populations below an inherent threshold decline towards extinction, while those above the threshold persist towards their carrying capacity. Analytically, we show that there are no periodic orbits, find a perpetually locally stable equilibrium at the extinction state (0,0), and define parameter ranges within which positive equilibria must occur. Numerically, we explore one- and two-dimensional steady-state bifurcation structures, varying dispersal rates and the Allee threshold. At high Allee thresholds, we uncover large parameter ranges in which the extinction state is the only fixed point. These regions have not been shown to occur with asymmetric dispersal rates with no mortality during migration; we therefore focus on the bifurcations that lead to this surprising extinction state. It is crucial to understand this behavior in efforts to restore endangered species that exhibit an Allee effect and asymmetric dispersal."
Because the paper is in preparation, I am not able to attach a copy at this time.
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A Deeper Analysis of the FitzHugh-Nagumo Neuron Model:
Bifurcations and Synchronization |
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description
As a final project for a mathematical modeling course I took in 2016, my partner Ellen Nguyen and I chose to study the dynamics of the well-known FitzHugh-Nagumo model of neurons and action potentials. We derived the model and studied its phase portrait and the Hopf bifurcation that occurs at certain values of the incoming voltage into the neuron. We also discussed the biological relevancy of these Hopf bifurcations and examined the activity of coupled neurons using a modifed FitzHugh-Nagumo model. my contribution Given my expertise on the mathematical side, and Ms. Nguyen's expertise in neuron function and a broader background in neuroscience, our duties were well-defined. Model derivation (Section 2), and bifurcation diagrams and analysis (Section 3) were authored by myself. My partner's contribution was in the introduction and motivation (Section 1), and discussion and future directions (Section 4). | ||
The Russian Movie Theater Project: Natural Language Processing & User Interface
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description
In the summer of 2016, I was afforded an incredible opportunity to study abroad for 6 weeks in St. Petersburg, Russia. While there, I conducted an interview with local resident Ekaterina Lebedeva for the Russian Movie Theater Project, a project focused on the history of Russian cinema and native Russians' memories of Soviet movie theaters. On the project, we analyzed the structure of the interviewees' natural language in conjunction with more cultural references and underpinnings. my contribution Along with my interview, I worked in a group to transcribe and translate into English the interviews captured that summer. After transcription, we used XML to 'tag' mentions of places, names, and films in the interviews for later natural language analysis. This tagging was formerly a manual process, and was prone to many errors as most students weren't familiar with proper XML nesting formats. I approached my professors about semi-automating the process using Python, and they enthusiastically embraced the idea. I worked with a fellow Russian student to code an interactive module to assist the researchers in this tagging and analysis, and built a web page from scratch to explain our process. |
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