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Research Summary

My general research area is physical applied mathematics with an emphasis on modern applications. One of my research goals has been to combine asymptotic and singular perturbation techniques with numerical methods in order to analyze nonlinear boundary-value problems for ordinary and partial differential equations arising in various applied fields. The areas of application include combustion theory, semiconductor device modeling, diffusion in biophysical problems with localized traps, nonlinear eigenvalue problems of MEMS, biological and chemical pattern formation with localized spots and stripes, interfacial dyanmics for reaction-diffusion systems, and metastable processes for chemotaxis, microwave heating, flame-fronts, and models of slow phase separation.

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.

For a recent survey of some of my work on linear and nonlinear diffusion processes in patching domains and reaction-diffusion systems with spatially localized structures please see:

  • Spots, Traps, and Patches: Asymptotic Analysis of Localized Solutions to some Linear and Nonlinear Diffusive Systems,
    (Nonlinearity, Vol. 31, No. 8, (2018), R189).

    Research Areas and Some Highlights