Non-monotone traveling waves for reaction-diffusion equations with delay

本文研究了一类时滞退化反应扩散方程的非单调行波解的稳定性，其中包括描述具有年龄结构的单种群种群（如澳大利亚绿头苍蝇）的Nicholson方程和Mackay-Glass方程两个典型模型.这些方程具有一类重要的解，即所谓的行波。振荡行波的稳定性是一个全新的课题，相关的研究工作也非常有限。当出生率函数为非单调函数且时滞较大时，或方程的退化性较强时，行波被检验为振荡的。从数学和生物学的角度来看，研究这类非单调波的渐近稳定性是非常有意义的，但也是具有挑战性的，因为这些非单调波刻画了这类时滞反应扩散方程的本质特征和结构.波浪稳定性是一个重要的问题，也是人们十分关注的研究热点之一。 工作方程和目标波前的退化性和非单调性使得研究变得更加复杂和困难，许多未解决的问题仍然像谜一样，如非临界振荡波前的全局稳定性，特别是临界振荡波前的全局稳定性，这些波前的原始解的收敛速度，具有很大时滞的解的分支，由于现有的方法不能直接应用于解决这些问题，我们需要寻找一些新思路和新策略。解决这些问题将是本建议的主要关注点。**

This research project concerns the stability of non-monotone traveling waves for a class of time-delayed degenerate reaction-diffusion equations, including two typical models of Nicholson's blowflies equation and Mackay-Glass equation, which describe the population dynamics of the single species with age-structure, such as Australian blowflies. These equations possess a class of important solutions, the so-called traveling waves. The topic for the stability of oscillatory traveling waves is brand new, and the related research work is also very limited. When the birth rate function is non-monotone and the time-delay is large, or the degeneracy of the equation is strong, the traveling waves are tested to be oscillating. From both mathematical and biological points of view, it is very significant but also challenging to investigate the asymptotic stability of these non-montone waves, because these waves portray the essential feature and structure of this class of reaction-diffusion equations with delay. The wave stability is an important issue and one of hot research spots by paying great attention. The degeneracy and the non-monotonicity of both the working equations and the targeted wavefronts make the study more complicated and difficult, and many open questions are still like mystery, such as global stability for non-critical oscillating wavefronts, particularly the global stability for critical oscillating wavefronts, the convergence rates of the original solutions to these wavefronts, bifurcation of solutions with a really large time-delay, numerical computations, and so on. Since the existing methods cannot be directly applied to solve these questions, we need to look for some new ideas and new strategies. To attack these problems will be the main concerning in this proposal.**