This research involves applications of interacting particle systems to biology. In particular, the investigator will construct models to explain biological phenomena such as the temporal oscillations of measles and the patchiness of plankton in the sea. Another area of interest is the connection with nonlinear partial differential equations. By mixing the particles at a fast rate and scaling space, the system converges to a nonlinear partial differential equation. This makes it possible to derive partial differential equations that more accurately model the biological systems and also to prove results about the particle systems with fast mixing. Interacting particle systems have a structure that makes them appropriate for use in a variety of contexts in physics, chemistry, and biology. There is a grid of sites, each of which can have a fixed set of values. For example, a city of individuals can be susceptible to, infected with, or immune to measles. Each site changes its value at a rate that depends on the values of its neighbors, and this research is interested in determining what happens to the system over time. These models should eventually provide a better understanding of global ecological problems.
