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PREFACE: ANALYSIS OF CROSS-DIFFUSION SYSTEMS

机译：前言：交叉扩散系统分析

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Directed motion of biological organisms in response to changes of chemical cues in their environment is an ubiquitous mechanism which occurs at all levels of complexity in biological systems, starting from a sub-cellular level through tissue levels up to the interspecies interactions on ecological or social levels. In the 1970s, Keller and Segel proposed their celebrated model for chemotaxis describing movement of unicellular microscopic organisms in response to the gradient of a chemical signal secreted by the organisms themselves and potentially leading to their eventual aggregation in small spatial regions. The Keller-Segel model and most of its extensions modelling various phenomena related to tumour growth or interspecies interactions in ecology belong to the class of quasilinear parabolic systems with triangular main part but nontrivial off-diagonal entries, also referred to as cross-diffusion systems. During the last few decades many efforts of mathematicians have been focused on the understanding of the singularity formation in nite time in such systems which is related to the process of aggregation and pattern formation. On the other hand, many works were devoted to nding conditions which prevent such nite-time blow-up, thus guaranteeing the existence of global-in-time solutions. Depending on the time scale of biological processes involved some models couple the parabolic equation describing the main component of a system, including the characteristic advective cross-diffusion part, with elliptic equations or even ODEs describing the evolution of the respectively remaining components. Some of the articles included in this volume contain rigorous analysis of new models in mathematical biology describing e.g. tumor growth or virus infection. Other contributions concentrate on new analytical results describing qualitative properties of well-established models.
机译：生物生物的定向运动响应于其环境中化学线索的变化是一种普遍存在的机制，其在生物系统中的所有水平中发生，从亚细胞水平通过组织水平达到生态或社会层面的互动的互动。 。在20世纪70年代，Keller和Segel提出了他们庆祝的模型，用于描述单细胞微观生物的运动，以应对生物体本身分泌的化学信号的梯度，并且可能导致其在小空间区域中的最终聚集。 Keller-Segel模型和其大部分扩展建模与肿瘤生长或生态学中的间隙相互作用的各种现象属于具有三角形主要部分但非竞争非对角线条目的Quasilinear抛物线系统的类别，也称为交叉扩散系统。在过去的几十年里，数学家的许多努力都集中在与聚集和模式形成过程有关的这种系统中的情况下的奇点形成的理解。另一方面，许多作品都致力于Nding条件，这防止了这种爆炸，从而保证了全球数据解决方案的存在。取决于生物过程的时间规模涉及一些模型，将描述系统的主要成分的抛物线方程耦合，包括特征方程式交叉扩散部分，其具有椭圆形式甚至描述分别剩余部件的演变的odes。本卷中包含的一些物品包含对描述例如描述的数学生物学的新模型进行严格分析。肿瘤生长或病毒感染。其他贡献集中在新的分析结果上，描述了熟悉的型号的定性特性。

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