My research broadly involves studying how stochasticity (i.e., randomness) influences ecological processes. Below, I briefly describe two major components of my research.
- Using quasi-potentials to analyze stability in stochastic ecological sytems
Stochastic influences are ubiquitous in ecological systems. In systems with alternative stable states, these random variations can cause abrupt shifts from one stable state to another. The results can be catastrophic (e.g., if an ecosystem transitions from a state corresponding to species coexistence to a state corresponding to the extinction of one or more species). Predicting how resilient an ecosystem is in the face of stochastic perturbations is a key question for biologists.
In physics, potential functions can be used to analyze stochastic dynamics. The state of a system can be viewed as the position of a ball rolling on a surface specified by the potential function. The bottoms of valleys correspond to stable states. Random perturbations jostle the ball as it rolls; occasionally these cause the ball to roll from one valley to another. The potential function can be used to make predictions about stationary probability distributions, expected first passage times, and transition frequencies.
Unfortunately, potential functions only exist for gradient systems, which, although common in physics, are almost never found in ecology. Fortunately, Freidlin-Wentzell theory created the concept of quasi-potential functions, which generalize potential functions for non-gradient systems. In my research, I use quasi-potential functions to make predicitons about the stability and resilience of ecological systems in the face of stochastic perturbations. This type of analysis is described in my paper with Dr. Abbott, "Balls, cups, and quasi-potentials: quantifying stability in stochastic systems". I worked with a team to design an R software package called QPot to calculate quasi-potentials. This software is available here, and is described in our recent paper "QPot: An R package for stochastic differential equation quasi-potential analysis".
- Using stochastic search strategies to model how organisms forage for food
Organisms ranging in size from E. coli to great white sharks are confronted with a common challenge: what is the best way to search for food resources? When these organisms do not have a priori information about the location of food items, they must implement a set of probabilistic movement rules called a stochastic search strategy. My research involves studying search strategies using a combination of mathematical analysis, computer simulation, and field observation. In our recent paper "Composite random search strategies based on non-proximate sensory cues", my coauthors and I explore how organisms can use a process called kinesis to locate resources.
I endeavor to be a broad and versatile mathematical biologist, both in terms of the mathematical techniques I employ and the biological topics I address. Mathematically, I am interested in stochastic proccesses in general (and stochastic differential equations in particular), partial differential equations, dynamical systems, and spatial statistics. Biologically, I am interested in population dynamics, spatial ecology, epidemiology, community ecology, ecosystem ecology, animal behavior, evolution, and physiology. I am always excited to broaden my horizons, so please contact me if you are interested in a research collaboration.