1. Environmental geometry and species diversity. One of the oldest and most celebrated lines of inquiry in ecology seeks to explain the fascinating prevalence of large-scale patterns in biological diversity. Most attention has been paid to why there are more species in the tropics, but similar questions can be asked of elevational gradients along mountainsides, depth transects in the ocean, or even microbial gradients within the human body. A multitude of hypotheses have been proposed to explain these biodiversity gradients, and arbitrating among these hypotheses requires an accurate understanding of the predictions that each makes. Towards this end, MS student Andrew Snyder-Beattie and I constructed a mathematical model that synthesizes a subset of these hypotheses that fundamentally rely on the geometry of environmental variation, such as latitudinal or elevational temperature gradients. Surprisingly, our model shows that together these geometric hypotheses predict more subtle and nuanced fine structure in biodiversity gradients than any single geometric hypothesis does alone. Encouragingly, these predictions seem to match fine structure that has been noted in multiple published empirical gradients, although that fine structure has rarely received deep attention (perhaps because of a lack of theory that anticipates it). A first paper describing our model appeared in The American Naturalist in 2016.
2. Science of science. Contemporary science can be viewed as a complex dynamical system, with many participants embedded in an increasingly connected social network, and each acting according to their own philosophies and incentives. How does such a community of scholars allocate scarce resources efficiently to expedite the collective acquisition of knowledge? I am interested in using mathematical models to understand this question and others in the burgeoning field of the science of science. My first foray into this discipline was a successful collaboration with Carl Bergstrom's group that used the mathematics of random walks to exolore how publication practices facilitated or impeded the collective evaluation of scientific claims of modest import (such as whether a particular dietary practice changes the chance of a particular health outcome). Our work will appear in eLife in late 2016.
3. Stability, resilience, and change on coral reefs. Coral-reef ecosystems are both spectacular and spectacularly diverse, yet reef ecosystems as we know them are imperiled by rapid environmental change. Together with my collaborators Peter Edmunds and Bob Carpenter, I am developing mathematical tools to help understand what continued environmental change portends for coral reefs and the ecosystems they support. We are particularly interested in understanding the effects of ocean acidification on coral reefs, and how the impacts of ocean acidification may interact with other stressors such as warmer seas and more frequent and/or more intense typhoons. Our work builds from experiments that Peter and Bob are leading at the Gump Environmental Research Station in Moorea, French Polynesia. In previous work, Peter and I used vector autoregressive models (statistical models for time-series data) to estimate metrics of ecosystem stability for coral communities in St. John, USVI.
4. Transgenic pest control. Genetic technologies offer new promises for regulating abundances of unwanted pest species. Mathematical modeling will be a key component of the design of successful pest-control strategies using transgenic methods. The Genetic Engineering and Society Center at NCSU, in which I participate, brings together scholars to examine the social, biological, and technical aspects of the use of genetic technology for various objectives, including the management of unwanted pests. PhD student Gregory Backus has developed the first mathematical models for exploring how transgenic technology might be used to eradicate invasive species from island ecosystems in ways that are more environmentally benign than other contemporary alternatives.