Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces.

Area of Science:

• Computational biology
• Mathematical modeling
• Biophysics

Background:

• Biological pattern formation is crucial for development.
• Reaction-diffusion (RD) systems are key models for pattern generation.
• Analyzing RD systems on complex geometries is computationally challenging.

Purpose of the Study:

• To develop computational techniques for studying biological pattern formation on arbitrary surfaces.
• To investigate the impact of surface geometry on reaction-diffusion systems.
• To extend existing RD analysis methods to large-scale surface meshes.

Main Methods:

• Utilized spectral techniques and linear stability analysis for pattern characterization.
• Implemented surface finite element methods and eigenanalysis of the Laplace-Beltrami operator.
• Employed numerical continuation and a multiresolution approach for nonlinear RD equations.

Main Results:

• Developed a framework to analyze reaction-diffusion systems on arbitrary surfaces.
• Demonstrated the ability to trace solution branches efficiently on large meshes.
• Showcased the framework's application to Brusselator and chemotactic models.

Conclusions:

• The developed computational framework enables the study of geometric effects on biological patterns.
• This approach extends pattern formation analysis to complex, real-world surface geometries.
• Provides new insights into how physical surface properties shape biological structures.