| Abstract: |
| Collective pattern formation arises in many biological systems, including cell aggregation, tissue organization, animal skin patterns, and interacting populations. Reaction-diffusion models provide an effective framework for describing the emergence and evolution of such spatial structures, but numerical simulation becomes challenging when solutions develop localized patterns, sharp gradients, and multiscale features. In this talk, I present moving mesh methods as an efficient computational approach for biological reaction-diffusion systems. The main idea is to adaptively redistribute mesh points according to the evolving solution profile, so that computational effort is concentrated near regions with significant spatial variation while unnecessary resolution is avoided elsewhere. I will discuss the basic idea and implementation of the moving mesh framework combined with finite difference discretizations, and illustrate its performance through examples exhibiting spike-type patterns, transition layers, and self-organized structures. These results show that moving mesh methods provide a practical and accurate tool for analyzing complex pattern formation phenomena in mathematical biology. |
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