Sharp traveling waves for degenerate diffusion equations with delay

本文研究了生态学中一类退化的时滞扩散方程--偏微分方程，该方程描述了具有年龄结构的单种群的种群动态，如澳大利亚绿头苍蝇。由于种群竞争或环境条件恶化而引起的退化扩散是生态学中种群扩散模型的一个常见特征，然而，与已有的线性扩散模型相比，退化扩散方程的波动动力学行为还存在许多不足.简并性通常会导致解失去它们的解，并提高了尖锐型行波的可能性，其中种群密度在有限点处减小到零，而不是渐近衰减到零。另一方面，大的时滞往往会导致解的振荡。这两个事实使得解的结构更加复杂，相关的研究也更具挑战性。 本研究的目的是了解扩散的时滞和退化对种群动力学模型的影响。主要目的包括：一是行波的结构。受扩散的时滞和简并性的影响，我们对尖行波、振荡行波和振荡尖波前的分类特别感兴趣。我们将试图证明这些波的存在/不存在，并澄清什么将是标准和机制的人口动力系统拥有这些波前。第二个问题是试图研究这些波前的稳定性，特别是尖波和振荡尖波。因为，尖波的结构完全不同于单调/非单调行波，并且这些波前的渐近稳定性是相当具有挑战性的，并且从未处理过。 该研究项目的提出将进一步丰富和发展退化时滞扩散方程的理论，为研究尖波前稳定性引入新的数学思想和方法，并为培养博士生提供良好的机会。学生和博士后通过执行这个项目。

This research proposal concerns the study of partial differential equations arising from ecology, a class of degenerate diffusion equations with delay, which describe the population dynamics of the single species with age-structure, such as Australian blowflies. Degenerate diffusion caused by competition between conspecifics or deteriorating environmental conditions is a common feature of population spreading modeling in ecology, however, there are numerous gaps in our knowledge about the wave dynamical behaviors of degenerate diffusion equations compared with the well-studied linear diffusion case. The degeneracy usually causes the solutions to lose their regularities, and raises the possibility of sharp type traveling waves, where the population density decreases to zero at a finite point, rather than decaying to zero asymptotically. On the other hand, the large time-delay often causes the solutions to be oscillating. These two facts make the structure of solutions more complicated and the relevant study more challenging.    The purpose of this research project is to understand the effects of time-delay and degeneracy of diffusion for the population dynamic models. The main objectives include: One is about the structure of traveling waves. Affected by the time-delay and degeneracy of diffusion, we are particularly interested in the classification of sharp traveling waves, oscillatory traveling waves, and oscillatory sharp wavefronts. We will try to prove the existence/non-existence of these waves, and clarify what will be the criteria and mechanism for the population dynamic system to possess these wavefronts. The second issue is to try to study the stability of these wavefronts, in particular, the sharp waves and oscillating sharp waves. Because, the structure of sharp waves is totally different from the monotone/non-monotone traveling waves, and the asymptotic stability of these wavefronts is quite challenging and never treated.    The proposed research program is expected to advance and enrich the theory of degenerate diffusion equations with time-delay, to introduce new mathematical ideas and methods to study stability of sharp wavefronts, and also to provide good opportunity to train Ph.D. students and postdocs by carrying out this project.