非局部时滞反应扩散系统的分支和周期解

A large number of diffusion systems with nonlocal delays have been proposed in the area of population ecology, epidemiology etc, and this kind of systems can better describe spatial nonuniformity, nonlocal effects and time delay. The bifurcation problem and periodic solutions of the mentioned systems above have important biological and theoretical meanings, however, time delay and spatial nonlocal properties will induce the difficulties on the study of mathematical theory. Based on the above reasons, this work will be devoted to investigating the bifurcations and periodic solutions of reaction-diffusion system with nonlocal delays. By considering a kind of reaction-diffusion system with nonlocal delay, the effect of nonlocal delay on the existence and stability of bifurcation periodic solutions of the system will be illustrated. Then the application of the obtained results in population ecology and epidemiology will be discussed. From mathematics point of view, the difficulties induced by nonlocal delay and the absence of compactness on unbounded domain will be especially concerned, subsequently, the methods for overcoming the difficulties will be explored.

在种群生态学和流行病学等学科领域中，大量具有非局部时滞的反应扩散系统被建立，其能够更好地描述空间非均匀性、非局部作用以及时间滞后现象。这类系统的分支问题和周期解具有重要的生物意义及理论意义，然而，系统中的时间滞后和空间非局部性将导致其在数学理论研究上的困难。基于此，本项目将致力于非局部时滞反应扩散系统的分支和周期解的研究。通过具体考虑一类非局部时滞反应扩散系统来说明非局部时滞对系统分支周期解的存在性和稳定性的影响，并探讨这些结果在生态学和流行病学中的应用。在数学上，将特别考虑由空间非局部作用和时间时滞所导致的困难以及在无界区域上“紧性”的缺失等，并寻找克服这些困难的方法。

研究了一类非局部时滞反应扩散系统的正常数平衡解的稳定性和分支周期解的存在性，并且给出了非局部时滞对系统稳定性的影响。我们主要利用分支理论和摄动方法考虑了其分支问题。此外，正在研究非局部时滞反应扩散系统分支周期解的稳定性，已得到一些初步结果。