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Analysis of a reaction-diffusion system with local and nonlocal diffusion terms.

机译：具有局部和非局部扩散项的反应扩散系统的分析。

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摘要

Reaction-diffusion describes the process in which multiple participating chemicals or agents react with each other, while simultaneously diffusing or spreading through a liquid or gaseous medium. Typically, these processes are studied for their ability to produce nontrivial patterns that evolve over time. These patterns, often referred to as Turing structures or Turing patterns, are diffusion driven. In the presence of diffusion, the Turing patterns are observable, but are not present in the absence of diffusion. It is important for reaction-diffusion models to replicate the behavior that is experimentally observed. That is to say that the models must be able to produce solutions with traits, such as pattern type, that are similar to experimentally observed traits. Mathematically, we seek to explain certain aspects of the models such as pattern selection in the hope of broadening our understanding of the underlying process for which the model represents.;I analyze a mixed reaction-diffusion system containing an instability that results in nontrivial Turing structures. This system uses a homotopy parameter beta to vary the effect of both local (beta = 1) and nonlocal (beta = 0) diffusion. Furthermore, I consider epsilon-scaled kernels J such that epsilonthetaJ is epsilon-independent for theta ∈ R . For theta 1 and 0 beta ≤ 1, I show that the generated Turing patterns are explained using only finite number of eigenfunctions corresponding to the most unstable eigenvalues of the linearization. However, for theta = 1 and beta 1, I show how the nonlinearity is no longer bounded above by an epsilon-dependent bound that ensures the smallness of the nonlinearity as in the theta 1 case. The lack of this critical bound allows for a greater influence of the nonlinearity. Consequently, the unstable eigenfunctions of the linearization do not describe the solutions as well as they do for the solutions of the theta 1 case. The numerics provided show little agreement between the solutions and their linearized counterparts as a consequence of greater influence of the nonlinearity.;The thesis is concluded with numerical pattern studies of the local and nonlocal reaction-diffusion systems. The patterns are studied as the values of various parameters of the reaction-diffusion system are changed. These numerical experiments reveal typical patterns such as stripes and spots, as well as irregular snakelike patterns. Furthermore, solutions for the local system subject to homogeneous Neumann boundary conditions are compared to the solutions of the local system subject to periodic boundary conditions. For some cases, the solutions for both systems are quite similar.
机译：反应扩散描述了一种过程，在该过程中，多种参与的化学物质或试剂相互反应，同时在液体或气体介质中扩散或扩散。通常，研究这些过程的能力以产生随时间变化的非平凡模式。这些模式（通常称为图灵结构或图灵模式）是扩散驱动的。在存在扩散的情况下，可观察到图灵图案，但在不存在扩散的情况下则不存在。对于反应扩散模型，重要的是要复制实验观察到的行为。也就是说，模型必须能够产生具有类似于实验观察到的特征的特征（例如模式类型）的解决方案。在数学上，我们试图解释模型的某些方面，例如模式选择，以期拓宽对模型所代表的基本过程的理解。我分析了包含不稳定性的混合反应扩散系统，该不稳定性导致非平凡的图灵结构。该系统使用同位参数β来改变局部（beta = 1）和非局部（beta = 0）扩散的影响。此外，我考虑了epsilon缩放的核J，使得epsilonthetaJ对于theta∈R独立于epsilon。对于theta <1和0

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  • 作者

    Tatum, Richard D.;

  • 作者单位

    George Mason University.;

  • 授予单位 George Mason University.;
  • 学科 Applied Mathematics.;Chemistry Physical.
  • 学位 Ph.D.
  • 年度 2010
  • 页码 229 p.
  • 总页数 229
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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