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Global dynamics of the local and nonlocal Patlak-Keller-Segel chemotaxis systems.

机译：局部和非局部Patlak-Keller-Segel趋化性系统的全局动力学。

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摘要

Chemotaxis is the directed movement of cells in response to chemical gradients. This thesis is devoted to studying two generic classes of two coupled parabolic equations modeling chemotaxis: a classical (local) Patlak-Keller-Segel chemotaxis model with/without growth (the proliferation and the reduction due to birth and death of cells) and a nonlocal gradient P-K-S chemotaxis model. Here, we first obtain new and deep characterizations of blowup mechanism for the chemotaxis models with/without growth. Then, under simple conditions on the growth source, we use these criteria to establish the global boundedness of the underlying models, which serves as a fundamental step to understand the dynamics of these models. Hence, it is possible to study the convergence of such solutions to bounded steady states with striking features such as spikes and transition layers. For the nonlocal model, part of our results follows this direction. More importantly, our results provide a full picture on how the sampling radius affects pattern formation and stability.;For the P-K-S chemotaxis systems with/without growth in n-D, it is recently known that blow-up is possible even in the presence of linear birth and superlinear degradation. Here, we derive new and interesting characterizations on the growth versus the boundedness. We show that the L r-boundedness of the cell density can guarantee its Linfinity-boundedness and hence its global boundedness, where r = n + epsilonor n/2 + epsilon, depending on whether the growth source is essentially linear (including no growth) or superlinear. Hence, a blowup solution also blows up in Lp-norm for any suitably large p. More detailed information on how the growth source affects the boundedness of the solution is derived. Later on, we use these criteria to derive some global boundedness and existence results: logistic growth in 2-D, cubic growth as initially proposed by Mimura and Tsujikawa in 3-D, (n--1) st growth in n-D with n ≥ 4, or cubic growth and convex domain in n-D with n ≥ 1. Therefore, in a chemotaxis-growth model, blow-up is impossible if the growth source is suitably strong. Finally, we stress that these results remove the commonly assumed convexity on the domain.;For the nonlocal chemotaxis model, we first correct the nonlocal gradient modified by Hillen-Painter-Schemiser from Othmer-Hillen, which fails to take care of the "boundary effects." Subsequently, we obtain the boundedness and the global existence of its solution in 1-D. As a byproduct, it offers a justification for the common belief that P-K-S models normally have no blow-ups in 1-D. Then we obtain the limiting equations when the sampling radius rho→ 0, as well as convergence to steady states when time t → infinity. Next, we show that the model has the ability to give rise to pattern formation if the chemotactic coefficient is larger than an expressible bifurcation value. Interestingly, the smaller the cell, the more likely pattern formation will occur. Then we establish a characterization of the limiting profiles for the nonconstant steady states as either spiky or of transition layer type. Finally, we obtain the full stability information for the nonconstant bifurcating solutions. Surprisingly, the stability is independent of the net creation rate of the chemical and is closely related to the cell radius. As a result, a critical degradation rate phenomenon is found: if the cell degradation rate lies below (above) a threshold/stabilizing value, the cell is stable (unstable). This threshold value is an increasing function of the cell radius. The large cells can compensate their degradation of the chemical, and become stable; however, for small cells to be stable, their degradation rate must be less than a threshold value.
机译：趋化性是细胞响应化学梯度的定向运动。本论文致力于研究两个耦合的抛物线方程的两个通用类别，它们建模了趋化性：具有（或不具有）增长（由于细胞的出生和死亡引起的增殖和减少）和非局部性的经典（局部）Patlak-Keller-Segel趋化性模型梯度PKS趋化模型。在这里，我们首先获得具有/不具有增长的趋化模型的爆破机理的新的和更深的特征。然后，在增长源的简单条件下，我们使用这些标准来建立基础模型的全局有界性，这是理解这些模型的动态性的基本步骤。因此，可以研究具有尖峰特征（例如尖峰和过渡层）的有界稳态的此类解的收敛性。对于非局部模型，我们的部分结果遵循该方向。更重要的是，我们的结果提供了有关采样半径如何影响图形形成和稳定性的完整描述。对于具有/不具有nD增长的PKS趋化系统，最近知道即使在线性出生的情况下也可能发生爆炸和超线性退化。在这里，我们得出关于增长与有界性的有趣的新特征。我们表明，细胞密度的L r有界可以保证其Linfinity有界，因此可以保证其整体有界，其中r = n + epsilonor n / 2 + epsilon，这取决于生长源是否基本上是线性的（包括无生长）或超线性。因此，对于任何适当的大p，爆破解决方案也会以Lp范数爆破。得出有关增长源如何影响解的有界性的更多详细信息。后来，我们使用这些标准得出一些全局有界性和存在性结果：二维的逻辑增长，三村和津川最初在3-D中提出的立方增长，n的（n-1）st增长，n≥ 4或nD中n≥1的立方增长和凸域。因此，在趋化生长模型中，如果增长源适当强壮，则不可能发生爆炸。最后，我们强调这些结果消除了该域上通常假定的凸度。对于非局部趋化模型，我们首先纠正了由Othmer-Hillen的Hillen-Painter-Schemiser修改的非局部梯度，该梯度未能解决“边界”问题。效果。”随后，我们获得了其一维解的有界性和整体存在性。作为副产品，它为人们普遍认为P-K-S模型通常不会在1-D中发生爆炸提供了理由。然后，当采样半径rho→0时，我们得到了极限方程；当时间t→无穷时，我们又收敛到稳态。接下来，我们表明如果趋化系数大于可表达的分叉值，则该模型具有引起图案形成的能力。有趣的是，单元越小，越可能发生图案形成。然后，我们建立了针对非恒定稳态的尖峰或过渡层类型的极限轮廓的表征。最后，我们获得了非恒定分叉解的完整稳定性信息。出人意料的是，稳定性与化学物质的净生成率无关，并且与细胞半径密切相关。结果，发现了严重的降解速率现象：如果细胞降解速率低于（高于）阈值/稳定值，则细胞稳定（不稳定）。该阈值是像元半径的增加函数。大电池可以补偿其化学降解，并变得稳定；但是，为了使小小区稳定，它们的退化率必须小于阈值。

著录项

  • 作者

    Xiang, Tian.;

  • 作者单位

    Tulane University School of Science and Engineering.;

  • 授予单位 Tulane University School of Science and Engineering.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2014
  • 页码 206 p.
  • 总页数 206
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 物理化学（理论化学）、化学物理学;
  • 关键词

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