Research Overview

Motivated by problems of environmental change, I study how far dynamic structures persist in the face of disturbances and uncertainty. Since the field’s celestial origins, dynamical systems have proved useful for describing and predicting many phenomena on Earth, from the flow of nutrients through an ecosystem to the balance between the planet’s incoming solar energy and outgoing heat radiation. As dynamicists, we often define a system’s structure based on its long-term behavior and the topology of its invariant sets. However, today’s most pressing environmental questions require an expanded mathematical perspective. Disturbances from both human activities and natural processes (e.g. glacial meltwater, fertilizer run-off, fires) perturb natural systems away from long-term attractors. In such systems, interactions between disturbances and short-term recovery processes drive outcomes. Broadening the scope of dynamical systems theory from long-term to transient behaviors leads to new and challenging mathematical questions.

My work probes aspects of attractor strength that shape the interplay between specific disturbances and recovery, as well as dynamic structures that constrain behavior under multiple disturbance scenarios. In addition to applying established mathematical theory to questions of environmental change, I develop new theory when existing tools do not suffice. The tools that I create support prediction and management of environmental impacts ranging from climate change to habitat destruction and nutrient pollution. My projects and interests thus span the full spectrum from pure mathematics to applied and even interdisciplinary research.

Current Projects

Doctoral Thesis

My thesis work with Dick McGehee puts a number on how intensely an attractor in a continuous time dynamical system pulls on its domain of attraction. This "intensity of attraction" provides a potential indicator of an attracting regime's resilience to disturbances. It also dictates in metric terms how far an attractor must persist as a vector field changes, and thus introduces a quantitative descriptor for the qualitative concept of robustness. This work extends theory that McGehee developed for discrete time to continuous time. Key tools include reachable sets in non-autonmous control systems, attractor blocks, and dynamical systems lacking forward uniqueness in time.

Flow-kick Modeling of Disturbance and Resilience

Given the ubiquity of both planned and unplanned ecosystem disruptions from climate change, nutrient loading, and habitat destruction, promoting the resilience of desired ecological regimes and services to perturbations has emerged as key goal in natural resource management. Measuring what we wish to manage requires methods for resilience quantification. This project develops resilience metrics tailored to repeated, discrete perturbations such as nutrient pulses, harvests, and storms. Representing disturbances as instantaneous "kicks" to state space interspersed with flows allows analysis of disturbed dynamics via a discrete map, whose equilibria give insight into disturbed behavior and resilience. This is joint work through the Math Climate Research Network with Mary Lou Zeeman, Alanna Hoyer-Leitzel, Sarah Iams, and students Stephen Ligtenberg, Ian Klasky, Victoria Lee, and Erika Bussmann.