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

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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.