论文标题
图灵的不稳定性分析奇异的交叉扩散问题
Turing instability analysis of a singular cross-diffusion problem
论文作者
论文摘要
Busenberg和Travis的种群模型是生态学和肿瘤建模的范式模型，因为它能够捕获有趣现象（例如种群的隔离）。它的奇异数学结构强制考虑正规化问题，以推断出与解决方案存在一样基本的属性。在本文中，我们对一般的正规化问题进行了弱非线性稳定性分析，以研究正则化参数极限的不稳定性模式的收敛性。我们用一些具体的示例证明，由于振荡的振幅衰减，在限制问题中观察到的模式形成（无界波数）在极限问题中不存在。我们还通过对问题的直接有限元仿真来检查稳定性分析的结果。
The population model of Busenberg and Travis is a paradigmatic model in ecology and tumour modelling due to its ability to capture interesting phenomena like the segregation of populations. Its singular mathematical structure enforces the consideration of regularized problems to deduce properties as fundamental as the existence of solutions. In this article we perform a weakly nonlinear stability analisys of a general class of regularized problems to study the convergence of the instability modes in the limit of the regularization parameter. We demonstrate with some specific examples that the pattern formation observed in the regularized problems, with unbounded wave numbers, is not present in the limit problem due to the amplitude decay of the oscillations. We also check the results of the stability analysis with direct finite element simulations of the problem.
