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MBI Seminar: Chuan Xue - Spatial Pattern Formation in Reaction-Diffusion Models: A Computational Approach

Chuan Xue
February 11, 2020
All Day
MBI Auditorium, Jennings Hall 355

Time: 10:20-11:15 a.m.
Event Host: Mathematical Biosciences Institute (MBI)
Short Description: In this lecture, Chuan Xue, MBI associate director and associate professor in the Department of Mathematics, presents a systematic method to compute multiple nonuniform steady states for reaction-diffusion models and determine their dependence on model parameters.


Reaction-diffusion equations have been widely used to describe biological pattern formation. Nonuniform steady states of reaction-diffusion models correspond to stationary spatial patterns supported by these models. Frequently these steady states are not unique and correspond to various spatial patterns observed in biology. Traditionally, time-marching methods or steady state solvers based on Newton's method were used to compute such solutions. However, the solutions that these methods converge to highly depend on the initial conditions or guesses.

In this lecture, Chuan Xue, MBI associate director and associate professor in the Department of Mathematics, presents a systematic method to compute multiple nonuniform steady states for reaction-diffusion models and determine their dependence on model parameters. The method is based on homotopy continuation techniques and involves mesh refinement, which significantly reduces computational cost. The method generates one-parameter steady state bifurcation diagrams that may contain multiple unconnected components, as well as two-parameter solution maps that divide the parameter space into different regions according to the number of steady states.

Xue will discuss how the method was applied to two classic reaction-diffusion models and compare the results with available theoretical analysis in the literature. The first is the Schnakenberg model which has been used to describe biological pattern formation due to diffusion-driven instability. The second is the Gray-Scott model which was proposed in the 1980s to describe autocatalytic glycolysis reactions. In each case, the method uncovers many, if not all, nonuniform steady states and their stabilities. (Joint work with Wenrui Hao from Penn State University.)

This talk is free and open to the public. Please visit this webpage for more information.

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