PIMS-UNBC Distinguished Colloquium: Michael Ward
Topic
Pattern Formation and Synchrony for Reaction-Diffusion Systems with Dynamic Boundary Conditions
Speakers
Details
Abstract: A new frontier for the modeling and analysis of reaction-diffusion PDE systems is where the diffusing species are nonlinearly coupled through dynamic interactions on the domain boundaries. Some specific examples of such systems, including bulk-membrane and bulk-cell reaction-diffusion systems, will be discussed, highlighting their importance in applications and the new solution behavior observed. For a specific such system, consisting of a scalar diffusion field that is coupled spatially in R^2 to a collection of small dynamically active "cells'', we will present a new memory-dependent ODE integro-differential system that characterizes how intracellular oscillations in the collection of cells are coupled through the PDE bulk-diffusion field. By using a fast numerical approach relying on the sum-of-exponentials method to derive a time-marching scheme for this nonlocal system, diffusion induced synchrony, as measured by the Kuramoto order parameter, is examined for various spatial arrangements of cells. This theoretical modeling framework, relevant when spatially localized nonlinear oscillators are coupled through a PDE diffusion field, is distinct from the traditional Kuramoto paradigm for studying oscillator synchronization on networks or graphs.