A primary focus of my work since 2000 has involved analyzing classes of reaction-diffusion systems exhibiting localized solutions in the form of either spots or stripes. The goal of the analysis is to study the stability and dynamics of these localized structures and to classify the different instability mechanisms through, largely, the study of nonlocal eigenvalue problems. Two of the primary models that have been studied in detail are the Gierer-Meinhardt and Gray-Scott reaction-diffusion systems. The types of instabilities that have been classified in this way are self-replicating instabilities, whereby spots of stripes undergo a division process, breathing instabilities of spots, and competition instabilities leading to spot self-annihilation. The analysis required for studying the stability of localized patterns is significantly different from the usual Turing stability analysis based on the linearization around a spatially inhomogeneous solution. Much of this work has been joint with Juncheng Wei of the Chinese University of Hong Kong, along with my current and former graduate students.
Since 2006 a new focus of my work has involved the analysis and modeling of biological diffusion problems with either small signalling compartments or in the presence of localized traps on cell surfaces. One direction of this work has been to give precise asymptotic estimates on the mean first passage time for diffusion either inside or on the surface of a cell in the presence of localized traps. My collaborators in this area are Ronny Straube of the Max-Planck Institute in Magdeburg, Germany, together with Dan Coombs and Anthony Peirce at UBC and one of my former graduate students.