Dynamics of some evolution equations with temporal delay and/or spatial diff

在本研究中，我打算研究一些演化方程的动力学模型所产生的结构化人口，传染病的传播，病原体和细胞的相互作用在免疫系统中，神经网络的激活和其他相关领域。这种模型方程的一个重要和新颖的特点是存在时间延迟和/或空间扩散/色散，以及空间非局部性，反映了空间和时间的双重方面。这种非局部性可能是由于不成熟个体在种群动态中的扩散，或者是潜在个体在模拟传染病时的流动性的结果，其中潜伏期在空间异质性环境中传播。这样的模型方程更真实地描述了演化过程，可以帮助更好地理解真实的世界中的过程，但也提出了更具挑战性和有趣的数学问题。主要关注的是时间延迟，空间扩散/分散，和空间非局部非线性的全球动态和模式，这样的方程的个人和/或联合影响。研究的主题包括：（i）行波解，它可以解释种群生物学中的域入侵，稳态之间的过渡过程和模式的传播;（ii）稳态及其稳定性;（iii）全局动力学和此类方程的模式。我们建议开发一些新的工具和技术，可能有助于建立一个新的理论，这类新的方程，从而丰富现有的理论发展方程。 我们期望这些理论结果在相关领域具有潜在的应用价值。例如，将所得结果应用于由这类方程表示的一些流行病模型，期望能够更好地理解疾病的传播机制，从而为如何控制流行病提供一些有用的、有价值的指导和建议。

In this research, I propose to investigate the dynamics of some evolution equations arising from modeling structured population, spread of infectious diseases, interaction of pathogens and cells in immune system, activation of neural networks and other related areas. One important and novel feature of such model equations is the presence of time delay and/or spatial diffusion/dispersion, as well as the spatial non-locality reflecting the dual aspect of the space and time. Such a non-locality could be a result of the dispersal of the immature individuals in population dynamics, or a consequence of mobility of latent individuals in modeling an infectious disease with latency spreading over a spatially heterogeneous environment. Such model equations more realistically describe the evolution processes and may help better understand the processes in the real world, and yet raise more challenging and interesting mathematical problems. The main concern is the individual and/or joint impact of the temporal delay, spatial diffusion/dispersion, and the spatial non-local nonlinearities on the global dynamics of and patterns in such equations. Among the topics of investigation are (i) traveling wave solutions which could account for the domain invasion in population biology, transition process between steady states, and propagation of patterns; (ii) steady states and their stability; (iii) global dynamics and patterns of such equations. We propose to develop some new tools and techniques that may help establish a new theory for this new class of equations, and thereby enrich the existing theory for evolution equations. We expect that the theoretic results will have potential applications in related areas. For example, by applying the obtained results to some epidemic disease models expressed by such equations, we expect to be able to better understand the mechanism of spread of the diseases, and thereby, provide some useful and valuable guidance and suggestions on how to control the epidemic diseases.