# Research

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## Research Interests

A list of my research interests include:

- Developing mathematical models which help explain fundamental questions in evolution and ecology, such as
- Are ecological constraints able to shape patterns of evolution? If ecological and evolutionary dynamics occur on the same temporal scales (so-called eco-evolutionary dynamics), what can we understand and even influence about the system?
- Can we understand species coexistence through pairwise species interactions?
- Many phenotypes are almost continuous in nature, but there are countless examples in nature in which distinct, discrete phenotypes arose (such as the life history strategies of anadromous trout). What causes a potentially infinite dimensional set of possible strategies to collapse to a low-dimensional space of strategies? How can we describe evolution as exploring this space of strategies?

- Applying theory to specific areas of concern and interest:
- Restored grasslands harbor significantly less biodiversity than virgin prairies. Can we use a theoretical knowledge of species invasions and priority effects to help restore systems with higher levels of biodiversity?
- Rapid anthropogenic changes in fundamental nutrient cycles (in particular, the massive production of chemical fertilizers) cause significant damages including algal blooms which lead to dead zones in oceans and seas. Has this resulted in subsequent evolutionary changes on the primary producers, and what impacts does this have on ecosystems?
- Modern agriculture is performed on a systematic, industrial scale to optimize for yields above most other factors. Lower-intensity forms of agriculture which use fewer petrochemicals and pesticides may be much friendlier to coexisting with wildlife, but is not commonly practiced. Are there ways to design agricultural systems which achieve a different optimal point--considering a plethora of factors such as food production, human well-being, regional cultures and economies, fundamental ecosystem services, biodiversity, and the long-term health of land and communities?

With a broad foundation in mathematics, I believe that there are many tools which can be applied to solve these problems, from probability and statistics to computational simulations with high performance computing, dynamical systems. and data science--and some tools which may not be so obvious or discovered yet.

## Current Research

I am currently a rotation student in the lab of Noah Rosenberg. I am studying mathematical phylogenetic trees constructed through evolutionary processes, and how they change over time.

## Past Research

As an undergraduate, I worked on two main research projects:

- (Advised by Corina Tarnita): Abstract: Interactions between bacteria which produce antibiotics and those which produce degrading enzymes to counteract the antibiotics are common in nature. They have been proposed as a mechanism enabling robust species coexistence and the stability of communities with many species. Current approaches to this rely on complex simulations, without effective mathematical treatment or insight. As a possible approach to better understand this system, I describe these bacterial interactions as a Wright-Fisher evolutionary model on a randomly colored graph. By taking an infinite-population limit, I derive a replicator equation, which is studied at varying degrees of simplicity. I show that the interaction can be captured by a three-strategy matrix game which exhibits stability. Three-strategy matrix games are then studied more generally, and I prove a theorem connecting coexistence of all species in the replicator equation to the ability of smaller sub-communities to be invaded.
- (Advised by Simon Levin): Nitrogen fixation by symbiotic plant-bacterial interactions plays a crucial role in global geochemical systems. Nitrogen fixation is a costly action performed by plants that allows them to exchange sugars for bacterially-fixed nitrogen in a usable form. However, nitrogen fixers can enrich the soil nitrogen of plants in their vicinity, thus contributing to a public good. Hence, they are enabling their competitors who do not fix nitrogen. To explore this paradox, I developed a continuous spatial model incorporating plant and nutrient dynamics to examine forest systems with nitrogen fixation.