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的稳定性，包括描述具有年龄结构的单物种种群动态的Nicholson苍蝇方程和Mackay-Glass方程的两个典型模型。这些方程有一类重要的解，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且时滞较大或方程简并性较强时，行波被检验为振荡。从数学和生物学的角度来看，研究这些非单调波动的渐近稳定性是非常有意义的，但也是具有挑战性的，因为这些波动刻画了这类时滞反应扩散方程的基本特征和结构。波浪稳定性是一个重要问题，也是近年来备受关注的研究热点之一。工作方程和目标波前的简并性和非单调性使得研究变得更加复杂和困难，许多悬而未决的问题仍然像谜一样神秘，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，这些波前的原解的收敛速度，具有很大时滞的解的分歧，数值计算等等。由于现有的方法不能直接用于解决这些问题，我们需要寻找一些新的思路和新的策略。解决这些问题将是这项提案的主要关注点。

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.**