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

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February 11, 2020
10:20AM - 11:15AM
Location
MBI Auditorium, Jennings Hall 355

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Add to Calendar 2020-02-11 10:20:00 2020-02-11 11:15:00 Seminar: Chuan Xue - Spatial Pattern Formation in Reaction-Diffusion Models: A Computational Approach

Chuan Xue

MBI Associate Director, Associate Professor, Department of Mathematics, The Ohio State University


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 paper, we present 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. We applied the method to two classic reaction-diffusion models and compared our 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.

MBI Auditorium, Jennings Hall 355 Mathematical Biosciences Institute mbi-webmaster@osu.edu America/New_York public
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Chuan Xue

MBI Associate Director, Associate Professor, Department of Mathematics, The Ohio State University


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 paper, we present 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. We applied the method to two classic reaction-diffusion models and compared our 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.

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